Research Article A Novel Multiobjective Optimization

Research ArticleA Novel Multiobjective Optimization Algorithm forHome Energy Management System in Smart Grid

Yanyu Zhang123 Peng Zeng1 Shuhui Li4 Chuanzhi Zang1 and Hepeng Li1

1Key Laboratory of Networked Control System Shenyang Institute of Automation Chinese Academy of SciencesShenyang 110016 China2School of Computer amp Information Engineering Henan University Kaifeng 475004 China3University of Chinese Academy of Sciences Beijing 100049 China4Department of Electrical amp Computer Engineering University of Alabama Tuscaloosa AL 35487 USA

Correspondence should be addressed to Yanyu Zhang zyyhenueducn and Peng Zeng zpsiacn

Received 21 July 2014 Revised 21 December 2014 Accepted 18 January 2015

Academic Editor Hari M Srivastava

Copyright copy 2015 Yanyu Zhang 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

Demand response (DR) is an effective method to lower peak-to-average ratio of demand facilitate the integration of renewableresources (eg wind and solar) and plug-in hybrid electric vehicles and strengthen the reliability of power system In smart gridimplementing DR through home energy management system (HEMS) in residential sector has a great significance However analgorithm that only optimally controls parts of HEMS rather than the overall system cannot obtain the best results In additionsingle objective optimization algorithm that minimizes electricity cost cannot quantify userrsquos comfort level and cannot take atradeoff between electricity cost and comfort level conveniently To tackle these problems this paper proposes a framework ofHEMS that consists of grid load renewable resource (ie solar resource) and battery In this framework a user has the ability tosell electricity to utility grid for revenue Different comfort level indicators are proposed for different home appliances according totheir characteristics and user preferences Based on these comfort level indicators this paper proposes amultiobjective optimizationalgorithm for HEMS that minimizes electricity cost and maximizes userrsquos comfort level simultaneously Simulation results indicatethat the algorithm can reduce userrsquos electricity cost significantly ensure userrsquos comfort level and take a tradeoff between the costand comfort level conveniently

1 Introduction

Over the past several decades with the development ofeconomy the electricity demandof thewholeworld increaseddramatically which makes the power system encounterstress conditions frequently At the same time the pressureof natural resources and environmental problems haveattracted great attention to incorporation of clean renewablegeneration sources such as wind and solar power [1 2] How-ever due to the variable and uncertainty characteristics theincreasing penetration of renewable generation sources intro-duces further challenges to the power system [3] In additionto these factors the emergence of a large number of plug-inhybrid electric vehicles (PHEVs) has the potential toincrease peak demand significantly overload distributionlines degenerate distribution transformers and threaten the

reliability of the power system [4ndash6] To alleviate theseproblems demand response (DR) plays an important role

DR is an effective method to lower peak-to-averageratio of demand improve the utilization of power assetsstrengthen the reliability of power system and facilitate theintegration of renewable resources and PHEVs [7ndash11] DR hasbeen successfully applied in industrial and commercial sec-tors On the other hand in residential sector it is difficult toimplement DR effectively because of little power consump-tion of single residential customer large numbers of customersand the lack of corresponding technologies and incentives inconventional power grid However take theUSA as exampleaccording to the report of the US Energy InformationAdministration (EIA) the residential sector consumes 20of the total energy supply and dominates 60 of peak load incertain parts of the country [12] From this point of view

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 807527 19 pageshttpdxdoiorg1011552015807527

2 Mathematical Problems in Engineering

implementing DR in the residential sector has a great signif-icance

In recent years with the emergence of smart grid DRin residential sector is getting more and more attentionDifferent from the conventional grid the smart grid has two-way energy and information flows which provides the base toimplement DR in the residential sector Residents can controlthe operations of their home appliances batteries PHEVsand distributed generation through home energy manage-ment system (HEMS) Optimal scheduling algorithm is oneof the key components ofHEMS and a hot topic in smart grid

Many papers have been published about the optimalscheduling algorithms in HEMSMost papers [1 13ndash21] focuson minimizing usersrsquo electricity usage cost while retainingusersrsquo comfort level at a predefined range For example in[16] the authors developed a smart-grid strategy thatmatchesrenewable energy generation (ie wind and solar power)withthe heating ventilation and air conditioning (HVAC) loadIn [19] the thermostatically controlled household loads arescheduled based on electricity price and energy consumptionforecasts by considering usersrsquo comfort level to minimizeelectricity cost In order to minimize the energy paymentthe authors in [20] proposed amultistage optimization-basedreal-time residential load management algorithm that takesinto account load uncertainty These algorithms are singleobjective optimization algorithms where user comfort istransformed into a set of constraintsThese algorithms do notquantitatively consider usersrsquo comfort level during operationHowever from the usersrsquo point of view in addition to mon-etary expense high comfort level is another objective thatthey pursue Unfortunately the objectives of cost and comfortlevel are conflicting Therefore compared with single objec-tive optimization algorithms that only consider energy pay-ment the multiobjective optimization algorithm that notonly minimizes monetary expense but also maximizes com-fort level simultaneously is more attractive and natural

For the moment the multiobjective optimal schedulingalgorithm for HEMS has not been well investigated In [22]the authors proposed an optimal residential energy consump-tion scheduling framework to achieve a tradeoff betweenminimizing payment and waiting time for the operation ofeach household appliance In [22] waiting time is used toindicate userrsquos comfort level Although thismethod is suitableto washing machine (WM) clothes dryer (CD) and dish-washer (DW) it is meaningless for HVAC and electric waterheater (EWH) where the userrsquos concern is temperature In[23] the authors developed a multiobjective air condition-ing control algorithm based on immune clonal selectionprogramming to determine the day-ahead 24 h temperatureschedule for air conditioning In addition to electricity costthe expected error for the desired indoor temperature isintroduced as a user comfort level indicator and optimized asan objective of the algorithm However this indicator cannotreflect usersrsquo temperature preferences in different seasons

In smart grid a residential customer not only hashome appliances that consume electricity but also may havedistributed renewable generation (eg wind and PV) andbatteries which may have extra electricity sold to the grid forrevenue as presented [1 14] However the optimal scheduling

algorithms proposed in those papers did not consider thepower distribution relationships among loads batteries dis-tributed renewable generation and grid For example thealgorithms in [17ndash23] only schedule the operation of loadsThe systems proposed in [13ndash16] include load storage systemand distributed renewable generation where the ability to sellelectricity to the grid is not considered Although the systemdescribed in [1 14] consists of loads batteries and renewablegeneration and the user has the ability to sell electricity tothe grid the power distribution relationships among differentcomponents of HEMS are not thoroughly investigated

This paper proposes a HEMS framework that includesloads batteries and renewable generation interconnectedwith the grid through a smart mater In this framework theuser can sell the electricity generated by renewable sources orstored in batteries to the grid for profit A set of comfort levelindicators are proposed for different home appliances Basedon these indicators a multiobjective optimal schedule modelis built which minimizes monetary expense and maximizesuser comfort level simultaneously An improved hybriddiscrete particle swarm optimization (PSO) is employed tosolve themodel and amultiobjective optimization algorithmforHEMS is proposedThe algorithm schedules the operationof home appliances batteries and renewable generation (iePV) as well as the optimal power distribution among loadsbatteries renewable generation and grids

The rest of this paper is organized as follows Section 2proposes a framework of HEMS and presents models andconstraints of home appliances and batteries Section 3 des-cribes the comfort level indicators for different home appli-ances and the formulation of the HEMS optimization prob-lem including the optimization objectives and constraintsSection 4 presents a multiobjective optimization algorithmbased on an improved hybrid discrete PSO techniqueSection 5 provides a case study and compares the simulationresults of the proposed algorithm with other algorithmsproposed in the literature Section 6 concludes the paper

2 System Model

21 DR Abilities of Different Home Appliances In terms ofthe schedulability home appliances can be divided into twocategories schedulable appliances (SAs) and nonschedulableappliances (NSAs) SAs refer to the appliances whose opera-tions can be scheduled to a certain extent without reducinguserrsquos comfort level such as WM CD DW HVAC andEWH On the contrary NSAs are the appliances whose oper-ations must be started immediately when users need theirservices such as computer television microwave oven andlight NSAs are also called critical loads SAs can be furtherclassified into two groups interruptible loads and uninter-ruptible loads For example PHEVs are interruptible loadsUninterruptible loads refer to the appliances whose opera-tions can be delayed but after being started theymust be keptworking until the tasks are completed such as WM and DW

Although several papers investigate the controllingmeth-ods for low power consumption appliances such as refrig-erators and coffee maker they are not evident for DR due

Mathematical Problems in Engineering 3

Controller

Internet

BAU

Smart grid

Energy flowInformation flow

PHEV

Smart meter

PV

Battery

Schedulable appliancesNonschedulable appliances

Figure 1 Framework of the HEMS in smart grid

to their low power consumption compared with the overallhousehold power consumption [24] In this paper we chooseHVAC EWH PHEV WM CD and DW as schedulingobjects to implement DR in household

Battery is another important kind of interruptible SAThe battery can be used to store surplus energy generated byrenewable sources (eg PV) or the energy from the gridwhenthe electricity price is low The stored electric energy can besupplied to loads or sold back to the grid for revenuewhen theprice is high Therefore battery is another scheduling objectin this paper

22 Framework of the HEMS The framework of HEMSproposed in this paper is shown in Figure 1 As mentionedabove the framework includes loads (ie NSAs SAs andPHEV) home energy storage battery and renewable gener-ation (ie PV) It must be pointed out that PHEV is a specialkind of loads Different from other loads a PHEV not onlycan act as a load that absorbs electricity from the grid orrenewable generation but also in some special cases acts as apower source through the vehicle-to-grid (V2G) or vehicle-to-home (V2H) function [25 26] In this paper the PHEV isconsidered as a load only

The two-way energy and information exchange betweenHEMS and the grid is realized through a smart meter Thesmartmeter is responsible for transmitting consumption datafromhome appliances to the utility company and it also relaysthe electricity price signal from the utility company backto a controller For the moment there are different time-based pricing tariffs including time-of-use (TOU) tariffcritical peak pricing (CPP) tariff and real-time pricing (RTP)tariff [27] This paper uses day-ahead RTP in other wordsthe electricity price of every hour is published by the utilitycompany to residents one day ahead

The controller is the kernel of HEMS Each HEMScomponent communicates with the controller over a homearea network (HAN) which can be realized through Zigbeecommunication technology [28] The controller connectswith the Internet by a broadband access unit (BAU) and getsweather information including outdoor temperature forecastvia the Internet Through the controller the user can setparameters and configure the system

In the framework the schedulable appliances (ie HVACEWHWM CD and DW) PHEV and battery are scheduledby the controller to minimize the electricity cost and maxi-mize user comfort level according to the electricity price userpreferences and PV power output

23 Power Distribution Relationships of the HEMS In theproposed framework the power distribution relationshipsamong loads battery PV and grid are shown in Figure 2 Inthis figure the components enclosed by the dash line that isloads PV and battery are owned by the user and they are thescheduling objects of the proposed algorithm 119901G2L

119905is the

power that is transmitted from grid to loads at time 119905 Sim-ilarly 119901P2L

119905and 119901

B2L119905

are the power transmitted from PV andbattery to loads 119901G2B

119905and 119901

P2B119905

represent the power transmit-ted to the battery from grid and PV 119901P2G

119905 and 119901

B2G119905

stand forthe power transmitted to grid from PV and battery at time 119905respectivelyThe values of these power distributions are equalto or greater than zero

Under the RTP scheme the scheduling of loads andpower distribution among different components affect thetotal monetary expense As a result besides load manage-ment optimal control of the power distributions among gridloads battery and PV is another important issue for theoptimization goals This problem will be discussed in detailin Section 3


Grid Loads

PV BatterypP2Bt

pB2Lt

pG2Lt

pB2Gt

pG2Bt

pP2Lt

pP2Gt

Figure 2 Power distribution relationships of the HEMS in smartgrid

24 Models and Constraints of Individual Appliances Themodel of each appliance used in this paper and the corre-sponding constraints are described as follows

241 HVAC TheHVACmodel developed in [29] is adoptedin this paper In this model the room temperature is calcu-lated as

119879Room119905+1

= 119879Room119905

+ Δ119905 sdot119866119905

Δ119888+ Δ119905 sdot

119862HVACΔ119888

sdot 119878HVAC119905

(1)

where 119879Room119905

and 119879Room119905+1

are the room temperatures (∘F) intime slots 119905 and 119905+1 respectivelyΔ119905 is the length of time slot 119905in hours 119866

119905is the heat gain rate of the house in Btuh during

time slot 119905 Δ119888 is the energy needed to change the temper-ature of the air in the room by 1∘F (Btu∘F) 119862HVAC is thecoolingheating capacity of HVAC in Btuh positive for heat-ing and negative for cooling 119878HVAC

