a hybrid constraints handling strategy for...

Research ArticleA Hybrid Constraints Handling Strategy forMulticonstrained Multiobjective Optimization Problem ofMicrogrid EconomicalEnvironmental Dispatch

Xin Li1 Jingang Lai23 and Ruoli Tang1

1School of Energy and Power Engineering Wuhan University of Technology Wuhan 430063 China2School of Electrical and Electronic Engineering Huazhong University of Science and Technology Wuhan 430074 China3School of Engineering RMIT University Melbourne VIC 3001 Australia

Correspondence should be addressed to Xin Li xinli0503hotmailcom

Received 25 July 2017 Accepted 4 December 2017 Published 19 December 2017

Academic Editor Dimitri Volchenkov

Copyright copy 2017 Xin Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Microgrid (MG) economicalenvironmental dispatch (MGEED) problem is a complex multiobjective optimization topic Sincethe generators are diversified and the operation mode changes frequently the MGEEED problem always has different types ofconstraints such as the load balance constraints and the ramp rates constraints whichmake it a nonlinear nonconvex optimizationproblem In this paper the mathematical model of a typical MG system applied in northwest China is established Then a hybridconstraints handling strategy (HCHS) based on nondominated sorting genetic algorithm II (NSGAII) is proposed to deal withthe typical constraints by which the constraints violations can be removed in several steps during the evolutionary process Adimensionality reduction method is introduced to simplify the optimization model And an individual repair approach is designedfor the violations of ramp rates constraints In order to balance the weights of various types of constraints the process of constraintshandling in standard NSGAII is revised Thereafter HCHS-NSGAII is applied to some typical MGEED problems considering allkinds of typical constraints The results show that HCHS-NSGAII can obtain feasible Pareto sets with satisfactory convergenceand distribution which is efficient in handling complex practical industrial MGEED problems with the change of constraintscombinations

1 Introduction

With the energy shortage and environmental pollutionbecoming serious [1ndash4] the technologies of microgrid (MG)with distributed energy resources (DERs) have been devel-oped rapidly [5ndash7] In remote areas of northwest China theenergy requirement is diversified but the level of energysupply is low [8] Thus it is meaningful to establish MGsystems by utilizing the local renewable energy such as thewind power and solar energy Usually the combined heatand power (CHP) systems like microturbines (MTs) are alsoneeded to provide electricity and heat As a result the applica-tions of MGs may cause uncertainty of energy efficiency andenvironmental pollution if the DERs cannot be dispatchedproperly Therefore solving the multiobjective optimizationproblem (MOP) of MG economicalenvironmental dispatch

(MGEED) is an important topic to save energy and reduceemissions simultaneously [9ndash12]

The complexity of the MG system makes MGEED anonlinear nonconvex mathematical optimization problemOne of the reasons is that in a practical MG system thereare various types of constraints caused by different dis-tributed generators (DGs) distributed energy storage systems(DESSs) and the whole MG system Thus as a MOP theseconstraints such as the equality and inequality constraintsand multivariable constraints make the feasible solutionsregion distributes unevenly in a high dimension search spacewhich may make it difficult for the optimization algorithmsto reach feasible Pareto fronts [12 13]

Many researchers have studied the nonlinear optimiza-tion methods and their application onMG systems economi-calenvironmental dispatch problems [14ndash19] In recent years

HindawiComplexityVolume 2017 Article ID 6249432 12 pageshttpsdoiorg10115520176249432

