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Research ArticleA Comfort-Aware Energy Efficient HVAC SystemBased on the Subspace Identification Method
O Tsakiridis1 D Sklavounos2 E Zervas1 and J Stonham2
1Department of Electronics TEI of Athens Egaleo 12210 Athens Greece2Department of Engineering and Design Brunel University Kingston Lane Middlesex UB8 3PH UK
Correspondence should be addressed to O Tsakiridis odytsakteiathgr
Received 12 October 2015 Revised 12 January 2016 Accepted 8 February 2016
Academic Editor Aleksander Zidansek
Copyright copy 2016 O Tsakiridis et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A proactive heating method is presented aiming at reducing the energy consumption in a HVAC system while maintainingthe thermal comfort of the occupants The proposed technique fuses time predictions for the zonesrsquo temperatures based on adeterministic subspace identification method and zonesrsquo occupancy predictions based on a mobility model in a decision schemethat is capable of regulating the balance between the total energy consumed and the total discomfort cost Simulation results forvarious occupation-mobility models demonstrate the efficiency of the proposed technique
1 Introduction
As the control and limitation of the energy consumptionremain a field with an exceptional technological and eco-nomical interest areas that have been considered as highenergy consumers comprise very challenging research issuesAccording to the US Energy Information Administrationfrom 2013 through 2040 the electricity consumption in thecommercial and residential sectors will be increasing by05 and 08 per year [1] It is well established and widelyaccepted through research studies that the main energyconsumers in the commercial and residential buildings arethe Heat Ventilation and Air Conditioning (HVAC) systemsas well as the lighting systems
Buildings contributed a 41 (or 40 quadrillion btu) to thetotal US energy consumption in 2014 On an average about43 of the energy consumption in a commercial and resi-dential building is due to HVAC systems [2] Therefore dueto the high rate of energy consumption the necessity of thedemand-driven control in the HVAC systems has becomeinevitable Inmodern buildings several sophisticated systemshave been applied aiming at providing this type of controlin the HVAC systems The state-of-the-art technology of theHVAC control considers the occupancy of the zones a very
important parameter playing a key role inmethods aiming atreducing energy consumption The detection and predictionof the zonesrsquo occupancy are a very challenging research fieldand several techniques have been proposed based on historicstatistical data as well as on probabilistic models
Another equally important factor taken into account inadvanced HVAC control systems is the thermal comfort ofthe occupants The objective of modern HVAC control sys-tems is to reduce energy consumptionwithout compromisingthe comfort of the occupants Wireless sensor networksequipped with temperature humidity and occupancy detec-tion sensor nodes are nowadays the basic platform to buildautomated HVAC control systems A number of methodsaiming to maintain the thermal comfort while saving energyhave been proposed Some of them are described in Section 2Towards this direction a novel technique is proposed in thispaper which aims at balancing the comfort and energy costsin a multizone system The decisions on heating the zones ornot may be taken either centrally or in a distributed mannerbywireless sensor nodes scattered in themultizone system Inany case temperature and zone occupancy information mustbe exchanged between a node responsible for a zone and itsneighboring nodes The decision process itself relies on twokinds of predictions (a) temperature-time predictions for
Hindawi Publishing CorporationJournal of EnergyVolume 2016 Article ID 5074846 13 pageshttpdxdoiorg10115520165074846
2 Journal of Energy
the zones and (b) the zonesrsquo occupancy profile The emphasisin this paper is on the zonesrsquo temperature predictions andto this end a deterministic subspace identification methodis used for modelling the thermal dynamics of each zoneThat is each zone is modelled by a simple state-space modelcapable of producing accurate predictions based on thesurrounding temperatures the heating power of the zone andthe current state that summarizes the temperature history ofthe zone For the zonesrsquo occupancy predictions we consider asemi-Markov model where occupants (moving as a swarm)stay in a zone for a random period of time and then moveto adjacent zones with given probabilities What is needed bythe decision process is the distribution of the first entrancetime to unoccupied zones Aiming at a proactive action theproposedmethod periodically computes the risk of activatingthe heater or not and decides in favor of the action thatproduces the smaller risk The computation of the risks relieson the relative weights of the energy and discomfort costs sothat the balance between the total energy consumed and thetotal discomfort cost may be regulated
The structure of the paper is as follows In Section 2 priorand related work of the field of the demand-driven HVACsystems that utilize occupancy and prediction methods ispresented In Section 3 the proposed comfort-aware energyefficient mechanism is described along with the subspaceidentification method used for obtaining temperature-timepredictions and a discussion on the distribution of thefirst entrance time to unoccupied zones Section 4 containssimulation results and assesses the effectiveness of the pro-posed algorithm Finally Section 5 summarizes the paper andproposes research directions for future work
2 Prior Work
The necessity of demand-driven HVAC systems for energyefficient solutions has orientated the researchers towardsoccupancy-based activated systems In multizone spacesthe reactive and proactive activation of the heatingcoolingof the zones contributes to significant energy savings andimproves the thermal comfort for the occupants Severalresearch works with valuable results of energy savings haveutilized the occupancy detection and prediction in order tocontrol the HVAC system appropriately For the occupancydetection several types of sensors are used (CO
2 motion
etc) while for the prediction combinational systems areusually applied utilizing mathematical predictive models(eg Markov chains) alongside with detected real data
The authors in [3] proposed an automatic thermostatcontrol system which is based on an occupancy predictionscheme that predicts the destination and arrival time of theoccupants in the air-conditioned areas in order to providea comfortable environment For the occupantsrsquo mobilityprediction the cell tower information of the mobile tele-phony system is utilized and the arrival time prediction isbased on historical patterns and route classification For thedestination prediction of locations very close to each other(intracell) the time-aided order-Markov predictor is usedThe authors in [4] proposed a closed-loop system for opti-mally controllingHVACsystems in buildings based on actual
occupancy levels named the Power-Efficient Occupancy-Based Energy Management (POEM) System In order toaccurately detect occupantsrsquo transition they deployed a wire-less network comprising two parts the occupancy estimationsystem (OPTNet) consisting of 22 camera nodes and a passiveinfrared (PIR) sensors system (BONet) By fusing the sensingdata from the WSN (OPTNet and BONet) with the outputof an occupancy transition model in a particle filter moreaccurate estimation of the current occupancy in each room isachieved Then according to the current occupancy in eachroom and the predicted one from the transition model acontrol schedule of the HVAC system takes over the preheatof the areas to the target temperature In [5] the authorsused real world data gathered from a wireless network of16 smart cameras called Smart Camera Occupancy PositionEstimation System (SCOPES) and in this way they developedoccupancy models Three types of Markov chain (MC)occupancy models were tested and these are the single MCthe closest distance and finally the blended (BMC) Theauthors concluded that BMC is the most efficient and thusthey embodied it as the occupancy predictionmethod in theirproposed ldquoOBSERVErdquo algorithm which is a temperaturecontrol strategy for HVAC systems
The authors in [6] have developed an integrated systemcalled SENTINEL that is a control system for HVAC systemsutilizing occupancy information For the occupancy detec-tion and localisation the system utilizes the existing Wi-Fi network and the clientsrsquo smart phones The occupancylocalization mechanism is based on the access point (AP)communication with a clientrsquos smart phone so that if anoccupantrsquos phone sends packets to an AP then he is locatedwithin the range of the AP By classifying the building areasinto twomain categories namely the personal and the sharedspaces the proposed mechanism activates the HVAC systemwhen an owner of a personal space (eg office) has beendetectedwithin an areawhere hisher office is located orwhenoccupants are detected within shared spaces In [7] an inte-grated heating and cooling control systemof a building is pre-sented aiming to reduce energy consumptionThe occupancybehavior prediction as well as the weather forecast as inputsto a virtual (software based) building model determinesthe control of the HVAC system The occupancy detectiontechnique utilizes Gaussian Mixture Models (GMM) for thecategorization of selected features yielding the highest infor-mation gain according to the different number of occupantsThis categorization was used for observation to a HiddenMarkovModel for the estimation of the number of occupantsA semi-Markov model was developed based on patternscomprised by sensory data of CO
2 acoustics motion and
light changes to estimate the duration of occupants in thespace The work in [8] proposes a model predictive control(MPC) technique aiming to reduce energy consumption ina HVAC system while maintaining comfortable environmentfor the occupants The occupancy predictive model is basedon the two-state Markov chain with the states modelling theoccupied and occupied condition of the areas The authorsin [9] propose a feedback control algorithm for a variableair volume (VAV) HVAC system for full actuated (zonesconsisting of one room) and underactuated (zones consisting
Journal of Energy 3
of more than one room) zones The proposed algorithmis called MOBSua (Measured Occupancy-Based Setback forunderactuated zones) and it utilizes real-time occupancydata through a WSN for optimum energy efficiency andthermal comfort of the occupants Moreover the algorithmcan be applied on conventional control systems with no needof occupancy information and it is scalable to arbitrary sizedbuildings
3 Comfort-Aware Energy Efficient Mechanism
We consider a multizone HVAC system consisting of119885 zones119911119894 119894 = 1 2 119885 Each zone is equipped with a wireless
sensor node capable of conveying the zonersquos temperatureand occupancy information to its neighbors or to a centralprocessing unit The occupancy information may be verysimple that is binary information indicating the presence orabsence of individuals in the zone or more advanced like thenumber of occupants in the zoneTheobjective is tominimizethe total energy and discomfort cost defined as
E = Eenergy +Ediscomfort
=
119885
sum
119894=1
int
infin
minusinfin
119882119894119868 (119867119894= on) 119889119905
+
119885
sum
119894=1
int
infin
minusinfin
119862119894119868 (119879119894lt 119879119888) 119889119905
(1)
Note that only the heating problem is addressed in this paperThus the case of spending energy for cooling the zonesand keeping the temperature 119879
119894within a certain comfort
zone is not treated for simplicity In (1) 119868(119860) denotes theindicator function which takes the value 1 or 0 dependingon whether condition 119860 is true or false 119882
119894is the power
of a heater that covers zone 119911119894and 119867
119894is the state of the
heater that is the heater is on or off Thus the first termof (1) is the total energy consumed by the multizone systemFor the discomfort cost we define the cost per unit time 119862
119894
and the target comfort threshold 119879119888 As long as the zonersquos
temperature 119879119894is higher than 119879
119888the occupants do not feel
discomfort The parameter 119862119894may depend on the zone the
number of the occupants (eg the total systemrsquos discomfortcost is proportional to the number of people experiencingdiscomfort) and the difference of the zonersquos temperature 119879
119894
and the comfort threshold 119879119888 It is worth noting that there is