119905is the working status of

HVAC in time slot 119905 1 for on and 0 for off For simplicity itis assumed that the HVAC runs with its rated power 119875HVAC

(kW) when it is turned onIn order to ensure the comfort preference HVAC should

regulate the room temperature within the certain rangeprespecified by the user This constraint is depicted in

119879Roommin le 119879

Room119905

le 119879Roommax (2)

where 119879Roommin and 119879

Roommax stand for the minimum and maxi-

mum room temperatures respectively

242 EWH The model presented in [2] is employed in thispaper to calculate the temperature of hot water inside theEWH tank

119879EWH119905+1

= 119879EWH119905

sdot 119890minus(1(119877

1015840

119905sdot119862))sdotΔ119905

+ 119866EWH

sdot 1198771015840

119905sdot 119879

EWHenv119905

+ 119861119905sdot 1198771015840

119905sdot 119879

EWHin119905

+ 119876119905sdot 1198771015840

119905 sdot [1 minus 119890

minus(1(1198771015840

119905sdot119862))sdotΔ119905

]

(3)

where 119879EWH119905

and 119879EWH119905+1

are the hot water temperatures (∘F)inside the EWH tank in time slots 119905 and 119905 + 1 respectively

119879EWHenv119905

and 119879EWHin119905

are the temperatures (∘F) of ambientenvironment and inlet water in time slot 119905 respectively 119862 isthe equivalent thermal mass (Btu∘F) Δ119905 is the length of atime slot in hours 119866EWH is the ratio of the surface area ofEWH to the thermal resistance of the tank 119861

119905 1198771015840119905 and119876

119905are

calculated according to the following equations respectively

119861119905= 119889water times 119865

119905times 119862119901 (4)

1198771015840

119905=

1

(119866EWH + 119861119905) (5)

119876119905= 34121 times 10

minus8times 119875

EWHtimes 119878

EWH119905

(6)

where 119889water is the density of water 119862119901is the specific heat

of water and 119865119905represents hot water flow rate in time slot 119905

119875EWH is the rated power of the EWH (kW) and 119878

EWH119905

is thestatus of the EWH in time slot 119905

The EWH has two statuses on and off When turnedon it runs with the rated power and the hot water tem-perature increases exponentially otherwise the temperaturedecreases Similar to HVAC the EWH should maintainthe hot water temperature within a prespecified range[119879

EWHmin 119879

EWHmax ] where 119879EWH

min and 119879EWHmax are the minimum and

maximumwater temperatures set by a user respectivelyThisconstraint is expressed by

119879EWHmin le 119879

EWH119905

le 119879EWHmax (7)

243 PHEV The battery state-of-charge (SOC) relationshipof PHEV [29] is calculated according to

SOCPHEV119905+1

= SOCPHEV119905

+ 119901PHEV119905

sdotΔ119905

119862PHEVsize

(8)

where SOCPHEV119905

and SOCPHEV119905+1

are the PHEV battery SOC intime slots 119905 and 119905 + 1 respectively 119862PHEV

size is the rated capacityof PHEV battery (kWsdoth) Δ119905 is the length of a time slot inhours 119901PHEV

119905is the charging power of PHEV in time slot 119905

(kW) in this paper it is assumed that the charging power isconstant that is the rated charging power of PHEV 119875PHEV

(kW) Therefore 119901PHEV119905

is determined by

119901PHEV119905

= 119875PHEV

sdot 119878PHEV119905

(9)

where 119878PHEV119905

is the charging status of PHEV in time slot 119905 with1 representing on and 0 representing off

To protect the PHEV battery from damage thebattery SOC should be maintained in a safe range of[SOCPHEV

min SOCPHEVmax ] where SOCPHEV

min and SOCPHEVmax are

the minimum and maximum allowable PHEV battery SOCTo satisfy userrsquos transportation demand the PHEV batteryshould be charged each day with a SOC equal to or greaterthan the prespecified SOC SOCPHEV

final These two constraintsare depicted in the following respectively

SOCPHEVmin le SOCPHEV

119905le SOCPHEV

max (10)

SOCPHEVfinal le SOCPHEV

119873slotle SOCPHEV

max (11)


where SOCPHEV119873slot

is the PHEV battery SOC in the final timeslot (ie time slot 119873slot) of the scheduling horizon 119873slotdenotes the total number of the time slots in the schedulinghorizon

244 WM CD and DW In this paper WM CD and DWare taken as noninterruptible appliances in other wordsthese appliances have two statuses on and off Once they areturned on they must keep working with the rated poweruntil their tasks are completedThe task starting time and thenumber of time slots that are needed for completing the taskof each appliance are set by the user They should meet theconstraints

1 le 119873119886

start le 119873slot minus 119873119886

task

119878119886

119905=

1 119905 = 119873119886

start 119873119886

start + 1 119873119886

start + 119873119886

task minus 1

0 otherwise

119873slot

sum

119905=1

119878119886

119905= 119873119886

task

(12)

where 119886 isin WMCDDW119873119886start is the time slot in which thetask of home appliance 119886 is started119873119886task is the number of timeslots that are needed to complete the task of appliance 119886 119878119886

119905

is the working status of appliance 119886 in time slot 119905 with 1representing on and 0 representing off

The power of appliance 119886 in time slot 119905 119901119886119905 is calculated

as

119901119886

119905= 119875119886sdot 119878119886

119905 (13)

where 119875119886 is the rated power of appliance 119886

245 Battery In order to prevent the home energy stor-age battery from overcharge and overdischarge the bat-tery SOC should always be maintained within a specifiedrange [SOCBat

min SOCBatmax] where SOCBat

min and SOCBatmax are

the minimum and maximum allowable SOC of the batteryrespectively This constraint is depicted in

SOCBatmin le SOCBat

119905le SOCBat

max (14)

The battery SOC associatedwith the charge and dischargeof the battery is calculated by the following equationsrespectively

SOCBat119905+1

= SOCBat119905

+(119901

Batch119905

sdot Δ119905 sdot 120578Batch )

119862Batsize

(15)

SOCBat119905+1

= SOCBat119905

minus(119901

Batdisch119905

sdot Δ119905)

(120578Batdisch sdot 119862Batsize)

(16)

where 119862Batsize is the rated battery capacity (kWsdoth) SOCBat

119905and

SOCBat119905+1

are the battery SOC in time slots 119905 and 119905 + 1 respec-tively 119901Batch

119905and 119901

Batdisch119905

represent the charging and dis-charging power in time slot 119905 respectively 120578Batch is the chargingefficiency and 120578

Batdisch is the discharging efficiency

To ensure the safety of operation the charging anddischarging powers should be controlled to be equal to orsmaller than the maximum allowable values As shown inFigure 2 the battery can be charged by power from the grid119901G2B119905

and PV 119901P2B119905

on the other hand the battery can alsosupply electricity to loads 119901B2L

119905and grid 119901

B2G119905

Consequentlythe constraints about charging and discharging power areformulated as follows respectively

0 le 119901Batch119905

= 119901P2B119905

+ 119901G2B119905

le 119875Batchmax (17)

0 le 119901Batdisch119905

= 119901B2L119905

+ 119901B2G119905

le 119875Batdischmax (18)

where 119875Batchmax and 119875

Batdischmax are the maximum allowable charg-

ing and discharging power of the battery respectivelyThe battery is not allowed to supply electricity to loads or

the grid when it is in charging state or to be charged when itis supplying electricity to loads or the grid This constraint isformulated as

(119901B2L119905

+ 119901B2G119905

) sdot (119901P2B119905

+ 119901G2B119905

) = 0

forall119905 isin 1 2 119873slot (19)

246 Other Constraints Besides the constraints mentionedabove other constraints are described as follows

119901P2L119905

+ 119901P2B119905

+ 119901P2G119905

= 119901PV119905

sum

119886isinA119901119886

119905+ 119901

CL119905

= 119901P2L119905

+ 119901B2L119905

+ 119901G2L119905

A = HVACEWHPHEVWMCDDW

(20)

where 119901PV119905

is the power output of PV in time slot 119905 in kW and119901CL119905

is the total power of the critical loads (ie nonschedulableloads) in time slot 119905 in kW

3 Multiobjective Optimization of HEMS

31 Electricity Cost In this paper minimizing the overallelectricity cost over the next 24 hours (ie next day) is oneof optimization objectives based on the forecasted outdoortemperature and power output of PV over the next 24 hourswhich can be obtained using the corresponding predictionalgorithms [30 31] However these algorithms have limita-tions due to the prediction accuracy In this study scenariosare used to capture the uncertainties of forecasted outdoortemperature and power output of the PV The overall netelectricity cost over the scheduling horizon is formulated as(21) which consists of three parts the first item represents theoverall electricity cost of buying electricity from the grid thesecond item stands for the degradation cost ($) of the home


TRoommin TRoom

max

TRoomset

Comfortable zone

Tolerable zone

Intolerable zone

TRoomset + ΔTRoom

UTRoomset minus ΔTRoom

L

ΔTRoomL ΔTRoom

U

Figure 3 Relationships among parameters of HVAC comfort levelindicator

energy storage battery and the third item denotes the overallrevenue of selling electricity to the grid

119865cost = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042 sdot

119873slot

sum

119905=1

[(119901G2L119904

11199042

119905+ 119901

G2B11990411199042

119905)

times Δ119905 times 119888grid119905

]

+

119873slot

sum

119905=1

[(119901P2B11990411199042

119905+ 119901

G2B11990411199042

119905+ 119901

B2L11990411199042

119905+ 119901

B2G11990411199042

119905)

times Δ119905 times 11988811990411199042

Deg] minus

119873slot

sum

119905=1

[(119901P2G119904

11199042

119905+ 119901

B2G11990411199042

119905) times Δ119905

times 119888sell119905

]

(21)

where stemp and sPV denote the outdoor temperature andPV power output scenario sets respectively 119904

1represents an

outdoor temperature scenario and 1199042stands for a PV power

output scenario 1199011199041 and 1199011199042 denote the occurrence possibility

of scenario 1199041and 1199042 respectively 119888grid

119905is the electricity price

in time slot 119905when the user buys electricity from the grid and119888sell119905

is the electricity price in time slot 119905 when the user sellselectricity to the grid 119901G2Ls

11199042

119905denotes the power transmitted

from the grid to loads in the scenario when 1199041and 1199042occur

at the same time and other symbols have similar meanings11988811990411199042

Deg is the battery degradation cost ($kWsdoth) the detaileddescription and calculation method of it can be found in [3]

32 Comfort Level Indicator for HomeAppliances In practicea residential user has different concerns for different homeappliances For example for HVAC and EWH the user paysmore attention to temperature however for PHEV WMCD and DW the user focuses on when the tasks of theseappliances are completed Therefore a set of comfort levelindicators are proposed based on appliance type

321 HVAC To quantify a userrsquos comfort level under theoperation of HVAC this paper proposes a comfort levelindicator whose definition is based on the assumption that

when the room temperature is equal to the user settingtemperature the user is most comfortable if the room tem-perature deviates from the setting value to a certain extentthe userrsquos comfort level will be decreased [32] The indicatoris calculated as

119868HVAC =100

119873slot

119873slot

sum

119905=1

119889Room119905

Δ119879Roommax

(22)

where Δ119879Roommax = max119879Room

set minus 119879Roommin 119879

Roommax minus 119879

Roomset 119889Room

119905

is determined by

119889Room119905

=

119879Roomset minus 119879

Room119905

119879Roommin le 119879

Room119905

lt 119879Roomset minus Δ119879

Room119871

0 119879Roomset minus Δ119879

Room119871

le 119879Room119905

le 119879Roomset + Δ119879

Room119880

119879Room119905

minus 119879Roomset 119879

Roomset + Δ119879

Room119880

lt 119879Room119905

le 119879Roommax

Δ119879Roommax otherwise

(23)

where119879Roomset is the desired indoor temperature set by the user

and Δ119879Room119871

and Δ119879Room119880

are two parameters that are relatedto the temperature deadband of HVAC and the userrsquos prefer-ence For example in summer the user prefers cool Conse-quently in this case Δ119879Room

119871gt Δ119879

Room119880

gt 0 According tothe definition of 119868HVAC it is within [0 100]

The relationships among parameters 119879Roommin 119879

Roommax

119879Roomset Δ119879Room

119871 and Δ119879

Room119880

are demonstrated in Figure 3The indoor temperature horizon is divided into three zonesby these parameters comfortable zone tolerable zone andintolerable zone