2 Complexity

multiobjective evolutionary algorithms (MOEAs) have beenintroduced to deal with theMGEEDproblems which ismoreefficient for solving nonconvex objective functions and moreflexible in handling the constraints [10 18 20] Howeverresearchersmainly focused on the accuracy and the efficiencyof algorithms instead of constraints handling strategiesPenalty functionmethod (PFM) which is a common simpleand efficient constraint handling method for constrainedoptimization problems is generally used in MOEAs In[19] the PEM was applied to convert the constrained MGcost optimization problem into an unconstrained one bymodifying the objective function with related penalty itemsThe simulation results showed that PEM could help theoptimization approach to find satisfied feasible solutionsHowever in that study the amount of the constraints wasnot large and the economical and environmental objectiveswere combined as only one optimization task In [21] theauthors introduced a method to handle the constraints inmultiobjective problems taking account of both feasibilityand domination which is called ldquoDebrsquos constraints handlingcriteriardquo It was used in some of the studies [13 22ndash24] andthe results showed that by using thismethod considering bothfeasibility and domination the proportion of the infeasiblesolutions could be reduced evidently [13 23] However thismethod needed to add the overall constraints which wereactually of different types and could not be easily quantifiedIn addition [20] proposed two frameworks based onMOEAsto solve the multiconstrained MGEED problems and bothof the strategies obtained the feasible Pareto sets Howeverthe authors only considered some of the common usedconstraints and the constraint handling approaches werenot customized which may reduce the efficiency when theoptimization frameworks were applied to other MGEEDproblems with different combinations of constraints

From the above researches it is obvious that the con-straints handling problems ofMGEEDhave not been system-atically studied Most of the studies utilized a general methodto deal with the violations However as a practical MOP theMGEED problems have serious challenges to the MOEAs forthe following reasons(1) MGEED problems have various types of constraintssuch as the equality constraints and inequality constraints(2) The dimension of the search space is high Sincealmost every variable in a solution should meet at leastone constraint the amount of the constraints are alwaysvery large which may make the feasible region in unevendistribution(3) When using MOEA as the optimization tool thegeneration of constraints violations may appear in differentsteps of the optimization process(4) As a practical MOP the combinations of the con-straints may change in different scenarios

Considering the above challenges the single and staticconstraint handling strategies may not always adapt to theMGEED problems especially when the constraints are com-plexTherefore it is necessary to study the hybrid approachesin dealing with all kinds of MG constraints

In remainder of this paper the system models andobjective functions are established in Section 2 The hybrid

constraints handling strategy (HCHS) based on nondom-inated sorting genetic algorithm II (NSGAII) is designedin Section 3 Thereafter the HCHS-NSGAII is applied toseveral practical MGEED problems with different constraintscombinations to study the efficiency of the proposed hybridconstraints handling strategy in Section 4 Finally the con-clusions of the study are presented

2 Modeling of MGEED

21 MG System Description TheMGEED problem is to allo-cate the output of every distributed component to meet thepredicted electricity and heat load demand without prejudiceto any of the constraints through the whole 24-hour processwhile maximizing the financial and ecological benefit of theMG system In this section the proposed objective functionsand constraints are discussedThe structure of the typicalMGused in this paper is shown in Figure 1 It can be seen thattwo kinds of uncontrollable DGs namely photovoltaic cells(PVs) and wind turbines (WTs) are considered of which themathematical models and related parameters are describedin [22 24] Two microturbines (MTs) and two fuel cells(FCs) with different parameter settings are considered as thecontrollableDGs to supply electricity powerAbattery bank isalso included as theDESS Besides the power exchangedwiththemain grid is consideredwhen theMG system turns to gridconnected mode by the PCC (point of common coupling)

22 Objective Functions In this paper the total operatingcost is considered as one of the objectives to be minimizedwhich contains the fuel cost the start-up cost the mainte-nance cost and the outcome by introducing the power fromthemain gridThus the objective function forminimum costcan be described below

min119862 (X)= 119879sum119905=1

119873119892sum119894=1

(CF119866119894119905 (X) + STC119866119894119905 (X) +OM119866119894119905 (X))

+ 119873119904sum119895=1

(OM119878119895119905 (X)) + 119862grid119905 (X)

