Research Article Greenhouse Modeling Using Continuous ...downloads.hindawi.com/journals/mpe/2013/639306.pdf · Research Article Greenhouse Modeling Using Continuous Timed Petri Nets

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 639306 9 pageshttpdxdoiorg1011552013639306

Research ArticleGreenhouse Modeling Using Continuous Timed Petri Nets

Joseacute Luis Tovany1 Roberto Ross-Leoacuten2 Javier Ruiz-Leoacuten2

Antonio Ramiacuterez-Trevintildeo2 and Ofelia Begovich2

1 ITESM Campus Guadalajara Avenida General Ramon Corona 2514 Colonia Nuevo Mexico 45201 Zapopan JAL Mexico2 CINVESTAV-IPN Unidad Guadalajara Avenida del Bosque 1145 45019 Zapopan JAL Mexico

Correspondence should be addressed to Javier Ruiz-Leon jruizgdlcinvestavmx

Received 5 April 2013 Revised 21 June 2013 Accepted 21 June 2013

Academic Editor Hamid Reza Karimi

Copyright copy 2013 Jose Luis Tovany et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a continuous timed Petri nets (ContPNs) based greenhouse modeling methodology The presentedmethodology is based on the definition of elementary ContPNmodules which are designed to capture the components of a generalenergy and mass balance differential equation like parts that are reducing or increasing variables such as heat CO

2concentration

and humidityThe semantics of ContPN is also extended in order to deal with variables depending on external greenhouse variablessuch as solar radiation Each external variable is represented by a place whose marking depends on an a priori known functionfor instance the solar radiation function of the greenhouse site which can be obtained statistically The modeling methodology isillustrated with a greenhouse modeling example

1 Introduction

Greenhouses allow increasing the quantity and improve thequality of the crops produced inside them The automationof greenhouses has been one of the main topics regardinggreenhouse functioning and production since using controlloops tuned as the agronomist and biologist researcherspropose improves the use of water energy and fertiliz-ers Simultaneously the volume and quality of crops areincreased

One of the main problems in controlling greenhousesis obtaining a fine mathematical greenhouse model captur-ing the actual greenhouse behavior since the models arerepresented by nonlinear differential equations includingdisturbances where the parameters are time variant Thederived models lead to very complex differential equationsand they are hard to obtain

In order to obtain a greenhouse model researchers usemany approaches most of them are based on heat and massbalance equations In [1] a greenhouse model includingnatural ventilation and evaporative cooling is presentedTheauthors use heat and mass balance equations to derive themodel Since that approach includes a linearization stage

the model is valid only around the operating point Anotherapproach deals with linear and nonlinear identification of thegreenhouse behavior using neural networks [2]This methoduses however a large amount of data samples due to theirlarge number of degrees of freedom and it also requires alarge computation time for training the neuronal networkIn [3] a robust method for nonlinear identification of aclimate system using evolutionary algorithms was proposedAlthough the model is validated the convergence of thealgorithm could be too long In [4] a fuzzy model ofa greenhouse by taking heat and water measurements isproposed However the number of fuzzy rules needed tocompute an actual greenhouse model is too large and it is notclear how to find out the rules

The approach herein presented uses continuous timedPetri nets (ContPNs) [5 6] to capture the greenhouse dynam-ics We propose a bottom-up modeling methodology firstContPNelementarymodules (balance generation consump-tion and fluid balance) are defined to represent the basiccomponents of an energy and mass balance equation such asstorage source loss generation and consumption of mass orenergy A balance module representing any energy or massbalance equation is obtained by merging these elementary

2 Mathematical Problems in Engineering

Heatsource Heat

storage

Heatsurroundings

Figure 1 Heat balance

t1

b

c p2

t2

d

a

p1

Figure 2 ContPN representation of tokens exchange

ContPN modules Then a ContPN model is constructed foreach greenhouse state variable adding as many elementarymodules as components exist in the energy and mass balanceequation Afterwards themodel parameters are identified andrepresented by the ContPN parameters such as marking andtransition firing rates

The greenhouse ContPN modeling methodology pre-sented in this paper provides a pictorial representation ofvariables which allows easy understanding of the interactionbetween places (variables) Also the ContPN model allowshaving a modular model where elements can be added orremoved as necessaryThe lack of negative values in Petri netsdoes not affect the system modeling because the greenhouseclimate (temperature water vapor concentration and CO

2

concentration) is a positive systemThis work is organized as follows In Section 2 some

concepts about Petri nets are presented and the extendedsemantics is proposed In Section 3 a Petri net modelingprocedure for greenhouses is proposed Section 4 presents anexample of a greenhouse modeled with ContPN Finally inSection 5 some conclusions are given

2 Preliminaries

21 Petri Nets Concepts This subsection introduces basicconcepts on continuous timed Petri nets In order to havemore detailed information an interested reader may alsoconsult [7ndash10]

Definition 1 A continuous Petri net (CPN) is a pair (119873m0)

where 119873 = (119875 119879PrePost) is a Petri net structure (PN) andm0isin R+ cup 0

|119875| is the initial marking and 119875 = 1199011 119901

119899

and 119879 = 1199051 119905

119896 are finite sets of elements named places

and transitions respectively PrePost isin N cup 0|119875|times|119879| are the

pre- and postincidence matrices where Pre[119894 119895](Post[119894 119895])represents the weight of the arc going from 119901

119894to 119905119895(from 119905

119895

to 119901119894)

pd tb1 pvar tb2

Figure 3 Balance module

pd tg pvar

Figure 4 Generation module

The incidence matrix denoted by C is defined by C =

Post minus Pre Right and left annullers of C are called 119879- and119875-flows respectively

