Research ArticleDynamic Modeling of Steam Condenser and Design ofPI Controller Based on Grey Wolf Optimizer
Shu-Xia Li1 and Jie-Sheng Wang12
1School of Electronic and Information Engineering University of Science amp Technology Liaoning Anshan 114044 China2National Financial Security and System Equipment Engineering Research Center University of Science amp Technology LiaoningAnshan 114044 China
Correspondence should be addressed to Jie-Sheng Wang wang jiesheng126com
Received 15 September 2015 Revised 16 November 2015 Accepted 18 November 2015
Academic Editor Salvatore Alfonzetti
Copyright copy 2015 S-X Li and J-S Wang 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
Shell-and-tube condenser is a heat exchanger for cooling steam with high temperature and pressure which is one of the mainkinds of heat exchange equipment in thermal nuclear and marine power plant Based on the lumped parameter modelingmethod the dynamic mathematical model of the simplified steam condenser is established Then the pressure PI control systemof steam condenser based on the MatlabSimulink simulation platform is designed In order to obtain better performance a newmetaheuristic intelligent algorithm grey wolf optimizer (GWO) is used to realize the fine-tuning of PI controller parametersOn the other hand the Z-N engineering tuning method genetic algorithm and particle swarm algorithm are adopted for tuningPI controller parameters and compared with GWO algorithm Simulation results show that GWO algorithm has better controlperformance than other four algorithms
1 Introduction
The condenser is one of the important kinds of equipmentin thermal power plant nuclear power plants and marinepower plant The reliability of condenser running directlyaffects the safety and economic operation of the entire powerplant or power system [1] A steam condenser is a piece ofmachinery that turns steam into water Many steam-basedsystems use a circuit of water to maximize their efficiencyWater is heated into steam the steam provides motivation fora process a steam condenser turns it back into water and thecycle begins again [2]The failure of the condenser may causethe boiler or steam turbine unit to overheat which endangersthe safety of the whole generating unit or power plantThe power plant has strict requirements on the reliabilityand the tightness of the condenser In addition to safetyconsiderations the condensation process of steam in thecondenser is an important part of the system thermodynamiccycle which greatly affects the economic performance of thesystem
Therefore through the computer simulation experi-ments the establishment of the dynamic model and under-standing the dynamic characteristics of the condenser havea great significance on improving the safety and economicoperation level of the steamcondenser [3] Ahybridmodelingapproach is proposed to describe the dynamic behavior ofthe two-phase flow condensers used in air-conditioning andrefrigeration systems based on fundamental energy andmassbalance governing equations and thermodynamic principlesThe model validation studies on an experimental systemshow that the model predicts the system dynamic well [4]A method to improve the load change capacity is proposedfor the water cooled power plants through controlling thecooling water flow Then the CCWCS (condenser coolingwater control system) is put forward to execute this methodon the premise of unit safety [5] A robust strategy for onlinefault detection and optimal control of condenser coolingwater systems is proposedThe optimal control strategy is for-mulated using a model-based approach in which simplifiedmodels and a hybrid quick search (HQS) method are used to
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 120975 9 pageshttpdxdoiorg1011552015120975
2 Mathematical Problems in Engineering
optimize the performance of the overall system by changingthe settings of the local process controllers [6] In the fieldof computer simulation the simulation technology is usedto simulate the operation of the condenser system and studyits working performanceThe optimization of the parametersof PID controller can not only satisfy the static accuracy butalso make the system have better stabilization The accuratemathematical model can accurately and comprehensivelyrepresent all working conditions in the real running processof steam condenser system (normal conditions fault condi-tions and transient conditions) andmake them reappear Onthe same time it can be used to carry through the researchon the dynamical characteristic to meet the requirements ofdifferent engineering projects
At present people by simulating biotic population andtheir evolution process in nature have developed a variety ofintelligence algorithms particle swarm optimization (PSO)algorithm ant colony optimization (ACO) algorithm artifi-cial bee colony (ABC) algorithm shuffled frog leaping (SFL)algorithm cuckoo search (CS) algorithm Dolphin partneroptimization (DPO) and Firefly algorithm (FA)
GA is the most typical algorithm of evolutionary branchThis algorithm simulates Darwinian evolution concepts [7]Each new population is created by the combination andmutation of the individuals in the previous generationParticle swarm optimization (PSO) was inspired from thesocial behavior of birds flocking [8] The PSO algorithmemploys multiple particles that chase the position of the bestparticle and their own best positions obtained so far Artificialbee colony (ABC) algorithm mimics the collective behaviorof bees in finding food sources [9] There are three typesof bees in ABC scout onlooker and employed bees Thescout bees are responsible for exploring the search spacewhereas onlooker and employed bees exploit the promisingsolutions found by scout bees Ant colony optimization(ACO) algorithm was inspired by the social behavior of antsin an ant colony In fact the social intelligence of ants infinding the shortest path between the nest and a source offood is themain inspiration of ACO [10] Cuckoo search (CS)algorithm was put forward by Yang and Deb in 2009 [11]This algorithm is mainly based on two aspects the cuckoorsquosparasitic reproduction mechanism and Levy flights searchprinciple Assume each cuckoo only lays one egg at a timeand randomly chooses birdrsquos nest to hatch the egg
The grey wolf optimizer (GWO) as a new swarm intel-ligent algorithm is put forward by Seyedali Mirjalili and soforth in 2014 whichmainlymimics wolf leadership hierarchyand huntingmechanism in nature Seyedali andMirjalili andso forth have proved that the search performance of basicwolf algorithm is superior to that of PSO GSA DE and FEPalgorithm Due to the wolvesrsquo algorithm with the advantagesof being simple in principle having fast seeking speed andhigh search precision and being easy to realize it is moreeasily combined with the practical engineering problems
Many kinds of swarm intelligence are inspired by hunt-ing and search behaviors of a population However GWOalgorithm simulates internal leadership hierarchy of wolvesthus in the searching process the position of best solutioncan be comprehensively assessed by three solutions So the
Cooling water inlet Steam inlet
Cooling Condensatewater outlet water outlet
Figure 1 Structure diagram of shell-and-tube condenser
GWO algorithm is able to greatly reduce the probabilityof producing premature phenomenon and falling into localoptimum
Aiming at the steam condenser pressure control problembased on the MatlabSimulink simulation platform and theestablished mathematical model a closed-loop condenserpressure control system is designed The GWO algorithm isadopted to optimize the parameters of the PI controller Sim-ulation results show the effectiveness of the proposed controlstrategy The paper is organized as follows In Section 2 thetechnique and dynamicmodeling of the shell-and-tube steamcondenser are introduced The GWO algorithm is illustratedin Section 3 The parameter optimization of PI controllerbased on GWO algorithm is described and the simulationexperiments and results analysis are introduced in detail inSection 4 Finally the conclusion illustrates the last part
2 Dynamic Modeling of Steam Condenser
21 Structure Characteristics and Working Principle of SteamCondenser The shell of shell-and-tube type condenser isusually cylindrical or elliptical as shown in Figure 1 which isconnected with end closures for constituting the water cham-bers Between the end closures and the shells a perforatedtube plate is fixed in which a lot of cooling water pipes arearranged hierarchicallyThe entrance pipe of steam is locatedin the upper part of the condenser shell which is directly orindirectly connected with the exhaust equipment through thecompensator In the lower part of the shell there is a gatheringtank (or a hot well water tank) of the condensed waterThe airoutlet port is positioned at the lower part of shell and the airis drawn through this nozzle
Theworking principle of the steam condenser is shown inFigure 2
Steam goes into steam field of the condenser throughsteam admission pipe Steamgets in touchwith the condensertube wall to begin the radiate condensation at the same timethe latent heat is transferred to the cooling water through thesurface of the cooling water pipe Cooling water with inlettemperature is fed into water chamber through the coolingwater pipe where the cooling water is assigned to all pipesof the first procedure in the lower part of the condensershell The cooling water flows into another water chamberalong the first six cooling water pipes and then enters thenext flow pipes and carries through heat exchange withthe steam Through such several procedures in return the
Mathematical Problems in Engineering 3
Steam inlet
Cooling Cooling
Condensate water outlet
Gc Tc1 Gc Tc2
Gh2 Th2
Gh1 Th1
water inlet water outlet
Figure 2 Schematic diagram of condenser
cooling water with the outlet temperature is discharged fromthe outlet pipes Due to the lack of system sealing propertythe air is drawn out from the condenser constantly to ensurethe requirements of the systemrsquos vacuum degree The drawngas contains the air and steam In the beginning of thecondensation the air volume is very smaller than the totalamount of steam With the steam and air flowing toward theexhaust port steam is continuously condensed down Thenthe steam quality in the mixture gradually decreases On thecontrary the relative content of the air increases graduallyUntil the relative content of air fed into the cooling zone airhas reached a great numerical extent the steam condensationprocess terminates
22 Mathematical Model
221 Mathematical Model of Shell Side of Condenser
(1) Steam Zone
(i) Steam mass equation is as follows
119889119866119904
119889119905= 119866st + 119866ost minus 119866119888 minus 119866ss (1)
where 119866119904is the steam content in the shell side of the
condenser (kgs) 119866st is the exhaust volume of steam turbine(kgs)119866ost is the other steam inlet of the condenser (kgs)119866
119888
is the main steam condensate (kgs) and119866ss is the amount ofsteam drawn out by vacuum pumping equipment (kgs)119866ss and 119866119888 are calculated by the following equations