322 EWH The definition of comfort level indicator forEWH is similar to that for HVAC and it is calculated as

119868EWH =100

119873slot

119873slot

sum

119905=1

119889EWH119905

Δ119879EWHmax

119889EWH119905

=

119879EWHset minus 119879

EWH119905

119879EWHmin le 119879

EWH119905

lt 119879EWHset minus Δ119879

EWH119871

0 119879EWHset minus Δ119879

EWH119871

le 119879EWH119905

le 119879EWHset + Δ119879

EWH119880

119879EWH119905

minus 119879EWHset 119879

EWHset + Δ119879

EWH119880

lt 119879EWH119905

le 119879EWHmax

Δ119879EWHmax otherwise

(24)

where Δ119879EWHmax = max119879EWH

set minus 119879EWHmin 119879

EWHmax minus 119879

EWHset and

119879EWHset Δ119879EWH

119871 andΔ119879

EWH119880

are user setting parameters whosemeanings are similar to the parameters of HVAC comfortlevel indicator 119868EWH is in the range of [0 100] too

323 PHEV A userrsquos comfort level about PHEV is deter-mined by the charging finish time The user is most satisfiedin the case where once a PHEV is plugged in and it is keptcharging until the PHEV battery SOC reaches the specifiedvalue In reality the user can tolerate some delay of thecharging finish time which makes the PHEV a flexible loadHowever this tolerance is limited if the delay is too long theuser will be unsatisfied Based on these facts the comfort levelindicator for PHEV is calculated as


119868PHEV =119889PHEV

119905end minus (119905plug + 119905charge + 119905delay)times 100

119889PHEV

=

0 119905plug + 119905charge le 119905PHEVfsoc le 119905plug + 119905charge + 119905delay

119905PHEVfsoc minus (119905plug + 119905charge + 119905delay) 119905

PHEVfsoc gt 119905plug + 119905charge + 119905delay

(25)

where 119905plug is the time shot when the PHEV is plugged in119905charge is the number of time slots that are needed to complete acharging task 119905delay is themaximum tolerant delay of chargingcompletion time in time slot 119905PHEV

fsoc is the time slot in whichthe PHEV battery SOC reaches the specified value 119905end isthe last time slot of the scheduling horizon According to

the definition 119868PHEV is within [0 100] Figure 4 illustrates therelationships among these parameters

324 WM CD and DW The comfort level indicators forWM CD and DW share the same expression and they arecalculated as

119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886

start minus (119905119886

ideal + 119905119886

delay)

(119905119886max minus 119905119886work) minus (119905119886ideal + 119905119886delay)times 100 119905

119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886

max specify the valid working interval forappliance 119886 119905119886ideal is the ideal time slot in which the appliance119886 is started and it is set by a user 119905119886start is the actual startingtime slot of appliance 119886 119905119886work denotes the number of timeslots that are needed by appliance 119886 to complete its task 119905119886delayis the tolerant delay of task finish time The relationships ofthese parameters are depicted in Figure 5 119868

119886is in the range of

[0 100] tooFor home appliance 119886 if its task is started before or in the

time slot 119905119886ideal + 119905119886

delay the user will be satisfied if the task isstarted in the range of [119905119886ideal + 119905

119886

delay 119905119886

max minus 119905119886

work] the task canbe completed before the deadline however the userrsquos comfortlevel will be decreased

It must be pointed out that according to the comfortlevel indicatorsrsquo definitions the smaller these indicators themore comfortable the user For example the user is mostcomfortable when 119868HVAC is equal to zero

Based on the above definitions the userrsquos overall comfortlevel during the scheduling horizon is formulated as

119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886

denotes the comfort level indicator value of appli-ance 119886 in the scenario when outdoor temperature scenario 119904

1

and PV power output scenario 1199042occur at the same time and

119873|A| is the number of appliances in schedule home appliance

setA119865comfort is within [0 100] When119865comfort is zero the useris most comfortable

33 Multiobjective Optimization Model The multiobjectiveoptimization model for HEMS is formulated as

min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)

In this model the working status of HVAC EWH andPHEV in each time slot and the task starting times ofWM CD and DW are decision variables Minimizing elec-tricity cost and maximizing user comfort level are the twoobjectives

4 Algorithm Design

41 Model Transformation For simplicity the multiobjectiveoptimization model proposed in Section 33 is transformedinto a single objective optimization model by weightingmethod

min 119865total = 120572119865cost + (1 minus 120572) 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)

where 120572 is called user preference factor 0 le 120572 le 1 throughwhich a user can take a tradeoff between the electricity costand the comfort level conveniently

To handle the constraints of model (29) the penalty func-tion method is used Model (29) is further transformed into


tcharge tdelay

tplug tPHEVfsoc tend

Comfortable zone

Tolerable zone

Figure 4 Relationships among parameters of PHEV comfort levelindicator

tawork tadelay

tamin taideal tastart tamax

Comfortable zone

Tolerable zone

Invalid zone

Figure 5 Relationships among parameters of CW CD and DWcomfort level indicators

(30) which is a nonconstraint single objective optimizationmodel and easy to solve

min 119865final = 119865total + 119872 sdot 119865viol (30)

where 119865final is the final objective function 119872 is a positivefigure that is big enough and 119865viol is the overall violationvalue

119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905

minus 119879Roommax 119879

Roommin minus 119879

Room119905

)

119879Roommax minus 119879Room

min

+

119873slot

sum

119905=1

max (0 119879EWH119905

minus 119879EWHmax 119879

EWHmin minus 119879

EWH119905

)

119879EWHmax minus 119879EWH

min

+

119873slot

sum

119905=1

max (0 SOCPHEV119905

minus SOCPHEVmax SOCPHEV

min minus SOCPHEV119905

)

SOCPHEVmax minus SOCPHEV

min

(31)

119865viol only considers constraints (2) (7) and (10) Otherconstraints can be guaranteed to be satisfied by other meth-ods presented in Section 43

Model (30) is solved by an improved high-dimensionalhybrid discrete particle swam optimization algorithm pre-sented in the following section

42 Improved Particle SwarmOptimization PSO is originallyproposed by Kennedy to solve unconstrained continuous

single objective optimization problems [33] and it is a kindof stochastic search algorithms Due to its simplicity strongsearch ability and robustness PSO has been explored indepth many improved PSO algorithms have been proposedin the literature and its application has not been limited tocontinuous optimization problems anymore Algorithmsbased on PSO have been applied in many engineering opti-mization fields [34ndash37]Therefore in this paper an improvedPSO is employed to solve the optimization model depicted in(30)

Although the basic PSO has many advantages it hasthe drawback of premature convergence and local optima[38] The performance of PSO algorithm can be improvedby identifying the particles which fell into local optimal areaand performing crossover mutation local search reset orreinitialization on these particles during operation [39] Forthis purpose [40] assigns a counter for each particle in thepopulation at each iteration the fitness value of each particleis compared with the global best particlersquos fitness valueIf the absolute value of the fitness value difference betweena particle and the global best particle is smaller than apredefined threshold the particlersquos counter is increased by 1and then the counter is checked as to whether it reaches thespecified maximum value if so the particlersquos position andvelocity vectors are initialized and the corresponding counteris resetThismethod is effective in some cases However if theparticle which fell into local optimal area is not in the samearea with the global best particle that keeps evolving theabsolute value of the fitness value difference may be keptgreater than the specified threshold As a result the particlewhich fell into local optimal area cannot be identified

To overcome this problem a novel ldquoworst particlesrdquo iden-tificationmethod is proposed to find out the particles that fellinto local optimal area based on the update of each particlersquospersonal best position vector and the sorted particle fitnessvalues This method is described as follows

Similar to [40] each particle in the population has acounter However different form [40] the counter is used torecord the number of iterations in which the particlersquospersonal best position vector is not updated successively Thecounter is named personal-best-update counter and updatedas

119862119894(119896) =

119862119894(119896 minus 1) + 1 P119894best (119896) = P119894best (119896 minus 1)

0 P119894best (119896) = P119894best (119896 minus 1) (32)

where119862119894(119896) and119862

119894(119896minus1) denote the counter values of particle

119894 at iterations 119896 and 119896 minus 1 respectively and P119894best(119896) andP119894best(119896 minus 1) are the personal best position vectors of particle 119894at iterations 119896 and 119896 minus 1 respectively

After updating each particlersquos counter the particles in thepopulation are sorted decreasingly according to their fitnessvalues If the optimization is minimization the smaller thefunction value is the greater the fitness value will be Withinthe last 119873

119888sorted particles the particles whose counters are

equal to or greater than the specified threshold 119862th areselected to create a particle set namedworst-particle-setΛ(119896)

denotes the worst-particle-set created at iteration 119896 and the


Start

Calculate the overall user demand in time slot t including the critical load and theload of schedulable home appliancesThe overall demand is denoted bysum

apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat

Let Δp1 = pPVt minussum

apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t

Let Δp2 = sum apat minus pPV

t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo

Δp1 lt PBat chmax Δp2 lt PBat disch

max

YesNoNoYes Yes No Yes NoYes

YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0

pP2Gt = Δp1 minus PBat ch

max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B

t = 0pG2Lt = Δp2 minus PBat disch

max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch

max pB2Gt = PBat disch

max minus Δp2 pB2Gt = 0 pG2B

t = 0 pG2Bt = PBat ch

max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max

Figure 6 Flowchart of power flow vectors calculation method

number of the particles in the set is |Λ(119896)| 0 le |Λ(119896)| le 119873119888 If

|Λ(119896)| = 0 the last particle of the sorted particles is selected astheworst particle if |Λ(119896)| = 1 the particle inΛ(119896) is selectedif |Λ(119896)| gt 1 a particle is selected randomly from Λ(119896) as theworst particle Finally the selected worst particle is initializedand the corresponding counter is reset

43 Algorithm Based on Improved Hybrid PSO Based onthe improved PSO algorithm described in Section 42 amultiobjective optimization algorithm forHEMS is describedin this section

In this optimization algorithm the 119894th decision vector ofthe optimization model (30) is depicted as

X119889119894= [SHVAC SEWH SPHEVTstart] (33)

where

SHVAC = [119878HVAC1

119878HVAC2

119878HVAC119873slot

]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]

SPHEV = [119878PHEV119905plug

119878PHEV119905plug+1

119878PHEV119873slot

]

Tstart = [119905CWstart 119905

CDstart 119905

DWstart]

(34)

As shown SHVAC SEWH and SPHEV are binary vectors andTstart is a discrete vector Therefore the decision vector Xd

119894is

a typical hybrid vector with a dimension of 3119873slot + 4 minus 119905plugThe dimension is related to the time slot when the PHEVis plugged in For example in the case that the schedulinghorizon is 24 hours and is evenly divided into 120 time slots(ie the length of each time slot is 12 minutes) and the PHEVis plugged in at time slot 91 the dimension of (33) is 273 Inthis paper it is assumed that the PHEV can be guaranteed tobe plugged in power grid through HEMS during [119905plug 119873slot]

In the algorithm the position vector of each particlerepresents a decision vector of model (30) The principalprocedure of the proposed multiobjective optimization algo-rithm for HEMS in smart grid is described as follows

Step 1 Get electricity price and forecasted weather informa-tion of the scheduling horizon from the utility company andthe Internet respectively Based on the weather forecastinformation the power output of PV during the schedulinghorizon is predicted

Step 2 Generate a set of outdoor temperature scenarios andPV power output scenarios using Monte Carlo simulations

Step 3 Set parameters of PSO such as population size 119873popmaximum iteration number 119896max maximum velocity weight119908max minimum velocity weight119908min thresholds119862th and119873

119888

Step 4 Set user preference parameters such as preferencefactor 120572 119879Room

min 119879Roommax 119879Room

set Δ119879Room119871

Δ119879Room119880

and 119905plug


Step 5 Initialize particle population

(1) Initialize position vector X119894 velocity vector V119894 andpersonal best vector P119894best

(2) Calculate power distribution vectorsPG2LPG2BPP2LPP2B PP2G PB2G and PB2L based on electricity pricePV power output and X119894 (ie [SHVAC SEWH

SPHEVTstart]) Here PG2L= [119901

G2L1

119901G2L2

119901G2L119873slot

]and the other power distribution vectors have similarforms The calculation procedure of these vectors isdescribed in Section 44

(3) Calculate the objective function value according to(30)

(4) After initializing all particles in the population ini-tialize the global best particlersquos position vector

(5) Reset iteration counter 119896 and personal-best-updatecounter of each particle

Step 6 Compute the velocity weight 119908(119896) [38] at iteration 119896

as

119908 (119896) = 119908max minus119896 (119908max minus 119908min)

119896max (35)

Step 7 For each particle in the population perform thefollowing operations

(1) Update the velocity vector as

V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V

119894119895(119896 + 1) denote the values of the 119895th

dimension of particle 119894 at iterations 119896 and 119896 + 1 respectively1199031and 119903

2are two random uniform distribution stochastic

variables within [0 1] 1198881= 1198882= 2 are two learning factors

119901119894119895is the 119895th dimension of the personal best vector of particle