Each place 119901119894has a marking denoted by 119898

119894isin R+ cup 0

Let 119909119894 119909119895

isin 119875 cup 119879 then the set ∙119909119894= 119909119895

| Pre[119895 119894] gt 0(119909119894∙ = 119909

119895| Post[119895 119894] gt 0) is the preset (postset) of 119909

119894

A transition 119905119895isin 119879 is enabled at markingm if and only if

for all 119901119894isin ∙119905119895 119898119894gt 0 Its enabling degree is given by

enab (119905119895m) = min

119901119894isin∙119905119895

119898[119901119894]

Pre [119894 119895] (1)

The enabling degree determines themaximum amount of119905119895that can be fired at marking m leading to a new marking

thusm1015840 = m + 120572119862[∙ 119895] where 0 lt 120572 lt enab(119905119895m)

If m is reachable from m0by firing the finite sequence

120590 of enabled transitions then m = m0+ C is named the

CPN state equation where isin R+ cup 0|119879| is the firing count

vector that is 119895is the cumulative amount of firing of 119905

119895in

the sequence 120590The set of all reachable markings from m

0is called the

reachability set and it is denoted by RS(119873m0) In the case of

a CPN system RS(119873m0) is a convex set [11]

A CPN is bounded when every place is bounded that isfor all 119901 isin 119875 exist119887

119901isin R st 119898[119901] le 119887

119901at every reachable

markingm and it is live when every transition is live (it canultimately be fired from every reachable marking) [8]

Definition 2 A continuous timed Petri net is a 3-tupleContPN = (119873 120582m

0) where (119873m

0) is a CPN and 120582

119879 rarr R+|119879| is a function associating a firing rate with each

transitionThe state equation of a ContPN is

m (120591) = Cf (120591) (2)

where 120591 is the time variable f(120591) = (120591)

Definition 3 A ContPN is called infinite server semanticContPN if the flow of a transition 119905

119894is

119891119894= 120582119894sdot enab (119905

119894m) = 120582

119894sdot min119901isin∙119905119894

119898 (119901)

Pre [119901 119905119894] (3)

where 119891119894is the flow of transition 119905

119894and the 119894th entry of the

vector f

Mathematical Problems in Engineering 3

pd tc pvar

Figure 5 Consumption module

pd tfb2tfb1 pvar

pconv

Figure 6 Fluid balance module

Notice that ContPN under infinite server semantics canactually be considered as a piecewise linear system (a class ofhybrid systems) due to the119898119894119899119894119898119906119898 operator that appears inthe enabling function in the flow definition Equation (2) canbe expressed as a piecewise linear system given by

m = CΛΠ (m) sdot m (4)

The firing rate matrix is denoted by Λ = diag(1205821

120582|119879|

) A configuration of a ContPN at 119898 is a set of (119901 119905) arcsdescribing the effective flow of all transitions

Π (m) [119894 119895] =

1

Pre [119894 119895]if119901119894is constraining 119905

119895

0 otherwise(5)

Definition 4 A ContPN is called product server semanticContPN if the flow of a transition 119905

119894is

119891119894= 120582119894sdot prod119901isin∙119905119894

119898 (119901)

Pre [119901 119905119894] (6)

In order to apply a control action in (2) a subtractingterm u such that 0 le 119906

119894le 119891119894 is added to every transition

119905119894to indicate that its flow can be reduced This control action

is adequate because it captures the real behavior that themaximum machine throughput can only be reduced Thusthe controlled flow of transition 119905

119894becomes 119908

119894= 119891119894minus 119906119894

Then introducing f = ΛΠ(m) sdot m and u in (2) the forcedstate equation is

m = C [f minus u] = Cw

0 le 119906119894le 119891119894

(7)

In order to obtain a simplified version of the stateequation the input vector u is rewritten as u = I

119906ΛΠ(m) sdotm

where I119906

= diag(1198681199061 119868

119906|119879|) and 0 le 119868

119906119894le 1 Then the

matrix I119888= I minus I

119906is constructed and the state equation can

be rewritten as

m = CI119888f = Cw (8)

Solar radiation

Humidifier

Lateral

Ventilation

Figure 7 A greenhouse example

p6p4 t5 t6

t7 t9

t14t12

t1t2 t3

p2 p1 p5

p3

p7

t13 t15 t8 t4 p10

p9

p8

t10t11

t16

t19

t23t22

t20 t21t17 t18

Figure 8 ContPN model of greenhouse temperature and water va-por concentration

0 1 2 3 4 5 6 7 8

285

290

295

300

305

310

315

320

325

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)

Figure 9 Greenhouse temperature dynamics using sine functionsfor disturbances


0 1 2 3 4 5 6 7 82

3

4

5

6

7

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

65

55

45

35

25

Figure 10 Water vapor concentration dynamics using sine func-tions for disturbances

Notice that 0 le 119868119888119894le 1 A transition is called noncontrollable

when its flow cannot be reduced Every non-controllabletransition 119905

119895has associated a constant input control 119868

119888119895= 1

22 ContPN with Extended Semantics Regular ContPNmodels do not include disturbances and nonlinearitiestherefore it is required to add semantics that allow us toincorporate them

Definition 5 Aplace is called a function place if its marking attime 120591 is determined by the actual marking of other places orexternal disturbances Thus the marking of a function place119901 is described by

119898[119901] (120591) = ℎ (m 119863) (9)

where 119898[119901](120591) is the marking of place 119901 at time 120591 ℎ(∙) is aknown function and 119863 is a measurable disturbance

Notice that the marking of function places is not deter-mined directly by the differential equations Also since func-tion places are mainly seen as disturbances their markings donot represent controllable variables

Definition 6 AContPN that includes function places is calleda ContPN with extended semantics

From now on all ContPNs in this paper are consideredwith extended semantics

23 Mass and Energy Balance Equations From experienceit is well known that matter and energy may change theirform but they cannot be created or destroyed This notion isexpressed in the general mass and energy balance equation