where 119866ao is the quality of the gas mixture pumped by thepumping unit (kgs) 119877 is the share air ratio in the condenser119867119904is the average enthalpy of steam (kJkg) and 119867cw is the
enthalpy of saturated water corresponding to the condenserpressure (kJkg)
where 119867gp is the bubbling oxygen exhaust steam enthalpy(kJkg) 119867
119908is the enthalpy of hot well water (kJkg) and
119867cw is the enthalpy of saturated water corresponding to thecondenser pressure (kJkg)
222 Mathematical Model of Condenser Tube Side Thedynamic heat balance equation of the circulating water isdescribed as follows
119872119908119862119908
1198891198792
119889119905= 119876 minus 119876
119908= 119880119860Δ119905
119898minus 119865cw119862119901 (119879 minus 119879cw) (11)
where 119872119908is the circulating water quality (kg) 119862
119901is the
circulating water heat capacity (kJ(kg lowast C)) 119876 is the steamoutlet heat (kJ) 119876
119908is the circulating water heat absorption
quantity (kJ) 119880 is the condenser heat transfer coefficient(W(m2 lowast ∘C)) Δ119905
119898is the logarithmic mean temperature
difference (∘C) 119860 is the condenser heat transfer area (m2)119865cw is the circulating water flow (kgs) 119879cw is the circulatingwater inlet temperature (∘C) and 119879 is the circulating wateroutlet temperature (∘C)
The logarithmic heat transfer temperature difference iscalculated by
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
where 119879119888is the saturated water temperature of the vapor
pressure in the condenser (∘C) and 119879119908is the condenser hot
well water temperature (∘C)The heat transfer error of the condenser is calculated by
120575119905= 119879119904minus 119879 (14)
where 119879119904is the saturated gas temperature corresponding to
saturation pressure in condenser (∘C)
3 Grey Wolf Optimizer
The grey wolf optimizer is a new metaheuristic intelligentalgorithm proposed by Mirjalili et al in 2014 [12] which issuccessfully applied in many fields such as security smartgrid power system management [13] parameter estimation[14] reactive power dispatch problem [15] flow shop schedul-ing problem [16] combined heat and power dispatch [17] andautomatic control [18] For immigrating the wolvesrsquo internalleadership hierarchy the wolves are divided into four typesalpha (120572) beta (120573) delta (120575) and omega (Ω) According
Xlowast minus X
(Xlowast Y)(Xlowast minus X Y)
(X Y)
YlowastminusY
(X Ylowast)
(Xlowast Ylowast)
(Xlowast minus X Ylowast)
(Xlowast minus X Ylowast minus Y)
(Xlowast Ylowast minus Y)
(X Ylowast minus Y)
Figure 3 Position vector of wolf and next moving position in two-dimensional space
to the principle of the wolves hunting the hunt process isdivided into three stages searching prey surrounding preyand attacking prey In the four groups of wolves 120572 120573 and 120575are seen as the three best wolves they guide the otherwolf (Ω)trending in the search space in the best region In the iterativesearching process 120572 120573 120575 and Ω wolves are used to realizethe assessment of prey possible positions in the optimizationprocess The positions of wolves are updated in accordancewith the following equations
=100381610038161003816100381610038161003816 sdot997888997888rarr119883119875(119905) minus (119905)
100381610038161003816100381610038161003816
(119905 + 1) =997888997888rarr119883119875(119905) minus sdot
(15)
where 119905 is the current iteration number and are thecoefficient vector 997888997888rarr119883
119875is the position vector of the prey and
is the position of the wolf The vectors and are expressedas follows
= 2119886 sdot997888rarr1199031minus 119886
= 2 sdot997888rarr1199032
(16)
where the coefficient 119886 linearly increases from 2 to 0 with theincrease of the iteration number and997888rarr119903
1and997888rarr1199032are the random
vector located in the scope [0 1]The principle and concept of the position update (15) are
described in Figure 3 [7]It can be seen from Figure 3 that the wolf in the position
(119883 119884) can be arranged a new location on the basis of theabove formula While Figure 3 shows only the 7 possiblepositions of the wolf the randomly adjusting parameters and can make the wolf move to anywhere in a continuousspace In GWO algorithm the positions of wolves 120572 120573 and120575 are likely the prey (optimal) position In the searchingprocess the previous three best solutions are assumed to be
Mathematical Problems in Engineering 5
Alpha
Beta
Delta Omega
Move
D120572
D120573
D120575
Figure 4 Sketch map of position update of the wolves
120572 120573 and 120575 and then the others are regarded as theΩ wolvesThe positions of 120572 120573 and 120575 are used to update their positionsThe following mathematical formulae are used to adjust theposition of Ω wolf again and the position update schematicgraph is shown in Figures 4 and 5
where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883
120575are the position of the wolves 120572 120573
and 120575 respectively 997888rarr1198621 997888rarr1198622and 997888rarr119862
3are random vectors and
represents the position of the current solution Equations (17)are used to calculate the approximate distance between thecurrent position and 120572 120573 and 120575 respectively After definingthe distance between them the final position of the currentsolution is calculated by the following
where 997888rarr1198601 997888rarr1198602 and 997888rarr119860
3are random vectors and 119905 represents
the number of iterationsSeen from the above equations (17) define the step size
of the wolf Ω tending to the wolves 120572 120573 and 120575 Equations(19)ndash(21) define the final position of Ω wolf
Unexplored
Unexplored
Explored
Explored
If |A|lt 1
If |A|ge 1
Figure 5 Exploration and development of wolves
It can be seen from Figure 4 that the random and adaptivevectors and can be used to balance the explorationand development capabilities of the GWO algorithm When|| gt 1 the wolf has detection ability On the other handwhen the value of vector is greater than 1 it can alsopromote the enhancement of the detection ability of the wolfIn contrast when || lt 1 and 119862 lt 1 the wolf rsquos informationmining capacity is enhanced In order to enhance the abilityof the wolf with the increase of the iteration number isdecreased linearly However is randomly generated in thewhole optimization process which can make the detectionand exploitation ability of the wolf to reach equilibrium at anystage especially in the final stage of the iteration and preventthe algorithm from falling into local optimum
The procedure of the GWO algorithm is described asfollows
Step 1 Initialize the wolves Randomly initialize the positionof the wolves119883
119894(119894 = 1 2 119899) and parameters 119886 119860 and 119862
Step 2 Calculate the fitness of eachwolf and choose the threewolves with best fitness as wolves 120572 120573 and 120575
Step 3 Update positions Based on (17)ndash(21) update thepositions of the other wolves that is to say the positions oftheΩ wolves
Step 4 Update parameters 119886 119860 and 119862
Step 5 Judge whether to meet the termination conditions ornot If satisfied the position of 120572 wolf and the fitness valueare the optimal output If the termination condition is notsatisfied return to Step 2
4 Parameter Optimization of PI ControllerBased on GWO Algorithm
41 Dynamic Model of Steam Condenser Based on Mat-labSimulink Simulation Software The establishment of the
6 Mathematical Problems in Engineering
dynamic mathematical model of the condenser is based onthe following assumption that the total amount of condensa-tion the circulating water flow and the condenser volume arecertain So it is set up based on the dynamic heat balance andmass balance of the condenser water
411 Dynamic Heat Balance In the dynamic heat balance itis assumed that the total amount of condensation is certainand that the input steam and the output condensate aresaturated Therefore the heat from steam to the circulatingwater and the steam potential heat are equal So the steamreleased heat can be approximated calculated by the follow-ing equations
119876 asymp 119880119860Δ119905119898
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
(22)
The heat transfer coefficient 119880 and the heat transfer area119860 can be replaced by the following exponential equationapproximately
1
119880119860= 1205721119865cwminus08
+ 1205722 (23)
where 1205721and 120572
2are constants
When 119865cwrarrinfin 1205722 is determined by 119880119860 In this case 119880119860is determined by the heat transfer ratio between the steamand the tube wall and the thermal resistance of the tube wallThus by assuming the outlet temperature of the circulatingwater 120572
2can be determined Based on the above assumptions
and equations the dynamic heat balance equation of thecondenser is described as follows
119889119879
119889119905=119865cw119872cw
(119879cw minus 119879) +119876
119872cw119862119901 (24)
412 Mass Balance Themass balance of steam and conden-sate is based on the assumption that the space 119881 is constantand the volume of the steam and air is constant That is tosay in order to maintain the vapor condensation level ofthe condenser (certain vacuum degree) the output flow ofcondensatewater needs to be controlled in a certain range Soin order to simplify the model we assume that the inlet andoutlet of the condensate are saturatedTherefore the ideal gasmodel equation is expressed as
119889119875
119889119905=119877119879119888
119881(119865119904minus 119865119888) (25)
where 119865119904is the steam flow (kgs) and 119865
119888is the condensate
water flow (kgs)Among them the condensationwater temperature119879
119888and
the condenser pressure 119875 have a unique relationship In orderto simplify the model it is approximated by the followinglinear relationship equation
119879119888= 120572119875 + 120573 (26)
The steam condenser model given above has five equa-tions among which two are dynamic equations Here there
Table 1 Parameters of steam condenser
Parameter Parameter value Unit119877 0461526 kJkgK119881 3 m3
Condensate water flow 4 kgs119865cw Cooling water flow 1078881 kgs119875 Condenser pressure 90 kPa119879 Circulating water outlet temperature 80 ∘C119879cw Circulating water inlet temperature 60 ∘C119879119888
Saturated water temperature 965538 ∘C119876 Steam heat 90626 kW
are eight variables (119865119904 119865119888 119865cw 119875 119879 119879cw 119879119888 and 119876) and ten
parameters (119877119881120582119880119860119872cw1198621199011205721205731205721 and1205722)The valuesof ten parameters are shown in Table 1 where 120572
1and 120572
2are
determined under 119879 = 90∘C (119865cw rarr infin) The values of theeight variables are shown in Table 2 under the assumptionthat the system is stable