a tradeoff between energy savings and discomfort cost Wemay preheat all zones to the comfort level thus renderingthe discomfort cost equal to zero This policy is inefficientdue to the large amount of energy consumed In the otherextreme we could act reactively by heating a zone only upondetecting occupants in it In this case the energy cost wouldbe as small as possible but the discomfort cost could increasedramatically In the sequel we will describe a method thatacts proactively by taking periodically decisions on whetherto heat a zone or not If the entrance to a zone is delayedthen we postpone heating until the next decision epoch Ifon the contrary it is anticipated that a zone will be occupiedbefore the zonersquos temperature reaches the comfort level thenwe preheat the zone
The decision of whether or not the heater of an unoccu-pied zone 119911
119894should be turned on depends on (a) the current
state of the zone (b) the relative value of the energy cost (119882119894)
and discomfort cost (119862119894) per unit time (c) the temperatures
of the surrounding zones and (d) an estimate of the time thatthe zone will become occupied This decision may be takeneither centrally from the central processing unit that gathersinformation by all zones or in a distributed manner by eachzonersquos node after collecting the relative information There isalso the possibility of dividing the decision process function-ality between the central unit and the wireless sensor networknodes For example the prediction of the first entrancetimes to unoccupied zones may be taken by the central unitthat is aware of the location of people in the multizonesystem and then the predicted values may be communicatedback to the zonesrsquo nodes for the final decision The specificimplementation of the decision process is irrelevant to thispaper and it will not be further analyzed
We assume that time is discretized 119905 = 119898119879119904119879119904is the sam-
pling period and that each node takes its decisions periodi-cally every119863 sampling periods Note that there is no need forthe nodes to be synchronized to each other Let 119884
119894(119905) denote
the random variable thatmodels the remaining time from thecurrent time instant 119905 until entrance to the unoccupied zone119911119894 It is obvious that the randomvariables119884
119896(119905) for the various
unoccupied zones 119911119896 do not follow the same distributionWe
further assume that the node is capable ofmaking predictionsfor the time it takes to exceed the comfort threshold 119879
119888 To
this end let 119878on119894(119905) denote the time period that is required to
exceed 119879119888if the heater is turned on immediately Similarly
we define 119878off119894(119905) as the predicted time to reach the comfort
threshold if the heater remains off for a period 119863 and thenat the next decision epoch it is turned onThe relation of theaforementioned parameters is shown in Figure 1
Suppose now that at time 119905 the node of the unoccupiedzone 119911
119894has to take a decisionwhether to turn the zonersquos heater
on or remain in the off state At the decision epoch 119905 the riskto turn the heater on is119877on119894(119905)
= 119882119894DP 119878off
119894(119905) le 119884
119894(119905)
+ 119882119894(119884119894(119905) minus 119878
on119894(119905)) 119875 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)
(2)
That is if 119878off119894(119905) le 119884
119894(119905) there is plenty of time to reach the
comfort threshold even if we start heating at the next decisionepoch and therefore we consume unnecessarily 119882
119894119863 units
of energy (this is the case depicted in Figure 1) Similarly if119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) we waste 119882
119894(119884119894(119905) minus 119878
on119894(119905)) units of
energy Note that there is no risk of turning on the heater ifthe entrance to the zone happens sooner than it is predictedIn a similar fashion we calculate the risk of keeping the heateroff In this case
119877off119894(119905)
= 119862119894DP 119884
119894(119905) le 119878
on119894(119905)
+ 119862119894(119878
off119894(119905) minus 119884
119894(119905)) 119875 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)
(3)
4 Journal of Energy
Yi(t)
Soni (t)
Soffi (t)
Ti
t
DTs
t + DTs
Tc
Figure 1 Predicted time intervals related to the decision process
Using (2) and (3) the decision criterion takes the form
119877on119894(119905)
on≶
off119877off119894(119905) (4)
Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones
Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884
119894(119905) le 119878
on119894(119905) then the heater
is turned on if 119878off119894(119905) lt 119884
119894(119905) the heater remains off and
if 119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) the heater will be turned onoff
depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882
119894and 119862
119894
If 119884119894(119905) is random then criterion (4) needs to be slightly
modified as the value of 119884119894(119905) is unknown In this case we
replace 119884119894(119905) with its conditional expected value given that
119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) Therefore in criterion (4) we
substitute 119884119894(119905) by 119864[119884
119894(119905) | 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)]
From the previous discussion it becomes clear that thedecision process needs the estimates 119878on
119894(119905) 119878off119894(119905) and the
distribution of the process 119884119894(119905) The former is the subject of
the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32
31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone
is treated separately and is represented by a discrete timelinear time invariant state-space model That is
119909119896+1
= 119860119909119896+ 119861119906119896+ 119908119896 (5)
119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)
where 119910119896is the zone temperature at time 119896 and is in general
an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910
119896is scalar
V119896represents the measurement noise due to imperfections
of the sensor 119906119896is a vector of input measurements and its
dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909
119896is the state vector of the
process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the
zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863
from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances
In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model
After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace
Journal of Energy 5
identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone
Following the notation and the derivation in [10] wedefine the block Hankel matrices
119880119901= 1198800|119894minus1
=(
(
1199060
1199061sdot sdot sdot 119906
119895minus1
1199061
1199062sdot sdot sdot 119906
119895
119906119894minus1
119906119894sdot sdot sdot 119906119894+119895minus2
)
)
119880119891= 119880119894|2119894minus1
=(
(
119906119894
119906119894+1
sdot sdot sdot 119906119894+119895minus1
119906119894+1
119906119894+2
sdot sdot sdot 119906119894+119895
1199062119894minus1
1199062119894
sdot sdot sdot 1199062119894+119895minus2
)
)
(7)
which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880
119901on top of119880
119891we have
the block Hankel matrix
1198800|2119894minus1
= (1198800|119894minus1
119880119894|2119894minus1
) = (
119880119901
119880119891
) (8)
which can also be partitioned as
1198800|2119894minus1
= (1198800|119894
119880119894+1|2119894minus1
) = (
119880+
119901
119880minus
119891
) (9)
by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884
0|2119894minus1
119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices
119882119901= 1198820|119894minus1
= (1198800|119894minus1
1198840|119894minus1
) = (
119880119901
119884119901
)
119882+
119901= (
119880+
119901
119884+119901
)
(10)
The state sequence 119883119898is defined as
119883119898= (119909119898 119909
119898+1sdot sdot sdot 119909119898+119895minus1) (11)
and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830
and 119883119891= 119883119894 respectively With the state-space model (13)
(15) we associate the extended observability matrix Γ119894and the
reversed extended controllability matrix Δ119894 where
Γ119894=
((((
(
119862
119862119860
1198621198602
119862119860119894minus1
))))
)
Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)
(12)
The subspace identificationmethod relies on determining thestate sequence 119883
119891and the extended observability matrix Γ
119894
directly from the input-output data 119906119896119910119896and based on them
in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884
119891
along the row space of 119880119891on the row space of119882
119901
O119894= 119884119891119880119891119882119901= [119884119891119880perp
119891] [119882119901119880perp
119891]dagger
119882119901 (13)
where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is
119860119861perp= 119868 minus 119860119861
119879(119861119861119879)dagger
119861 (14)
As it is proved in [10] the matrix O119894is equal to the product
of the extended observability matrix and the ldquofuturerdquo statevector
O119894= Γ119894119883119891 (15)
Having in our disposal the matrixO119894we use its singular value
decomposition
O119894= (1198801 119880
2) (
11987810
0 0)(
119881119879
1
119881119879
2
) = 11988011198781119881119879
1(16)
and we identify the extended observation matrix Γ119894and the
state vector119883119891as
Γ119894= 119880111987812
1119879
119883119891= 119879minus111987812
1119881119879
1= Γdagger
119894O119894
(17)
where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation
One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection
O119894minus1
= 119884minus
119891119880minus119891119882dagger
119901= Γ119894minus1119883119894+1 (18)
where Γ119894minus1
is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1
in our case) Similar to the previous derivation we have
119883119894+1
= Γdagger
119894minus1O119894minus1
(19)
and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system
(
119883119894+1
119884119894|119894
) = (
119860 119861
119862 119863)(
119883119894
119880119894|119894
) (20)
After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909
0 they update
6 Journal of Energy
the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909
119905 The node bases its
decision on the estimates 119878on119894(119905) and 119878off
119894(119905) Recall that 119878on
119894(119905)
is the time needed to reach the comfort level 119879119888if the heater
is on That is
119878on119894(119905) = min ℓ 119910
119905+ℓge 119879119888 (21)
The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906
119905
(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909
119905repeats
(5) (6) until 119910119905+119896
exceeds the value 119879119888 Note that
119910119905+119896
= 119862119860119896119909119905+ 119862119860119896minus1
119861119906119905+ 119862119860119896minus2
119861119906119905+1
+ sdot sdot sdot
+ 119862119860119861119906119905+119896minus2
+ 119862119861119906119905+119896minus1
+ 119863119906119905+119896
(22)
and for 119906119905= 119906119905+1
= sdot sdot sdot = 119906119905+119896
we obtain
119910119905+119896
= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860
119896minus1) 119861119906119905+ 119863119906119905
= 119862119860119896119909119905+ 119862 (119868 minus 119860)
minus1(119868 minus 119860
119896) 119861119906119905+ 119863119906119905
(23)
If we set 119910119905+119896
= 119879119888and
119908 = 119879119888minus 119862 (119868 minus 119860)
minus1119861119906119905minus 119863119906119905 (24)
then (23) takes the form
119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)
minus1119860119896119861119906119905
= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)
minus1119881Λ119896119881minus1119861119906119905
(25)
where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881
minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off
119894(119905) In this case
however we first repeat (5) 119863 times with input-vector 119906119896
reflecting the fact that the heater is off and then starting fromthe new state 119909
119905+119863we estimate the time needed to reach 119879
119888if
the heater is on
32 On the Distribution of 119884119899(119905) The random variable 119884
119899(119905)
expresses the remaining time from the current time instant 119905until entrance to zone 119911
119899 That is we assume that occupants
move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911
119899 Each time the occupants visit a zone 119911
ℓthey
znz120001
Figure 2 Movement of occupants from zone 119911ℓto the adjacent
unoccupied zone 119911119899
stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865
ℓ(119909) We use the variable 119905
to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865
ℓ(119909) may be
given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911
ℓ
1199091 1199092 119909
119873 is available then
ℓ(119909) =
1
119873
119873
sum
119898=1
119868 (119909119898isin [0 119909]) (26)
where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones
119911ℓ 119911119899with zone 119911
ℓbeing occupied and 119911
119899unoccupied as it is
shown in Figure 2 We assume that the occupants remain tozone 119911
ℓfor a random period of time 119883
ℓwhich is distributed