119894 119901119892119895

is the 119895th dimension of the global best position vector

(2) Update the position vector X119894

The first part of X119894 that is [SHVAC SEWH SPHEV] isupdated as

119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)

where 1199033is a random uniform distribution stochastic variable

within [0 1]The second part of X119894 that is Tstart is updated as

119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil

119895 = 3119873slot + 2 minus 119905plug 3119873slot + 4 minus 119905plug(38)

where the symbol lceilrceil denotes rounding to zero

(3) Calculate power distribution vectorsPG2LPG2BPP2LPP2B PP2G PB2G and PB2L


(5) Update the personal best position vector according to(39) and the personal-best-update counter accordingto (32)

P119894best (119896 + 1)

=

X119894 (119896 + 1) 119865final (X119894 (119896 + 1)) lt 119865final (P119894best (119896))

P119894best (119896) 119865final (X119894 (119896 + 1)) ge 119865final (P119894best (119896))

(39)

(6) Update the global best position vector

P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)

Step 8 Identify theworst particlewhich fell into local optimalarea as follows

(1) Create the worst-particle-set following the methoddescribed in Section 42

(2) Select the worst particle and reinitialize it and resetthe corresponding personal-best-update counter

Step 9 Increase the iteration counter by 1 Check whether theiteration counter reaches the maximum number 119896max or notIf not jump to Step 6 if so continue Step 10

Step 10 Calculate the electricity cost indicator and comfortlevel indicator as follows

(1) Output the global best position vector(2) Calculate the power distribution vectors(3) Calculate the electricity cost indicator and comfort

level indicator according to (21) and (27) respectivelyCheck whether the user is satisfied with the indicatorsor not If not jump to Step 4 if so continue Step 11

Step 11 Control the operations of HVAC EWH PHEVWMCD and DW according to the vector P

119892 the final global

best particlersquos position vector In each time slot the powerdistribution among loads battery renewable generation andpower grid is determined through the method described inSection 44

44 Calculation Methods of Power Distribution Vectors Intime slot 119905 the values of 119901G2L

119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905

are determined following the flowchart shown inFigure 6

As shown in Figure 6 the power output of PV is suppliedto loads first If there is extra energy it is used to charge thehome energy storage battery or sold to the utility grid for rev-enue if the PV power output cannot meet the load demandthe shortage is met by the home energy storage battery or the


10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)

Figure 7 Electricity price

grid In this paper the production cost of PV is assumed tobe negligible

119888Bat119905

denotes the price of the energy stored in the homeenergy storage battery in time slot 119905 its value is determinedby charging operation and it is not affected by discharging119888Bat119905

is calculated as follows [1]

119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905

) sdot Δ119905 sdot 120578Batch

(41)

where 119864Bat119905

and 119864Bat119905+1

denote the energy in kWsdoth stored inthe home energy storage battery in time slots 119905 and 119905 + 1respectively

119888Batsell119905

stands for a threshold of grid electricity price intime slot 119905 If the grid electricity price 119888

grid119905

is greater than119888Batsell119905

the home energy storage battery is allowed to sell thestored energy to the grid for revenue 119888Batsell

119905is calculated as

119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2

119888grid119873slot

0 lt 120573sell

lt 1

(42)

120573sell controls the possibility of selling the energy stored in

the home energy storage battery to the grid in time slot 119905 thesmaller 120573sell is the greater the possibility will be

119888Batbuy119905

is another threshold of grid electricity price in timeslot 119905 If the grid electricity price 119888

grid119905

is smaller than 119888Batbuy119905

the home energy storage battery is allowed to buy electricityfrom the grid for charging 119888Batbuy


119888Batbuy119905

= 120573buy

sdotmax 119888grid119905

119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)

Figure 8 Forecasted outdoor temperature

120573buy controls the possibility of buying electricity from the

grid to charge the home energy storage battery in time slot 119905The greater the 120573buy is the greater the possibility will be

From the algorithm described in Section 43 and thepower distribution vector calculation method presented inthis section it can the said that the constraints of (30) thatis constraints (11)-(12) (14) and (17)ndash(20) are ensured to besatisfied

5 Case Studies

In this section to validate the effectiveness of the algorithmproposed in this paper simulations were performed exten-sively All simulation programs were coded in C language inthe environment of Microsoft Visual Studio 2008 and wererun on aWindows 7 (32 bit) Intel Core i7-3540M300GHzcomputer with an 800GB memory

51 Input Data and Parameter Setting The scheduling hori-zon is 24 hours and it is divided evenly into 120 time slotswith each time slot being 12 minutes that is 119873slot = 120Δ119905 = 02 h

The data of real-time electricity price and outdoor tem-perature from [41] hot water usage from [42] and criticalload from [43] are utilized The electricity price outdoortemperature hot water usage and critical load are shown inFigures 7 8 9 and 10 respectively

In this paper it is assumed that the capacity of the PVsystem in Figure 1 is 575 kWand the PVpower output duringscheduling horizon is as shown in Figure 11 In each time slotthe price of selling electricity to the power grid by a user 119888sell

119905

is assumed to be equal to the price that heshe pays for buyingelectricity from the grid 119888grid

119905

Based on the forecasted outdoor temperature and PVpower output two sets of stochastic scenarios were generatedwhich are shown in Figures 12 and 13 respectively

The parameters of house and HVAC of [24] are used inthis paper which are listed in Table 1


10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)

Figure 9 Hot water usage profile

10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)

Figure 10 Critical load profile

10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)

Figure 11 Forecasted PV power output

s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)

Figure 12 Scenarios of outdoor temperature over schedulinghorizon

45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10

Figure 13 Scenarios of PV power output over scheduling horizon

Consequently in model (1) 119866119905= 50082 times (119879

Outdoor119905

minus

119879Room119905

) (Btuh) where 119879Outdoor119905

is the outdoor temperature intime slot 119905 Δ119888 = 340008 (Btu∘F) Δ119905 = 02 h and the ratedpower is 2352 (kW) The user preference parameters aboutHVAC 119879

Roommin 119879Room

max 119879Roomset Δ119879Room

119871 and Δ119879

Room119880

are 72 8076 3 and 2 (∘F) respectively

The parameters of EWH are as follows The volume ofEWH tank is 50 gallons 119866

EWH is 10 (Btu(hlowast∘F)) 119862 is


Table 1 Parameter of the house under study

Parameter Value Unit

House size 2000 + 500basement ft2

119860floor 119860 ceiling 119860wall and 119860window(the area of the floor ceilingwall and window)

2000 20002600 520 ft2

119877ceiling 119877wall and 119877window (theheat resistance of the ceilingwall and window)

49 13 2 ft2lowast∘F(Btuh)

119862HVAC (capacity of the AC unit) 34000 Btuh

Table 2 Parameter setting of CW CD and DW

Parameter CW CD DW119905119886

min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886

delay (time slots) 5 5 5Rated power (kW) 05 4 1

41988 (Btu∘F) the rated power is 45 (kW) 119879EWHenv119905

isequal to 119879

Room119905

and 119879EWHin119905

is 60 (∘F) The user preferenceparameters 119879EWH

min 119879EWHmax 119879EWH

set Δ119879EWH119871

and Δ119879EWH119880


As for the PHEV 119862PHEVsize is 16 (kWsdoth) SOCPHEV

0is 05

119875PHEV is 33 (kW) and SOCPHEV

min SOCPHEVmax and SOCPHEV

final are02 095 and 085 respectively It is assumed that the PHEVis plugged in the grid in time slot 91 that is 119905plug = 91 119905chargeis 9 119905delay is 3 and 119905end = 119873slot = 120

The parameters ofWM CD andDWare listed in Table 2As for the home energy storage battery the rated battery

capacity 119862Batsize is 1344 (kWsdoth) 119864Bat

0is 672 (kWsdoth) SOCBat

minis 02 SOCBat

max is 1 119875Batchmax = 119875

Batdischmax = 20 (kW) and

120578Batch = 120578

Batdisch = 09 It is assumed that the battery is lead-

acid battery the capital cost is 2176 ($) The degradation costfunction coefficients in [44] are used

As described in Section 44 120573sell and 120573buy are two impor-

tant parameters that affect the power distributions betweenthe grid and home energy storage battery In the simulationsthey are 06 and 08 respectively

The parameters of PSO are as followsThe population size119873pop is 30 and the maximum iteration number 119896max is 3000119908min 119908max 119862th and 119873c are 02 09 20 and 5 respectivelyAccording to the values of 119873slot and 119905plug the dimension ofposition vector and velocity vector of each particle is 273

52 Case Description For comparison different cases werestudied and the input data and parameter setting in all casesare the same as those presented in Section 51These cases aresummarized as follows

Case 1 It is the multiobjective optimization algorithm forHEMS proposed in this paper and the scheduling objects are

PHEV EWH PHEVWMCDDW and home energy storagebattery In this case the user has the ability to sell electricityto the grid for revenue The power distributions among gridloads PV and battery are demonstrated in Figure 2

Case 2 It is a single objective optimization algorithm thatminimizes the electricity cost Other aspects of Case 2 are thesame as those in Case 1

Case 3 It is a single objective optimization algorithm thatminimizes the electricity cost too However the user doesnot have the ability to sell electricity to the grid for revenueThe home energy storage battery is used to store the excessiveenergy generated by the PV In this case if the power outputfrom PV is greater than the sum of load demand and thecharging power of home energy storage battery the excessiveenergy will be discarded

Case 4 This case is similar to Case 3 except that in thiscase the operations of home appliances are not optimallyscheduled by any algorithmand the user uses these applianceswithout considering the electricity price PV power outputand forecasted outdoor temperature

53 Cost and Comfort Level The simulation results of thesefour cases are listed in Table 3

For Cases 1 2 and 3 the algorithms were run inde-pendently 30 times with different initializations The resultspresented in Table 2 are the average values of different runsFor Case 1 the user preference factor 120572 is 05

FromTable 3 we can see that the user ismost comfortablein Case 4 However in this case the userrsquos net electricity costis the highest because of not considering the electricity priceand PV power output During the operations of the PHEVand CD the electricity price is high and the PV power outputis lowwhich increases the electricity costThepower fromPVis supplied to the loads and the home energy storage batteryWhen the PV power output is high (eg time slots 60ndash90)the excessive power is discarded due to the inability to sellelectricity to the utility grid for revenue Therefore the PVutilization efficiency is very low

In Case 3 the schedulable home appliances and the homeenergy storage battery are optimally scheduled according toelectricity price PV power output forecasted outdoor tem-perature and user preferences The power consumption onpeak hours is shifted to off-peak hours or hours when the PVpower output is high more demand of loads is met by the PVand the PV utilization efficiency is improved significantly Asa result the amount of electricity purchased from the grid andthe overall net cost are reduced Compared with Case 4 theelectricity purchase cost and net cost of Case 3 are reduced by3276 and 1827 respectively at the cost of user comfortlevel

In Cases 1 and 2 the user has the ability to sell electricityto grid for revenue Therefore the excessive power of PVcan be sold to the utility grid and there is no power wastedIn addition the home energy storage battery can absorbelectricity from the grid when the electricity price is low andsupply the stored energy to loads or sell back to the utility grid


Table 3 Simulation results of different cases

Case Purchase cost (cents) Battery cost (cents) Sales revenue (cents) Net cost (cents) Comfort levelCase 1 8467 10889 5729 13627 2487Case 2 7790 10741 5867 12635 3677Case 3 6882 8079 mdash 14961 3413Case 4 10235 8070 mdash 18305 00

for profit when the electricity price is high These two factorscontribute to overall net cost reduction On the other handbecause the home energy storage battery absorbs electricityfrom the grid the costs of purchasing electricity from the gridin Cases 2 and 1 are higher than that of Case 3 Meanwhilemore energy is charged to or discharged from the homeenergy storage battery therefore the battery degradationcosts in the two cases are higher than those in Cases 3 and4 However due to the gain of selling electricity to the utilitygrid compared with Case 3 the net costs of Cases 2 and 1are reduced by 1555 and 892 respectively Consequentlywe can say that the framework of HEMS proposed in thispaper is better than other frameworks that only include partsof these components or without the ability to sell electricityto the utility grid

Compared with Case 2 the net cost of Case 1 is increasedby 785 However the comfort level of Case 1 is improvedby 3236 Besides this improvement in Case 1 the usercan take a tradeoff between the electricity cost and comfortlevel through the preference factor 120572 conveniently and theuserrsquos comfort level is quantified Based on these results wecan conclude that the multiobjective optimization algorithmproposed in this paper is superior to the single objectivealgorithms in Cases 2 3 and 4

54 Operation States of Schedulable Home Appliances Whenthe forecasted outdoor temperature scenario is 119904

6and PV

power output scenario is 1199048 the operation states of schedu-

lable home appliances under the control of the algorithmproposed in this paper are shown in Figure 14

As shown in Figure 14 the temperatures of room andwater in the tank of EWHarewithin their prespecified rangesMost charge power for PHEV is shifted from high price hoursto low price hours and the final SOC of the PHEV battery hasreached the specified value WM CD and DW finish theirtasks before their deadlines