(1)

where X is the decision variable vector 119879 is the number ofthe time intervals 119873119892 and 119873119904 are the total amounts of theDGs and DESSs respectively CF119866119894119905 and STC119866119894119905 are the fuelcost and the start-up cost of the 119894th controllable DG at time119905 respectively OM119866119894119905 and OM119878119895119905 are the maintenance costsfor the 119894th controllable DG and the 119895thDESS at time 119905 respec-tively119862grid119905 is the cost of the purchased power from themaingrid The model functions of CF119866119894119905 OM119866119894119905 OM119878119895119905 119862grid119905can be found in [22 23] and the start-up cost function isdescribed below

STC119866119894119905 = 120590119894 + 120575119894 [1 minus exp(minus119879off 119894119905120591119894 )] (1 minus 119906119894 (119905)) (2)

where 120590119894 and 120575119894 are the hot start-up cost and cold start-upcost of the 119894th controllable DG respectively 119879off 119894(119905) is the

Complexity 3

Circuit breaker

Circuit breaker

PV

MT

Circuit breaker

Circuit breaker

BAT

MT

WT

Circuit breaker

Sensitive load

Interruptible load

Adjustable load

Heat load

PCC

Feeder A

Feeder B

Feeder C

FC

Main grid

Figure 1 The structure of a typical MG

time during which the 119894th controllable DG has been off atthe beginning of the 119905th scheduling period 120591119894 and 119906119894(119905) arethe cooling time constant and the onoff status of the 119894thcontrollable DG respectively

Another objective to be optimized is the emission fromthe MG system which is shown below

min119864 (X) = 119879sum119905=1

119873119892sum119894=1

119864119866119894119905 +119873119904sum119895=1

119864119878119895119905 + 119864grid119905 (3)

where119864119866119894119905 119864119878119895119905 and119864grid119905 represent the emission of the 119894thcontrollable DG the 119895th DESS and the main grid at time 119905respectively In this paper only the emission production ofthe controllable DGs is considered as shown below

119864119866119894119905 = 120572119894 sdot PW1198661198941199052 + 120573119894 sdot PW119866119894119905 + 120574119894 (4)

where 120572119894 120573119894 120574119894 are the emission coefficients and PW119866119894119905 isthe power output of the power generators The values of theemission coefficients can be found in [25]

23 Typical Constraints As a practical multiobjective opti-mization problem the MGEED problem may have varioustypes of constraints In this paper some typical ones aremainly introduced as follows

(1) Rated Power Constraints Each controllable DG has itsmaximum and minimum output power constraints Simi-larly the power from the grid is also limited The constraintsare shown below

PW119866119894min le PW119866119894119905 le PW119866119894max (5)

PWgridmin le PWgrid119905 le PWgridmax (6)

where PWgrid119905 is the power exchanged with the main grid

(2) Electricity Power Balance Constraints The electricitypower generated by all the components from the MG system

should exactly meet the total load demands sum119873119871119897=1

119875119864119871119897(119905) attime 119905 which can be described as

119873119892sum119894=1

PW119866119894119905 +119873119904sum119895=1

PW119878119895119905 + 119875grid (119905) minus119873119871sum119897=1

119875119864119871119897 (119905) = 0 (7)

where PW119878119895119905 is the 119895th chargeddischarged power of theDESSs at time 119905 and 119873119871 is the number of electricity loaddemands In this paper119873119871 = 119873119878 = 1(3) Heat Power Balance Constraints The thermal power119876ℎ119900119896119905 from the MTs should exactly meet the heat loaddemand 119875HL119897(119905) which is shown below

119873119898sum119896=1

119876ℎ119900119896119905 + 119875HL119897 (119905) = 0 (8)

where119876ℎ119900119896119905 is the quantity of the exhaust heat of the 119896thMTat time 119905 and119873119898 is the number of theMTs in theMG systemThemathematicalmodel and parameter settings of119876ℎ119900119896119905 canbe found in [22]

(4) State of Charge Constraints The battery bank (BAT)cannot be overcharged or overused so the limits of the stateof charge (SOC) of the battery bank are as follows

SOCmin le SOC119905 le SOCmax (9)