according to 119865ℓ(119909) and then upon departure they enter zone
119911119899 In this case 119884
119899(119905) expresses the remaining waiting time
to zone 119911ℓand follows the residual life distribution which is
defined by
119877119884119899(119905)
(119910) = 1 minus 119877119884119899(119905)
(119910) = 119875 119883ℓgt 120591 + 119910 | 119883
ℓgt 120591
=119865ℓ(120591 + 119910)
119865ℓ(120591)
(27)
Next we consider the case that occupants are currently ina zone (call it 119911
0) and upon departure they visit in cascade
zones 1199111 1199112 119911
119899as it is shown in Figure 3 Then
119884119899(119905) = 119884
0+ 1198831+ 1198832+ sdot sdot sdot + 119883
119899minus1 (28)
where 1198840is the remaining time in zone 119911
0and 119883
119894is the
occupation period of zone 119911119894 It is well known that the
distribution of 119884119899(119905) is the convolution of the distributions of
the random variables 1198840 1198831 119883
119899minus1 That is
119865119884119899(119905)
(119910) = 119875 119884119899(119905) le 119910
= 1198770(119910) ⋆ 119865
1(119910) ⋆ 119865
2(119910) ⋆ sdot sdot sdot ⋆ 119865
119899minus1(119910)
(29)
with 1198770(119910) = 1 minus 119865
0(120591 + 119910)119865
0(120591) A more compact form is
obtained by considering a transform of the distributions that
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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2 Journal of Energy
the zones and (b) the zonesrsquo occupancy profile The emphasisin this paper is on the zonesrsquo temperature predictions andto this end a deterministic subspace identification methodis used for modelling the thermal dynamics of each zoneThat is each zone is modelled by a simple state-space modelcapable of producing accurate predictions based on thesurrounding temperatures the heating power of the zone andthe current state that summarizes the temperature history ofthe zone For the zonesrsquo occupancy predictions we consider asemi-Markov model where occupants (moving as a swarm)stay in a zone for a random period of time and then moveto adjacent zones with given probabilities What is needed bythe decision process is the distribution of the first entrancetime to unoccupied zones Aiming at a proactive action theproposedmethod periodically computes the risk of activatingthe heater or not and decides in favor of the action thatproduces the smaller risk The computation of the risks relieson the relative weights of the energy and discomfort costs sothat the balance between the total energy consumed and thetotal discomfort cost may be regulated
The structure of the paper is as follows In Section 2 priorand related work of the field of the demand-driven HVACsystems that utilize occupancy and prediction methods ispresented In Section 3 the proposed comfort-aware energyefficient mechanism is described along with the subspaceidentification method used for obtaining temperature-timepredictions and a discussion on the distribution of thefirst entrance time to unoccupied zones Section 4 containssimulation results and assesses the effectiveness of the pro-posed algorithm Finally Section 5 summarizes the paper andproposes research directions for future work
2 Prior Work
The necessity of demand-driven HVAC systems for energyefficient solutions has orientated the researchers towardsoccupancy-based activated systems In multizone spacesthe reactive and proactive activation of the heatingcoolingof the zones contributes to significant energy savings andimproves the thermal comfort for the occupants Severalresearch works with valuable results of energy savings haveutilized the occupancy detection and prediction in order tocontrol the HVAC system appropriately For the occupancydetection several types of sensors are used (CO
2 motion
etc) while for the prediction combinational systems areusually applied utilizing mathematical predictive models(eg Markov chains) alongside with detected real data
The authors in [3] proposed an automatic thermostatcontrol system which is based on an occupancy predictionscheme that predicts the destination and arrival time of theoccupants in the air-conditioned areas in order to providea comfortable environment For the occupantsrsquo mobilityprediction the cell tower information of the mobile tele-phony system is utilized and the arrival time prediction isbased on historical patterns and route classification For thedestination prediction of locations very close to each other(intracell) the time-aided order-Markov predictor is usedThe authors in [4] proposed a closed-loop system for opti-mally controllingHVACsystems in buildings based on actual
occupancy levels named the Power-Efficient Occupancy-Based Energy Management (POEM) System In order toaccurately detect occupantsrsquo transition they deployed a wire-less network comprising two parts the occupancy estimationsystem (OPTNet) consisting of 22 camera nodes and a passiveinfrared (PIR) sensors system (BONet) By fusing the sensingdata from the WSN (OPTNet and BONet) with the outputof an occupancy transition model in a particle filter moreaccurate estimation of the current occupancy in each room isachieved Then according to the current occupancy in eachroom and the predicted one from the transition model acontrol schedule of the HVAC system takes over the preheatof the areas to the target temperature In [5] the authorsused real world data gathered from a wireless network of16 smart cameras called Smart Camera Occupancy PositionEstimation System (SCOPES) and in this way they developedoccupancy models Three types of Markov chain (MC)occupancy models were tested and these are the single MCthe closest distance and finally the blended (BMC) Theauthors concluded that BMC is the most efficient and thusthey embodied it as the occupancy predictionmethod in theirproposed ldquoOBSERVErdquo algorithm which is a temperaturecontrol strategy for HVAC systems
The authors in [6] have developed an integrated systemcalled SENTINEL that is a control system for HVAC systemsutilizing occupancy information For the occupancy detec-tion and localisation the system utilizes the existing Wi-Fi network and the clientsrsquo smart phones The occupancylocalization mechanism is based on the access point (AP)communication with a clientrsquos smart phone so that if anoccupantrsquos phone sends packets to an AP then he is locatedwithin the range of the AP By classifying the building areasinto twomain categories namely the personal and the sharedspaces the proposed mechanism activates the HVAC systemwhen an owner of a personal space (eg office) has beendetectedwithin an areawhere hisher office is located orwhenoccupants are detected within shared spaces In [7] an inte-grated heating and cooling control systemof a building is pre-sented aiming to reduce energy consumptionThe occupancybehavior prediction as well as the weather forecast as inputsto a virtual (software based) building model determinesthe control of the HVAC system The occupancy detectiontechnique utilizes Gaussian Mixture Models (GMM) for thecategorization of selected features yielding the highest infor-mation gain according to the different number of occupantsThis categorization was used for observation to a HiddenMarkovModel for the estimation of the number of occupantsA semi-Markov model was developed based on patternscomprised by sensory data of CO
2 acoustics motion and
light changes to estimate the duration of occupants in thespace The work in [8] proposes a model predictive control(MPC) technique aiming to reduce energy consumption ina HVAC system while maintaining comfortable environmentfor the occupants The occupancy predictive model is basedon the two-state Markov chain with the states modelling theoccupied and occupied condition of the areas The authorsin [9] propose a feedback control algorithm for a variableair volume (VAV) HVAC system for full actuated (zonesconsisting of one room) and underactuated (zones consisting
Journal of Energy 3
of more than one room) zones The proposed algorithmis called MOBSua (Measured Occupancy-Based Setback forunderactuated zones) and it utilizes real-time occupancydata through a WSN for optimum energy efficiency andthermal comfort of the occupants Moreover the algorithmcan be applied on conventional control systems with no needof occupancy information and it is scalable to arbitrary sizedbuildings
3 Comfort-Aware Energy Efficient Mechanism
We consider a multizone HVAC system consisting of119885 zones119911119894 119894 = 1 2 119885 Each zone is equipped with a wireless
sensor node capable of conveying the zonersquos temperatureand occupancy information to its neighbors or to a centralprocessing unit The occupancy information may be verysimple that is binary information indicating the presence orabsence of individuals in the zone or more advanced like thenumber of occupants in the zoneTheobjective is tominimizethe total energy and discomfort cost defined as
E = Eenergy +Ediscomfort
=
119885
sum
119894=1
int
infin
minusinfin
119882119894119868 (119867119894= on) 119889119905
+
119885
sum
119894=1
int
infin
minusinfin
119862119894119868 (119879119894lt 119879119888) 119889119905
(1)
Note that only the heating problem is addressed in this paperThus the case of spending energy for cooling the zonesand keeping the temperature 119879
119894within a certain comfort
zone is not treated for simplicity In (1) 119868(119860) denotes theindicator function which takes the value 1 or 0 dependingon whether condition 119860 is true or false 119882
119894is the power
of a heater that covers zone 119911119894and 119867
119894is the state of the
heater that is the heater is on or off Thus the first termof (1) is the total energy consumed by the multizone systemFor the discomfort cost we define the cost per unit time 119862
119894
and the target comfort threshold 119879119888 As long as the zonersquos
temperature 119879119894is higher than 119879
119888the occupants do not feel
discomfort The parameter 119862119894may depend on the zone the
number of the occupants (eg the total systemrsquos discomfortcost is proportional to the number of people experiencingdiscomfort) and the difference of the zonersquos temperature 119879
119894
and the comfort threshold 119879119888 It is worth noting that there is
a tradeoff between energy savings and discomfort cost Wemay preheat all zones to the comfort level thus renderingthe discomfort cost equal to zero This policy is inefficientdue to the large amount of energy consumed In the otherextreme we could act reactively by heating a zone only upondetecting occupants in it In this case the energy cost wouldbe as small as possible but the discomfort cost could increasedramatically In the sequel we will describe a method thatacts proactively by taking periodically decisions on whetherto heat a zone or not If the entrance to a zone is delayedthen we postpone heating until the next decision epoch Ifon the contrary it is anticipated that a zone will be occupiedbefore the zonersquos temperature reaches the comfort level thenwe preheat the zone
The decision of whether or not the heater of an unoccu-pied zone 119911
119894should be turned on depends on (a) the current
state of the zone (b) the relative value of the energy cost (119882119894)
and discomfort cost (119862119894) per unit time (c) the temperatures
of the surrounding zones and (d) an estimate of the time thatthe zone will become occupied This decision may be takeneither centrally from the central processing unit that gathersinformation by all zones or in a distributed manner by eachzonersquos node after collecting the relative information There isalso the possibility of dividing the decision process function-ality between the central unit and the wireless sensor networknodes For example the prediction of the first entrancetimes to unoccupied zones may be taken by the central unitthat is aware of the location of people in the multizonesystem and then the predicted values may be communicatedback to the zonesrsquo nodes for the final decision The specificimplementation of the decision process is irrelevant to thispaper and it will not be further analyzed
We assume that time is discretized 119905 = 119898119879119904119879119904is the sam-
pling period and that each node takes its decisions periodi-cally every119863 sampling periods Note that there is no need forthe nodes to be synchronized to each other Let 119884
119894(119905) denote
the random variable thatmodels the remaining time from thecurrent time instant 119905 until entrance to the unoccupied zone119911119894 It is obvious that the randomvariables119884
119896(119905) for the various
unoccupied zones 119911119896 do not follow the same distributionWe
further assume that the node is capable ofmaking predictionsfor the time it takes to exceed the comfort threshold 119879
119888 To
this end let 119878on119894(119905) denote the time period that is required to
exceed 119879119888if the heater is turned on immediately Similarly
we define 119878off119894(119905) as the predicted time to reach the comfort