55 Power Distributions among Grid Load PV and BatteryWhen the operation states of schedulable home appliances areas shown in Figure 14 the corresponding power distributionsamong grid load PV and battery are demonstrated asfollows

The power distributions between the grid and the homeenergy storage battery are shown in Figure 15

From Figure 15 we can find that when the electricity priceis low that is between time slot 1 and time slot 62 the homeenergy storage battery absorbs electricity from the grid whenthe price is high that is between time slot 63 and time slot 110it sells the stored electricity to the grid for profit

The SOC charging power and discharging power of thehome energy storage battery are shown in Figure 16

Figure 16 shows that the SOC of the home energy storagebattery is within the specified range during the schedulinghorizon Between time slot 1 and time slot 62 most chargingpower is from grid between time slot 63 and time slot 95most charging power is from PV Before time slot 50 thepower discharged from the home energy storage battery issupplied to loads between time slots 51 and 120 and thedischarged power is supplied to loads and sold to the grid

Figure 17 displays the power distributions from PV toloads battery and grid

Figure 18 depicts the power distribution that is trans-mitted from the grid to HEMS that is the power that ispurchased by the user in each time slot

As shown in Figure 18 HEMS purchases electricity fromthe utility grid when the electricity price is low When theprice is high the amount of electricity purchased from theutility grid is reduced or even does not buy electricity anymore For example from time slot 61 to time slot 90 HEMSdoes not buy electricity from the grid because of the highelectricity price

56 Parameter Analysis To demonstrate the effect of prefer-ence factor to optimization results the input data and otherparameters are kept the same as those in Case 1 and themultiobjective optimization algorithmwas run with differentpreference factors The optimization results are shown inFigure 19

As shown in Figure 19 when the user preference factorincreases from 0 to 1 the electricity cost decreases from 16918cents to 12634 cents and the comfort level indicator increasesfrom 1355 to 3677 In other words with the increase ofpreference factor the electricity cost increases and the userfeels less comfortable

The optimization algorithm runs with different 120573sell whilethe other parameters and input data are kept the same as thosein Case 1 Figure 20 shows the relationship between the totalenergy that is transmitted from battery to the grid and 120573

sellAs shown in Figure 20 with the increase of 120573sell the total

amount of energy that is transmitted from the home energystorage battery to grid decreasesThe reason is that the greater120573sell decreases the possibility of selling the stored energy to the

grid for revenueFigure 21 depicts the relationship between the total

energy that is transmitted from the grid to the home energystorage battery and 120573

buyAs shown in Figure 21 with the increase of 120573buy the total

amount energy that is transmitted from the grid to the home


Maximum tolerabletemperature

Minimum tolerabletemperature

Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)

(a) Room temperature



Water temperaturein tank

140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)

(b) Water temperature in tank of EWH

Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120

(c) Charging power of PHEV

Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC

(d) PHEV battery SOC

CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power

Figure 14 Operation states of schedulable home appliances

Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Maximum chargingpower

(a) Power transmitted from grid to battery

Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Maximum dischargingpower

(b) Power transmitted from battery to grid

Figure 15 Power distributions between the grid and the home energy storage battery


Maximum battery SOCBattery SOC

Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot

(b) Charging power of battery

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120

Maximum charging powerD

ischa

rgin

g po

wer

(kW

)

Time slot

(c) Discharging power of battery

Figure 16 SOC charging power and discharging power of the home energy storage battery

4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)

(a) Power output of PV

4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)

(b) Power transmitted from PV to loads

4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Maximum charging power of battery

Pow

er (k

W)

(c) Power transmitted from PV to battery

4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)

(d) Power transmitted from PV to grid

Figure 17 Power distributions from PV to loads battery and grid

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10

Figure 18 Purchased power from grid

energy storage battery increases The reason is that greater120573buy means more chances of buying electricity from the grid

to charge the home energy storage batterywhen the electricityprice is low

57 Runtime and Robustness of Algorithm The parametersand input data were kept as described in Section 51 and themaximum iteration number was set to 3000 The algorithmsin Cases 1 2 and 3 were run 30 times respectively and thestatistics of runtime and final fitness value were obtainedTheresults are displayed in Table 4

As presented in Table 4 the runtimes in Cases 1 and 2are longer than that in Case 3 The reason is that the formertwo cases need to determine the power distribution amongloads PV battery and grid through the method described in


Table 4 Statistics of runtime and fitness value

Case Parameter Minimum Maximum Mean Standard deviation

Case 1 Runtime (s) 819630 831010 824493 03515Fitness value 784738 821968 805712 12511



120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or

User preference factor

Comfort levelElectricity cost

Figure 19 Relationship between optimization results and prefer-ence factor

0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor

Figure 20 Relationship between total energy transmitted from thehome energy storage battery to grid and 120573

sell

Section 44 Because the proposed multiobjective optimiza-tion algorithm for HEMS is an offline optimization algo-rithm the runtime is acceptable for practical applicationsIn fact compared to the algorithm in [1] it is very fast Inaddition from the standard deviations of runtime and fitnessvalue in Table 4 we can say that the proposed algorithm isrobust enough for application

0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor

Figure 21 Relationship between total energy transmitted from thegrid to the home energy storage battery and 120573

buy

6 Conclusion

In smart grid implementing DR in residential sectorthrough HEMS has attracted much interest of both academicresearchers and industrial engineers This paper first pro-poses a framework of HEMS including grid load PV andhome energy storage battery In this framework a user has theability to sell electricity to utility grid for revenue In order toquantify userrsquos comfort level during the operation of homeappliances a set of comfort level indicators are proposedbased on home appliancersquos characteristics and userrsquos pref-erences A novel multiobjective optimization algorithm forHEMS is proposed which minimizes the electricity cost andmaximizes the comfort level of the user simultaneously Thealgorithm optimally controls the operations of schedulablehome appliances such as PEHV EWH WM CD and DWand the power distributions among grid load PV andbattery according to the electricity price power output of PVforecasted outdoor temperature and user preferences Thestochastic natures of PV generation and outdoor temperatureare considered andmodeled by scenario methodThe controlof both home appliances and power distributions amongdifferent components of HEMS is investigated in depth bysimulations The computation time and robustness of theproposed algorithm are acceptable for practical applicationsThe results demonstrate that compared with other frame-work of HEMS and algorithms the framework of HEMS and


algorithm proposed in this paper can reduce electricity costand improve a userrsquos comfort level significantly provides theuser with an efficient method of taking a tradeoff betweenthe electricity cost and the comfort level and facilitates theimplementation of DR in residential sector in smart grid

In this paper the PHEV is considered as a pure loadwhich absorbs electricity for transportation However insmart grid when the grid electricity price is high it can alsosupply the stored electricity to other home appliances or crit-ical loads through the V2H function to reduce the amount ofelectricity purchased from grid and home ownerrsquos electricitycost In the future we will integrate this function into themodel proposed in this paper In this case determining thepower distribution among loads PV battery PHEV and gridwill become more challenging

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch and Development of China (2011AA040103) theNatural Science Foundation of China (61233007 61300215)and the Important National Science and Technology SpecificProject (2013ZX03005004) The authors would like to sin-cerely thank the anonymous reviewer whose comments havehelped them improve the paper significantly

References

[1] Z Wu S Zhou J Li and X-P Zhang ldquoReal-time schedulingof residential appliances via conditional risk-at-valuerdquo IEEETransactions on Smart Grid vol 5 no 3 pp 1282ndash1291 2014

[2] S A Pourmousavi S N Patrick and M H Nehrir ldquoReal-timedemand response through aggregate electric water heaters forload shifting andbalancingwind generationrdquo IEEETransactionson Smart Grid vol 5 no 2 pp 769ndash778 2014

[3] W Su J Wang and J Roh ldquoStochastic energy scheduling inmicrogrids with intermittent renewable energy resourcesrdquo IEEETransactions on Smart Grid 2013

[4] W Su and M-Y Chow ldquoPerformance evaluation of an EDA-based large-scale plug-in hybrid electric vehicle charging algo-rithmrdquo IEEE Transactions on Smart Grid vol 3 no 1 pp 308ndash315 2012

[5] S N Shao M Pipattanasomporn and S Rahman ldquoGridintegration of electric vehicles and demand response withcustomer choicerdquo IEEE Transactions on Smart Grid vol 3 no1 pp 543ndash550 2012

[6] K Clement-Nyns E Haesen and J Driesen ldquoThe impactof Charging plug-in hybrid electric vehicles on a residentialdistribution gridrdquo IEEE Transactions on Power Systems vol 25no 1 pp 371ndash380 2010

[7] P Siano ldquoDemand response and smart gridsmdasha surveyrdquoRenew-able and Sustainable Energy Reviews vol 30 pp 461ndash478 2014

[8] J Aghaei and M-I Alizadeh ldquoDemand response in smartelectricity grids equipped with renewable energy sources a

reviewrdquo Renewable and Sustainable Energy Reviews vol 18 pp64ndash72 2013

[9] RM Oviedo Z Fan S Gormus and P Kulkarni ldquoA residentialPHEV load coordination mechanism with renewable sourcesin smart gridsrdquo International Journal of Electrical Power andEnergy Systems vol 55 pp 511ndash521 2014

[10] US Department of Energy ldquoBenefits of demand response inelectricity markets and recommendations for achieving themrdquoTech Rep US Department of Energy 2006

[11] S Shao M Pipattanasomporn and S Rahman ldquoAn approachfor demand response to alleviate power system stress condi-tionsrdquo in Proceedings of the IEEE PES General Meeting TheElectrification of Transportation and the Grid of the Future pp1ndash7 Detroit Mich USA July 2011

[12] U S E I Administration httpwwweiagov[13] X Chen T Wei and S Hu ldquoUncertainty-aware household

appliance scheduling considering dynamic electricity pricing insmart homerdquo IEEE Transactions on Smart Grid vol 4 no 2 pp932ndash941 2013

[14] T Hubert and S Grijalva ldquoModeling for residential electricityoptimization in dynamic pricing environmentsrdquo IEEE Transac-tions on Smart Grid vol 3 no 4 pp 2224ndash2231 2012

[15] F de Angelis M Boaro D Fuselli S Squartini F Piazza andQWei ldquoOptimal home energy management under dynamic elec-trical and thermal constraintsrdquo IEEE Transactions on IndustrialInformatics vol 9 no 3 pp 1518ndash1527 2013

[16] A Arabali M Ghofrani M Etezadi-Amoli M S Fadali andY Baghzouz ldquoGenetic-algorithm-based optimization approachfor energy managementrdquo IEEE Transactions on Power Deliveryvol 28 no 1 pp 162ndash170 2013

[17] A Aswani N Master J Taneja D Culler and C TomlinldquoReducing transient and steady state electricity consumptionin HVAC using learning-based model-predictive controlrdquo Pro-ceedings of the IEEE vol 100 no 1 pp 240ndash253 2012

[18] Z Zhao W C Lee Y Shin and K B Song ldquoAn optimalpower schedulingmethod for demand response in home energymanagement systemrdquo IEEE Transactions on Smart Grid vol 4no 3 pp 1391ndash1400 2013

[19] P Du and N Lu ldquoAppliance commitment for household loadschedulingrdquo IEEE Transactions on Smart Grid vol 2 no 2 pp411ndash419 2011

[20] P Samadi H Mohsenian-Rad V W S Wong and R SchoberldquoTackling the load uncertainty challenges for energy consump-tion scheduling in smart gridrdquo IEEE Transactions on SmartGrid vol 4 no 2 pp 1007ndash1016 2013

[21] T T Kim and H V Poor ldquoScheduling power consumption withprice uncertaintyrdquo IEEE Transactions on Smart Grid vol 2 no3 pp 519ndash527 2011

[22] A-H Mohsenian-Rad and A Leon-Garcia ldquoOptimal residen-tial load control with price prediction in real-time electricitypricing environmentsrdquo IEEE Transactions on Smart Grid vol1 no 2 pp 120ndash133 2010

[23] Y Y Hong J K Lin C P Wu and C C Chuang ldquoMulti-objective air-conditioning control considering fuzzy parame-ters using immune clonal selection programmingrdquo IEEE Trans-actions on Smart Grid vol 3 no 4 pp 1603ndash1610 2012

[24] M PipattanasompornMKuzlu and S Rahman ldquoAn algorithmfor intelligent home energy management and demand responseanalysisrdquo IEEE Transactions on Smart Grid vol 3 no 4 pp2166ndash2173 2012


[25] C Guille and G Gross ldquoA conceptual framework for thevehicle-to-grid (V2G) implementationrdquo Energy Policy vol 37no 11 pp 4379ndash4390 2009

[26] F Berthold B Blunier D Bouquain S Williamson and AMiraoui ldquoOffline and online optimization of Plug-in HybridElectric Vehicle energy usage (home-to-vehicle and vehicle-to-home)rdquo in Proceedings of the IEEE Transportation ElectrificationConference and Expo (ITEC rsquo12) pp 1ndash6 June 2012