(5) BAT ChargeDischarge Constraints The chargingdis-charging power of the BAT (119875char119875dischar) is limited in orderto protect the devices which can be described as

119875char (119905) le 119875charmax

119875dischar (119905) le 119875discharmax (10)

4 Complexity

The relation between the state of charge and the charg-ingdischarging power of the BAT mentioned above can beexpressed as

SOC119905 = SOC119905minus1 + 120578char119875charΔ119905 minus 1120578dischar119875discharΔ119905 (11)

where 120578char and 120578dischar are the charging and dischargingefficiencies of the BAT and Δ119905 is the time interval

(6) Ramp Rates Constraints The increasedecrease of outputpower of MTs in unit time is called ramp rate which reflectsthe performance of the DGs The ramp rates cannot exceed acertain value which can be expressed as

1003816100381610038161003816PW119866119894119905 minus PW119866119894119905minus11003816100381610038161003816 le PW119866119894ramp (12)

3 HCHS-NSGAII

It can be seen from Section 2 that the MOP of MGEED hascomplicated solution spaces due to the complexity of thevariable vectors the objective functions and the constraintswhich makes it difficult for the optimization algorithms tofind the optimal Pareto set Especially there is a wide varietyof constraints which has strong coupling and nonlinearityTherefore one of the keys to solve the MOP of MGEEDis to design high efficient optimization algorithms whichcan accurately handle the multiple constraints In this paperNSGAII is introduced as the core optimization tool to solvethe MOP of MGEED Furthermore NSGAII is improvedwith an MG-multiconstraint handling approach to adapt thechallenges in this specific multiconstrained MGEED

31 Standard NSGAII NSGAII was proposed by Deb et alin 2002 [21] which is one of the most efficient dominance-based MOEAs The main features of NSGAII are describedas follows (1) elitist based strategy in this way the elitistindividuals are kept during the evolution procedure (2) fastnondominated sorting by utilizing this method the compu-tational complexity can be reduced (3) crowding distancecalculation by sorting the individuals in the same Paretolevel according to the crowding distance the diversity of thepopulation is well protected NSGAII has strong robustnessin dealing with complexMOPs and can obtain solutions withgood diversities quicklyThe process of standard NSGAII canbe found in [21]

32 Hybrid Constraint Handling Strategy As mentionedin Section 1 when solving MOPs using NSGAII Debrsquosconstraints handling criteria are based on the individualfeasibility and the violations of the overall constraints in thesorting procedure which is widely used in many studiesDuring the constraints handling process the solutions withsmaller overall constraints will be selected if neither of thecandidates is feasible However in the practical MGEEDdifferent constraints cannot be combined directly and themethod in [21] is not able to handle the constraints related tovariables generation process Therefore this paper proposeda hybrid constraint handling strategy (HCHS) to improve

the performance of NSGAII in dealing with the complexconstraints which is described as follows

321 Dimensionality Reduction for the Equality ConstraintsThe equality constraints violations such as the violationsof electricity power and heat power balance constraints aredifficult for the method in NSGAII to completely eliminatebecause of the generating ways of the variables in the equali-ties Thus this paper suggests that the output of MT1 and thepower exchanged with the grid should be selected to transferthe equalities into inequalities with its own limits whichdecreases the difficulties of dealing with the constraintsTake the electricity power balance constraints handling as anexample The transformation procedure is shown below

First according to (7) PW1198921(119905) can be expressed as

PW1198921 (119905) = 119891minus11 (119875HL119897 (119905) minus119873119898sum119896=2

119891119896 (PW119892119896 (119905))) (13)

Then according to (7) and (13) 119875grid(119905) can be describedas

119875grid (119905)= 119875119864119871 (119905)minus 119873119892sum119894=2

PW119866119894119905 + 119891minus11 (119875HL119897 (119905) minus119873119898sum119896=2

119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905

(14)