119876 = int120591119891

1205910

(119902in minus 119902out + 119902gen minus 119902con) 119889120591 (10)

where 119876 is the accumulated quantity (final amount ofquantity minus initial amount of quantity) inside the system

boundary during the time interval [1205910 120591119891] 119902in is the amount

of quantity entering the system through the system boundary119902out is the amount of quantity leaving the system through thesystem boundary 119902gen is the amount of quantity generated(ie formed) inside the system boundary 119902con is the amountof quantity consumed (ie converted to another form) insidethe system boundary and a quantity may be in any mass orheat unit

3 Modeling Methodology

31 Greenhouse System A greenhouse is a building whichisolates the crop from the outside environment prevent-ing it from hazards such as extreme climate changes andplagues Also it improves the crop production by meansof the greenhouse climate manipulation provided throughsome components that can be added temperature can bemanipulated by means of ventilation heating systems andwater sprinklers water concentration can be manipulated bymeans of humidifiers water sprinklers ventilation and fansluminosity can bemanipulated bymeans of shadedmesh andlight bulbs carbon dioxide concentration can bemanipulatedby means of CO

2injectors Notice that some components

affect more than one climate variable The selection ofcomponents varies depending on the geographical area andeconomical factors

Nevertheless all environmental influences over a green-housemanipulated or not fulfill the energy andmass balanceequation (10) For example a simple heat balance equation isdepicted in Figure 1 where the heat source for instance solarradiation is the heat entering (119902in) into a greenhouse systemthe heat storage the greenhouse itself is the heat absorbed(119876) by the greenhouse system and the heat to surroundingsfor instance by ventilation is the heat loss (119902out) outside thegreenhouse system

For the generated and consumed flows some examplesare as follows the energy gained from condensation is part ofthe generated heat 119902gen the energy consumed by evaporativecooling is part of the heat consumed 119902con

Therefore we propose a modeling approach based onthe construction of ContPN modules that represent eachcomponent of the balance equation

32 ElementaryModules Somemodules are defined in orderto represent the flows in the balance equation A firstapproach to a balance module is obtained from Figure 2

The ContPN of Figure 2 has the following matrices

C = [minus119886 119889

119888 minus119888] Π =

[[

[

1

1198860

01

119888

]]

]

Λ = [1205821

0

0 1205822

]

(11)Thus the marking equations are given by

1= minus12058211198981+

119889

11988812058221198982

2=

119887

11988612058211198981minus 12058221198982

(12)


0 2 4 6 8285

290

295

300

Time (hr)

Out

side t

empe

ratu

re (K

)

(a)

0 2 4 6 80

500

1000

1500

Time (hr)

Sola

r rad

iatio

n (W

m2)

(b)

0 2 4 6 86

7

8

Time (hr)

Out

side v

apor

conc

entr

atio

n (k

gm

3)

times10minus3

(c)

0 2 4 6 80

1

2

Time (hr)

Win

d sp

eed

(ms

)

15

05

(d)

Figure 11 Measured disturbances

The balance of the marking is given when the steadystate markings of 119898

1and 119898

2are equal So the equilibrium

points of the previous equations must be 1198981= 1198982 Thus the

required relationships are

119887

119886=

119888

119889=

1205822

1205821

(13)

Replacing the latter relationships the equations of themarking are

1= minus12058211198981+ 12058211198982

2= 12058221198981minus 12058221198982

(14)

For example given a temperature in 1199011and a different

temperature in 1199012 the difference between 120582

1and 120582

2is given

by the heat capacity of each system

In order to prove that the equilibrium points are stablethe Lyapunov function 119881(m) = (12120582

1)11989821+ (12120582

2)11989822is

used where the derivative of 119881(m) is given by

(m) =1

1205821

11989811+

1

1205822

11989822

= minus 1198982

1+ 211989821198981minus 1198982

2

= minus (1198981minus 1198982)2

(15)

which is negative for any 1198981

= 1198982 so the equilibrium points

are stableThe number of tokens in the steady state depends on

the initial values 1198981(0) and 119898

2(0) The marking at the

equilibrium point can be separated in three cases 1205821

lt 1205822

1205821gt 1205822 and 120582

1= 1205822

If 1205821

lt 1205822 1198981gains (or losses) tokens faster than 119898

2

losses (or gains) them so the steady state marking value is


0 1 2 3 4 5 6 7 8

290

300

310

320

330

340

350

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)

Figure 12 Greenhouse temperature dynamics with measured dis-turbances

0 1 2 3 4 5 6 7 82

3

4

5

6

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

55

45

35

25

Figure 13Water vapor concentration dynamics with measured dis-turbances

closer to 1198981(0) If 120582

1gt 1205822 1198981gains (or losses) tokens faster

than 1198982losses (or gains) them so the steady state marking

value is closer to 1198982(0) In case 120582

1= 1205822 the steady state

marking is given by (1198981(0) + 119898

2(0))2

The balance module as presented in Figure 2 with restric-tions (13) can be represented as the ContPN of Figure 3 whenone of the variables is measured (119901

119889is a function place) and

its dynamics are notmodeled In order to represent a balancethe transitions 119905

1198871and 1199051198872have the same firing rate 120582

1198871= 1205821198872

Thus the equation of the ContPN of Figure 3 is

var = minus1205821198871119898var + 120582

1198871119898119889 (16)

For generation and consumption flows the ContPNs ofFigures 4 and 5 are used respectively

The heat and mass balance can be carried out by a fluidthat affects proportionally the transfer between variables Inthat case a ContPN as in Figure 6 is used This ContPNis defined with product semantics in order to represent theproduct of the fluid 119901conv with the variables 119901var and 119901

119889

There are modules dependent on a device but the devicedynamics is considered to be faster than the greenhousedynamics so the dynamics of the devices are not modeledThe only difference is that transitions related to devices arecontrollable that is the transitions of a device module havethe form 119868