[27] F E R Commission ldquoAssessment of demand response ampadvanced meteringrdquo 2012

[28] D-MHan and J-H Lim ldquoDesign and implementation of smarthome energy management systems based on ZigBeerdquo IEEETransactions on Consumer Electronics vol 56 no 3 pp 1417ndash1425 2010

[29] S Shao M Pipattanasomporn and S Rahman ldquoDevelopmentof physical-based demand response-enabled residential loadmodelsrdquo IEEE Transactions on Power Systems vol 28 no 2 pp607ndash614 2013

[30] J ShiW-J Lee Y Liu Y Yang and PWang ldquoForecasting poweroutput of photovoltaic systems based on weather classificationand support vector machinesrdquo IEEE Transactions on IndustryApplications vol 48 no 3 pp 1064ndash1069 2012

[31] G Capizzi C Napoli and F Bonanno ldquoInnovative second-generation wavelets construction with recurrent neural net-works for solar radiation forecastingrdquo IEEE Transactions onNeural Networks and Learning Systems vol 23 no 11 pp 1805ndash1815 2012

[32] Y Y Zhang P Zeng and C Z Zang ldquoMulti-objective optimalcontrol algorithm for HVAC based on particle swarm opti-mizationrdquo in Proceedings of the 5th International Conferenceon Intelligent Control and Information Processing pp 417ndash423Dalian China August 2014

[33] J Kennedy ldquoParticle swarm optimizationrdquo in Encyclopedia ofMachine Learning pp 760ndash766 Springer 2010

[34] H Yin C Zhang B Zhang Y Guo and T Liu ldquoA hybridmultiobjective discrete particle swarm optimization algorithmfor a sla-aware service composition problemrdquo MathematicalProblems in Engineering vol 2014 no 14 Article ID 2529342014

[35] Y J Gong J Zhang H S Chung et al ldquoAn efficient resourceallocation scheme using particle swarm optimizationrdquo IEEETransactions on Evolutionary Computation vol 16 no 6 pp801ndash816 2012

[36] P Faria J Soares Z Vale H Morais and T Sousa ldquoModifiedparticle swarm optimization applied to integrated demandresponse and DG resources schedulingrdquo IEEE Transactions onSmart Grid vol 4 no 1 pp 606ndash616 2013

[37] P Siano and G Mokryani ldquoAssessing wind turbines placementin a distribution market environment by using particle swarmoptimizationrdquo IEEE Transactions on Power Systems vol 28 no4 pp 3852ndash3864 2013

[38] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 July 1999

[39] Z-H Zhan J Zhang Y Li and H S-H Chung ldquoAdaptiveparticle swarm optimizationrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 39 no 6 pp1362ndash1381 2009

[40] X F Xie W J Zhang and Z L Yang ldquoAdaptive particle swarmoptimization on individual levelrdquo in Proceedings of the 6thInternational Conference on Signal Processing vol 2 pp 1215ndash1218 IEEE Beijing China August 2002

[41] J W Black Integrating demand into the US electric power sys-tem technicaleconomic and regulatory frameworks for respon-sive load [PhD thesis] Engineering System Division Mas-sachusetts Institute of Technology Cambridge Mass USA2005

[42] The simulation of building-integrated fuel cell and othercogeneration systems httpwwwecbcsorgdocsAnnex 42Domestic Energy Profilespdf

[43] Residential Solar-Storage Systems Critical Load Analysis andIncremental Cost of Energy Storage httpwwwcunyeduaboutresourcessustainabilitysolar-summitagenda2013Kema-NYC Solar Summit Kleinbergpdf

[44] C Zhou K QianM Allan andW Zhou ldquoModeling of the costof EV battery wear due to V2G application in power systemsrdquoIEEE Transactions on Energy Conversion vol 26 no 4 pp 1041ndash1050 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


implementing DR in the residential sector has a great signif-icance

In recent years with the emergence of smart grid DRin residential sector is getting more and more attentionDifferent from the conventional grid the smart grid has two-way energy and information flows which provides the base toimplement DR in the residential sector Residents can controlthe operations of their home appliances batteries PHEVsand distributed generation through home energy manage-ment system (HEMS) Optimal scheduling algorithm is oneof the key components ofHEMS and a hot topic in smart grid

Many papers have been published about the optimalscheduling algorithms in HEMSMost papers [1 13ndash21] focuson minimizing usersrsquo electricity usage cost while retainingusersrsquo comfort level at a predefined range For example in[16] the authors developed a smart-grid strategy thatmatchesrenewable energy generation (ie wind and solar power)withthe heating ventilation and air conditioning (HVAC) loadIn [19] the thermostatically controlled household loads arescheduled based on electricity price and energy consumptionforecasts by considering usersrsquo comfort level to minimizeelectricity cost In order to minimize the energy paymentthe authors in [20] proposed amultistage optimization-basedreal-time residential load management algorithm that takesinto account load uncertainty These algorithms are singleobjective optimization algorithms where user comfort istransformed into a set of constraintsThese algorithms do notquantitatively consider usersrsquo comfort level during operationHowever from the usersrsquo point of view in addition to mon-etary expense high comfort level is another objective thatthey pursue Unfortunately the objectives of cost and comfortlevel are conflicting Therefore compared with single objec-tive optimization algorithms that only consider energy pay-ment the multiobjective optimization algorithm that notonly minimizes monetary expense but also maximizes com-fort level simultaneously is more attractive and natural

For the moment the multiobjective optimal schedulingalgorithm for HEMS has not been well investigated In [22]the authors proposed an optimal residential energy consump-tion scheduling framework to achieve a tradeoff betweenminimizing payment and waiting time for the operation ofeach household appliance In [22] waiting time is used toindicate userrsquos comfort level Although thismethod is suitableto washing machine (WM) clothes dryer (CD) and dish-washer (DW) it is meaningless for HVAC and electric waterheater (EWH) where the userrsquos concern is temperature In[23] the authors developed a multiobjective air condition-ing control algorithm based on immune clonal selectionprogramming to determine the day-ahead 24 h temperatureschedule for air conditioning In addition to electricity costthe expected error for the desired indoor temperature isintroduced as a user comfort level indicator and optimized asan objective of the algorithm However this indicator cannotreflect usersrsquo temperature preferences in different seasons

In smart grid a residential customer not only hashome appliances that consume electricity but also may havedistributed renewable generation (eg wind and PV) andbatteries which may have extra electricity sold to the grid forrevenue as presented [1 14] However the optimal scheduling

algorithms proposed in those papers did not consider thepower distribution relationships among loads batteries dis-tributed renewable generation and grid For example thealgorithms in [17ndash23] only schedule the operation of loadsThe systems proposed in [13ndash16] include load storage systemand distributed renewable generation where the ability to sellelectricity to the grid is not considered Although the systemdescribed in [1 14] consists of loads batteries and renewablegeneration and the user has the ability to sell electricity tothe grid the power distribution relationships among differentcomponents of HEMS are not thoroughly investigated

This paper proposes a HEMS framework that includesloads batteries and renewable generation interconnectedwith the grid through a smart mater In this framework theuser can sell the electricity generated by renewable sources orstored in batteries to the grid for profit A set of comfort levelindicators are proposed for different home appliances Basedon these indicators a multiobjective optimal schedule modelis built which minimizes monetary expense and maximizesuser comfort level simultaneously An improved hybriddiscrete particle swarm optimization (PSO) is employed tosolve themodel and amultiobjective optimization algorithmforHEMS is proposedThe algorithm schedules the operationof home appliances batteries and renewable generation (iePV) as well as the optimal power distribution among loadsbatteries renewable generation and grids

The rest of this paper is organized as follows Section 2proposes a framework of HEMS and presents models andconstraints of home appliances and batteries Section 3 des-cribes the comfort level indicators for different home appli-ances and the formulation of the HEMS optimization prob-lem including the optimization objectives and constraintsSection 4 presents a multiobjective optimization algorithmbased on an improved hybrid discrete PSO techniqueSection 5 provides a case study and compares the simulationresults of the proposed algorithm with other algorithmsproposed in the literature Section 6 concludes the paper

2 System Model

21 DR Abilities of Different Home Appliances In terms ofthe schedulability home appliances can be divided into twocategories schedulable appliances (SAs) and nonschedulableappliances (NSAs) SAs refer to the appliances whose opera-tions can be scheduled to a certain extent without reducinguserrsquos comfort level such as WM CD DW HVAC andEWH On the contrary NSAs are the appliances whose oper-ations must be started immediately when users need theirservices such as computer television microwave oven andlight NSAs are also called critical loads SAs can be furtherclassified into two groups interruptible loads and uninter-ruptible loads For example PHEVs are interruptible loadsUninterruptible loads refer to the appliances whose opera-tions can be delayed but after being started theymust be keptworking until the tasks are completed such as WM and DW

Although several papers investigate the controllingmeth-ods for low power consumption appliances such as refrig-erators and coffee maker they are not evident for DR due


Controller

Internet

BAU

Smart grid


PHEV

Smart meter

PV

Battery










119905is the


119905and 119901

B2L119905


119905and 119901

P2B119905


119905 and 119901

B2G119905




Grid Loads

PV BatterypP2Bt

pB2Lt

pG2Lt

pB2Gt

pG2Bt

pP2Lt

pP2Gt




119879Room119905+1

= 119879Room119905

+ Δ119905 sdot119866119905

Δ119888+ Δ119905 sdot

119862HVACΔ119888

sdot 119878HVAC119905

(1)


and 119879Room119905+1








119879Roommin le 119879

Room119905






119879EWH119905+1

= 119879EWH119905

sdot 119890minus(1(119877

1015840

119905sdot119862))sdotΔ119905

+ 119866EWH

sdot 1198771015840

119905sdot 119879

EWHenv119905

+ 119861119905sdot 1198771015840

119905sdot 119879

EWHin119905

+ 119876119905sdot 1198771015840

119905 sdot [1 minus 119890

minus(1(1198771015840

119905sdot119862))sdotΔ119905

]

(3)

where 119879EWH119905

and 119879EWH119905+1


119879EWHenv119905

and 119879EWHin119905


119905 1198771015840119905 and119876

119905are


119861119905= 119889water times 119865

119905times 119862119901 (4)

1198771015840

119905=

1

(119866EWH + 119861119905) (5)

119876119905= 34121 times 10

minus8times 119875

EWHtimes 119878

EWH119905

(6)




EWH119905



EWHmin 119879




119879EWHmin le 119879

EWH119905

le 119879EWHmax (7)


SOCPHEV119905+1

= SOCPHEV119905

+ 119901PHEV119905

sdotΔ119905

119862PHEVsize

(8)

where SOCPHEV119905

and SOCPHEV119905+1






is determined by

119901PHEV119905

= 119875PHEV

sdot 119878PHEV119905

(9)









119905le SOCPHEV

max (10)



max (11)





1 le 119873119886


task

119878119886

119905=

1 119905 = 119873119886

start 119873119886

start + 1 119873119886

start + 119873119886

task minus 1

0 otherwise

119873slot

sum

119905=1

119878119886

119905= 119873119886

task

(12)


119905



as

119901119886

119905= 119875119886sdot 119878119886

119905 (13)






SOCBatmin le SOCBat

119905le SOCBat

max (14)


SOCBat119905+1

= SOCBat119905

+(119901

Batch119905


119862Batsize

(15)

SOCBat119905+1

= SOCBat119905

minus(119901

Batdisch119905

sdot Δ119905)


(16)


119905and

SOCBat119905+1


119905and 119901

Batdisch119905




and PV 119901P2B119905


119905and grid 119901

B2G119905


0 le 119901Batch119905

= 119901P2B119905

+ 119901G2B119905



= 119901B2L119905

+ 119901B2G119905






(119901B2L119905

+ 119901B2G119905

) sdot (119901P2B119905

+ 119901G2B119905

) = 0

forall119905 isin 1 2 119873slot (19)


119901P2L119905

+ 119901P2B119905

+ 119901P2G119905

= 119901PV119905

sum

119886isinA119901119886

119905+ 119901

CL119905

= 119901P2L119905

+ 119901B2L119905

+ 119901G2L119905


(20)

where 119901PV119905






TRoommin TRoom

max

TRoomset

Comfortable zone

Tolerable zone

Intolerable zone

TRoomset + ΔTRoom


L

ΔTRoomL ΔTRoom

U



119865cost = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042 sdot

119873slot

sum

119905=1

[(119901G2L119904

11199042

119905+ 119901

G2B11990411199042

119905)


]

+

119873slot

sum

119905=1

[(119901P2B11990411199042

119905+ 119901

G2B11990411199042

119905+ 119901

B2L11990411199042

119905+ 119901

B2G11990411199042

119905)

times Δ119905 times 11988811990411199042

Deg] minus

119873slot

sum

119905=1

[(119901P2G119904

11199042

119905+ 119901

B2G11990411199042

119905) times Δ119905


]

(21)