threshold if the heater remains off for a period 119863 and thenat the next decision epoch it is turned onThe relation of theaforementioned parameters is shown in Figure 1
Suppose now that at time 119905 the node of the unoccupiedzone 119911
119894has to take a decisionwhether to turn the zonersquos heater
on or remain in the off state At the decision epoch 119905 the riskto turn the heater on is119877on119894(119905)
= 119882119894DP 119878off
119894(119905) le 119884
119894(119905)
+ 119882119894(119884119894(119905) minus 119878
on119894(119905)) 119875 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)
(2)
That is if 119878off119894(119905) le 119884
119894(119905) there is plenty of time to reach the
comfort threshold even if we start heating at the next decisionepoch and therefore we consume unnecessarily 119882
119894119863 units
of energy (this is the case depicted in Figure 1) Similarly if119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) we waste 119882
119894(119884119894(119905) minus 119878
on119894(119905)) units of
energy Note that there is no risk of turning on the heater ifthe entrance to the zone happens sooner than it is predictedIn a similar fashion we calculate the risk of keeping the heateroff In this case
119877off119894(119905)
= 119862119894DP 119884
119894(119905) le 119878
on119894(119905)
+ 119862119894(119878
off119894(119905) minus 119884
119894(119905)) 119875 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)
(3)
4 Journal of Energy
Yi(t)
Soni (t)
Soffi (t)
Ti
t
DTs
t + DTs
Tc
Figure 1 Predicted time intervals related to the decision process
Using (2) and (3) the decision criterion takes the form
119877on119894(119905)
on≶
off119877off119894(119905) (4)
Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones
Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884
119894(119905) le 119878
on119894(119905) then the heater
is turned on if 119878off119894(119905) lt 119884
119894(119905) the heater remains off and
if 119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) the heater will be turned onoff
depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882
119894and 119862
119894
If 119884119894(119905) is random then criterion (4) needs to be slightly
modified as the value of 119884119894(119905) is unknown In this case we
replace 119884119894(119905) with its conditional expected value given that
119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) Therefore in criterion (4) we
substitute 119884119894(119905) by 119864[119884
119894(119905) | 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)]
From the previous discussion it becomes clear that thedecision process needs the estimates 119878on
119894(119905) 119878off119894(119905) and the
distribution of the process 119884119894(119905) The former is the subject of
the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32
31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone
is treated separately and is represented by a discrete timelinear time invariant state-space model That is
119909119896+1
= 119860119909119896+ 119861119906119896+ 119908119896 (5)
119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)
where 119910119896is the zone temperature at time 119896 and is in general
an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910
119896is scalar
V119896represents the measurement noise due to imperfections
of the sensor 119906119896is a vector of input measurements and its
dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909
119896is the state vector of the
process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the
zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863
from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances
In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model
After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace
Journal of Energy 5
identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone
Following the notation and the derivation in [10] wedefine the block Hankel matrices
119880119901= 1198800|119894minus1
=(
(
1199060
1199061sdot sdot sdot 119906
119895minus1
1199061
1199062sdot sdot sdot 119906
119895
119906119894minus1
119906119894sdot sdot sdot 119906119894+119895minus2
)
)
119880119891= 119880119894|2119894minus1
=(
(
119906119894
119906119894+1
sdot sdot sdot 119906119894+119895minus1
119906119894+1
119906119894+2
sdot sdot sdot 119906119894+119895
1199062119894minus1
1199062119894
sdot sdot sdot 1199062119894+119895minus2
)
)
(7)
which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880
119901on top of119880
119891we have
the block Hankel matrix
1198800|2119894minus1
= (1198800|119894minus1
119880119894|2119894minus1
) = (
119880119901
119880119891
) (8)
which can also be partitioned as
1198800|2119894minus1
= (1198800|119894
119880119894+1|2119894minus1
) = (
119880+
119901
119880minus
119891
) (9)
by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884
0|2119894minus1
119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices
119882119901= 1198820|119894minus1
= (1198800|119894minus1
1198840|119894minus1
) = (
119880119901
119884119901
)
119882+
119901= (
119880+
119901
119884+119901
)
(10)
The state sequence 119883119898is defined as
119883119898= (119909119898 119909
119898+1sdot sdot sdot 119909119898+119895minus1) (11)
and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830
and 119883119891= 119883119894 respectively With the state-space model (13)
(15) we associate the extended observability matrix Γ119894and the
reversed extended controllability matrix Δ119894 where
Γ119894=
((((
(
119862
119862119860
1198621198602
119862119860119894minus1
))))
)
Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)
(12)
The subspace identificationmethod relies on determining thestate sequence 119883
119891and the extended observability matrix Γ
119894
directly from the input-output data 119906119896119910119896and based on them
in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884
119891
along the row space of 119880119891on the row space of119882
119901
O119894= 119884119891119880119891119882119901= [119884119891119880perp
119891] [119882119901119880perp
119891]dagger
119882119901 (13)
where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is
119860119861perp= 119868 minus 119860119861
119879(119861119861119879)dagger
119861 (14)
As it is proved in [10] the matrix O119894is equal to the product
of the extended observability matrix and the ldquofuturerdquo statevector
O119894= Γ119894119883119891 (15)
Having in our disposal the matrixO119894we use its singular value
decomposition
O119894= (1198801 119880
2) (
11987810
0 0)(
119881119879
1
119881119879
2
) = 11988011198781119881119879
1(16)
and we identify the extended observation matrix Γ119894and the
state vector119883119891as
Γ119894= 119880111987812
1119879
119883119891= 119879minus111987812
1119881119879
1= Γdagger
119894O119894
(17)
where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation
One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection
O119894minus1
= 119884minus
119891119880minus119891119882dagger
119901= Γ119894minus1119883119894+1 (18)
where Γ119894minus1
is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1
in our case) Similar to the previous derivation we have
119883119894+1
= Γdagger
119894minus1O119894minus1
(19)
and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system
(
119883119894+1
119884119894|119894
) = (
119860 119861
119862 119863)(
119883119894
119880119894|119894
) (20)
After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909
0 they update
6 Journal of Energy
the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909
119905 The node bases its
decision on the estimates 119878on119894(119905) and 119878off
119894(119905) Recall that 119878on
119894(119905)
is the time needed to reach the comfort level 119879119888if the heater
is on That is
119878on119894(119905) = min ℓ 119910
119905+ℓge 119879119888 (21)
The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906
119905
(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909
119905repeats
(5) (6) until 119910119905+119896
exceeds the value 119879119888 Note that
119910119905+119896
= 119862119860119896119909119905+ 119862119860119896minus1
119861119906119905+ 119862119860119896minus2
119861119906119905+1
+ sdot sdot sdot
+ 119862119860119861119906119905+119896minus2
+ 119862119861119906119905+119896minus1
+ 119863119906119905+119896
(22)
and for 119906119905= 119906119905+1
= sdot sdot sdot = 119906119905+119896
we obtain
119910119905+119896
= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860
119896minus1) 119861119906119905+ 119863119906119905
= 119862119860119896119909119905+ 119862 (119868 minus 119860)
minus1(119868 minus 119860
119896) 119861119906119905+ 119863119906119905
(23)
If we set 119910119905+119896
= 119879119888and
119908 = 119879119888minus 119862 (119868 minus 119860)
minus1119861119906119905minus 119863119906119905 (24)
then (23) takes the form
119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)
minus1119860119896119861119906119905
= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)
minus1119881Λ119896119881minus1119861119906119905
(25)
where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881
minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off
119894(119905) In this case
however we first repeat (5) 119863 times with input-vector 119906119896
reflecting the fact that the heater is off and then starting fromthe new state 119909
119905+119863we estimate the time needed to reach 119879
119888if
the heater is on
32 On the Distribution of 119884119899(119905) The random variable 119884
119899(119905)
expresses the remaining time from the current time instant 119905until entrance to zone 119911
119899 That is we assume that occupants
move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911
119899 Each time the occupants visit a zone 119911
ℓthey
znz120001
Figure 2 Movement of occupants from zone 119911ℓto the adjacent
unoccupied zone 119911119899
stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865
ℓ(119909) We use the variable 119905
to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865
ℓ(119909) may be
given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911
ℓ
1199091 1199092 119909
119873 is available then
ℓ(119909) =
1
119873
119873
sum
119898=1
119868 (119909119898isin [0 119909]) (26)
where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones
119911ℓ 119911119899with zone 119911
ℓbeing occupied and 119911
119899unoccupied as it is
shown in Figure 2 We assume that the occupants remain tozone 119911
ℓfor a random period of time 119883
ℓwhich is distributed
according to 119865ℓ(119909) and then upon departure they enter zone
119911119899 In this case 119884
119899(119905) expresses the remaining waiting time
to zone 119911ℓand follows the residual life distribution which is
defined by
119877119884119899(119905)
(119910) = 1 minus 119877119884119899(119905)
(119910) = 119875 119883ℓgt 120591 + 119910 | 119883
ℓgt 120591
=119865ℓ(120591 + 119910)
119865ℓ(120591)
(27)
Next we consider the case that occupants are currently ina zone (call it 119911
0) and upon departure they visit in cascade
zones 1199111 1199112 119911
119899as it is shown in Figure 3 Then
119884119899(119905) = 119884
0+ 1198831+ 1198832+ sdot sdot sdot + 119883
119899minus1 (28)
where 1198840is the remaining time in zone 119911
0and 119883
119894is the
occupation period of zone 119911119894 It is well known that the
distribution of 119884119899(119905) is the convolution of the distributions of
the random variables 1198840 1198831 119883
119899minus1 That is
119865119884119899(119905)
(119910) = 119875 119884119899(119905) le 119910
= 1198770(119910) ⋆ 119865
1(119910) ⋆ 119865
2(119910) ⋆ sdot sdot sdot ⋆ 119865
119899minus1(119910)
(29)
with 1198770(119910) = 1 minus 119865
0(120591 + 119910)119865
0(120591) A more compact form is
obtained by considering a transform of the distributions that
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
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Journal of Energy 3
of more than one room) zones The proposed algorithmis called MOBSua (Measured Occupancy-Based Setback forunderactuated zones) and it utilizes real-time occupancydata through a WSN for optimum energy efficiency andthermal comfort of the occupants Moreover the algorithmcan be applied on conventional control systems with no needof occupancy information and it is scalable to arbitrary sizedbuildings
3 Comfort-Aware Energy Efficient Mechanism
We consider a multizone HVAC system consisting of119885 zones119911119894 119894 = 1 2 119885 Each zone is equipped with a wireless
sensor node capable of conveying the zonersquos temperatureand occupancy information to its neighbors or to a centralprocessing unit The occupancy information may be verysimple that is binary information indicating the presence orabsence of individuals in the zone or more advanced like thenumber of occupants in the zoneTheobjective is tominimizethe total energy and discomfort cost defined as
E = Eenergy +Ediscomfort
=
119885
sum
119894=1
int
infin
minusinfin
119882119894119868 (119867119894= on) 119889119905
+
119885
sum
119894=1