Based on the above model equations and the softwareMatlabSimulink a simulation model of PI controller forcondenser pressure closed-loop control system is establishedas shown in Figure 6 which includes a first-order delay unitused to represent the actuator with a time constant 120591 = 10 (s)and a lag unit caused by the pressure sensor with the timeconstant 120591 = 5 (s)
42 Encoding and Fitness Function Because the design ofthe PID controller is actually a multidimensional functionoptimization problem the GWO algorithm adopts the realnumber coding So for the parameters optimization of the PIcontroller each wolf can be directly coded as (119870
119901 119870119894)
119883 = 119870119901 119870119894 (27)
The control parameter optimization is designed to makethe control error tend to zero and has a faster response speedand smaller overshoot So the evaluation of the performanceof each set of control parameters is good or bad the integralof the product of absolute error and time is selected as thefitness function
ITAE = intinfin
0
119905 |119890 (119905)| 119889119905 (28)
Mathematical Problems in Engineering 7
Click here to tune the PID controller
60
Reaction curvePID tuning
Steam condenserScope
Pressure setpoint
PID
PID controller
Input step test
4
⟨Fcw⟩⟨T⟩
⟨Q⟩⟨P⟩Tcw
Tcw
Fs
Fs y =
Fcw
+++ minus
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
optimize the performance of the overall system by changingthe settings of the local process controllers [6] In the fieldof computer simulation the simulation technology is usedto simulate the operation of the condenser system and studyits working performanceThe optimization of the parametersof PID controller can not only satisfy the static accuracy butalso make the system have better stabilization The accuratemathematical model can accurately and comprehensivelyrepresent all working conditions in the real running processof steam condenser system (normal conditions fault condi-tions and transient conditions) andmake them reappear Onthe same time it can be used to carry through the researchon the dynamical characteristic to meet the requirements ofdifferent engineering projects
At present people by simulating biotic population andtheir evolution process in nature have developed a variety ofintelligence algorithms particle swarm optimization (PSO)algorithm ant colony optimization (ACO) algorithm artifi-cial bee colony (ABC) algorithm shuffled frog leaping (SFL)algorithm cuckoo search (CS) algorithm Dolphin partneroptimization (DPO) and Firefly algorithm (FA)
GA is the most typical algorithm of evolutionary branchThis algorithm simulates Darwinian evolution concepts [7]Each new population is created by the combination andmutation of the individuals in the previous generationParticle swarm optimization (PSO) was inspired from thesocial behavior of birds flocking [8] The PSO algorithmemploys multiple particles that chase the position of the bestparticle and their own best positions obtained so far Artificialbee colony (ABC) algorithm mimics the collective behaviorof bees in finding food sources [9] There are three typesof bees in ABC scout onlooker and employed bees Thescout bees are responsible for exploring the search spacewhereas onlooker and employed bees exploit the promisingsolutions found by scout bees Ant colony optimization(ACO) algorithm was inspired by the social behavior of antsin an ant colony In fact the social intelligence of ants infinding the shortest path between the nest and a source offood is themain inspiration of ACO [10] Cuckoo search (CS)algorithm was put forward by Yang and Deb in 2009 [11]This algorithm is mainly based on two aspects the cuckoorsquosparasitic reproduction mechanism and Levy flights searchprinciple Assume each cuckoo only lays one egg at a timeand randomly chooses birdrsquos nest to hatch the egg
The grey wolf optimizer (GWO) as a new swarm intel-ligent algorithm is put forward by Seyedali Mirjalili and soforth in 2014 whichmainlymimics wolf leadership hierarchyand huntingmechanism in nature Seyedali andMirjalili andso forth have proved that the search performance of basicwolf algorithm is superior to that of PSO GSA DE and FEPalgorithm Due to the wolvesrsquo algorithm with the advantagesof being simple in principle having fast seeking speed andhigh search precision and being easy to realize it is moreeasily combined with the practical engineering problems
Many kinds of swarm intelligence are inspired by hunt-ing and search behaviors of a population However GWOalgorithm simulates internal leadership hierarchy of wolvesthus in the searching process the position of best solutioncan be comprehensively assessed by three solutions So the
Cooling water inlet Steam inlet
Cooling Condensatewater outlet water outlet
Figure 1 Structure diagram of shell-and-tube condenser
GWO algorithm is able to greatly reduce the probabilityof producing premature phenomenon and falling into localoptimum
Aiming at the steam condenser pressure control problembased on the MatlabSimulink simulation platform and theestablished mathematical model a closed-loop condenserpressure control system is designed The GWO algorithm isadopted to optimize the parameters of the PI controller Sim-ulation results show the effectiveness of the proposed controlstrategy The paper is organized as follows In Section 2 thetechnique and dynamicmodeling of the shell-and-tube steamcondenser are introduced The GWO algorithm is illustratedin Section 3 The parameter optimization of PI controllerbased on GWO algorithm is described and the simulationexperiments and results analysis are introduced in detail inSection 4 Finally the conclusion illustrates the last part
2 Dynamic Modeling of Steam Condenser
21 Structure Characteristics and Working Principle of SteamCondenser The shell of shell-and-tube type condenser isusually cylindrical or elliptical as shown in Figure 1 which isconnected with end closures for constituting the water cham-bers Between the end closures and the shells a perforatedtube plate is fixed in which a lot of cooling water pipes arearranged hierarchicallyThe entrance pipe of steam is locatedin the upper part of the condenser shell which is directly orindirectly connected with the exhaust equipment through thecompensator In the lower part of the shell there is a gatheringtank (or a hot well water tank) of the condensed waterThe airoutlet port is positioned at the lower part of shell and the airis drawn through this nozzle
Theworking principle of the steam condenser is shown inFigure 2
Steam goes into steam field of the condenser throughsteam admission pipe Steamgets in touchwith the condensertube wall to begin the radiate condensation at the same timethe latent heat is transferred to the cooling water through thesurface of the cooling water pipe Cooling water with inlettemperature is fed into water chamber through the coolingwater pipe where the cooling water is assigned to all pipesof the first procedure in the lower part of the condensershell The cooling water flows into another water chamberalong the first six cooling water pipes and then enters thenext flow pipes and carries through heat exchange withthe steam Through such several procedures in return the
Mathematical Problems in Engineering 3
Steam inlet
Cooling Cooling
Condensate water outlet
Gc Tc1 Gc Tc2
Gh2 Th2
Gh1 Th1
water inlet water outlet
Figure 2 Schematic diagram of condenser
cooling water with the outlet temperature is discharged fromthe outlet pipes Due to the lack of system sealing propertythe air is drawn out from the condenser constantly to ensurethe requirements of the systemrsquos vacuum degree The drawngas contains the air and steam In the beginning of thecondensation the air volume is very smaller than the totalamount of steam With the steam and air flowing toward theexhaust port steam is continuously condensed down Thenthe steam quality in the mixture gradually decreases On thecontrary the relative content of the air increases graduallyUntil the relative content of air fed into the cooling zone airhas reached a great numerical extent the steam condensationprocess terminates
22 Mathematical Model
221 Mathematical Model of Shell Side of Condenser
(1) Steam Zone
(i) Steam mass equation is as follows
119889119866119904
119889119905= 119866st + 119866ost minus 119866119888 minus 119866ss (1)
where 119866119904is the steam content in the shell side of the
condenser (kgs) 119866st is the exhaust volume of steam turbine(kgs)119866ost is the other steam inlet of the condenser (kgs)119866
119888
is the main steam condensate (kgs) and119866ss is the amount ofsteam drawn out by vacuum pumping equipment (kgs)119866ss and 119866119888 are calculated by the following equations
where 119866ao is the quality of the gas mixture pumped by thepumping unit (kgs) 119877 is the share air ratio in the condenser119867119904is the average enthalpy of steam (kJkg) and 119867cw is the
enthalpy of saturated water corresponding to the condenserpressure (kJkg)
where 119867gp is the bubbling oxygen exhaust steam enthalpy(kJkg) 119867
119908is the enthalpy of hot well water (kJkg) and
119867cw is the enthalpy of saturated water corresponding to thecondenser pressure (kJkg)
222 Mathematical Model of Condenser Tube Side Thedynamic heat balance equation of the circulating water isdescribed as follows
119872119908119862119908
1198891198792
119889119905= 119876 minus 119876
119908= 119880119860Δ119905
119898minus 119865cw119862119901 (119879 minus 119879cw) (11)
where 119872119908is the circulating water quality (kg) 119862
119901is the
circulating water heat capacity (kJ(kg lowast C)) 119876 is the steamoutlet heat (kJ) 119876
119908is the circulating water heat absorption
quantity (kJ) 119880 is the condenser heat transfer coefficient(W(m2 lowast ∘C)) Δ119905
119898is the logarithmic mean temperature
difference (∘C) 119860 is the condenser heat transfer area (m2)119865cw is the circulating water flow (kgs) 119879cw is the circulatingwater inlet temperature (∘C) and 119879 is the circulating wateroutlet temperature (∘C)
The logarithmic heat transfer temperature difference iscalculated by
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
where 119879119888is the saturated water temperature of the vapor
pressure in the condenser (∘C) and 119879119908is the condenser hot
well water temperature (∘C)The heat transfer error of the condenser is calculated by
120575119905= 119879119904minus 119879 (14)
where 119879119904is the saturated gas temperature corresponding to
saturation pressure in condenser (∘C)
3 Grey Wolf Optimizer
The grey wolf optimizer is a new metaheuristic intelligentalgorithm proposed by Mirjalili et al in 2014 [12] which issuccessfully applied in many fields such as security smartgrid power system management [13] parameter estimation[14] reactive power dispatch problem [15] flow shop schedul-ing problem [16] combined heat and power dispatch [17] andautomatic control [18] For immigrating the wolvesrsquo internalleadership hierarchy the wolves are divided into four typesalpha (120572) beta (120573) delta (120575) and omega (Ω) According
Xlowast minus X
(Xlowast Y)(Xlowast minus X Y)
(X Y)
YlowastminusY
(X Ylowast)
(Xlowast Ylowast)
(Xlowast minus X Ylowast)