1represents an







11199042








119868HVAC =100

119873slot

119873slot

sum

119905=1

119889Room119905

Δ119879Roommax

(22)




Roomset 119889Room

119905

is determined by

119889Room119905

=


Room119905

119879Roommin le 119879

Room119905


Room119871


Room119871

le 119879Room119905

le 119879Roomset + Δ119879

Room119880

119879Room119905


Roomset + Δ119879

Room119880

lt 119879Room119905

le 119879Roommax


(23)


and Δ119879Room119871

and Δ119879Room119880


119871gt Δ119879

Room119880



Roommax


119871 and Δ119879

Room119880



119868EWH =100

119873slot

119873slot

sum

119905=1

119889EWH119905

Δ119879EWHmax

119889EWH119905

=


EWH119905

119879EWHmin le 119879

EWH119905


EWH119871


EWH119871

le 119879EWH119905

le 119879EWHset + Δ119879

EWH119880

119879EWH119905


EWHset + Δ119879

EWH119880

lt 119879EWH119905

le 119879EWHmax


(24)



EWHmax minus 119879

EWHset and


119871 andΔ119879

EWH119880




119868PHEV =119889PHEV


119889PHEV

=




(25)





119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886


ideal + 119905119886

delay)


119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886




time slot 119905119886ideal + 119905119886


119886

delay 119905119886

max minus 119905119886




119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886


1





min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)


4 Algorithm Design



st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)




tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Controller

Internet

BAU

Smart grid


PHEV

Smart meter

PV

Battery










119905is the


119905and 119901

B2L119905


119905and 119901

P2B119905


119905 and 119901

B2G119905




Grid Loads

PV BatterypP2Bt

pB2Lt

pG2Lt

pB2Gt

pG2Bt

pP2Lt

pP2Gt




119879Room119905+1

= 119879Room119905

+ Δ119905 sdot119866119905

Δ119888+ Δ119905 sdot

119862HVACΔ119888

sdot 119878HVAC119905

(1)


and 119879Room119905+1








119879Roommin le 119879

Room119905






119879EWH119905+1

= 119879EWH119905

sdot 119890minus(1(119877

1015840

119905sdot119862))sdotΔ119905

+ 119866EWH

sdot 1198771015840

119905sdot 119879

EWHenv119905

+ 119861119905sdot 1198771015840

119905sdot 119879

EWHin119905

+ 119876119905sdot 1198771015840

119905 sdot [1 minus 119890

minus(1(1198771015840

119905sdot119862))sdotΔ119905

]

(3)

where 119879EWH119905

and 119879EWH119905+1


119879EWHenv119905

and 119879EWHin119905


119905 1198771015840119905 and119876

119905are


119861119905= 119889water times 119865

119905times 119862119901 (4)

1198771015840

119905=

1

(119866EWH + 119861119905) (5)

119876119905= 34121 times 10

minus8times 119875

EWHtimes 119878

EWH119905

(6)




EWH119905



EWHmin 119879




119879EWHmin le 119879

EWH119905

le 119879EWHmax (7)


SOCPHEV119905+1

= SOCPHEV119905

+ 119901PHEV119905

sdotΔ119905

119862PHEVsize

(8)

where SOCPHEV119905

and SOCPHEV119905+1






is determined by

119901PHEV119905

= 119875PHEV

sdot 119878PHEV119905

(9)









119905le SOCPHEV

max (10)



max (11)





1 le 119873119886


task

119878119886

119905=

1 119905 = 119873119886

start 119873119886

start + 1 119873119886

start + 119873119886

task minus 1

0 otherwise

119873slot

sum

119905=1

119878119886

119905= 119873119886

task

(12)


119905



as

119901119886

119905= 119875119886sdot 119878119886

119905 (13)






SOCBatmin le SOCBat

119905le SOCBat

max (14)


SOCBat119905+1

= SOCBat119905

+(119901

Batch119905


119862Batsize

(15)

SOCBat119905+1

= SOCBat119905

minus(119901

Batdisch119905

sdot Δ119905)


(16)


119905and

SOCBat119905+1


119905and 119901

Batdisch119905




and PV 119901P2B119905


119905and grid 119901

B2G119905


0 le 119901Batch119905

= 119901P2B119905

+ 119901G2B119905



= 119901B2L119905

+ 119901B2G119905






(119901B2L119905

+ 119901B2G119905

) sdot (119901P2B119905

+ 119901G2B119905

) = 0

forall119905 isin 1 2 119873slot (19)


119901P2L119905

+ 119901P2B119905

+ 119901P2G119905

= 119901PV119905

sum

119886isinA119901119886

119905+ 119901

CL119905

= 119901P2L119905

+ 119901B2L119905

+ 119901G2L119905


(20)

where 119901PV119905






TRoommin TRoom

max

TRoomset

Comfortable zone

Tolerable zone

Intolerable zone

TRoomset + ΔTRoom


L

ΔTRoomL ΔTRoom

U



119865cost = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042 sdot

119873slot

sum

119905=1

[(119901G2L119904

11199042

119905+ 119901

G2B11990411199042

119905)


]

+

119873slot

sum

119905=1

[(119901P2B11990411199042

119905+ 119901

G2B11990411199042

119905+ 119901

B2L11990411199042

119905+ 119901

B2G11990411199042

119905)

times Δ119905 times 11988811990411199042

Deg] minus

119873slot

sum

119905=1

[(119901P2G119904

11199042

119905+ 119901

B2G11990411199042

119905) times Δ119905


]

(21)


1represents an







11199042








119868HVAC =100

119873slot

119873slot

sum

119905=1

119889Room119905

Δ119879Roommax

(22)




Roomset 119889Room

119905

is determined by

119889Room119905

=


Room119905

119879Roommin le 119879

Room119905


Room119871


Room119871

le 119879Room119905

le 119879Roomset + Δ119879

Room119880

119879Room119905


Roomset + Δ119879

Room119880

lt 119879Room119905

le 119879Roommax


(23)


and Δ119879Room119871

and Δ119879Room119880


119871gt Δ119879

Room119880



Roommax


119871 and Δ119879

Room119880



119868EWH =100

119873slot

119873slot

sum

119905=1

119889EWH119905

Δ119879EWHmax

119889EWH119905

=


EWH119905

119879EWHmin le 119879

EWH119905


EWH119871


EWH119871

le 119879EWH119905

le 119879EWHset + Δ119879

EWH119880

119879EWH119905


EWHset + Δ119879

EWH119880

lt 119879EWH119905

le 119879EWHmax


(24)



EWHmax minus 119879

EWHset and


119871 andΔ119879

EWH119880




119868PHEV =119889PHEV


119889PHEV

=




(25)





119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886


ideal + 119905119886

delay)


119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886




time slot 119905119886ideal + 119905119886


119886

delay 119905119886

max minus 119905119886




119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886


1





min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)


4 Algorithm Design



st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)




tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Grid Loads

PV BatterypP2Bt

pB2Lt

pG2Lt

pB2Gt

pG2Bt

pP2Lt

pP2Gt




119879Room119905+1

= 119879Room119905

+ Δ119905 sdot119866119905

Δ119888+ Δ119905 sdot

119862HVACΔ119888

sdot 119878HVAC119905

(1)


and 119879Room119905+1








119879Roommin le 119879

Room119905






119879EWH119905+1

= 119879EWH119905

sdot 119890minus(1(119877

1015840

119905sdot119862))sdotΔ119905

+ 119866EWH

sdot 1198771015840

119905sdot 119879

EWHenv119905

+ 119861119905sdot 1198771015840

119905sdot 119879

EWHin119905

+ 119876119905sdot 1198771015840

119905 sdot [1 minus 119890

minus(1(1198771015840

119905sdot119862))sdotΔ119905

]

(3)

where 119879EWH119905

and 119879EWH119905+1


119879EWHenv119905

and 119879EWHin119905


119905 1198771015840119905 and119876

119905are


119861119905= 119889water times 119865

119905times 119862119901 (4)

1198771015840

119905=

1

(119866EWH + 119861119905) (5)

119876119905= 34121 times 10

minus8times 119875

EWHtimes 119878

EWH119905

(6)




EWH119905



EWHmin 119879




119879EWHmin le 119879

EWH119905

le 119879EWHmax (7)


SOCPHEV119905+1

= SOCPHEV119905

+ 119901PHEV119905

sdotΔ119905

119862PHEVsize

(8)

where SOCPHEV119905

and SOCPHEV119905+1






is determined by

119901PHEV119905

= 119875PHEV

sdot 119878PHEV119905

(9)









119905le SOCPHEV

max (10)



max (11)





1 le 119873119886


task

119878119886

119905=

1 119905 = 119873119886

start 119873119886

start + 1 119873119886

start + 119873119886

task minus 1

0 otherwise

119873slot

sum

119905=1

119878119886

119905= 119873119886

task

(12)


119905



as

119901119886

119905= 119875119886sdot 119878119886

119905 (13)






SOCBatmin le SOCBat

119905le SOCBat

max (14)


SOCBat119905+1

= SOCBat119905

+(119901

Batch119905


119862Batsize

(15)

SOCBat119905+1

= SOCBat119905

minus(119901

Batdisch119905

sdot Δ119905)


(16)


119905and

SOCBat119905+1


119905and 119901

Batdisch119905




and PV 119901P2B119905


119905and grid 119901

B2G119905


0 le 119901Batch119905

= 119901P2B119905

+ 119901G2B119905



= 119901B2L119905

+ 119901B2G119905






(119901B2L119905

+ 119901B2G119905

) sdot (119901P2B119905

+ 119901G2B119905

) = 0

forall119905 isin 1 2 119873slot (19)


119901P2L119905

+ 119901P2B119905

+ 119901P2G119905

= 119901PV119905

sum

119886isinA119901119886

119905+ 119901

CL119905

= 119901P2L119905

+ 119901B2L119905

+ 119901G2L119905


(20)

where 119901PV119905






TRoommin TRoom

max

TRoomset

Comfortable zone

Tolerable zone

Intolerable zone

TRoomset + ΔTRoom


L

ΔTRoomL ΔTRoom

U



119865cost = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042 sdot

119873slot

sum

119905=1

[(119901G2L119904

11199042

119905+ 119901

G2B11990411199042

119905)


]

+

119873slot

sum

119905=1

[(119901P2B11990411199042

119905+ 119901

G2B11990411199042

119905+ 119901

B2L11990411199042

119905+ 119901

B2G11990411199042

119905)

times Δ119905 times 11988811990411199042

Deg] minus

119873slot

sum

119905=1

[(119901P2G119904

11199042

119905+ 119901

B2G11990411199042

119905) times Δ119905


]

(21)


1represents an







11199042








119868HVAC =100

119873slot

119873slot

sum

119905=1

119889Room119905

Δ119879Roommax

(22)




Roomset 119889Room

119905

is determined by

119889Room119905

=


Room119905

119879Roommin le 119879

Room119905


Room119871


Room119871

le 119879Room119905

le 119879Roomset + Δ119879

Room119880

119879Room119905


Roomset + Δ119879

Room119880

lt 119879Room119905

le 119879Roommax


(23)


and Δ119879Room119871

and Δ119879Room119880


119871gt Δ119879

Room119880



Roommax


119871 and Δ119879

Room119880



119868EWH =100

119873slot

119873slot

sum

119905=1

119889EWH119905

Δ119879EWHmax

119889EWH119905

=


EWH119905

119879EWHmin le 119879

EWH119905


EWH119871


EWH119871

le 119879EWH119905

le 119879EWHset + Δ119879

EWH119880

119879EWH119905


EWHset + Δ119879

EWH119880

lt 119879EWH119905

le 119879EWHmax


(24)



EWHmax minus 119879

EWHset and


119871 andΔ119879

EWH119880




119868PHEV =119889PHEV


119889PHEV

=




(25)





119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886


ideal + 119905119886

delay)


119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886




time slot 119905119886ideal + 119905119886


119886

delay 119905119886

max minus 119905119886




119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886


1





min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)


4 Algorithm Design



st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)




tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 le 119873119886


task

119878119886

119905=

1 119905 = 119873119886

start 119873119886

start + 1 119873119886

start + 119873119886

task minus 1

0 otherwise

119873slot

sum

119905=1

119878119886

119905= 119873119886

task

(12)


119905



as

119901119886

119905= 119875119886sdot 119878119886

119905 (13)






SOCBatmin le SOCBat

119905le SOCBat

max (14)


SOCBat119905+1

= SOCBat119905

+(119901

Batch119905


119862Batsize

(15)

SOCBat119905+1

= SOCBat119905

minus(119901

Batdisch119905

sdot Δ119905)


(16)


119905and

SOCBat119905+1


119905and 119901

Batdisch119905




and PV 119901P2B119905


119905and grid 119901

B2G119905


0 le 119901Batch119905

= 119901P2B119905

+ 119901G2B119905



= 119901B2L119905

+ 119901B2G119905






(119901B2L119905

+ 119901B2G119905

) sdot (119901P2B119905

+ 119901G2B119905

) = 0

forall119905 isin 1 2 119873slot (19)