int
infin
minusinfin
119862119894119868 (119879119894lt 119879119888) 119889119905
(1)
Note that only the heating problem is addressed in this paperThus the case of spending energy for cooling the zonesand keeping the temperature 119879
119894within a certain comfort
zone is not treated for simplicity In (1) 119868(119860) denotes theindicator function which takes the value 1 or 0 dependingon whether condition 119860 is true or false 119882
119894is the power
of a heater that covers zone 119911119894and 119867
119894is the state of the
heater that is the heater is on or off Thus the first termof (1) is the total energy consumed by the multizone systemFor the discomfort cost we define the cost per unit time 119862
119894
and the target comfort threshold 119879119888 As long as the zonersquos
temperature 119879119894is higher than 119879
119888the occupants do not feel
discomfort The parameter 119862119894may depend on the zone the
number of the occupants (eg the total systemrsquos discomfortcost is proportional to the number of people experiencingdiscomfort) and the difference of the zonersquos temperature 119879
119894
and the comfort threshold 119879119888 It is worth noting that there is
a tradeoff between energy savings and discomfort cost Wemay preheat all zones to the comfort level thus renderingthe discomfort cost equal to zero This policy is inefficientdue to the large amount of energy consumed In the otherextreme we could act reactively by heating a zone only upondetecting occupants in it In this case the energy cost wouldbe as small as possible but the discomfort cost could increasedramatically In the sequel we will describe a method thatacts proactively by taking periodically decisions on whetherto heat a zone or not If the entrance to a zone is delayedthen we postpone heating until the next decision epoch Ifon the contrary it is anticipated that a zone will be occupiedbefore the zonersquos temperature reaches the comfort level thenwe preheat the zone
The decision of whether or not the heater of an unoccu-pied zone 119911
119894should be turned on depends on (a) the current
state of the zone (b) the relative value of the energy cost (119882119894)
and discomfort cost (119862119894) per unit time (c) the temperatures
of the surrounding zones and (d) an estimate of the time thatthe zone will become occupied This decision may be takeneither centrally from the central processing unit that gathersinformation by all zones or in a distributed manner by eachzonersquos node after collecting the relative information There isalso the possibility of dividing the decision process function-ality between the central unit and the wireless sensor networknodes For example the prediction of the first entrancetimes to unoccupied zones may be taken by the central unitthat is aware of the location of people in the multizonesystem and then the predicted values may be communicatedback to the zonesrsquo nodes for the final decision The specificimplementation of the decision process is irrelevant to thispaper and it will not be further analyzed
We assume that time is discretized 119905 = 119898119879119904119879119904is the sam-
pling period and that each node takes its decisions periodi-cally every119863 sampling periods Note that there is no need forthe nodes to be synchronized to each other Let 119884
119894(119905) denote
the random variable thatmodels the remaining time from thecurrent time instant 119905 until entrance to the unoccupied zone119911119894 It is obvious that the randomvariables119884
119896(119905) for the various
unoccupied zones 119911119896 do not follow the same distributionWe
further assume that the node is capable ofmaking predictionsfor the time it takes to exceed the comfort threshold 119879
119888 To
this end let 119878on119894(119905) denote the time period that is required to
exceed 119879119888if the heater is turned on immediately Similarly
we define 119878off119894(119905) as the predicted time to reach the comfort
threshold if the heater remains off for a period 119863 and thenat the next decision epoch it is turned onThe relation of theaforementioned parameters is shown in Figure 1
Suppose now that at time 119905 the node of the unoccupiedzone 119911
119894has to take a decisionwhether to turn the zonersquos heater
on or remain in the off state At the decision epoch 119905 the riskto turn the heater on is119877on119894(119905)
= 119882119894DP 119878off
119894(119905) le 119884
119894(119905)
+ 119882119894(119884119894(119905) minus 119878
on119894(119905)) 119875 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)
(2)
That is if 119878off119894(119905) le 119884
119894(119905) there is plenty of time to reach the
comfort threshold even if we start heating at the next decisionepoch and therefore we consume unnecessarily 119882
119894119863 units
of energy (this is the case depicted in Figure 1) Similarly if119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) we waste 119882
119894(119884119894(119905) minus 119878
on119894(119905)) units of
energy Note that there is no risk of turning on the heater ifthe entrance to the zone happens sooner than it is predictedIn a similar fashion we calculate the risk of keeping the heateroff In this case
119877off119894(119905)
= 119862119894DP 119884
119894(119905) le 119878
on119894(119905)
+ 119862119894(119878
off119894(119905) minus 119884
119894(119905)) 119875 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)
(3)
4 Journal of Energy
Yi(t)
Soni (t)
Soffi (t)
Ti
t
DTs
t + DTs
Tc
Figure 1 Predicted time intervals related to the decision process
Using (2) and (3) the decision criterion takes the form
119877on119894(119905)
on≶
off119877off119894(119905) (4)
Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones
Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884
119894(119905) le 119878
on119894(119905) then the heater
is turned on if 119878off119894(119905) lt 119884
119894(119905) the heater remains off and
if 119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) the heater will be turned onoff
depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882
119894and 119862
119894
If 119884119894(119905) is random then criterion (4) needs to be slightly
modified as the value of 119884119894(119905) is unknown In this case we
replace 119884119894(119905) with its conditional expected value given that
119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) Therefore in criterion (4) we
substitute 119884119894(119905) by 119864[119884
119894(119905) | 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)]
From the previous discussion it becomes clear that thedecision process needs the estimates 119878on
119894(119905) 119878off119894(119905) and the
distribution of the process 119884119894(119905) The former is the subject of
the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32
31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone
is treated separately and is represented by a discrete timelinear time invariant state-space model That is
119909119896+1
= 119860119909119896+ 119861119906119896+ 119908119896 (5)
119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)
where 119910119896is the zone temperature at time 119896 and is in general
an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910
119896is scalar
V119896represents the measurement noise due to imperfections
of the sensor 119906119896is a vector of input measurements and its
dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909
119896is the state vector of the
process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the
zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863
from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances
In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model
After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace
Journal of Energy 5
identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone
Following the notation and the derivation in [10] wedefine the block Hankel matrices
119880119901= 1198800|119894minus1
=(
(
1199060
1199061sdot sdot sdot 119906
119895minus1
1199061
1199062sdot sdot sdot 119906
119895
119906119894minus1
119906119894sdot sdot sdot 119906119894+119895minus2
)
)
119880119891= 119880119894|2119894minus1
=(
(
119906119894
119906119894+1
sdot sdot sdot 119906119894+119895minus1
119906119894+1
119906119894+2
sdot sdot sdot 119906119894+119895
1199062119894minus1
1199062119894
sdot sdot sdot 1199062119894+119895minus2
)
)
(7)
which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880
119901on top of119880
119891we have
the block Hankel matrix
1198800|2119894minus1
= (1198800|119894minus1
119880119894|2119894minus1
) = (
119880119901
119880119891
) (8)
which can also be partitioned as
1198800|2119894minus1
= (1198800|119894
119880119894+1|2119894minus1
) = (
119880+
119901
119880minus
119891
) (9)
by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884
0|2119894minus1
119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices
119882119901= 1198820|119894minus1
= (1198800|119894minus1
1198840|119894minus1
) = (
119880119901
119884119901
)
119882+
119901= (
119880+
119901
119884+119901
)
(10)
The state sequence 119883119898is defined as
119883119898= (119909119898 119909
119898+1sdot sdot sdot 119909119898+119895minus1) (11)
and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830
and 119883119891= 119883119894 respectively With the state-space model (13)
(15) we associate the extended observability matrix Γ119894and the
reversed extended controllability matrix Δ119894 where
Γ119894=
((((
(
119862
119862119860
1198621198602
119862119860119894minus1
))))
)
Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)
(12)
The subspace identificationmethod relies on determining thestate sequence 119883
119891and the extended observability matrix Γ
119894
directly from the input-output data 119906119896119910119896and based on them
in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884
119891
along the row space of 119880119891on the row space of119882
119901
O119894= 119884119891119880119891119882119901= [119884119891119880perp
119891] [119882119901119880perp
119891]dagger
119882119901 (13)
where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is
119860119861perp= 119868 minus 119860119861
119879(119861119861119879)dagger
119861 (14)
As it is proved in [10] the matrix O119894is equal to the product
of the extended observability matrix and the ldquofuturerdquo statevector
O119894= Γ119894119883119891 (15)
Having in our disposal the matrixO119894we use its singular value
decomposition
O119894= (1198801 119880
2) (
11987810
0 0)(
119881119879
1
119881119879
2
) = 11988011198781119881119879
1(16)
and we identify the extended observation matrix Γ119894and the
state vector119883119891as
Γ119894= 119880111987812
1119879
119883119891= 119879minus111987812
1119881119879
1= Γdagger
119894O119894
(17)
where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation
One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection
O119894minus1
= 119884minus
119891119880minus119891119882dagger
119901= Γ119894minus1119883119894+1 (18)
where Γ119894minus1
is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1
in our case) Similar to the previous derivation we have
119883119894+1
= Γdagger
119894minus1O119894minus1
(19)
and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system
(
119883119894+1
119884119894|119894
) = (
119860 119861
119862 119863)(
119883119894
119880119894|119894
) (20)
After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909
0 they update
6 Journal of Energy
the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909
119905 The node bases its
decision on the estimates 119878on119894(119905) and 119878off
119894(119905) Recall that 119878on
119894(119905)
is the time needed to reach the comfort level 119879119888if the heater
is on That is
119878on119894(119905) = min ℓ 119910
119905+ℓge 119879119888 (21)
The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906
119905
(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909
119905repeats
(5) (6) until 119910119905+119896
exceeds the value 119879119888 Note that
119910119905+119896
= 119862119860119896119909119905+ 119862119860119896minus1
119861119906119905+ 119862119860119896minus2
119861119906119905+1
+ sdot sdot sdot
+ 119862119860119861119906119905+119896minus2
+ 119862119861119906119905+119896minus1
+ 119863119906119905+119896
(22)
and for 119906119905= 119906119905+1
= sdot sdot sdot = 119906119905+119896
we obtain
119910119905+119896
= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860
119896minus1) 119861119906119905+ 119863119906119905
= 119862119860119896119909119905+ 119862 (119868 minus 119860)
minus1(119868 minus 119860
119896) 119861119906119905+ 119863119906119905
(23)
If we set 119910119905+119896
= 119879119888and
119908 = 119879119888minus 119862 (119868 minus 119860)
minus1119861119906119905minus 119863119906119905 (24)
then (23) takes the form
119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)
minus1119860119896119861119906119905
= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)