119888119894120582119894as stated in Section 2

33 Greenhouse ContPN Model Since every greenhousephysical variable fulfills the energy and mass balance equa-tion we propose amodeling approach based on the construc-tion of modules as described in the following

331 Modeling Procedure

(1) Create places for variables and function places fordisturbancesVariable places capture the greenhouse variables(such as soil temperature air temeprature and CO

2

concentration) and function places capture externalvariables (such as solar radiation and external tem-perature)

(2) Construct amodule for each variable of interest in thegreenhouse

(2a) A balance module is associated with each phys-ical exchange (heat or mass) affecting the cor-responding variable (eg ventilation conduc-tion)

(2b) A generator module is associated to each physi-cal transformation inside the greenhouse whichincreases the corresponding variable (eg evap-otranspiration)

(2c) A consumption module is associated with eachphysical transformation inside the greenhousewhich decreases the corresponding variable(eg condensation evapotranspiration)

(2d) A fluid balance module is associated with eachphysical exchange (heat or mass) affecting thecorresponding variable with the proportionaleffect of a fluid (eg natural ventilation)

(3) Merge all constructed balance modules(4) Identify the model parameters

Following the previous procedure we obtain the greenhouse119862119900119899119905119875119873 model For a practical illustration we show in thenext section the greenhouseContPNmodeling of two climatevariables temperature and water vapor concentration

4 Greenhouse Modeling Example

Consider the greenhouse climate system of Figure 7 Wewant to obtain the greenhouse temperature and water vapor


concentration model According to step 1 of the modelingprocedure we have to associate places for the involvedvariables greenhouse temperature 119879

119892 soil temperature 119879

119904

and one for the vapor concentration119862H2O as shown inTable 1The function places associated with the other variables

are 1199014to solar radiation 119868

119900 1199015to outside temperature 119879

119900

1199016to subsoil temperature 119879ss 119901

7to outside water vapor

concentration 119862H2O119900 1199018 for wind speed V 1199019to humidifier

maximum water flow 119865hum and 11990110

to water condensation120593cons

Following step 2 for the greenhouse temperature 119879119892

we construct generation module for solar radiation (119902119868119900

in)and condensation 120593cons (119902

consin ) consumption module for the

humidifier 119865hum (119902humcon ) balance module for soil tempera-ture 119879

119904(119902119879119904

bal) leaks nondependent on wind and conductionthrough cover (119902

119879119900

bal119888) and fluid balance module for leaksdependent on wind (119902

119879119900

fbalVin) and controlled natural ventila-tion (119902

119879119900

fbalVcn)For the greenhouse humidity 119862H2O we construct genera-

tion module for humidifier 119865hum (120593humin ) consumption mod-

ule for condensation120593cons (120593condcon ) balancemodule for outside

humidity 119862H2O119900 leaks nondependent on wind (120593119862H2O119900bal ) and

fluid balance module for leaks dependent on wind (120593119862H2O119900fbalVin)

and controlled natural ventilation (120593119862H2O119900fbalVcn)

Since soil temperature is also modeled we constructbalance module for greenhouse temperature 119879

119892(119902119879119892

bal) andsubsoil temperature (119902

119879ssbal) Then modules for each variable

are constructed as shown in Table 2Then we merge all constructed modulesThus we obtain

the ContPN depicted in Figure 8 with state equations

1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)

Solar radiation 119868119900 soil temperature 119879

119904 outside temper-

ature 119879119900 water vapor condensation 120593cons humidifier 119865hum

and outside humidity concentration 119862H2O119900 are consideredas random albeit measurable This is because along the day

Table 1 Relation between variables and places

Variable Place119879119892

1199011

119879119904

1199012

119862H2O 1199013

Table 2 Relation between variables and function places

Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23

these environmental variables are changing nevertheless wecan add sensors in order to measure them

It has to be noted that the energy balance between1198981and

1198982is related by minus120582

71198981+12058281198982and since a balancemodule is

used1205827= 1205828 so the balance can be referred to as120582

8(1198982minus1198981)

which is an energy exchangeA similar procedure can be done for the remaining terms

of 1 1198984 and 119898

1are related by a balance module 119898

5and

1198981by two-fluid balance module (one is controllable but the

other is not) it gains energy from 1198984 it also gains and loses

water because of 119898119909and 119898

119910 respectively It has to be noted

that for a greenhouse temperature above 273∘K the tokens in1198981will be higher than the tokens in 119898

9and 119898

10

In the case of 2 the relations are only balance modules

between 1198982and119898

1or1198986 For

3 there is a balance module

between 1198983and 119898

7 1198987and 119898

3are related by two-fluid

balance module (one is controllable but the other is not) itgains and loses water because of 119898

119909and 119898

119910 respectively

The identification of the model parameters can be carriedout according to the preferred method In this example theleast square method is used The model proposed in [12] istaken as the real system and the ContPN model depicted inFigure 8 will be the identified model In order to simplify themethod the identification is carried out in two steps In thefirst one the firing of controllable transitions is avoided (iethe parameters associated with noncontrollable transitionsare computed) These parameters are fed to the second


identification step In this step the parameters associated withcontrollable transitions are derived and the whole ContPNmodel is obtained

We are using the parameters values presented in [12Chapter 7 pp 135ndash150] without any crop inside the green-house and heating pipes are not considered Besides weadd humidifier dynamics and external weather variables areconsidered as a sine function at different frequencies andamplitudes The identification was carried out using theleast squares method The simulation time for the originalmodel is 8 hours so the functions used to approximate theexternal variables are positive during the simulation timeThe following external variables were considered for theidentification

119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems

120593cons = 3 times 10minus10

+ 2 times 10minus10 sin (119905) kgm2s

(18)