Therefore (6) can be transformed as

PWgridmin

le 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905 le PWgridmax

(15)

where PW1198921(119905) and 119875HL119897(119905) are the electricity power outputof MT1 and the heat load demand respectively Equation (15)describes the final inequality constraint after transformation

By the transformation process above the equality con-straints violations are converted to inequality ones which areeasier to be removed Furthermore since one of the variablesin the equality has been replaced the dimensionality of thesearch space can be reduced In this way the optimizationmodel can be simplified

322 Repair Process after Generation of a New IndividualSince the individuals are generated using some heuristic-based stochastic methods in NSGA-II the constraint han-dling method in [21] cannot reduce the violation of some

Complexity 5

Input PW119866119894Output PW1015840119866119894PW1015840119866119894 larr []for 119905 larr 2 to 119879 do

if |PW119866119894119905minusPW119866119894119905minus1|gtPW119866119894rampif PW119866119894119905 gt PW119866119894119905minus1

PW1015840119866119894119905 larr min(PW119866119894maxPW119866119894119905 + PW119866119894ramp)else

PW1015840119866119894119905 larr max(PW119866119894minPW119866119894119905 minus PW119866119894ramp)end

endendreturn PW1015840119866119894119905

Algorithm 1 Repair process

constraints such as the ramp rates constraints and the ratedpower constraints related to variables generation processThus a repair process is needed after the variable ini-tialization process and the genetic operation process Thepseudocode of repairing the new individuals is presented inAlgorithm 1

It can be seen from Algorithm 1 that the repair processcan transfer the infeasible individuals which violate theramp rates constraints and the rated power constraints intofeasible individuals In this way the feasibility of the potentialsolutions is guaranteed However the repair process can onlyhandle the violations in the variable generation process andthe computational complexity may be high It is clear that atthe beginning of the evolutionary procedure the proportionof the infeasible solutions in the population is high while thepopulation is diversified because of the random generationprocess However at the end of the optimization the feasibil-ity of the solutions should be guaranteedTherefore the repairprobability on the infeasible potential solutions is designedto be dynamic depending on the evolutionary generations asshown below

119901repair (GENcur)

= (GENcurGENswi

)2 GENcur le GENswi

1 GENcur gt GENswi(16)

where GENcur and GENswi are the current generation andthe switch generation At the beginning of the evolutionaryprocess the value of 119901repair is small and the change speedis slow with the increase of the generation In this way asignificant number of infeasible solutions can be kept inthe population to protect the diversity which may lead thealgorithm to findmore feasible regionsWhenGENcur is closeto GENswi the increase speed of 119901repair is high and mostof the infeasible individuals could be repaired When thenumber of evolutionary generations is larger than GENswiall the infeasible solutions should be repaired to ensure thefeasibility of the final solutions In this paper GENswi equalshalf of the maximum generations

323 Normalization and Weighted Sum Process in SelectionFor the overall violations combined with different types ofconstraints such as the battery SOC constraints and thetransferred constraints in Section 321 each of the subitemsshould be normalized before being used which can becalculated as

V119897119896norm = V119897119896 minus V119896min

V119896max minus V119896min (17)

where V119897119896 and V119897119896norm are the actual and normalized violationvalues of the 119896th type of constraint in the 119897th individ-ual respectively V119896min and V119896max are the minimum andmaximum violation values of the 119896th type of constraint inthe population respectively Then the overall constraintsviolation of the 119897th individual can be obtained by

V119897 = 1199081 sdot V1198971norm + 1199082 sdot V1198972norm + sdot sdot sdot + 119908119888 sdot V119897119888norm (18)

where 119908119896 is the penalty weight of the 119896th type of constraintand 119888 is the number of the constraints types using method in[21]