minus1119881Λ119896119881minus1119861119906119905
(25)
where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881
minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off
119894(119905) In this case
however we first repeat (5) 119863 times with input-vector 119906119896
reflecting the fact that the heater is off and then starting fromthe new state 119909
119905+119863we estimate the time needed to reach 119879
119888if
the heater is on
32 On the Distribution of 119884119899(119905) The random variable 119884
119899(119905)
expresses the remaining time from the current time instant 119905until entrance to zone 119911
119899 That is we assume that occupants
move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911
119899 Each time the occupants visit a zone 119911
ℓthey
znz120001
Figure 2 Movement of occupants from zone 119911ℓto the adjacent
unoccupied zone 119911119899
stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865
ℓ(119909) We use the variable 119905
to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865
ℓ(119909) may be
given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911
ℓ
1199091 1199092 119909
119873 is available then
ℓ(119909) =
1
119873
119873
sum
119898=1
119868 (119909119898isin [0 119909]) (26)
where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones
119911ℓ 119911119899with zone 119911
ℓbeing occupied and 119911
119899unoccupied as it is
shown in Figure 2 We assume that the occupants remain tozone 119911
ℓfor a random period of time 119883
ℓwhich is distributed
according to 119865ℓ(119909) and then upon departure they enter zone
119911119899 In this case 119884
119899(119905) expresses the remaining waiting time
to zone 119911ℓand follows the residual life distribution which is
defined by
119877119884119899(119905)
(119910) = 1 minus 119877119884119899(119905)
(119910) = 119875 119883ℓgt 120591 + 119910 | 119883
ℓgt 120591
=119865ℓ(120591 + 119910)
119865ℓ(120591)
(27)
Next we consider the case that occupants are currently ina zone (call it 119911
0) and upon departure they visit in cascade
zones 1199111 1199112 119911
119899as it is shown in Figure 3 Then
119884119899(119905) = 119884
0+ 1198831+ 1198832+ sdot sdot sdot + 119883
119899minus1 (28)
where 1198840is the remaining time in zone 119911
0and 119883
119894is the
occupation period of zone 119911119894 It is well known that the
distribution of 119884119899(119905) is the convolution of the distributions of
the random variables 1198840 1198831 119883
119899minus1 That is
119865119884119899(119905)
(119910) = 119875 119884119899(119905) le 119910
= 1198770(119910) ⋆ 119865
1(119910) ⋆ 119865
2(119910) ⋆ sdot sdot sdot ⋆ 119865
119899minus1(119910)
(29)
with 1198770(119910) = 1 minus 119865
0(120591 + 119910)119865
0(120591) A more compact form is
obtained by considering a transform of the distributions that
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
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Submit your manuscripts athttpwwwhindawicom
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
4 Journal of Energy
Yi(t)
Soni (t)
Soffi (t)
Ti
t
DTs
t + DTs
Tc
Figure 1 Predicted time intervals related to the decision process
Using (2) and (3) the decision criterion takes the form
119877on119894(119905)
on≶
off119877off119894(119905) (4)
Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones
Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884
119894(119905) le 119878
on119894(119905) then the heater
is turned on if 119878off119894(119905) lt 119884
119894(119905) the heater remains off and
if 119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) the heater will be turned onoff
depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882
119894and 119862
119894
If 119884119894(119905) is random then criterion (4) needs to be slightly
modified as the value of 119884119894(119905) is unknown In this case we
replace 119884119894(119905) with its conditional expected value given that
119878on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905) Therefore in criterion (4) we
substitute 119884119894(119905) by 119864[119884
119894(119905) | 119878
on119894(119905) lt 119884
119894(119905) le 119878
off119894(119905)]
From the previous discussion it becomes clear that thedecision process needs the estimates 119878on
119894(119905) 119878off119894(119905) and the
distribution of the process 119884119894(119905) The former is the subject of
the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32
31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone
is treated separately and is represented by a discrete timelinear time invariant state-space model That is
119909119896+1
= 119860119909119896+ 119861119906119896+ 119908119896 (5)
119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)
where 119910119896is the zone temperature at time 119896 and is in general
an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910
119896is scalar
V119896represents the measurement noise due to imperfections
of the sensor 119906119896is a vector of input measurements and its
dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909
119896is the state vector of the
process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the
zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863
from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances
In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model
After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace
Journal of Energy 5
identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone
Following the notation and the derivation in [10] wedefine the block Hankel matrices
119880119901= 1198800|119894minus1
=(
(
1199060
1199061sdot sdot sdot 119906
119895minus1
1199061
1199062sdot sdot sdot 119906
119895
119906119894minus1
119906119894sdot sdot sdot 119906119894+119895minus2
)
)
119880119891= 119880119894|2119894minus1
=(
(
119906119894
119906119894+1
sdot sdot sdot 119906119894+119895minus1
119906119894+1
119906119894+2
sdot sdot sdot 119906119894+119895
1199062119894minus1
1199062119894
sdot sdot sdot 1199062119894+119895minus2
)
)
(7)
which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880
119901on top of119880
119891we have
the block Hankel matrix
1198800|2119894minus1
= (1198800|119894minus1
119880119894|2119894minus1
) = (
119880119901
119880119891
) (8)
which can also be partitioned as
1198800|2119894minus1
= (1198800|119894
119880119894+1|2119894minus1
) = (
119880+
119901
119880minus
119891
) (9)
by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884
0|2119894minus1
119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices
119882119901= 1198820|119894minus1
= (1198800|119894minus1
1198840|119894minus1
) = (
119880119901
119884119901
)
119882+
119901= (
119880+
119901
119884+119901
)
(10)
The state sequence 119883119898is defined as
119883119898= (119909119898 119909
119898+1sdot sdot sdot 119909119898+119895minus1) (11)
and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830
and 119883119891= 119883119894 respectively With the state-space model (13)
(15) we associate the extended observability matrix Γ119894and the
reversed extended controllability matrix Δ119894 where
Γ119894=
((((
(
119862
119862119860
1198621198602
119862119860119894minus1
))))
)
Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)
(12)
The subspace identificationmethod relies on determining thestate sequence 119883
119891and the extended observability matrix Γ
119894
directly from the input-output data 119906119896119910119896and based on them
in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884
119891
along the row space of 119880119891on the row space of119882
119901
O119894= 119884119891119880119891119882119901= [119884119891119880perp
119891] [119882119901119880perp
119891]dagger
119882119901 (13)
where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is
119860119861perp= 119868 minus 119860119861
119879(119861119861119879)dagger
119861 (14)
As it is proved in [10] the matrix O119894is equal to the product
of the extended observability matrix and the ldquofuturerdquo statevector
O119894= Γ119894119883119891 (15)
Having in our disposal the matrixO119894we use its singular value
decomposition
O119894= (1198801 119880
2) (
11987810
0 0)(
119881119879
1
119881119879
2
) = 11988011198781119881119879
1(16)
and we identify the extended observation matrix Γ119894and the
state vector119883119891as
Γ119894= 119880111987812
1119879
119883119891= 119879minus111987812
1119881119879
1= Γdagger
119894O119894
(17)
where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation
One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection
O119894minus1
= 119884minus
119891119880minus119891119882dagger
119901= Γ119894minus1119883119894+1 (18)
where Γ119894minus1
is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1
in our case) Similar to the previous derivation we have
119883119894+1
= Γdagger
119894minus1O119894minus1
(19)
and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system
(
119883119894+1
119884119894|119894
) = (
119860 119861
119862 119863)(
119883119894
119880119894|119894
) (20)
After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909
0 they update
6 Journal of Energy
the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909
119905 The node bases its
decision on the estimates 119878on119894(119905) and 119878off
119894(119905) Recall that 119878on
119894(119905)
is the time needed to reach the comfort level 119879119888if the heater
is on That is
119878on119894(119905) = min ℓ 119910
119905+ℓge 119879119888 (21)
The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906
119905
(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909
119905repeats
(5) (6) until 119910119905+119896
exceeds the value 119879119888 Note that
119910119905+119896
= 119862119860119896119909119905+ 119862119860119896minus1
119861119906119905+ 119862119860119896minus2
119861119906119905+1
+ sdot sdot sdot
+ 119862119860119861119906119905+119896minus2
+ 119862119861119906119905+119896minus1
+ 119863119906119905+119896
(22)
and for 119906119905= 119906119905+1
= sdot sdot sdot = 119906119905+119896
we obtain
119910119905+119896
= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860
119896minus1) 119861119906119905+ 119863119906119905
= 119862119860119896119909119905+ 119862 (119868 minus 119860)
minus1(119868 minus 119860
119896) 119861119906119905+ 119863119906119905
(23)
If we set 119910119905+119896
= 119879119888and
119908 = 119879119888minus 119862 (119868 minus 119860)
minus1119861119906119905minus 119863119906119905 (24)
then (23) takes the form
119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)
minus1119860119896119861119906119905
= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)
minus1119881Λ119896119881minus1119861119906119905
(25)
where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881
minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off
119894(119905) In this case
however we first repeat (5) 119863 times with input-vector 119906119896
reflecting the fact that the heater is off and then starting fromthe new state 119909
119905+119863we estimate the time needed to reach 119879
119888if
the heater is on
32 On the Distribution of 119884119899(119905) The random variable 119884
119899(119905)
expresses the remaining time from the current time instant 119905until entrance to zone 119911
119899 That is we assume that occupants
move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911
119899 Each time the occupants visit a zone 119911
ℓthey
znz120001
Figure 2 Movement of occupants from zone 119911ℓto the adjacent
unoccupied zone 119911119899
stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865
ℓ(119909) We use the variable 119905
to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865
ℓ(119909) may be
given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911
ℓ
1199091 1199092 119909
119873 is available then
ℓ(119909) =
1
119873
119873
sum
119898=1
119868 (119909119898isin [0 119909]) (26)
where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones
119911ℓ 119911119899with zone 119911
ℓbeing occupied and 119911
119899unoccupied as it is
shown in Figure 2 We assume that the occupants remain tozone 119911
ℓfor a random period of time 119883
ℓwhich is distributed
according to 119865ℓ(119909) and then upon departure they enter zone
119911119899 In this case 119884
119899(119905) expresses the remaining waiting time
to zone 119911ℓand follows the residual life distribution which is
defined by
119877119884119899(119905)
(119910) = 1 minus 119877119884119899(119905)
(119910) = 119875 119883ℓgt 120591 + 119910 | 119883
ℓgt 120591
=119865ℓ(120591 + 119910)
119865ℓ(120591)
(27)
Next we consider the case that occupants are currently ina zone (call it 119911
0) and upon departure they visit in cascade
zones 1199111 1199112 119911
119899as it is shown in Figure 3 Then
119884119899(119905) = 119884
0+ 1198831+ 1198832+ sdot sdot sdot + 119883
119899minus1 (28)
where 1198840is the remaining time in zone 119911
0and 119883
119894is the
occupation period of zone 119911119894 It is well known that the