The percentage of use of the actuators is presented asfollows

1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)

The initial conditions are

119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)

In order to validate the proposedmodelingmethodologywe nowpresent a comparison between ourmodel and the oneproposed by [12] All the simulations and identification werecarried out in MATLAB and Simulink

In Figure 9 a comparison between the ContPN green-house temperature model and the one used by [12] ispresented In Figure 10 a comparison between the ContPNgreenhouse humidity model and the one used by [12] ispresented From these figures it can be seen that the proposedmodeling methodology shows a good agreement with theoriginal system capturing in an accurate way the dynamicbehavior of the greenhouses variablesThe error (119890

119879119892= 119879119892orminus

119879119892id and 119890

119862H2O= 119862H2Oor minus 119862H2Oid) between the original

system and the identified system is less than 10minus3In order to demonstrate the accuracy of the proposed

modeling methodology under a real and severe scenarioanother identification is carried out using real data for 119868

119900

119879119900 119862H2O119900 and V (see Figure 11) in the winter of 2012 from

a greenhouse prototype located in Jalisco Mexico The otherexternal disturbances120593cons 1198681198881 and 119868

1198882are taken as in (18) and

(19) The initial conditions are the same as in (20) It can beseen in Figures 12 and 13 that the identified model has a smallerror in comparison to the original model which is still lessthan 10minus3

5 Conclusions

The greenhouse ContPN modeling methodology presentedin this paper provides a pictorial representation of variableswhich allows easy understanding of the interaction betweenthem The bounds in actuators are represented naturally bythe marking of a place as in the case of the humidifier Inthe case of the humidifier although the tokens flow from itsplace can be reduced with the control the representing placeis a source place because the tokens are constant and theyrepresent the maximum capacity of water flow

The most important point is that it allows having amodular model Thus elements can be added or removedas necessary Also the lack of negative values in PN do notaffect the system modeling because the greenhouse climate(temperature water vapor concentration and CO

2concen-

tration) is a positive systemThe simulation contains fixed parameters for the original

system but a greenhouse parameter may change accordingto certain variables which will provide bigger variations inthe model and the need to identify constantly in orderto change the model parameters that represent better thegreenhouse Future work will include the identification of areal greenhouse prototype and its control design

Acknowledgments

This work was supported by project no 107195 CONACyTMexico J L Tovany and R Ross-Leon were supported byCONACyT Grants nos 300891 and 13527 respectively

References

[1] T Boulard and A Baille ldquoA simple greenhouse climate controlmodel incorporating effects of ventilation and evaporativecoolingrdquo Agricultural and Forest Meteorology vol 65 no 3-4pp 145ndash157 1993

[2] J B Cunha ldquoGreenhouse climate models an overviewrdquo inProceedings of the 4th European Federation for Information Tech-nologies in Agriculture Food and the Environment Conference(EFITA rsquo03) Debrecen Hungary 2003

[3] J M Herrero X Blasco MMartınez C Ramos and J SanchisldquoRobust identification of non-linear greenhouse model usingevolutionary algorithmsrdquo Control Engineering Practice vol 16no 5 pp 515ndash530 2008

[4] P Salgado and J B Cunha ldquoGreenhouse climate hierarchicalfuzzy modellingrdquo Control Engineering Practice vol 13 no 5 pp613ndash628 2005

[5] M Kloetzer C Mahulea C Belta and M Silva ldquoAn automatedframework for formal verification of timed continuous petrinetsrdquo IEEE Transactions on Industrial Informatics vol 6 no 3pp 460ndash471 2010


[6] C R Vazquez and M Silva ldquoTiming-dependent boundednessand liveness in continuous petri netsrdquo in Proceedings of the 10thInternationalWorkshop on Discrete Event Systems (WODES rsquo10)pp 10ndash17 2010

[7] R David and H Alla Discrete Continuous and Hybrid PetriNets Springer Berlin Germany 2005

[8] J Desel and J Esparza Free Choice Petri Nets vol 40 ofCambridge Tracts in Theoretical Computer Science CambridgeUniversity Press Cambridge UK 1995

[9] C Mahulea A Ramırez-Trevino L Recalde and M SilvaldquoSteady-state control reference and token conservation laws incontinuous petri net systemsrdquo IEEE Transactions on Automa-tion Science and Engineering vol 5 no 2 pp 307ndash320 2008

[10] M Silva and L Recalde ldquoPetri nets and integrality relaxationsa view of continuous petri net modelsrdquo IEEE Transactions onSystemsMan andCybernetics C vol 32 no 4 pp 314ndash327 2002

[11] M Silva and L Recalde ldquoOn fluidification of Petri Nets fromdiscrete to hybrid and continuous modelsrdquo Annual Reviews inControl vol 28 no 2 pp 253ndash266 2004

[12] G van Straten G van Willigenburg E van Henten and R vanOoteghem Optimal Control of Greenhouse Cultivation CRCPress New York NY USA 2010

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

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

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


Heatsource Heat

storage

Heatsurroundings

Figure 1 Heat balance

t1

b

c p2

t2

d

a

p1

Figure 2 ContPN representation of tokens exchange

ContPN modules Then a ContPN model is constructed foreach greenhouse state variable adding as many elementarymodules as components exist in the energy and mass balanceequation Afterwards themodel parameters are identified andrepresented by the ContPN parameters such as marking andtransition firing rates

The greenhouse ContPN modeling methodology pre-sented in this paper provides a pictorial representation ofvariables which allows easy understanding of the interactionbetween places (variables) Also the ContPN model allowshaving a modular model where elements can be added orremoved as necessaryThe lack of negative values in Petri netsdoes not affect the system modeling because the greenhouseclimate (temperature water vapor concentration and CO