(Xlowast minus X Ylowast minus Y)
(Xlowast Ylowast minus Y)
(X Ylowast minus Y)
Figure 3 Position vector of wolf and next moving position in two-dimensional space
to the principle of the wolves hunting the hunt process isdivided into three stages searching prey surrounding preyand attacking prey In the four groups of wolves 120572 120573 and 120575are seen as the three best wolves they guide the otherwolf (Ω)trending in the search space in the best region In the iterativesearching process 120572 120573 120575 and Ω wolves are used to realizethe assessment of prey possible positions in the optimizationprocess The positions of wolves are updated in accordancewith the following equations
=100381610038161003816100381610038161003816 sdot997888997888rarr119883119875(119905) minus (119905)
100381610038161003816100381610038161003816
(119905 + 1) =997888997888rarr119883119875(119905) minus sdot
(15)
where 119905 is the current iteration number and are thecoefficient vector 997888997888rarr119883
119875is the position vector of the prey and
is the position of the wolf The vectors and are expressedas follows
= 2119886 sdot997888rarr1199031minus 119886
= 2 sdot997888rarr1199032
(16)
where the coefficient 119886 linearly increases from 2 to 0 with theincrease of the iteration number and997888rarr119903
1and997888rarr1199032are the random
vector located in the scope [0 1]The principle and concept of the position update (15) are
described in Figure 3 [7]It can be seen from Figure 3 that the wolf in the position
(119883 119884) can be arranged a new location on the basis of theabove formula While Figure 3 shows only the 7 possiblepositions of the wolf the randomly adjusting parameters and can make the wolf move to anywhere in a continuousspace In GWO algorithm the positions of wolves 120572 120573 and120575 are likely the prey (optimal) position In the searchingprocess the previous three best solutions are assumed to be
Mathematical Problems in Engineering 5
Alpha
Beta
Delta Omega
Move
D120572
D120573
D120575
Figure 4 Sketch map of position update of the wolves
120572 120573 and 120575 and then the others are regarded as theΩ wolvesThe positions of 120572 120573 and 120575 are used to update their positionsThe following mathematical formulae are used to adjust theposition of Ω wolf again and the position update schematicgraph is shown in Figures 4 and 5
where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883
120575are the position of the wolves 120572 120573
and 120575 respectively 997888rarr1198621 997888rarr1198622and 997888rarr119862
3are random vectors and
represents the position of the current solution Equations (17)are used to calculate the approximate distance between thecurrent position and 120572 120573 and 120575 respectively After definingthe distance between them the final position of the currentsolution is calculated by the following
where 997888rarr1198601 997888rarr1198602 and 997888rarr119860
3are random vectors and 119905 represents
the number of iterationsSeen from the above equations (17) define the step size
of the wolf Ω tending to the wolves 120572 120573 and 120575 Equations(19)ndash(21) define the final position of Ω wolf
Unexplored
Unexplored
Explored
Explored
If |A|lt 1
If |A|ge 1
Figure 5 Exploration and development of wolves
It can be seen from Figure 4 that the random and adaptivevectors and can be used to balance the explorationand development capabilities of the GWO algorithm When|| gt 1 the wolf has detection ability On the other handwhen the value of vector is greater than 1 it can alsopromote the enhancement of the detection ability of the wolfIn contrast when || lt 1 and 119862 lt 1 the wolf rsquos informationmining capacity is enhanced In order to enhance the abilityof the wolf with the increase of the iteration number isdecreased linearly However is randomly generated in thewhole optimization process which can make the detectionand exploitation ability of the wolf to reach equilibrium at anystage especially in the final stage of the iteration and preventthe algorithm from falling into local optimum
The procedure of the GWO algorithm is described asfollows
Step 1 Initialize the wolves Randomly initialize the positionof the wolves119883
119894(119894 = 1 2 119899) and parameters 119886 119860 and 119862
Step 2 Calculate the fitness of eachwolf and choose the threewolves with best fitness as wolves 120572 120573 and 120575
Step 3 Update positions Based on (17)ndash(21) update thepositions of the other wolves that is to say the positions oftheΩ wolves
Step 4 Update parameters 119886 119860 and 119862
Step 5 Judge whether to meet the termination conditions ornot If satisfied the position of 120572 wolf and the fitness valueare the optimal output If the termination condition is notsatisfied return to Step 2
4 Parameter Optimization of PI ControllerBased on GWO Algorithm
41 Dynamic Model of Steam Condenser Based on Mat-labSimulink Simulation Software The establishment of the
6 Mathematical Problems in Engineering
dynamic mathematical model of the condenser is based onthe following assumption that the total amount of condensa-tion the circulating water flow and the condenser volume arecertain So it is set up based on the dynamic heat balance andmass balance of the condenser water
411 Dynamic Heat Balance In the dynamic heat balance itis assumed that the total amount of condensation is certainand that the input steam and the output condensate aresaturated Therefore the heat from steam to the circulatingwater and the steam potential heat are equal So the steamreleased heat can be approximated calculated by the follow-ing equations
119876 asymp 119880119860Δ119905119898
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
(22)
The heat transfer coefficient 119880 and the heat transfer area119860 can be replaced by the following exponential equationapproximately
1
119880119860= 1205721119865cwminus08
+ 1205722 (23)
where 1205721and 120572
2are constants
When 119865cwrarrinfin 1205722 is determined by 119880119860 In this case 119880119860is determined by the heat transfer ratio between the steamand the tube wall and the thermal resistance of the tube wallThus by assuming the outlet temperature of the circulatingwater 120572
2can be determined Based on the above assumptions
and equations the dynamic heat balance equation of thecondenser is described as follows
119889119879
119889119905=119865cw119872cw
(119879cw minus 119879) +119876
119872cw119862119901 (24)
412 Mass Balance Themass balance of steam and conden-sate is based on the assumption that the space 119881 is constantand the volume of the steam and air is constant That is tosay in order to maintain the vapor condensation level ofthe condenser (certain vacuum degree) the output flow ofcondensatewater needs to be controlled in a certain range Soin order to simplify the model we assume that the inlet andoutlet of the condensate are saturatedTherefore the ideal gasmodel equation is expressed as
119889119875
119889119905=119877119879119888
119881(119865119904minus 119865119888) (25)
where 119865119904is the steam flow (kgs) and 119865
119888is the condensate
water flow (kgs)Among them the condensationwater temperature119879
119888and
the condenser pressure 119875 have a unique relationship In orderto simplify the model it is approximated by the followinglinear relationship equation
119879119888= 120572119875 + 120573 (26)
The steam condenser model given above has five equa-tions among which two are dynamic equations Here there
Table 1 Parameters of steam condenser
Parameter Parameter value Unit119877 0461526 kJkgK119881 3 m3
Condensate water flow 4 kgs119865cw Cooling water flow 1078881 kgs119875 Condenser pressure 90 kPa119879 Circulating water outlet temperature 80 ∘C119879cw Circulating water inlet temperature 60 ∘C119879119888
Saturated water temperature 965538 ∘C119876 Steam heat 90626 kW
are eight variables (119865119904 119865119888 119865cw 119875 119879 119879cw 119879119888 and 119876) and ten
parameters (119877119881120582119880119860119872cw1198621199011205721205731205721 and1205722)The valuesof ten parameters are shown in Table 1 where 120572
1and 120572
2are
determined under 119879 = 90∘C (119865cw rarr infin) The values of theeight variables are shown in Table 2 under the assumptionthat the system is stable
Based on the above model equations and the softwareMatlabSimulink a simulation model of PI controller forcondenser pressure closed-loop control system is establishedas shown in Figure 6 which includes a first-order delay unitused to represent the actuator with a time constant 120591 = 10 (s)and a lag unit caused by the pressure sensor with the timeconstant 120591 = 5 (s)
42 Encoding and Fitness Function Because the design ofthe PID controller is actually a multidimensional functionoptimization problem the GWO algorithm adopts the realnumber coding So for the parameters optimization of the PIcontroller each wolf can be directly coded as (119870
119901 119870119894)
119883 = 119870119901 119870119894 (27)
The control parameter optimization is designed to makethe control error tend to zero and has a faster response speedand smaller overshoot So the evaluation of the performanceof each set of control parameters is good or bad the integralof the product of absolute error and time is selected as thefitness function
ITAE = intinfin
0
119905 |119890 (119905)| 119889119905 (28)
Mathematical Problems in Engineering 7
Click here to tune the PID controller
60
Reaction curvePID tuning
Steam condenserScope
Pressure setpoint
PID
PID controller
Input step test
4
⟨Fcw⟩⟨T⟩
⟨Q⟩⟨P⟩Tcw
Tcw
Fs
Fs y =
Fcw
+++ minus
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
cooling water with the outlet temperature is discharged fromthe outlet pipes Due to the lack of system sealing propertythe air is drawn out from the condenser constantly to ensurethe requirements of the systemrsquos vacuum degree The drawngas contains the air and steam In the beginning of thecondensation the air volume is very smaller than the totalamount of steam With the steam and air flowing toward theexhaust port steam is continuously condensed down Thenthe steam quality in the mixture gradually decreases On thecontrary the relative content of the air increases graduallyUntil the relative content of air fed into the cooling zone airhas reached a great numerical extent the steam condensationprocess terminates
22 Mathematical Model
221 Mathematical Model of Shell Side of Condenser
(1) Steam Zone
(i) Steam mass equation is as follows
119889119866119904
119889119905= 119866st + 119866ost minus 119866119888 minus 119866ss (1)
where 119866119904is the steam content in the shell side of the
condenser (kgs) 119866st is the exhaust volume of steam turbine(kgs)119866ost is the other steam inlet of the condenser (kgs)119866
119888
is the main steam condensate (kgs) and119866ss is the amount ofsteam drawn out by vacuum pumping equipment (kgs)119866ss and 119866119888 are calculated by the following equations