324 Optimization Procedure of HCHS-NSGAII The threeapproaches designed above are combined with Debrsquos con-straints handling criteria to deal with all kinds of constraintsviolations in the MOPs of MGEED when using NSGAII Inpractical day-ahead MGEED the data may always changewith time Therefore the forecast load demands and the pre-dicted environmental data should be obtained first Besidesthe structure and the operation mode of the MG may alsochange [26 27] according to the operatorsrsquo requirementwhich would cause the change in the optimization modelsSo the optimization objectives and constraints should alsobe updated before applying the optimization algorithm Oneof the key parts is to classify the constraints using themethod proposed in this section which would guaranteethe efficiency of HCHS-NSGAII Then the optimizationprocedure of HCHS-NSGAII starts

Before running themain program the optimizationmod-els are simplified by the dimensionality reduction methodsThen after the population initial process the infeasible

6 Complexity

Table 1 The rates of the feasible solutions ()

Scenarios HCHS-NSGAII S-NSGAII PFM-NSGAIIBest Mean Best Mean Best Mean

Scenario One 100 98 87 85 78 74Scenario Two 100 92 33 29 26 23Scenario Three 98 88 16 13 12 11Scenario Four 94 86 7 5 2 1

Table 2 The best extreme feasible solutions for the minimum cost and emission using the three algorithms

Methods Scenario One Scenario Two Scenario Three Scenario Four

HCHS-NSGAII for obj11 (2012 279283) (8106 665581) (12247 864043) (18356 1012973)for obj2 (9872 0268) (17799 305079) (20147 721519) (24268 894012)

S-NSGAII for obj1 (2039 279347) (8365 663062) (12467 866023) (18539 1000891)for obj2 (9935 0295) (17074 323159) (15387 775686) (20578 942468)

PFM-NSGAII for obj1 (2021 279436) (8600 674194) (12699 858216) (18587 1000572)for obj2 (9866 3617) (18351 326766) (13956 805070) (19535 959465)

1obj1 and obj2 represent to the two optimization objectives namely the minimum cost and the minimum emission

individuals are partly repaired according to (16) After thatthe main program loop begins And the normalizationand weighted sum methods are applied when using Debrsquosconstraints handling criteria to deal with the violations of theinequality constraints After the nondominated sorting theselection and the genetic operation process the algorithmwill choose whether the partial repair or the total repairprocess is utilized for the new population according toGENcur Then the new population and the old populationwill be combined and the nondominated sorting and theselection will start again This procedure above is repeateduntil the termination condition is reached The flowchart ofHCHS-NSGAII is shown in Figure 2

4 Simulations and DiscussionIn this section the proposed HCHS-NSGAII is tested ona series of MGEED simulation problems For comparisonanother two most used constraints handling methods insolving practical MOPs are also applied The performancesof the three approaches above are discussed

41 Data and Parameter Settings The environmental dataincluding the wind speed irradiance and air temperaturecan be found in [23] The curves of heat and electricity loaddemand on a typical day are shown in Figure 3 In additionthe electricity prices and parameter settings of the objectivefunctions and constraints can be found in [22 23]

Besides as described in Section 1 the PFM is one ofthe most popular constraints handling methods in practicaloptimization problems And Debrsquos constraints handling cri-teria method applied in the standard NSGAII (S-NSGAII)is the most widely used constraints handling method insolving MOPsTherefore these two methods are also appliedto the following simulations here with NSGAII as theiroptimization algorithm The population size is 100 and themaximumgeneration number is 500 for the three algorithmsThe other parameter settings can be found in [21]

Comparing PFM-NSGAII with S-NSGAII the only dif-ference is that the fitness function of PFM-NSGAII is modi-fied as

119865119898 (x(119894)) = 119891119898 (x(119894)) + 119866119898 (x(119894)) (19)

where 119891119898(x(119894)) is the objective function value of the 119894thindividual for the119898th objective In this paper to compare theindividuals properly 119866119898(x(119894)) is calculated as