2

concentration) is a positive systemThis work is organized as follows In Section 2 some

concepts about Petri nets are presented and the extendedsemantics is proposed In Section 3 a Petri net modelingprocedure for greenhouses is proposed Section 4 presents anexample of a greenhouse modeled with ContPN Finally inSection 5 some conclusions are given

2 Preliminaries

21 Petri Nets Concepts This subsection introduces basicconcepts on continuous timed Petri nets In order to havemore detailed information an interested reader may alsoconsult [7ndash10]

Definition 1 A continuous Petri net (CPN) is a pair (119873m0)

where 119873 = (119875 119879PrePost) is a Petri net structure (PN) andm0isin R+ cup 0

|119875| is the initial marking and 119875 = 1199011 119901

119899

and 119879 = 1199051 119905

119896 are finite sets of elements named places

and transitions respectively PrePost isin N cup 0|119875|times|119879| are the

pre- and postincidence matrices where Pre[119894 119895](Post[119894 119895])represents the weight of the arc going from 119901

119894to 119905119895(from 119905

119895

to 119901119894)

pd tb1 pvar tb2

Figure 3 Balance module

pd tg pvar

Figure 4 Generation module

The incidence matrix denoted by C is defined by C =

Post minus Pre Right and left annullers of C are called 119879- and119875-flows respectively

Each place 119901119894has a marking denoted by 119898

119894isin R+ cup 0

Let 119909119894 119909119895

isin 119875 cup 119879 then the set ∙119909119894= 119909119895

| Pre[119895 119894] gt 0(119909119894∙ = 119909

119895| Post[119895 119894] gt 0) is the preset (postset) of 119909

119894

A transition 119905119895isin 119879 is enabled at markingm if and only if

for all 119901119894isin ∙119905119895 119898119894gt 0 Its enabling degree is given by

enab (119905119895m) = min

119901119894isin∙119905119895

119898[119901119894]

Pre [119894 119895] (1)

The enabling degree determines themaximum amount of119905119895that can be fired at marking m leading to a new marking

thusm1015840 = m + 120572119862[∙ 119895] where 0 lt 120572 lt enab(119905119895m)

If m is reachable from m0by firing the finite sequence

120590 of enabled transitions then m = m0+ C is named the

CPN state equation where isin R+ cup 0|119879| is the firing count

vector that is 119895is the cumulative amount of firing of 119905

119895in

the sequence 120590The set of all reachable markings from m

0is called the

reachability set and it is denoted by RS(119873m0) In the case of

a CPN system RS(119873m0) is a convex set [11]

A CPN is bounded when every place is bounded that isfor all 119901 isin 119875 exist119887

119901isin R st 119898[119901] le 119887

119901at every reachable

markingm and it is live when every transition is live (it canultimately be fired from every reachable marking) [8]

Definition 2 A continuous timed Petri net is a 3-tupleContPN = (119873 120582m

0) where (119873m

0) is a CPN and 120582

119879 rarr R+|119879| is a function associating a firing rate with each

transitionThe state equation of a ContPN is

m (120591) = Cf (120591) (2)

where 120591 is the time variable f(120591) = (120591)

Definition 3 A ContPN is called infinite server semanticContPN if the flow of a transition 119905

119894is

119891119894= 120582119894sdot enab (119905

119894m) = 120582

119894sdot min119901isin∙119905119894

119898 (119901)

Pre [119901 119905119894] (3)

where 119891119894is the flow of transition 119905

119894and the 119894th entry of the

vector f


pd tc pvar


pd tfb2tfb1 pvar

pconv





120582|119879|


Π (m) [119894 119895] =

1


119895

0 otherwise(5)


119894is

119891119894= 120582119894sdot prod119901isin∙119905119894

119898 (119901)

Pre [119901 119905119894] (6)





119894becomes 119908

119894= 119891119894minus 119906119894



0 le 119906119894le 119891119894

(7)


119906ΛΠ(m) sdotm

where I119906

= diag(1198681199061 119868

119906|119879|) and 0 le 119868

119906119894le 1 Then the



be rewritten as

m = CI119888f = Cw (8)

Solar radiation

Humidifier

Lateral

Ventilation


p6p4 t5 t6

t7 t9

t14t12

t1t2 t3

p2 p1 p5

p3

p7

t13 t15 t8 t4 p10

p9

p8

t10t11

t16

t19

t23t22

t20 t21t17 t18


0 1 2 3 4 5 6 7 8

285

290

295

300

305

310

315

320

325

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)



0 1 2 3 4 5 6 7 82

3

4

5

6

7

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

65

55

45

35

25





119888119895= 1



119898[119901] (120591) = ℎ (m 119863) (9)






119876 = int120591119891

1205910














C = [minus119886 119889

119888 minus119888] Π =

[[

[

1

1198860

01

119888

]]

]

Λ = [1205821

0

0 1205822

]


1= minus12058211198981+

119889

11988812058221198982

2=

119887

11988612058211198981minus 12058221198982

(12)


0 2 4 6 8285

290

295

300

Time (hr)

Out

side t

empe

ratu

re (K

)

(a)

0 2 4 6 80

500

1000

1500

Time (hr)

Sola

r rad

iatio

n (W

m2)

(b)

0 2 4 6 86

7

8

Time (hr)

Out

side v

apor

conc

entr

atio

n (k

gm

3)

times10minus3

(c)

0 2 4 6 80

1

2

Time (hr)

Win

d sp

eed

(ms

)

15

05

(d)



1and 119898




119887

119886=

119888

119889=

1205822

1205821

(13)


1= minus12058211198981+ 12058211198982

2= 12058221198981minus 12058221198982

(14)



1and 120582

2is given



1)11989821+ (12120582

2)11989822is


(m) =1

1205821

11989811+

1

1205822

11989822

= minus 1198982

1+ 211989821198981minus 1198982

2

= minus (1198981minus 1198982)2

(15)







lt 1205822

1205821gt 1205822 and 120582

1= 1205822

If 1205821


2



0 1 2 3 4 5 6 7 8

290

300

310

320

330

340

350

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)