where 119866ao is the quality of the gas mixture pumped by thepumping unit (kgs) 119877 is the share air ratio in the condenser119867119904is the average enthalpy of steam (kJkg) and 119867cw is the
enthalpy of saturated water corresponding to the condenserpressure (kJkg)
where 119867gp is the bubbling oxygen exhaust steam enthalpy(kJkg) 119867
119908is the enthalpy of hot well water (kJkg) and
119867cw is the enthalpy of saturated water corresponding to thecondenser pressure (kJkg)
222 Mathematical Model of Condenser Tube Side Thedynamic heat balance equation of the circulating water isdescribed as follows
119872119908119862119908
1198891198792
119889119905= 119876 minus 119876
119908= 119880119860Δ119905
119898minus 119865cw119862119901 (119879 minus 119879cw) (11)
where 119872119908is the circulating water quality (kg) 119862
119901is the
circulating water heat capacity (kJ(kg lowast C)) 119876 is the steamoutlet heat (kJ) 119876
119908is the circulating water heat absorption
quantity (kJ) 119880 is the condenser heat transfer coefficient(W(m2 lowast ∘C)) Δ119905
119898is the logarithmic mean temperature
difference (∘C) 119860 is the condenser heat transfer area (m2)119865cw is the circulating water flow (kgs) 119879cw is the circulatingwater inlet temperature (∘C) and 119879 is the circulating wateroutlet temperature (∘C)
The logarithmic heat transfer temperature difference iscalculated by
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
where 119879119888is the saturated water temperature of the vapor
pressure in the condenser (∘C) and 119879119908is the condenser hot
well water temperature (∘C)The heat transfer error of the condenser is calculated by
120575119905= 119879119904minus 119879 (14)
where 119879119904is the saturated gas temperature corresponding to
saturation pressure in condenser (∘C)
3 Grey Wolf Optimizer
The grey wolf optimizer is a new metaheuristic intelligentalgorithm proposed by Mirjalili et al in 2014 [12] which issuccessfully applied in many fields such as security smartgrid power system management [13] parameter estimation[14] reactive power dispatch problem [15] flow shop schedul-ing problem [16] combined heat and power dispatch [17] andautomatic control [18] For immigrating the wolvesrsquo internalleadership hierarchy the wolves are divided into four typesalpha (120572) beta (120573) delta (120575) and omega (Ω) According
Xlowast minus X
(Xlowast Y)(Xlowast minus X Y)
(X Y)
YlowastminusY
(X Ylowast)
(Xlowast Ylowast)
(Xlowast minus X Ylowast)
(Xlowast minus X Ylowast minus Y)
(Xlowast Ylowast minus Y)
(X Ylowast minus Y)
Figure 3 Position vector of wolf and next moving position in two-dimensional space
to the principle of the wolves hunting the hunt process isdivided into three stages searching prey surrounding preyand attacking prey In the four groups of wolves 120572 120573 and 120575are seen as the three best wolves they guide the otherwolf (Ω)trending in the search space in the best region In the iterativesearching process 120572 120573 120575 and Ω wolves are used to realizethe assessment of prey possible positions in the optimizationprocess The positions of wolves are updated in accordancewith the following equations
=100381610038161003816100381610038161003816 sdot997888997888rarr119883119875(119905) minus (119905)
100381610038161003816100381610038161003816
(119905 + 1) =997888997888rarr119883119875(119905) minus sdot
(15)
where 119905 is the current iteration number and are thecoefficient vector 997888997888rarr119883
119875is the position vector of the prey and
is the position of the wolf The vectors and are expressedas follows
= 2119886 sdot997888rarr1199031minus 119886
= 2 sdot997888rarr1199032
(16)
where the coefficient 119886 linearly increases from 2 to 0 with theincrease of the iteration number and997888rarr119903
1and997888rarr1199032are the random
vector located in the scope [0 1]The principle and concept of the position update (15) are
described in Figure 3 [7]It can be seen from Figure 3 that the wolf in the position
(119883 119884) can be arranged a new location on the basis of theabove formula While Figure 3 shows only the 7 possiblepositions of the wolf the randomly adjusting parameters and can make the wolf move to anywhere in a continuousspace In GWO algorithm the positions of wolves 120572 120573 and120575 are likely the prey (optimal) position In the searchingprocess the previous three best solutions are assumed to be
Mathematical Problems in Engineering 5
Alpha
Beta
Delta Omega
Move
D120572
D120573
D120575
Figure 4 Sketch map of position update of the wolves
120572 120573 and 120575 and then the others are regarded as theΩ wolvesThe positions of 120572 120573 and 120575 are used to update their positionsThe following mathematical formulae are used to adjust theposition of Ω wolf again and the position update schematicgraph is shown in Figures 4 and 5
where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883
120575are the position of the wolves 120572 120573
and 120575 respectively 997888rarr1198621 997888rarr1198622and 997888rarr119862
3are random vectors and
represents the position of the current solution Equations (17)are used to calculate the approximate distance between thecurrent position and 120572 120573 and 120575 respectively After definingthe distance between them the final position of the currentsolution is calculated by the following
where 997888rarr1198601 997888rarr1198602 and 997888rarr119860
3are random vectors and 119905 represents
the number of iterationsSeen from the above equations (17) define the step size
of the wolf Ω tending to the wolves 120572 120573 and 120575 Equations(19)ndash(21) define the final position of Ω wolf
Unexplored
Unexplored
Explored
Explored
If |A|lt 1
If |A|ge 1
Figure 5 Exploration and development of wolves
It can be seen from Figure 4 that the random and adaptivevectors and can be used to balance the explorationand development capabilities of the GWO algorithm When|| gt 1 the wolf has detection ability On the other handwhen the value of vector is greater than 1 it can alsopromote the enhancement of the detection ability of the wolfIn contrast when || lt 1 and 119862 lt 1 the wolf rsquos informationmining capacity is enhanced In order to enhance the abilityof the wolf with the increase of the iteration number isdecreased linearly However is randomly generated in thewhole optimization process which can make the detectionand exploitation ability of the wolf to reach equilibrium at anystage especially in the final stage of the iteration and preventthe algorithm from falling into local optimum
The procedure of the GWO algorithm is described asfollows
Step 1 Initialize the wolves Randomly initialize the positionof the wolves119883
119894(119894 = 1 2 119899) and parameters 119886 119860 and 119862
Step 2 Calculate the fitness of eachwolf and choose the threewolves with best fitness as wolves 120572 120573 and 120575
Step 3 Update positions Based on (17)ndash(21) update thepositions of the other wolves that is to say the positions oftheΩ wolves
Step 4 Update parameters 119886 119860 and 119862
Step 5 Judge whether to meet the termination conditions ornot If satisfied the position of 120572 wolf and the fitness valueare the optimal output If the termination condition is notsatisfied return to Step 2
4 Parameter Optimization of PI ControllerBased on GWO Algorithm
41 Dynamic Model of Steam Condenser Based on Mat-labSimulink Simulation Software The establishment of the
6 Mathematical Problems in Engineering
dynamic mathematical model of the condenser is based onthe following assumption that the total amount of condensa-tion the circulating water flow and the condenser volume arecertain So it is set up based on the dynamic heat balance andmass balance of the condenser water
411 Dynamic Heat Balance In the dynamic heat balance itis assumed that the total amount of condensation is certainand that the input steam and the output condensate aresaturated Therefore the heat from steam to the circulatingwater and the steam potential heat are equal So the steamreleased heat can be approximated calculated by the follow-ing equations
119876 asymp 119880119860Δ119905119898
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
(22)
The heat transfer coefficient 119880 and the heat transfer area119860 can be replaced by the following exponential equationapproximately
1
119880119860= 1205721119865cwminus08
+ 1205722 (23)
where 1205721and 120572
2are constants
When 119865cwrarrinfin 1205722 is determined by 119880119860 In this case 119880119860is determined by the heat transfer ratio between the steamand the tube wall and the thermal resistance of the tube wallThus by assuming the outlet temperature of the circulatingwater 120572
2can be determined Based on the above assumptions
and equations the dynamic heat balance equation of thecondenser is described as follows
119889119879
119889119905=119865cw119872cw
(119879cw minus 119879) +119876
119872cw119862119901 (24)
412 Mass Balance Themass balance of steam and conden-sate is based on the assumption that the space 119881 is constantand the volume of the steam and air is constant That is tosay in order to maintain the vapor condensation level ofthe condenser (certain vacuum degree) the output flow ofcondensatewater needs to be controlled in a certain range Soin order to simplify the model we assume that the inlet andoutlet of the condensate are saturatedTherefore the ideal gasmodel equation is expressed as
119889119875
119889119905=119877119879119888
119881(119865119904minus 119865119888) (25)
where 119865119904is the steam flow (kgs) and 119865
119888is the condensate
water flow (kgs)Among them the condensationwater temperature119879
119888and
the condenser pressure 119875 have a unique relationship In orderto simplify the model it is approximated by the followinglinear relationship equation
119879119888= 120572119875 + 120573 (26)
The steam condenser model given above has five equa-tions among which two are dynamic equations Here there
Table 1 Parameters of steam condenser
Parameter Parameter value Unit119877 0461526 kJkgK119881 3 m3
Condensate water flow 4 kgs119865cw Cooling water flow 1078881 kgs119875 Condenser pressure 90 kPa119879 Circulating water outlet temperature 80 ∘C119879cw Circulating water inlet temperature 60 ∘C119879119888
Saturated water temperature 965538 ∘C119876 Steam heat 90626 kW
are eight variables (119865119904 119865119888 119865cw 119875 119879 119879cw 119879119888 and 119876) and ten
parameters (119877119881120582119880119860119872cw1198621199011205721205731205721 and1205722)The valuesof ten parameters are shown in Table 1 where 120572
1and 120572
2are
determined under 119879 = 90∘C (119865cw rarr infin) The values of theeight variables are shown in Table 2 under the assumptionthat the system is stable
Based on the above model equations and the softwareMatlabSimulink a simulation model of PI controller forcondenser pressure closed-loop control system is establishedas shown in Figure 6 which includes a first-order delay unitused to represent the actuator with a time constant 120591 = 10 (s)and a lag unit caused by the pressure sensor with the timeconstant 120591 = 5 (s)