119866119898 (x(119894)) = 119877119898 (119891119898min + V (x(119894)) (119891119898max minus 119891119898min)) (20)

where 119877119898 is the penalty coefficient of the fitness function forthe 119898th objective V(x(119894)) is the overall violation value of the119894th individual which can be calculated by (18) 119891119898min and119891119898max are the minimum and maximum value for the 119898thobjective in the population In this paper 119877119898 = 1000042 Simulation Experiments In this subsection four MGoperation scenarios with different constraints are consideredIn Scenario One only constraints expressed in (5) (7) and(9) are taken into account which means the MTs onlygenerate electricity In Scenario Two the constraint in (8)is also used which shows the effects of heat load on thescheduling results In ScenarioThree besides the constraintsmentioned above the electricity power limit exchanged withthe main grid is considered (6) And in Scenario Four allthe constraints described in Section 2 are applied The lastpopulation and the feasible solutions in the objective spaceof the four scenarios utilizing the three algorithms are shownin Figures 4ndash7 respectively Eachmethodwill run 10 times forevery scenario and the best and average results are recordedThe rates of the feasible solutions are shown in Table 1 Thebest extreme feasible solutions for the two objectives usingthe three algorithms can be found in Table 2 And the averagefeasible results of the minimum cost and minimum emissionare shown in Table 3

Complexity 7

Dimensionality reduction to simplify the model

Start

Population initializationprocess

Partial repair process

GENcur = 0

Is the termination condition reached

Normalization and weighted sum process

Nondominated sorting individuals selection and genetic operation

Combination of the populations Nondominatedsorting and selection

End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi

Total repair process Partial repair process

No

Yes

Figure 2 Flowchart of HCHS-NSGAII

8 Complexity

Table 3 The average feasible results of the total cost and emission obtained by the three methods

Scenarios HCHS-NSGAII S-NSGAII PFM-NSGAIICost ($) Emission (kg) Cost ($) Emission (kg) Cost ($) Emission (kg)


Electricity loadHeat load

406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)

Figure 3 The heat load and electricity load curves

It can be seen from Figure 4 and Table 1 that when thereare only power output limits battery capacity limits andelectricity power balance constraint both HCHS-NSGAIIand S-NAGAII can solve the MGEED problem well whileHCHS-NAGAII obtains all the feasible solutions as its bestrecord PFM-NSGAII only gets 78 feasible solutions All ofthe three methods reached approximate Pareto fronts Theaverage amounts of feasible solutions are not much differentfrom their relative highest amounts which illustrates that thethree constraint handling methods have strong robustness indealing with the MGEED problem in Scenario One FromTables 2 and 3 it can be seen that the extreme solutionsobtained by the three algorithms are similar and PFM-NSGAII is even better in finding the minimum cost Thismeans that all of them can manage this MGEED problem

In Scenario Two another equality constraint is addedIt can be seen from Figures 5(a) and 5(b) that the solutionsby HCHS-NSGAII distribute uniformly and all of themare feasible at the best record However although some ofthe solutions by S-NSGAII can dominate those by HCHS-NSGAII (Figure 5(a)) they are actually infeasible And mostof the remaining solutions which are feasible are dominatedby those obtained by HCHS-NSGAII (Figure 5(b)) PFM-NSGAII obtains similar results as S-NSGAII while those byPFM-NSGAII are a little worse since it gets more infeasiblesolutions according to Figure 5 and Table 1 Tables 2 and 3also show that HCHS-NSGAII gains the best minimum costand emission values among the threemethods while the per-formances of S-NSGAII and PFM-NSGAII are getting worseespecially in the aspects of convergence and distributionThis implies that S-NSGAII and PFM-NSGAII cannot handlethe equality constraints violations well particularly when

there are more than one equality constraint However bysimplifying the optimization model with the dimensionalityreduction process HCHS-NSGAII loses much less feasiblesolutions than the other two methods