119901P2L119905

+ 119901P2B119905

+ 119901P2G119905

= 119901PV119905

sum

119886isinA119901119886

119905+ 119901

CL119905

= 119901P2L119905

+ 119901B2L119905

+ 119901G2L119905


(20)

where 119901PV119905






TRoommin TRoom

max

TRoomset

Comfortable zone

Tolerable zone

Intolerable zone

TRoomset + ΔTRoom


L

ΔTRoomL ΔTRoom

U



119865cost = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042 sdot

119873slot

sum

119905=1

[(119901G2L119904

11199042

119905+ 119901

G2B11990411199042

119905)


]

+

119873slot

sum

119905=1

[(119901P2B11990411199042

119905+ 119901

G2B11990411199042

119905+ 119901

B2L11990411199042

119905+ 119901

B2G11990411199042

119905)

times Δ119905 times 11988811990411199042

Deg] minus

119873slot

sum

119905=1

[(119901P2G119904

11199042

119905+ 119901

B2G11990411199042

119905) times Δ119905


]

(21)


1represents an







11199042








119868HVAC =100

119873slot

119873slot

sum

119905=1

119889Room119905

Δ119879Roommax

(22)




Roomset 119889Room

119905

is determined by

119889Room119905

=


Room119905

119879Roommin le 119879

Room119905


Room119871


Room119871

le 119879Room119905

le 119879Roomset + Δ119879

Room119880

119879Room119905


Roomset + Δ119879

Room119880

lt 119879Room119905

le 119879Roommax


(23)


and Δ119879Room119871

and Δ119879Room119880


119871gt Δ119879

Room119880



Roommax


119871 and Δ119879

Room119880



119868EWH =100

119873slot

119873slot

sum

119905=1

119889EWH119905

Δ119879EWHmax

119889EWH119905

=


EWH119905

119879EWHmin le 119879

EWH119905


EWH119871


EWH119871

le 119879EWH119905

le 119879EWHset + Δ119879

EWH119880

119879EWH119905


EWHset + Δ119879

EWH119880

lt 119879EWH119905

le 119879EWHmax


(24)



EWHmax minus 119879

EWHset and


119871 andΔ119879

EWH119880




119868PHEV =119889PHEV


119889PHEV

=




(25)





119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886


ideal + 119905119886

delay)


119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886




time slot 119905119886ideal + 119905119886


119886

delay 119905119886

max minus 119905119886




119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886


1





min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)


4 Algorithm Design



st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)




tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










TRoommin TRoom

max

TRoomset

Comfortable zone

Tolerable zone

Intolerable zone

TRoomset + ΔTRoom


L

ΔTRoomL ΔTRoom

U



119865cost = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042 sdot

119873slot

sum

119905=1

[(119901G2L119904

11199042

119905+ 119901

G2B11990411199042

119905)


]

+

119873slot

sum

119905=1

[(119901P2B11990411199042

119905+ 119901

G2B11990411199042

119905+ 119901

B2L11990411199042

119905+ 119901

B2G11990411199042

119905)

times Δ119905 times 11988811990411199042

Deg] minus

119873slot

sum

119905=1

[(119901P2G119904

11199042

119905+ 119901

B2G11990411199042

119905) times Δ119905


]

(21)


1represents an







11199042








119868HVAC =100

119873slot

119873slot

sum

119905=1

119889Room119905

Δ119879Roommax

(22)




Roomset 119889Room

119905

is determined by

119889Room119905

=


Room119905

119879Roommin le 119879

Room119905


Room119871


Room119871

le 119879Room119905

le 119879Roomset + Δ119879

Room119880

119879Room119905


Roomset + Δ119879

Room119880

lt 119879Room119905

le 119879Roommax


(23)


and Δ119879Room119871

and Δ119879Room119880


119871gt Δ119879

Room119880



Roommax


119871 and Δ119879

Room119880



119868EWH =100

119873slot

119873slot

sum

119905=1

119889EWH119905

Δ119879EWHmax

119889EWH119905

=


EWH119905

119879EWHmin le 119879

EWH119905


EWH119871


EWH119871

le 119879EWH119905

le 119879EWHset + Δ119879

EWH119880

119879EWH119905


EWHset + Δ119879

EWH119880

lt 119879EWH119905

le 119879EWHmax


(24)



EWHmax minus 119879

EWHset and


119871 andΔ119879

EWH119880




119868PHEV =119889PHEV


119889PHEV

=




(25)





119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886


ideal + 119905119886

delay)


119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886




time slot 119905119886ideal + 119905119886


119886

delay 119905119886

max minus 119905119886




119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886


1





min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)


4 Algorithm Design



st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)




tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119868PHEV =119889PHEV


119889PHEV

=




(25)





119868119886=

0 119905119886

min le 119905119886

start le 119905119886

ideal + 119905119886

delay

119905119886


ideal + 119905119886

delay)


119886

ideal + 119905119886

delay lt 119905119886

start le 119905119886

max minus 119905119886

work

119886 isin WMCDDW

(26)

where 119905119886

min and 119905119886




time slot 119905119886ideal + 119905119886


119886

delay 119905119886

max minus 119905119886




119865comfort = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

1

119873|A|

sum

119886isinA11986811990411199042

119886


(27)

where 11986811990411199042119886


1





min 119865cost

min 119865comfort

st (2) (7) (10)ndash(12) (14) (17)ndash(20)

(28)


4 Algorithm Design



st (2) (7) (10)ndash(12) (14) (17)ndash(20)(29)




tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










tcharge tdelay


Comfortable zone

Tolerable zone


tawork tadelay


Comfortable zone

Tolerable zone

Invalid zone





119865viol = sum

1199041isinStemp

sum

1199042isinSPV

11990111990411199011199042

sdot

119873slot

sum

119905=1

max (0 119879Room119905



Room119905

)


min

+

119873slot

sum

119905=1

max (0 119879EWH119905


EWHmin minus 119879

EWH119905

)


min

+

119873slot

sum

119905=1




)


min

(31)








119862119894(119896) =



where119862119894(119896) and119862








Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start


apat

sumapat lt pPV

t

SOCBatt lt SOCBat

max

pG2Lt = 0 pB2L

t = 0

pP2Lt = sum

apat


apat

pP2Bt = 0 pP2G

t = 0

pP2Lt = pPV

t


t

No

No

Yes

Yes

cgridt lt cBat

t

No

Yes

YesSOCBat

t lt SOCBatmax

pG2Bt = 0

pP2Bt = 0

pP2Gt = Δp1

pG2Lt = Δp2

pB2Gt = 0

pB2Lt = 0

cgridt lt c

Bat buyt

Yes

No

NoNo


max


YesNo

NoYes

cgridt lt c

Bat buyt c

gridt lt c

Bat buyt

pP2Bt = Δp1 pP2G

t = 0

pP2Bt = PBat ch

max

pB2Gt = 0


max

pG2Bt = 0

pB2Lt = Δp2

pG2Lt = 0

pG2Bt = 0

pB2Gt = 0 pG2B


max

pB2Lt = 0

pB2Gt = 0

pG2Lt = Δp2

pB2Gt = PBat disch

max pB2Gt = 0

pG2Bt = PBat ch

max minus Δp1

pB2Gt = 0

pG2Bt = 0

pB2Gt = 0

pG2Bt = PBat ch




max

End

SOCBatt gt SOCBat

min

pG2Bt = 0

cgridt gt cBat sell

t c

gridt gt cBat sell

t

pB2Lt = PBat disch

max







where


119878HVAC2


]

SEWH = [119878EWH1

119878EWH2

119878EWH119873slot

]


119878PHEV119905plug+1


]


CDstart 119905

DWstart]

(34)


119894is






119888



set Δ119879Room119871

Δ119879Room119880

and 119905plug






G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














G2L1

119901G2L2

119901G2L119873slot






as


119896max (35)



V119894119895(119896 + 1) = 119908 (119896) V

119894119895(119896) + 119888

11199031[119901119894119895

minus 119909119894119895(119896)]

+ 11988821199032[119901119892119895

minus 119909119894119895(119896)]

(36)

where V119894119895(119896) and V






119894 119901119892119895




119909119894119895(119896 + 1) =

1 1199033lt

10

(1 + exp (minusV119894119895(119896 + 1)))

0 otherwise

119895 = 1 2 3119873slot + 1 minus 119905plug

(37)



119909119894119895(119896 + 1) = 119909

119894119895(119896) + lceilV

119894119895(119896 + 1)rceil






P119894best (119896 + 1)

=



(39)


P119892(119896 + 1) = arg min

1le119894le119873pop[119865final (P

119894

best (119896 + 1))] (40)












119905 119901G2B119905

119901P2L119905

119901P2B119905

119901P2G119905

119901B2L119905

and 119901

B2G119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

8

9

Time slot

(cen

tskW

middoth)



119888Bat119905



119888Bat119905+1

=(119888

Bat119905

sdot 119864Bat119905

+ 119901G2B119905

sdot Δ119905 sdot 119888grid119905

)

119864Bat119905+1

119864Bat119905+1

= 119864Bat119905

+ (119901G2B119905

+ 119901P2B119905


(41)

where 119864Bat119905

and 119864Bat119905+1


119888Batsell119905


grid119905




119888Batsell119905

= 120573sell

sdotmax 119888grid1

119888grid2


0 lt 120573sell

lt 1

(42)



119888Batbuy119905


grid119905




119888Batbuy119905

= 120573buy


119888grid119905+1


0 lt 120573buy

lt 1

(43)

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)





5 Case Studies





119905


119905




10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10 20 30 40 50 60 70 80 90 100 110 1200

1

2

3

4

5

6

7

Time slot

Hot

wat

er u

sage

(gal

h)


10 20 30 40 50 60 70 80 90 100 110 1200

01

02

03

04

05

Time slot

Criti

cal l

oad

(kW

)


10 20 30 40 50 60 70 80 90 100 110 1200

05

1

15

2

25

3

35

4

45

Time slot

PV p

ower

out

put (

kW)


s1s2s3s4s5

s6s7s8s9s10

10 20 30 40 50 60 70 80 90 100 110 12070

75

80

85

90

95

100

Time slot

Out

door

tem

pera

ture

(∘F)


45

4

35

3

25

2

15

1

05

0

Pow

er o

utpu

t (Kw

)

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

s1s2s3s4s5

s6s7s8s9s10



Outdoor119905

minus

119879Room119905



Roommin 119879Room


119871 and Δ119879

Room119880









2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














2000 20002600 520 ft2






min (time slot) 46 96 46119905119886

max (time slot) 95 120 105119905119886

ideal (time slot) 60 100 70119905119886

work (time slots) 5 10 5119905119886



isequal to 119879

Room119905

and 119879EWHin119905



set Δ119879EWH119871

and Δ119879EWH119880



0is 05







minis 02 SOCBat



120578Batch = 120578























6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















6and PV
























Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Room temperature

Outdoortemperature

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Room

tem

pera

ture

(∘F)





140

130

120

110

100

9010 20 30 40 50 60 70 80 90 100 110 120

Time slot

Wat

er te

mpe

ratu

re (∘

F)


Time slot

Char

ge p

ower

(kW

)

0

1

2

3

4

95 100 105 110 115 120


Time slotPH

EV S

OC

0

05

1

95 100 105 110 115 120

Minimum battery SOC

Maximum battery SOC

PHEV battery SOC


CW p

ower

(kW

)

0

02

04

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(e) CW power

CD p

ower

(kW

)

0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(f) CD power

DW

pow

er (k

W)

0

05

1

0 10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(g) DW power


Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot



Pow

er (k

W)

0

05

1

15

2

25

10 20 30 40 50 60 70 80 90 100 110 120

Time slot






Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Minimum battery SOC

Batte

ry S

OC

0

05

1

10 20 30 40 50 60 70 80 90 100 110 120

Time slot

(a) Battery SOC

25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


Char

ging

pow

er (k

W)

Time slot


25

2

15

1

05

010 20 30 40 50 60 70 80 90 100 110 120


ischa

rgin

g po

wer

(kW

)

Time slot



4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot


Pow

er (k

W)


4

3

2

1

030 40 50 60 70 80 90 100 110

Time slot

Pow

er (k

W)



10 20 30 40 50 60 70 80 90 100 110 120

Time slot

Purc

hase

d po

wer

(kW

)

0

5

10












120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120

140

160

180

0 01 02 03 04 05 06 07 08 09 110

20

30

40

Elec

tric

ity co

st (c

ents)

Com

fort

leve

l ind

icat

or




0

2

4

6

8

10

12

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Sell factor


sell


0

1

2

3

4

5

6

0 01 02 03 04 05 06 07 08 09 1

Ener

gy (k

Wmiddoth

)

Buy factor


buy

6 Conclusion







Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Novel Multiobjective Optimization