42 Encoding and Fitness Function Because the design ofthe PID controller is actually a multidimensional functionoptimization problem the GWO algorithm adopts the realnumber coding So for the parameters optimization of the PIcontroller each wolf can be directly coded as (119870
119901 119870119894)
119883 = 119870119901 119870119894 (27)
The control parameter optimization is designed to makethe control error tend to zero and has a faster response speedand smaller overshoot So the evaluation of the performanceof each set of control parameters is good or bad the integralof the product of absolute error and time is selected as thefitness function
ITAE = intinfin
0
119905 |119890 (119905)| 119889119905 (28)
Mathematical Problems in Engineering 7
Click here to tune the PID controller
60
Reaction curvePID tuning
Steam condenserScope
Pressure setpoint
PID
PID controller
Input step test
4
⟨Fcw⟩⟨T⟩
⟨Q⟩⟨P⟩Tcw
Tcw
Fs
Fs y =
Fcw
+++ minus
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
where 119867gp is the bubbling oxygen exhaust steam enthalpy(kJkg) 119867
119908is the enthalpy of hot well water (kJkg) and
119867cw is the enthalpy of saturated water corresponding to thecondenser pressure (kJkg)
222 Mathematical Model of Condenser Tube Side Thedynamic heat balance equation of the circulating water isdescribed as follows
119872119908119862119908
1198891198792
119889119905= 119876 minus 119876
119908= 119880119860Δ119905
119898minus 119865cw119862119901 (119879 minus 119879cw) (11)
where 119872119908is the circulating water quality (kg) 119862
119901is the
circulating water heat capacity (kJ(kg lowast C)) 119876 is the steamoutlet heat (kJ) 119876
119908is the circulating water heat absorption
quantity (kJ) 119880 is the condenser heat transfer coefficient(W(m2 lowast ∘C)) Δ119905
119898is the logarithmic mean temperature
difference (∘C) 119860 is the condenser heat transfer area (m2)119865cw is the circulating water flow (kgs) 119879cw is the circulatingwater inlet temperature (∘C) and 119879 is the circulating wateroutlet temperature (∘C)
The logarithmic heat transfer temperature difference iscalculated by
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
where 119879119888is the saturated water temperature of the vapor
pressure in the condenser (∘C) and 119879119908is the condenser hot
well water temperature (∘C)The heat transfer error of the condenser is calculated by
120575119905= 119879119904minus 119879 (14)
where 119879119904is the saturated gas temperature corresponding to
saturation pressure in condenser (∘C)
3 Grey Wolf Optimizer
The grey wolf optimizer is a new metaheuristic intelligentalgorithm proposed by Mirjalili et al in 2014 [12] which issuccessfully applied in many fields such as security smartgrid power system management [13] parameter estimation[14] reactive power dispatch problem [15] flow shop schedul-ing problem [16] combined heat and power dispatch [17] andautomatic control [18] For immigrating the wolvesrsquo internalleadership hierarchy the wolves are divided into four typesalpha (120572) beta (120573) delta (120575) and omega (Ω) According
Xlowast minus X
(Xlowast Y)(Xlowast minus X Y)
(X Y)
YlowastminusY
(X Ylowast)
(Xlowast Ylowast)
(Xlowast minus X Ylowast)
(Xlowast minus X Ylowast minus Y)
(Xlowast Ylowast minus Y)
(X Ylowast minus Y)
Figure 3 Position vector of wolf and next moving position in two-dimensional space
to the principle of the wolves hunting the hunt process isdivided into three stages searching prey surrounding preyand attacking prey In the four groups of wolves 120572 120573 and 120575are seen as the three best wolves they guide the otherwolf (Ω)trending in the search space in the best region In the iterativesearching process 120572 120573 120575 and Ω wolves are used to realizethe assessment of prey possible positions in the optimizationprocess The positions of wolves are updated in accordancewith the following equations
=100381610038161003816100381610038161003816 sdot997888997888rarr119883119875(119905) minus (119905)
100381610038161003816100381610038161003816
(119905 + 1) =997888997888rarr119883119875(119905) minus sdot
(15)
where 119905 is the current iteration number and are thecoefficient vector 997888997888rarr119883
119875is the position vector of the prey and
is the position of the wolf The vectors and are expressedas follows
= 2119886 sdot997888rarr1199031minus 119886
= 2 sdot997888rarr1199032
(16)
where the coefficient 119886 linearly increases from 2 to 0 with theincrease of the iteration number and997888rarr119903
1and997888rarr1199032are the random
vector located in the scope [0 1]The principle and concept of the position update (15) are
described in Figure 3 [7]It can be seen from Figure 3 that the wolf in the position
(119883 119884) can be arranged a new location on the basis of theabove formula While Figure 3 shows only the 7 possiblepositions of the wolf the randomly adjusting parameters and can make the wolf move to anywhere in a continuousspace In GWO algorithm the positions of wolves 120572 120573 and120575 are likely the prey (optimal) position In the searchingprocess the previous three best solutions are assumed to be
Mathematical Problems in Engineering 5
Alpha
Beta
Delta Omega
Move
D120572
D120573
D120575
Figure 4 Sketch map of position update of the wolves
120572 120573 and 120575 and then the others are regarded as theΩ wolvesThe positions of 120572 120573 and 120575 are used to update their positionsThe following mathematical formulae are used to adjust theposition of Ω wolf again and the position update schematicgraph is shown in Figures 4 and 5
where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883
120575are the position of the wolves 120572 120573
and 120575 respectively 997888rarr1198621 997888rarr1198622and 997888rarr119862
3are random vectors and
represents the position of the current solution Equations (17)are used to calculate the approximate distance between thecurrent position and 120572 120573 and 120575 respectively After definingthe distance between them the final position of the currentsolution is calculated by the following
where 997888rarr1198601 997888rarr1198602 and 997888rarr119860
3are random vectors and 119905 represents
the number of iterationsSeen from the above equations (17) define the step size
of the wolf Ω tending to the wolves 120572 120573 and 120575 Equations(19)ndash(21) define the final position of Ω wolf
Unexplored
Unexplored
Explored
Explored
If |A|lt 1
If |A|ge 1
Figure 5 Exploration and development of wolves
It can be seen from Figure 4 that the random and adaptivevectors and can be used to balance the explorationand development capabilities of the GWO algorithm When|| gt 1 the wolf has detection ability On the other handwhen the value of vector is greater than 1 it can alsopromote the enhancement of the detection ability of the wolfIn contrast when || lt 1 and 119862 lt 1 the wolf rsquos informationmining capacity is enhanced In order to enhance the abilityof the wolf with the increase of the iteration number isdecreased linearly However is randomly generated in thewhole optimization process which can make the detectionand exploitation ability of the wolf to reach equilibrium at anystage especially in the final stage of the iteration and preventthe algorithm from falling into local optimum
The procedure of the GWO algorithm is described asfollows
Step 1 Initialize the wolves Randomly initialize the positionof the wolves119883
119894(119894 = 1 2 119899) and parameters 119886 119860 and 119862
Step 2 Calculate the fitness of eachwolf and choose the threewolves with best fitness as wolves 120572 120573 and 120575
Step 3 Update positions Based on (17)ndash(21) update thepositions of the other wolves that is to say the positions oftheΩ wolves
Step 4 Update parameters 119886 119860 and 119862
Step 5 Judge whether to meet the termination conditions ornot If satisfied the position of 120572 wolf and the fitness valueare the optimal output If the termination condition is notsatisfied return to Step 2
4 Parameter Optimization of PI ControllerBased on GWO Algorithm
41 Dynamic Model of Steam Condenser Based on Mat-labSimulink Simulation Software The establishment of the
6 Mathematical Problems in Engineering
dynamic mathematical model of the condenser is based onthe following assumption that the total amount of condensa-tion the circulating water flow and the condenser volume arecertain So it is set up based on the dynamic heat balance andmass balance of the condenser water
411 Dynamic Heat Balance In the dynamic heat balance itis assumed that the total amount of condensation is certainand that the input steam and the output condensate aresaturated Therefore the heat from steam to the circulatingwater and the steam potential heat are equal So the steamreleased heat can be approximated calculated by the follow-ing equations
119876 asymp 119880119860Δ119905119898
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
(22)
The heat transfer coefficient 119880 and the heat transfer area119860 can be replaced by the following exponential equationapproximately
1
119880119860= 1205721119865cwminus08
+ 1205722 (23)
where 1205721and 120572
2are constants
When 119865cwrarrinfin 1205722 is determined by 119880119860 In this case 119880119860is determined by the heat transfer ratio between the steamand the tube wall and the thermal resistance of the tube wallThus by assuming the outlet temperature of the circulatingwater 120572
2can be determined Based on the above assumptions
and equations the dynamic heat balance equation of thecondenser is described as follows
119889119879
119889119905=119865cw119872cw
(119879cw minus 119879) +119876
119872cw119862119901 (24)
412 Mass Balance Themass balance of steam and conden-sate is based on the assumption that the space 119881 is constantand the volume of the steam and air is constant That is tosay in order to maintain the vapor condensation level ofthe condenser (certain vacuum degree) the output flow ofcondensatewater needs to be controlled in a certain range Soin order to simplify the model we assume that the inlet andoutlet of the condensate are saturatedTherefore the ideal gasmodel equation is expressed as
119889119875
119889119905=119877119879119888
119881(119865119904minus 119865119888) (25)
where 119865119904is the steam flow (kgs) and 119865
119888is the condensate
water flow (kgs)Among them the condensationwater temperature119879
119888and
the condenser pressure 119875 have a unique relationship In orderto simplify the model it is approximated by the followinglinear relationship equation
119879119888= 120572119875 + 120573 (26)
The steam condenser model given above has five equa-tions among which two are dynamic equations Here there
Table 1 Parameters of steam condenser
Parameter Parameter value Unit119877 0461526 kJkgK119881 3 m3
Condensate water flow 4 kgs119865cw Cooling water flow 1078881 kgs119875 Condenser pressure 90 kPa119879 Circulating water outlet temperature 80 ∘C119879cw Circulating water inlet temperature 60 ∘C119879119888