distribution of 119884119899(119905) is the convolution of the distributions of
the random variables 1198840 1198831 119883
119899minus1 That is
119865119884119899(119905)
(119910) = 119875 119884119899(119905) le 119910
= 1198770(119910) ⋆ 119865
1(119910) ⋆ 119865
2(119910) ⋆ sdot sdot sdot ⋆ 119865
119899minus1(119910)
(29)
with 1198770(119910) = 1 minus 119865
0(120591 + 119910)119865
0(120591) A more compact form is
obtained by considering a transform of the distributions that
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of Energy 5
identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone
Following the notation and the derivation in [10] wedefine the block Hankel matrices
119880119901= 1198800|119894minus1
=(
(
1199060
1199061sdot sdot sdot 119906
119895minus1
1199061
1199062sdot sdot sdot 119906
119895
119906119894minus1
119906119894sdot sdot sdot 119906119894+119895minus2
)
)
119880119891= 119880119894|2119894minus1
=(
(
119906119894
119906119894+1
sdot sdot sdot 119906119894+119895minus1
119906119894+1
119906119894+2
sdot sdot sdot 119906119894+119895
1199062119894minus1
1199062119894
sdot sdot sdot 1199062119894+119895minus2
)
)
(7)
which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880
119901on top of119880
119891we have
the block Hankel matrix
1198800|2119894minus1
= (1198800|119894minus1
119880119894|2119894minus1
) = (
119880119901
119880119891
) (8)
which can also be partitioned as
1198800|2119894minus1
= (1198800|119894
119880119894+1|2119894minus1
) = (
119880+
119901
119880minus
119891
) (9)
by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884
0|2119894minus1
119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices
119882119901= 1198820|119894minus1
= (1198800|119894minus1
1198840|119894minus1
) = (
119880119901
119884119901
)
119882+
119901= (
119880+
119901
119884+119901
)
(10)
The state sequence 119883119898is defined as
119883119898= (119909119898 119909
119898+1sdot sdot sdot 119909119898+119895minus1) (11)
and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830
and 119883119891= 119883119894 respectively With the state-space model (13)
(15) we associate the extended observability matrix Γ119894and the
reversed extended controllability matrix Δ119894 where
Γ119894=
((((
(
119862
119862119860
1198621198602
119862119860119894minus1
))))
)
Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)
(12)
The subspace identificationmethod relies on determining thestate sequence 119883
119891and the extended observability matrix Γ
119894
directly from the input-output data 119906119896119910119896and based on them
in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884
119891
along the row space of 119880119891on the row space of119882
119901
O119894= 119884119891119880119891119882119901= [119884119891119880perp
119891] [119882119901119880perp
119891]dagger
119882119901 (13)
where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is
119860119861perp= 119868 minus 119860119861
119879(119861119861119879)dagger
119861 (14)
As it is proved in [10] the matrix O119894is equal to the product
of the extended observability matrix and the ldquofuturerdquo statevector
O119894= Γ119894119883119891 (15)
Having in our disposal the matrixO119894we use its singular value
decomposition
O119894= (1198801 119880
2) (
11987810
0 0)(
119881119879
1
119881119879
2
) = 11988011198781119881119879
1(16)
and we identify the extended observation matrix Γ119894and the
state vector119883119891as
Γ119894= 119880111987812
1119879
119883119891= 119879minus111987812
1119881119879
1= Γdagger
119894O119894
(17)
where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation
One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection
O119894minus1
= 119884minus
119891119880minus119891119882dagger
119901= Γ119894minus1119883119894+1 (18)
where Γ119894minus1
is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1
in our case) Similar to the previous derivation we have
119883119894+1
= Γdagger
119894minus1O119894minus1
(19)
and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system
(
119883119894+1
119884119894|119894
) = (
119860 119861
119862 119863)(
119883119894
119880119894|119894
) (20)
After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909
0 they update
6 Journal of Energy
the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909
119905 The node bases its
decision on the estimates 119878on119894(119905) and 119878off
119894(119905) Recall that 119878on
119894(119905)
is the time needed to reach the comfort level 119879119888if the heater
is on That is
119878on119894(119905) = min ℓ 119910
119905+ℓge 119879119888 (21)
The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906
119905
(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909
119905repeats
(5) (6) until 119910119905+119896
exceeds the value 119879119888 Note that
119910119905+119896
= 119862119860119896119909119905+ 119862119860119896minus1
119861119906119905+ 119862119860119896minus2
119861119906119905+1
+ sdot sdot sdot
+ 119862119860119861119906119905+119896minus2
+ 119862119861119906119905+119896minus1
+ 119863119906119905+119896
(22)
and for 119906119905= 119906119905+1
= sdot sdot sdot = 119906119905+119896
we obtain
119910119905+119896
= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860
119896minus1) 119861119906119905+ 119863119906119905
= 119862119860119896119909119905+ 119862 (119868 minus 119860)
minus1(119868 minus 119860
119896) 119861119906119905+ 119863119906119905
(23)
If we set 119910119905+119896
= 119879119888and
119908 = 119879119888minus 119862 (119868 minus 119860)
minus1119861119906119905minus 119863119906119905 (24)
then (23) takes the form
119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)
minus1119860119896119861119906119905
= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)
minus1119881Λ119896119881minus1119861119906119905
(25)
where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881
minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off
119894(119905) In this case
however we first repeat (5) 119863 times with input-vector 119906119896
reflecting the fact that the heater is off and then starting fromthe new state 119909
119905+119863we estimate the time needed to reach 119879
119888if
the heater is on
32 On the Distribution of 119884119899(119905) The random variable 119884
119899(119905)
expresses the remaining time from the current time instant 119905until entrance to zone 119911
119899 That is we assume that occupants
move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911
119899 Each time the occupants visit a zone 119911
ℓthey
znz120001
Figure 2 Movement of occupants from zone 119911ℓto the adjacent
unoccupied zone 119911119899
stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865
ℓ(119909) We use the variable 119905
to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865
ℓ(119909) may be
given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911
ℓ
1199091 1199092 119909
119873 is available then
ℓ(119909) =
1
119873
119873
sum
119898=1
119868 (119909119898isin [0 119909]) (26)
where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones
119911ℓ 119911119899with zone 119911
ℓbeing occupied and 119911
119899unoccupied as it is
shown in Figure 2 We assume that the occupants remain tozone 119911
ℓfor a random period of time 119883
ℓwhich is distributed
according to 119865ℓ(119909) and then upon departure they enter zone
119911119899 In this case 119884
119899(119905) expresses the remaining waiting time
to zone 119911ℓand follows the residual life distribution which is
defined by
119877119884119899(119905)
(119910) = 1 minus 119877119884119899(119905)
(119910) = 119875 119883ℓgt 120591 + 119910 | 119883
ℓgt 120591
=119865ℓ(120591 + 119910)
119865ℓ(120591)
(27)
Next we consider the case that occupants are currently ina zone (call it 119911
0) and upon departure they visit in cascade
zones 1199111 1199112 119911
119899as it is shown in Figure 3 Then
119884119899(119905) = 119884
0+ 1198831+ 1198832+ sdot sdot sdot + 119883
119899minus1 (28)
where 1198840is the remaining time in zone 119911
0and 119883
119894is the
occupation period of zone 119911119894 It is well known that the
distribution of 119884119899(119905) is the convolution of the distributions of
the random variables 1198840 1198831 119883
119899minus1 That is
119865119884119899(119905)
(119910) = 119875 119884119899(119905) le 119910
= 1198770(119910) ⋆ 119865
1(119910) ⋆ 119865
2(119910) ⋆ sdot sdot sdot ⋆ 119865
119899minus1(119910)
(29)
with 1198770(119910) = 1 minus 119865
0(120591 + 119910)119865
0(120591) A more compact form is
obtained by considering a transform of the distributions that
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
6 Journal of Energy
the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909
119905 The node bases its
decision on the estimates 119878on119894(119905) and 119878off
119894(119905) Recall that 119878on
119894(119905)
is the time needed to reach the comfort level 119879119888if the heater
is on That is
119878on119894(119905) = min ℓ 119910
119905+ℓge 119879119888 (21)
The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906
119905
(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909
119905repeats
(5) (6) until 119910119905+119896
exceeds the value 119879119888 Note that
119910119905+119896
= 119862119860119896119909119905+ 119862119860119896minus1
119861119906119905+ 119862119860119896minus2
119861119906119905+1
+ sdot sdot sdot
+ 119862119860119861119906119905+119896minus2
+ 119862119861119906119905+119896minus1
+ 119863119906119905+119896
(22)
and for 119906119905= 119906119905+1
= sdot sdot sdot = 119906119905+119896
we obtain
119910119905+119896
= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860
119896minus1) 119861119906119905+ 119863119906119905
= 119862119860119896119909119905+ 119862 (119868 minus 119860)
minus1(119868 minus 119860
119896) 119861119906119905+ 119863119906119905
(23)
If we set 119910119905+119896
= 119879119888and
119908 = 119879119888minus 119862 (119868 minus 119860)
minus1119861119906119905minus 119863119906119905 (24)
then (23) takes the form
119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)
minus1119860119896119861119906119905
= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)
minus1119881Λ119896119881minus1119861119906119905
(25)
where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881
minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off
119894(119905) In this case
however we first repeat (5) 119863 times with input-vector 119906119896
reflecting the fact that the heater is off and then starting fromthe new state 119909
119905+119863we estimate the time needed to reach 119879
119888if
the heater is on
32 On the Distribution of 119884119899(119905) The random variable 119884
119899(119905)
expresses the remaining time from the current time instant 119905until entrance to zone 119911
119899 That is we assume that occupants
move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911
119899 Each time the occupants visit a zone 119911
ℓthey
znz120001
Figure 2 Movement of occupants from zone 119911ℓto the adjacent
unoccupied zone 119911119899
stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865
ℓ(119909) We use the variable 119905
to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865
ℓ(119909) may be
given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911
ℓ
1199091 1199092 119909
119873 is available then
ℓ(119909) =
1
119873
119873
sum
119898=1
119868 (119909119898isin [0 119909]) (26)
where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones
119911ℓ 119911119899with zone 119911
ℓbeing occupied and 119911
119899unoccupied as it is
shown in Figure 2 We assume that the occupants remain tozone 119911
ℓfor a random period of time 119883
ℓwhich is distributed
according to 119865ℓ(119909) and then upon departure they enter zone
119911119899 In this case 119884
119899(119905) expresses the remaining waiting time
to zone 119911ℓand follows the residual life distribution which is
defined by
119877119884119899(119905)
(119910) = 1 minus 119877119884119899(119905)
(119910) = 119875 119883ℓgt 120591 + 119910 | 119883
ℓgt 120591
=119865ℓ(120591 + 119910)
119865ℓ(120591)
(27)
Next we consider the case that occupants are currently ina zone (call it 119911
0) and upon departure they visit in cascade
zones 1199111 1199112 119911
119899as it is shown in Figure 3 Then
119884119899(119905) = 119884
0+ 1198831+ 1198832+ sdot sdot sdot + 119883
119899minus1 (28)
where 1198840is the remaining time in zone 119911
0and 119883
119894is the
occupation period of zone 119911119894 It is well known that the
distribution of 119884119899(119905) is the convolution of the distributions of
the random variables 1198840 1198831 119883
119899minus1 That is
119865119884119899(119905)
(119910) = 119875 119884119899(119905) le 119910
= 1198770(119910) ⋆ 119865
1(119910) ⋆ 119865
2(119910) ⋆ sdot sdot sdot ⋆ 119865
119899minus1(119910)
(29)
with 1198770(119910) = 1 minus 119865
0(120591 + 119910)119865
0(120591) A more compact form is
obtained by considering a transform of the distributions that
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