0 1 2 3 4 5 6 7 82

3

4

5

6

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

55

45

35

25


closer to 1198981(0) If 120582






2(0))2





1198871= 1205821198872


var = minus1205821198871119898var + 120582

1198871119898119889 (16)



119889






2














119904




119900











119904(119902119879119904


119879119900


119879119900


119879119900








119892(119902119879119892





1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)







1199011

119879119904

1199012

119862H2O 1199013


Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23






8(1198982minus1198981)


of 1 1198984 and 119898


5and






9and 119898

10



1or1198986 For



7 1198987and 119898



119909and 119898

119910 respectively





119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










pd tc pvar


pd tfb2tfb1 pvar

pconv





120582|119879|


Π (m) [119894 119895] =

1


119895

0 otherwise(5)


119894is

119891119894= 120582119894sdot prod119901isin∙119905119894

119898 (119901)

Pre [119901 119905119894] (6)





119894becomes 119908

119894= 119891119894minus 119906119894



0 le 119906119894le 119891119894

(7)


119906ΛΠ(m) sdotm

where I119906

= diag(1198681199061 119868

119906|119879|) and 0 le 119868

119906119894le 1 Then the



be rewritten as

m = CI119888f = Cw (8)

Solar radiation

Humidifier

Lateral

Ventilation


p6p4 t5 t6

t7 t9

t14t12

t1t2 t3

p2 p1 p5

p3

p7

t13 t15 t8 t4 p10

p9

p8

t10t11

t16

t19

t23t22

t20 t21t17 t18


0 1 2 3 4 5 6 7 8

285

290

295

300

305

310

315

320

325

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)



0 1 2 3 4 5 6 7 82

3

4

5

6

7

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

65

55

45

35

25





119888119895= 1



119898[119901] (120591) = ℎ (m 119863) (9)






119876 = int120591119891

1205910














C = [minus119886 119889

119888 minus119888] Π =

[[

[

1

1198860

01

119888

]]

]

Λ = [1205821

0

0 1205822

]


1= minus12058211198981+

119889

11988812058221198982

2=

119887

11988612058211198981minus 12058221198982

(12)


0 2 4 6 8285

290

295

300

Time (hr)

Out

side t

empe

ratu

re (K

)

(a)

0 2 4 6 80

500

1000

1500

Time (hr)

Sola

r rad

iatio

n (W

m2)

(b)

0 2 4 6 86

7

8

Time (hr)

Out

side v

apor

conc

entr

atio

n (k

gm

3)

times10minus3

(c)

0 2 4 6 80

1

2

Time (hr)

Win

d sp

eed

(ms

)

15

05

(d)



1and 119898




119887

119886=

119888

119889=

1205822

1205821

(13)


1= minus12058211198981+ 12058211198982

2= 12058221198981minus 12058221198982

(14)



1and 120582

2is given



1)11989821+ (12120582

2)11989822is


(m) =1

1205821

11989811+

1

1205822

11989822

= minus 1198982

1+ 211989821198981minus 1198982

2

= minus (1198981minus 1198982)2

(15)







lt 1205822

1205821gt 1205822 and 120582

1= 1205822

If 1205821


2



0 1 2 3 4 5 6 7 8

290

300

310

320

330

340

350

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)


0 1 2 3 4 5 6 7 82

3

4

5

6

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

55

45

35

25


closer to 1198981(0) If 120582






2(0))2





1198871= 1205821198872


var = minus1205821198871119898var + 120582

1198871119898119889 (16)



119889






2














119904




119900











119904(119902119879119904


119879119900


119879119900


119879119900








119892(119902119879119892





1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)







1199011

119879119904

1199012

119862H2O 1199013


Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23






8(1198982minus1198981)


of 1 1198984 and 119898


5and






9and 119898

10



1or1198986 For



7 1198987and 119898



119909and 119898

119910 respectively





119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 82

3

4

5

6

7

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

65

55

45

35

25





119888119895= 1



119898[119901] (120591) = ℎ (m 119863) (9)






119876 = int120591119891

1205910














C = [minus119886 119889

119888 minus119888] Π =

[[

[

1

1198860

01

119888

]]

]

Λ = [1205821

0

0 1205822

]


1= minus12058211198981+

119889

11988812058221198982

2=

119887

11988612058211198981minus 12058221198982

(12)


0 2 4 6 8285

290

295

300

Time (hr)

Out

side t

empe

ratu

re (K

)

(a)

0 2 4 6 80

500

1000

1500

Time (hr)

Sola

r rad

iatio

n (W

m2)

(b)

0 2 4 6 86

7

8

Time (hr)

Out

side v

apor

conc

entr

atio

n (k

gm

3)

times10minus3

(c)

0 2 4 6 80

1

2

Time (hr)

Win

d sp

eed

(ms

)

15

05

(d)



1and 119898




119887

119886=

119888

119889=

1205822

1205821

(13)


1= minus12058211198981+ 12058211198982

2= 12058221198981minus 12058221198982

(14)



1and 120582

2is given



1)11989821+ (12120582

2)11989822is


(m) =1

1205821

11989811+

1

1205822

11989822

= minus 1198982

1+ 211989821198981minus 1198982

2

= minus (1198981minus 1198982)2

(15)







lt 1205822

1205821gt 1205822 and 120582

1= 1205822

If 1205821


2



0 1 2 3 4 5 6 7 8

290

300

310

320

330

340

350

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)


0 1 2 3 4 5 6 7 82

3

4

5

6

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

55

45

35

25


closer to 1198981(0) If 120582






2(0))2





1198871= 1205821198872


var = minus1205821198871119898var + 120582

1198871119898119889 (16)