Saturated water temperature 965538 ∘C119876 Steam heat 90626 kW
are eight variables (119865119904 119865119888 119865cw 119875 119879 119879cw 119879119888 and 119876) and ten
parameters (119877119881120582119880119860119872cw1198621199011205721205731205721 and1205722)The valuesof ten parameters are shown in Table 1 where 120572
1and 120572
2are
determined under 119879 = 90∘C (119865cw rarr infin) The values of theeight variables are shown in Table 2 under the assumptionthat the system is stable
Based on the above model equations and the softwareMatlabSimulink a simulation model of PI controller forcondenser pressure closed-loop control system is establishedas shown in Figure 6 which includes a first-order delay unitused to represent the actuator with a time constant 120591 = 10 (s)and a lag unit caused by the pressure sensor with the timeconstant 120591 = 5 (s)
42 Encoding and Fitness Function Because the design ofthe PID controller is actually a multidimensional functionoptimization problem the GWO algorithm adopts the realnumber coding So for the parameters optimization of the PIcontroller each wolf can be directly coded as (119870
119901 119870119894)
119883 = 119870119901 119870119894 (27)
The control parameter optimization is designed to makethe control error tend to zero and has a faster response speedand smaller overshoot So the evaluation of the performanceof each set of control parameters is good or bad the integralof the product of absolute error and time is selected as thefitness function
ITAE = intinfin
0
119905 |119890 (119905)| 119889119905 (28)
Mathematical Problems in Engineering 7
Click here to tune the PID controller
60
Reaction curvePID tuning
Steam condenserScope
Pressure setpoint
PID
PID controller
Input step test
4
⟨Fcw⟩⟨T⟩
⟨Q⟩⟨P⟩Tcw
Tcw
Fs
Fs y =
Fcw
+++ minus
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
Figure 4 Sketch map of position update of the wolves
120572 120573 and 120575 and then the others are regarded as theΩ wolvesThe positions of 120572 120573 and 120575 are used to update their positionsThe following mathematical formulae are used to adjust theposition of Ω wolf again and the position update schematicgraph is shown in Figures 4 and 5
where 997888rarr119883120572 997888rarr119883120573 and 997888rarr119883
120575are the position of the wolves 120572 120573
and 120575 respectively 997888rarr1198621 997888rarr1198622and 997888rarr119862
3are random vectors and
represents the position of the current solution Equations (17)are used to calculate the approximate distance between thecurrent position and 120572 120573 and 120575 respectively After definingthe distance between them the final position of the currentsolution is calculated by the following
where 997888rarr1198601 997888rarr1198602 and 997888rarr119860
3are random vectors and 119905 represents
the number of iterationsSeen from the above equations (17) define the step size
of the wolf Ω tending to the wolves 120572 120573 and 120575 Equations(19)ndash(21) define the final position of Ω wolf
Unexplored
Unexplored
Explored
Explored
If |A|lt 1
If |A|ge 1
Figure 5 Exploration and development of wolves
It can be seen from Figure 4 that the random and adaptivevectors and can be used to balance the explorationand development capabilities of the GWO algorithm When|| gt 1 the wolf has detection ability On the other handwhen the value of vector is greater than 1 it can alsopromote the enhancement of the detection ability of the wolfIn contrast when || lt 1 and 119862 lt 1 the wolf rsquos informationmining capacity is enhanced In order to enhance the abilityof the wolf with the increase of the iteration number isdecreased linearly However is randomly generated in thewhole optimization process which can make the detectionand exploitation ability of the wolf to reach equilibrium at anystage especially in the final stage of the iteration and preventthe algorithm from falling into local optimum
The procedure of the GWO algorithm is described asfollows
Step 1 Initialize the wolves Randomly initialize the positionof the wolves119883
119894(119894 = 1 2 119899) and parameters 119886 119860 and 119862
Step 2 Calculate the fitness of eachwolf and choose the threewolves with best fitness as wolves 120572 120573 and 120575
Step 3 Update positions Based on (17)ndash(21) update thepositions of the other wolves that is to say the positions oftheΩ wolves
Step 4 Update parameters 119886 119860 and 119862
Step 5 Judge whether to meet the termination conditions ornot If satisfied the position of 120572 wolf and the fitness valueare the optimal output If the termination condition is notsatisfied return to Step 2
4 Parameter Optimization of PI ControllerBased on GWO Algorithm
41 Dynamic Model of Steam Condenser Based on Mat-labSimulink Simulation Software The establishment of the
6 Mathematical Problems in Engineering
dynamic mathematical model of the condenser is based onthe following assumption that the total amount of condensa-tion the circulating water flow and the condenser volume arecertain So it is set up based on the dynamic heat balance andmass balance of the condenser water
411 Dynamic Heat Balance In the dynamic heat balance itis assumed that the total amount of condensation is certainand that the input steam and the output condensate aresaturated Therefore the heat from steam to the circulatingwater and the steam potential heat are equal So the steamreleased heat can be approximated calculated by the follow-ing equations
119876 asymp 119880119860Δ119905119898
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
(22)
The heat transfer coefficient 119880 and the heat transfer area119860 can be replaced by the following exponential equationapproximately
1
119880119860= 1205721119865cwminus08
+ 1205722 (23)
where 1205721and 120572
2are constants
When 119865cwrarrinfin 1205722 is determined by 119880119860 In this case 119880119860is determined by the heat transfer ratio between the steamand the tube wall and the thermal resistance of the tube wallThus by assuming the outlet temperature of the circulatingwater 120572
2can be determined Based on the above assumptions
and equations the dynamic heat balance equation of thecondenser is described as follows
119889119879
119889119905=119865cw119872cw
(119879cw minus 119879) +119876
119872cw119862119901 (24)
412 Mass Balance Themass balance of steam and conden-sate is based on the assumption that the space 119881 is constantand the volume of the steam and air is constant That is tosay in order to maintain the vapor condensation level ofthe condenser (certain vacuum degree) the output flow ofcondensatewater needs to be controlled in a certain range Soin order to simplify the model we assume that the inlet andoutlet of the condensate are saturatedTherefore the ideal gasmodel equation is expressed as
119889119875
119889119905=119877119879119888
119881(119865119904minus 119865119888) (25)
where 119865119904is the steam flow (kgs) and 119865
119888is the condensate
water flow (kgs)Among them the condensationwater temperature119879
119888and
the condenser pressure 119875 have a unique relationship In orderto simplify the model it is approximated by the followinglinear relationship equation
119879119888= 120572119875 + 120573 (26)
The steam condenser model given above has five equa-tions among which two are dynamic equations Here there
Table 1 Parameters of steam condenser
Parameter Parameter value Unit119877 0461526 kJkgK119881 3 m3
Condensate water flow 4 kgs119865cw Cooling water flow 1078881 kgs119875 Condenser pressure 90 kPa119879 Circulating water outlet temperature 80 ∘C119879cw Circulating water inlet temperature 60 ∘C119879119888
Saturated water temperature 965538 ∘C119876 Steam heat 90626 kW
are eight variables (119865119904 119865119888 119865cw 119875 119879 119879cw 119879119888 and 119876) and ten
parameters (119877119881120582119880119860119872cw1198621199011205721205731205721 and1205722)The valuesof ten parameters are shown in Table 1 where 120572
1and 120572
2are
determined under 119879 = 90∘C (119865cw rarr infin) The values of theeight variables are shown in Table 2 under the assumptionthat the system is stable
Based on the above model equations and the softwareMatlabSimulink a simulation model of PI controller forcondenser pressure closed-loop control system is establishedas shown in Figure 6 which includes a first-order delay unitused to represent the actuator with a time constant 120591 = 10 (s)and a lag unit caused by the pressure sensor with the timeconstant 120591 = 5 (s)
42 Encoding and Fitness Function Because the design ofthe PID controller is actually a multidimensional functionoptimization problem the GWO algorithm adopts the realnumber coding So for the parameters optimization of the PIcontroller each wolf can be directly coded as (119870
119901 119870119894)
119883 = 119870119901 119870119894 (27)
The control parameter optimization is designed to makethe control error tend to zero and has a faster response speedand smaller overshoot So the evaluation of the performanceof each set of control parameters is good or bad the integralof the product of absolute error and time is selected as thefitness function
ITAE = intinfin
0
119905 |119890 (119905)| 119889119905 (28)
Mathematical Problems in Engineering 7
Click here to tune the PID controller
60
Reaction curvePID tuning
Steam condenserScope
Pressure setpoint
PID
PID controller
Input step test
4
⟨Fcw⟩⟨T⟩
⟨Q⟩⟨P⟩Tcw
Tcw
Fs
Fs y =
Fcw
+++ minus
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
dynamic mathematical model of the condenser is based onthe following assumption that the total amount of condensa-tion the circulating water flow and the condenser volume arecertain So it is set up based on the dynamic heat balance andmass balance of the condenser water
411 Dynamic Heat Balance In the dynamic heat balance itis assumed that the total amount of condensation is certainand that the input steam and the output condensate aresaturated Therefore the heat from steam to the circulatingwater and the steam potential heat are equal So the steamreleased heat can be approximated calculated by the follow-ing equations
119876 asymp 119880119860Δ119905119898
Δ119905119898=
119879 minus 119879cwln ((119879
119904minus 119879cw) (119879119904 minus 119879))
(22)
The heat transfer coefficient 119880 and the heat transfer area119860 can be replaced by the following exponential equationapproximately
1
119880119860= 1205721119865cwminus08
+ 1205722 (23)
where 1205721and 120572
2are constants
When 119865cwrarrinfin 1205722 is determined by 119880119860 In this case 119880119860is determined by the heat transfer ratio between the steamand the tube wall and the thermal resistance of the tube wallThus by assuming the outlet temperature of the circulatingwater 120572
2can be determined Based on the above assumptions
and equations the dynamic heat balance equation of thecondenser is described as follows
119889119879
119889119905=119865cw119872cw
(119879cw minus 119879) +119876
119872cw119862119901 (24)