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Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Wind EnergyJournal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of Energy 7
zn
z0 z1
Figure 3 Movement of occupants in cascade from zone 1199110to zone
119911119899
is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865
119883(119909) of a nonnega-
tive random variable119883 is defined by
119865119883(119904) = int
infin
0
119890119904119909119889119865119883(119909) (30)
(29) takes the form
119865119884119899(119905)
(119904) = 1198770(119904) sdot 119865
1(119904) sdot 119865
2(119904) sdot sdot 119865
119899minus1(119904) (31)
In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865
119884119899(119905)(119910)
For example if the occupation periods 119883119894 119894 = 0 119899 minus 1
are exponentially distributed with corresponding parameters120582119894 then 119884
0is also exponentially distributed with parameter
1205820due to the memoryless property of the exponential
distribution and the probability density function of 119884119899(119905) is
(assuming that the parameters 120582119894are all distinct)
119891119884119899(119905)
(119909) =
119899minus1
sum
119894=0
1205820sdot sdot sdot 120582119899minus1
prod119899minus1
119895=0119895 =119894(120582119895minus 120582119894)
exp (minus119909120582119894) (32)
(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891
119884119899(119905)(119909) does not exist or it is difficult to be
obtained we resort to the saddlepoint approximation [11] toinvert 119865
119884119899(119905)(119904) To this end for a random variable 119883 with
cumulative distribution function (cdf) 119865119883(119909) we define the
cumulant generating function (cgf) 119870119883(119904) as the logarithm
of the corresponding mfg 119865119883(119904) that is
119870119883(119904) = log119865
119883(119904) (33)
Then the saddlepoint approximation of the probability den-sity function (pdf) 119891
119883(119909) is
119891119883(119909)
asymp (1
212058711987010158401015840119909( (119909))
)
12
exp 119870119883( (119909)) minus (119909) 119909
(34)
where (119909) is the solution to the saddlepoint equation
1198701015840
119883(119904) = 119909 (35)
Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865
119883(119904) Note that wemay
use the empirical mgf 119883(119904) instead so that a nonparametric
estimation of the pdf 119891119883(119909) is possible
The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901
119894119895 Given a transition from zone 119911
119894to
zone 119911119895 the occupation time in 119911
119894has distribution function
119865119894119895(119909) The matrix Q(119909) with elements 119876
119894119895(119909) = 119901
119894119895119865119894119895(119909)
is the so called semi-Markov kernel The occupation timedistribution in zone 119911
119894independent of the state to which a
transition is made is 119865119894(119909) = sum
119895119876119894119895(119909) Consider now the
case that at time 119905 occupants exist in zone 119911119894and the node of
zone 119911119899has tomake a decision onwhether to start heating the
zone or not For the decision process the distribution of timeuntil entrance to zone 119911
119899 119884119899(119905) is needed 119884
119899(119905) is the sum
of two terms The first term is the residual occupation timeat zone 119911
119894and the second term is the time to reach 119911
119899upon
departure from zone 119911119894 The distribution 119877
119894(119909) of the first
term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911
119894
Thus if zone 119911119894neighbors119873
119894zones namely 119911
119896 119896 = 1 119873
119894
then the time to reach zone 119911119899upon departure is distributed
according to
119866119894119899(119909) =
119873119894
sum
119896=1
119901119894119896119867119896119899(119909) (36)
where119867119896119899(119909) is the distribution of the first passage from zone
119911119896to zone 119911
119899 Pyke [12] andMason [13 14] provided solutions
for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867
119894119895(119909) is given by
119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]
minus1)119889minus1
(37)
where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865
119894119895(119909) Notation 119872
119889denotes the
matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ
119894119895= (minus1)
119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of
119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get
1198671119899(119904) =
Δ1198991(119904)
Δ119899119899(119904)
(38)
By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
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Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
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Nuclear InstallationsScience and Technology of
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
8 Journal of Energy
I
III
II
IV
V
IXVI
VIII
VII
To
To
To
To
Figure 4 Multizone model
In summary
119865119884119899(119905)
(119904) = 119877119894(119904) sdot 119866
119894119899(119904) = 119877
119894(119904) sdot (
119873119894
sum
119896=1
119901119894119896119867119896119899(119904)) (39)
with 119867119896119899(119904) computed using (38) Next the saddlepoint
approximation technique can be used to calculate the density119891119884119899(119905)
(119909)We have tomention that the process of estimating the first
passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865
119894119895(119904)
is obtained Having in our disposal the transforms 119865119894119895(119904) we
can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867
119894119895(119904) using (38) Finally numerical
methods may be used to invert these transforms in order toobtain the necessary probability density functions
4 Simulation Results
We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])
Table 1 Training and testing parameters for outer temperaturemodel and heat gain
Walterrsquos model Heat gainPhase 119879max 119879min 119879
119898119902(119905)
Training 15 2 7 600Testing 16 1 6 700
Table 2 Training and testing zonesrsquo target temperatures
Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21
5 10 15 20 250Time of day (h)
0
5
10
15
Tem
pera
ture
To
(∘C)
Figure 5 Empirical estimation of 119879119900
In Figure 4 119879119900denotes the outside temperature which
is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879
119898
need to be provided in order to estimate 119879119900 As an example if
119879max = 15∘C 119879min = 2
∘C and 119879119898= 7∘C the daily variations
of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm
41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879
119904=
1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892
to those of Table 2 (first row) with a margin equal to119879119898119892
= 1∘C That is for a specific zone the heater is on until
temperature119879119905119892+119879119898119892
is reached After this point the heater isturned off until temperature hits the lower threshold119879
119905119892minus119879119898119892
and the whole process is repeated
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
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FuelsJournal of
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Industrial EngineeringJournal of
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
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Nuclear InstallationsScience and Technology of
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of Energy 9
0 1 2 3 4 5 6 7 8 9 10 11Order
Zone VZone I
Zone II
10minus15
10minus10
10minus5
100
Sing
ular
val
ues
Figure 6 Singular values of zones I II and V
Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2
Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase
We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
02
4
6
8
10
12
14
16
18
20
Time (sec)
Zone I (real)
Zone I (predicted)
Zone IV (real)
Zone IV (predicted)
Tem
pera
ture
Figure 7 Real and predicted temperature of zones I and IV
zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model
Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is
119891119883(119909) =
120573120572
Γ (120572)119909120572minus1
119890minus120573119909
(40)
with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862
119894
The results are averages over 200 runs of the simulation Thecase of 119862
119894= 0 is the full reactive case where heating of
zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862
119894 the decision process
acts proactively by spending energy in order to reduce thediscomfort level
Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone
IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
10 Journal of Energy
0
100
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 8 Total comfort cost versus119862119894 The occupation time of zone
I is Gamma distributed
15
2
25
3
Tota
l ene
rgy
cost
(kW
h)
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
00
Comfort (C)
E[X] = 3600
E[X] = 7200
Figure 9 Total energy cost versus 119862119894 The occupation time of zone
I is Gamma distributed
In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty
The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is
119891119883(119909) =
120572119909120572
119898
119909120572+1 119909 ge 119909
119898
0 119909 lt 119909119898
(41)
Zone I
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Zone IV(C = 0)
Zone IV(C = 10000)
Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0
1000 2000 3000 4000 5000 6000 70000Time (sec)
0
05
1
15
2
25
3
35Pa
reto
times10
minus3
xm = 300
xm = 900
xm = 1800
xm = 3600
Figure 11 Pareto pdf for various values of 119909119898
with expected value
119864 [119883] =
120572119909119898
120572 minus 1 120572 gt 1
infin 120572 le 1
(42)
Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=
300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases
Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862
119894 As it is observed a 50 improvement on the
comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909
119898= 300) which is due
to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909
119898
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of Energy 11
200
300
400
500
600
700
800
900
Tota
l com
fort
cost
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
xm = 300xm = 900
xm = 3600
Figure 12 Total comfort cost for Pareto distributed occupationperiod
xm = 300
xm = 900
xm = 1800
xm = 3600
1
15
2
25
Tota
l ene
rgy
cost
(kW
h)
100 200 300 400 500 600 700 800 900 10000Comfort gain (C)
Figure 13 Total energy cost for Pareto distributed occupationperiod
and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13
Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution
Table 3 Parameters used in cascade movement scenario
Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20
Table 4 Energy and comfort costs for constant occupancy times
Comfort weight 119862 500 1000 1500 2000
119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020
119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025
I
IV
V
II
III VI
VII
VIII
IX
Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)
The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on
Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
12 Journal of Energy
Fixed proactivecomfort cost
Energy cost
Comfort cost
1000
1500
2000
2500
3000
3500
4000
4500
Tota
l com
fort
cost
600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)
8
85
9
95
10
105
11
Tota
l ene
rgy
cost
(kW
h)
proactiveFixed
energy cost
Figure 15 Total energy and comfort costs (cascade movement sce-nario)
Zone I
Zone IX
Zone II
Zone VIII
Zone V
Zone IV
2
4
6
8
10
12
14
16
18
20
Tem
pera
ture
5000 10000 150000Time (sec)
Figure 16 Temperature profiles of zones (cascade movement sce-nario)
Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process
5 Conclusions and Future Work
Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone
to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique
Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862
119894may depend
on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov
[2] US Energy Information Administration June 2015 httpwwweiagov
[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013
[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013
[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011
[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013
[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014
[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014
[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of Energy 13
[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996
[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954
[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961
[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953
[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956
[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005
[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015
[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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