119889






2














119904




119900











119904(119902119879119904


119879119900


119879119900


119879119900








119892(119902119879119892





1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)







1199011

119879119904

1199012

119862H2O 1199013


Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23






8(1198982minus1198981)


of 1 1198984 and 119898


5and






9and 119898

10



1or1198986 For



7 1198987and 119898



119909and 119898

119910 respectively





119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8285

290

295

300

Time (hr)

Out

side t

empe

ratu

re (K

)

(a)

0 2 4 6 80

500

1000

1500

Time (hr)

Sola

r rad

iatio

n (W

m2)

(b)

0 2 4 6 86

7

8

Time (hr)

Out

side v

apor

conc

entr

atio

n (k

gm

3)

times10minus3

(c)

0 2 4 6 80

1

2

Time (hr)

Win

d sp

eed

(ms

)

15

05

(d)



1and 119898




119887

119886=

119888

119889=

1205822

1205821

(13)


1= minus12058211198981+ 12058211198982

2= 12058221198981minus 12058221198982

(14)



1and 120582

2is given



1)11989821+ (12120582

2)11989822is


(m) =1

1205821

11989811+

1

1205822

11989822

= minus 1198982

1+ 211989821198981minus 1198982

2

= minus (1198981minus 1198982)2

(15)







lt 1205822

1205821gt 1205822 and 120582

1= 1205822

If 1205821


2



0 1 2 3 4 5 6 7 8

290

300

310

320

330

340

350

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)


0 1 2 3 4 5 6 7 82

3

4

5

6

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

55

45

35

25


closer to 1198981(0) If 120582






2(0))2





1198871= 1205821198872


var = minus1205821198871119898var + 120582

1198871119898119889 (16)



119889






2














119904




119900











119904(119902119879119904


119879119900


119879119900


119879119900








119892(119902119879119892





1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)







1199011

119879119904

1199012

119862H2O 1199013


Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23






8(1198982minus1198981)


of 1 1198984 and 119898


5and






9and 119898

10



1or1198986 For



7 1198987and 119898



119909and 119898

119910 respectively





119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8

290

300

310

320

330

340

350

Time (hr)

OriginalIdentified

Gre

enho

use t

empe

ratu

re (K

)


0 1 2 3 4 5 6 7 82

3

4

5

6

Time (hr)

OriginalIdentified

Vapo

r con

cent

ratio

n (k

gm

3)

times10minus3

55

45

35

25


closer to 1198981(0) If 120582






2(0))2





1198871= 1205821198872


var = minus1205821198871119898var + 120582

1198871119898119889 (16)



119889






2














119904




119900











119904(119902119879119904


119879119900


119879119900


119879119900








119892(119902119879119892





1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)







1199011

119879119904

1199012

119862H2O 1199013


Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23






8(1198982minus1198981)


of 1 1198984 and 119898


5and






9and 119898

10



1or1198986 For



7 1198987and 119898



119909and 119898

119910 respectively





119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119904




119900











119904(119902119879119904


119879119900


119879119900


119879119900








119892(119902119879119892





1= minus (120582

7+ 1205829)1198981minus (11986811988811205826+ 1205825)11989811198988+ 12058281198982

+ 12058211198984+ 12058241198985+ (11986811988811205823+ 1205822)11989851198988

+ 12058211119898119909minus 119868119888212058210119898119910

2= 120582121198981minus (12058213

+ 12058215)1198982+ 120582141198986

3= minus 120582

221198983minus (119868119888112058221

+ 12058220)11989831198988+ 120582191198987

+ (119868119888112058218

+ 12058217)11989871198988minus 12058223119898119909+ 119868119888212058216119898119910

119898119909= min (119898

1 11989810) 119898

119910= min (119898

1 1198989)

1198984= 119868119900(120591) 119898

5= 119879119900

1198986= 119879ss (120591) 119898

7= 119862H2O119900 (120591)

1198988= V (120591) 119898

9= 119865hum (120591) 119898

10= 120593cons (120591)

(17)







1199011

119879119904

1199012

119862H2O 1199013


Module 119875Var 119879

119902119868119900

in 1199014119868119900

1199051

119902119879119900

fbalVin 1199015119879119900

1199052and 1199055

119902119879119900

fbalVcn 1199015119879119900

1199053and 1199056

119902119879119900

bal119888 1199015119879119900

1199054and 1199057

119902119879119904

bal 1199012119879119904

1199058and 1199059

119902humcon 1199019119865hum 119905

10

119902consin 11990110120593cons 119905

11

119902119879119892

bal 1199011119879119892

11990512and 11990513

119902119879ssbal 119901

6119879ss 119905

14and 11990515

120593humin 119901

9119865hum 119905

16

120593119862H2O119900bal 119901

7119862H2O119900 119905

17and 11990520

120593119862H2O119900fbalVin 119901

7119862H2O119900 119905

18and 11990521

120593119862H2O119900fbalVcn 119901

7119862H2O119900 119905

19and 11990522

120593humcon 119901

10120593cons 119905

23






8(1198982minus1198981)


of 1 1198984 and 119898


5and






9and 119898

10



1or1198986 For



7 1198987and 119898



119909and 119898

119910 respectively





119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119868119900= 400sin (000011119905) Wm2

119879119900= 298 + 7sin (000011119905) K

119879ss = 29315 + 3sin (000011119905) K

119862H2O119900 = 00060692 + 0002sin (2119905) kgm3

V = ℎ (119905) for ℎ (119905) = 10sin (0001119905) ge 1

1 elsems



(18)


1198681198881

= 05 + 05sin (0001119905)

1198681198882

= 0133sin (000011119905) (19)


119879119892= 288K

119879119904= 298K

119862H2O = 00026 kgm3

(20)



119879119892= 119879119892orminus

119879119892id and 119890




119900





5 Conclusions



2concen-



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









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