412 Mass Balance Themass balance of steam and conden-sate is based on the assumption that the space 119881 is constantand the volume of the steam and air is constant That is tosay in order to maintain the vapor condensation level ofthe condenser (certain vacuum degree) the output flow ofcondensatewater needs to be controlled in a certain range Soin order to simplify the model we assume that the inlet andoutlet of the condensate are saturatedTherefore the ideal gasmodel equation is expressed as
119889119875
119889119905=119877119879119888
119881(119865119904minus 119865119888) (25)
where 119865119904is the steam flow (kgs) and 119865
119888is the condensate
water flow (kgs)Among them the condensationwater temperature119879
119888and
the condenser pressure 119875 have a unique relationship In orderto simplify the model it is approximated by the followinglinear relationship equation
119879119888= 120572119875 + 120573 (26)
The steam condenser model given above has five equa-tions among which two are dynamic equations Here there
Table 1 Parameters of steam condenser
Parameter Parameter value Unit119877 0461526 kJkgK119881 3 m3
Condensate water flow 4 kgs119865cw Cooling water flow 1078881 kgs119875 Condenser pressure 90 kPa119879 Circulating water outlet temperature 80 ∘C119879cw Circulating water inlet temperature 60 ∘C119879119888
Saturated water temperature 965538 ∘C119876 Steam heat 90626 kW
are eight variables (119865119904 119865119888 119865cw 119875 119879 119879cw 119879119888 and 119876) and ten
parameters (119877119881120582119880119860119872cw1198621199011205721205731205721 and1205722)The valuesof ten parameters are shown in Table 1 where 120572
1and 120572
2are
determined under 119879 = 90∘C (119865cw rarr infin) The values of theeight variables are shown in Table 2 under the assumptionthat the system is stable
Based on the above model equations and the softwareMatlabSimulink a simulation model of PI controller forcondenser pressure closed-loop control system is establishedas shown in Figure 6 which includes a first-order delay unitused to represent the actuator with a time constant 120591 = 10 (s)and a lag unit caused by the pressure sensor with the timeconstant 120591 = 5 (s)
42 Encoding and Fitness Function Because the design ofthe PID controller is actually a multidimensional functionoptimization problem the GWO algorithm adopts the realnumber coding So for the parameters optimization of the PIcontroller each wolf can be directly coded as (119870
119901 119870119894)
119883 = 119870119901 119870119894 (27)
The control parameter optimization is designed to makethe control error tend to zero and has a faster response speedand smaller overshoot So the evaluation of the performanceof each set of control parameters is good or bad the integralof the product of absolute error and time is selected as thefitness function
ITAE = intinfin
0
119905 |119890 (119905)| 119889119905 (28)
Mathematical Problems in Engineering 7
Click here to tune the PID controller
60
Reaction curvePID tuning
Steam condenserScope
Pressure setpoint
PID
PID controller
Input step test
4
⟨Fcw⟩⟨T⟩
⟨Q⟩⟨P⟩Tcw
Tcw
Fs
Fs y =
Fcw
+++ minus
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
Figure 6 MatlabSimulink simulation model of PI controller forcondenser pressure closed-loop control system
0 10 20 30 40 50785
79
795
80
Time (s)
Am
plitu
de
⟨T⟩
ZNGA
PSOGWO
Figure 7 Output response curves of outlet temperature of coolingwater under different algorithms
Table 3 Parameters of PI controller
PID parameters ZN GA PSO GWO119870119901
108 853 707 447119870119894
422 091 104 089
43 Simulation Experiments and Results Analysis of PI Con-troller On the basis of the above established model of steamcondenser the GWO algorithm is adopted to optimize theparameters of the adopted PI controller The self-tuningperformances are compared with the Z-N engineering tun-ing method genetic algorithm (GA) and particle swarmoptimization (PSO) algorithm Respectively run GWO PSOand GA algorithm 30 times and then select the best PIDparameters of each algorithm The output response curvesof cooling water outlet temperature circulating water flowsteam discharge heat and condenser pressure are shown inFigures 7ndash10 The parameters of PI controllers are listed inTable 3
0 10 20 30 40 5095
100
105
110
115
120
125
130
135
140
145
Am
plitu
de
Time (s)
ZNGA
PSOGWO
⟨Fcw⟩
Figure 8 Output response curves of circulating water flow underdifferent algorithms
As seen from the above simulation results the PI con-troller under the optimization by the proposed GWO algo-rithm has the best control performance that is to say smallovershoot and short rise time and adjustment time The Z-Nengineering self-tuning method has the worst performancewhere the overshoot is the largest and the rise time andadjustment time are the longest The GWO algorithm caneffectively improve the system control quality and achieve thedesired effect
Because the Z-N method belongs to engineering settingmethod setting the PID parameters depends on experience
8 Mathematical Problems in Engineering
0 10 20 30 40 5081
82
83
84
85
86
87
88
89
90
Am
plitu
de
⟨P⟩
Time (s)
ZNGA
PSOGWO
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
Figure 10 Output response curves of pressure under differentalgorithms
value so the effect of PID control is poor The GA algorithmand PSO algorithm are intelligent method so control effectof PID controller whose parameters are optimized by thesetwo methods has been greatly improved relative to the Z-Nsetting method But because of their inherent search modeand the defects the search accuracy is not high enough thusresult of parameters setting is poorer than that of grey wolfoptimizer It can be known from searching way of GWOalgorithm above that the grey wolf optimizer is going tosearch solutions in the comprehensive evaluation by threewolves Therefore the GWO algorithm has a high searchprecision which can make it sure to search a better solutionvalue Thus the GWO algorithm increases the probability ofsearching a better group of PID parameters
5 Conclusions
In this paper a dynamic model of steam condenser and theclosed-loop control system are established on the basis of thelumped parameter model of the condenser For the pressureclosed-loop control system in order to improve the systemperformance the GWO algorithm is adopted to optimize thePI parametersThrough the simulation experiments and per-formance comparison of cooling water outlet temperaturecirculating water flow steam discharge heat and condenserpressure the introduction of the GWO algorithm makes thesteam condenser PI controller have better control effect Inorder to have a better control effect of steam condenser wecan try to improve the design of PID controller Since theresearch of grey wolf optimizer is just in its infancy theGWO algorithm should be improved to obtain better PIDparameters in the further study Other superior algorithmsshould be exploited for PID parameter optimization
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
Shu-Xia Li participated in the data collection analysis algo-rithm simulation draft writing and critical revision of thispaper Jie-Sheng Wang participated in the concept designand interpretation and commented on the paper
Acknowledgments
This work is partially supported by National Key Technolo-gies R amp D Program of China (Grant no 2014BAF05B01)Project by National Natural Science Foundation of China(Grant no 21576127) Program for Liaoning Excellent Talentsin University (Grant no LR2014008) Project by LiaoningProvincial Natural Science Foundation of China (Grant no2014020177) Program for Research Special Foundation ofUniversity of Science and Technology of Liaoning (Grantno 2015TD04) and Opening Project of National FinancialSecurity and System Equipment Engineering Research Cen-ter (Grant no USTLKEC201401)
References
[1] T Tahri S A Abdul-Wahab A Bettahar M Douani H Al-Hinai and Y Al-Mulla ldquoSimulation of the condenser of theseawater greenhouse Part I theoretical developmentrdquo Journalof Thermal Analysis and Calorimetry vol 96 no 1 pp 35ndash422009
[2] P K Bansal and T C Chin ldquoModelling and optimisation ofwire-and-tube condenserrdquo International Journal of Refrigera-tion vol 26 no 5 pp 601ndash613 2003
[3] M R Malin ldquoModelling flow in an experimental marinecondenserrdquo International Communications in Heat and MassTransfer vol 24 no 5 pp 597ndash608 1997
[4] XDingWCai P Duan and J Yan ldquoHybrid dynamicmodelingfor two phase flow condensersrdquo Applied Thermal Engineeringvol 62 no 2 pp 830ndash837 2014
[5] WWangD Zeng J Liu YNiu andC Cui ldquoFeasibility analysisof changing turbine load in power plants using continuouscondenser pressure adjustmentrdquo Energy vol 64 pp 533ndash5402014
[6] Z Ma and S Wang ldquoOnline fault detection and robust controlof condenser cooling water systems in building central chillerplantsrdquo Energy and Buildings vol 43 no 1 pp 153ndash165 2011
[7] D Whitley ldquoAn executable model of a simple genetic algo-rithmrdquo Foundations of Genetic Algorithms vol 2 no 1519 pp45ndash62 2014
[8] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995
[9] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing vol 8no 1 pp 687ndash697 2008
Mathematical Problems in Engineering 9
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
[10] M Dorigo M Birattari and T Stutzle ldquoAnt colony optimiza-tionrdquo IEEE Computational Intelligence Magazine vol 1 no 4pp 28ndash39 2006
[11] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo in Pro-ceedings of the World Congress on Nature amp Biologically InspiredComputing (NABIC rsquo09) pp 210ndash214 IEEE Coimbatore IndiaDecember 2009
[12] S Mirjalili S M Mirjalili and A Lewis ldquoGrey wolf optimizerrdquoAdvances in Engineering Software vol 69 pp 46ndash61 2014
[13] B Mahdad and K Srairi ldquoBlackout risk prevention in a smartgrid based flexible optimal strategy using Grey Wolf-patternsearch algorithmsrdquo Energy Conversion and Management vol98 pp 411ndash429 2015
[14] X Song L Tang S Zhao et al ldquoGrey Wolf Optimizer forparameter estimation in surface wavesrdquo Soil Dynamics andEarthquake Engineering vol 75 pp 147ndash157 2015
[15] M H Sulaiman Z Mustaffa M R Mohamed and O AlimanldquoUsing the gray wolf optimizer for solving optimal reactivepower dispatch problemrdquo Applied Soft Computing vol 32 pp286ndash292 2015
[16] G M Komaki and V Kayvanfar ldquoGrey Wolf Optimizer algo-rithm for the two-stage assembly flow shop scheduling problemwith release timerdquo Journal of Computational Science vol 8 pp109ndash120 2015
[17] N Jayakumar S Subramanian S Ganesan and E BElanchezhian ldquoGrey wolf optimization for combined heatand power dispatch with cogeneration systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 74 pp252ndash264 2016
[18] Y Sharma and L C Saikia ldquoAutomatic generation controlof a multi-area STmdashthermal power system using Grey Wolfoptimizer algorithm based classical controllersrdquo InternationalJournal of Electrical Power amp Energy Systems vol 73 pp 853ndash862 2015