In Scenario Three where the MGEED problem becomesmore difficult by adding the exchanged electricity powerlimits 88 of the population by HCHS-NSGAII is still fea-sible within 500 generations according to the average resultin Table 1 whereas S-NSGAII and PFM-NSGAII just focuson part of the Pareto front according to Figure 6 and only16 and 12 of the populations are feasible This is becauseDebrsquos constraints handling criteria and the PFM cannot dealwith the different kinds of constraints simultaneously andmore infeasible solutions are kept by the elitist based strategyAs a result most of the solutions converge to an infeasiblezone As forHCHS-NSGAII the normalization andweightedsum process ensures that all of the constraints violations indifferent units are taken into account equally in case thatsome certain kinds of violations are ignored Therefore theapproximate Pareto front obtained by HCHS-NSGAII canhave good distribution and population diversity

In Scenario Four it can be seen from Figure 7 thatthe results of S-NSGAII and PFM-NSGAII are similar asthose in Scenario Three and only 5 and 1 of the pop-ulations are feasible according to the average results Theramp rates constraints violations are added directly in theoverall constraints by the above two constraints handlingapproaches Since the violations appear during the variablesgeneration process (which is a random and uncontrollableprocess) Debrsquos constraints handling criteria and the PFMcannot remove them completely Thus S-NSGAII and PFM-NSGAII cannot lead the algorithm to search a new directionto which the violations can be smallerTherefore they cannotdirectly converge to the feasible approximate Pareto frontFor HCHS-NSGAII according to Figure 7(b) and Table 1it can get 86 feasible solutions which is about 17 times and86 times the feasible solutions obtained by S-NSGAII andPFM-NSGAII respectively Obviously the proposed repairprocess has avoided the violations of ramp rates constraintsand ensured the population diversity

5 Conclusions

In this paper a hybrid constraints handling strategy basedon NSGAII is proposed for solving the MGEED with varioustypes of constraints In the HCHS-NSGAII framework thedimensionality reductionmethod is employed to simplify theoptimization model by transforming the equality constraints

Complexity 9

HCHS-NSGAIIS-NSGAIIPFM-NSGAII

0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)

(a) The last population


30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)

(b) The feasible solutions

Figure 4 The best optimization results by HCHS-NSGAII S-NSGAII and PFM-NSGAII in Scenario One


80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)


Figure 5 The best optimization results by HCHS-NSGAII S-NSGAII and PFM-NSGAII in Scenario Two

into inequality ones Meanwhile a repair process is suggestedto handle the ramp rates constraints after the generation ofnew individuals which is a dynamical process to ensure thepopulation diversity and the solutions feasibility In additionthe normalization and weighted sum approaches are intro-duced to balance the weights of different kinds of constraintsAnd then HCHS-NSGAII is applied to a series of MGEEDproblems with different combinations of MG constraintsand the results are compared with those obtained by S-NSGAII and PFM-NSGAIIThe results show that by utilizing

HCHS NSGAII can gain feasible Pareto sets with satisfactoryconvergence and distribution in different scenarios while thewidely used constraints handling methods lose the feasiblesolutions and fall into local optimum with the increase ofthe MOPs complexity It is evident that for a complicatedindustrial MGEED problem which may have various typesof constraints a hybrid constraints handling strategy is moreefficient than single methods And it is better to removethe violations of different constraints in several steps duringthe evolutionary process instead of converting them into

10 Complexity

HCHS-NSGAIIS-NSGAIIPFM-NSGA-II

720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)


Figure 6 The best optimization results by HCHS-NSGAII S-NSGAII and PFM-NSGAII in Scenario Three


880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)


Figure 7 The best optimization results by HCHS-NSGAII S-NSGAII and PFM-NSGAII in Scenario Four

an overall constraints violation Further studies are neededfor designing more efficient algorithms to solve the MGEEDproblems with the proposed HCHS

Nomenclature

MG MicrogridDERs Distributed energy resourcesCHP Combined heat and powerMT MicroturbineMOP Multiobjective optimization problemMGEED MG economicalenvironmental dispatchDG Distributed generator

DESS Distributed energy storage systemMOEA Multiobjective evolutionary

algorithmPFM Penalty function methodHCHS Hybrid constraints handling

strategyNSGAII Nondominated sorting genetic

algorithm IIPV PhotovoltaicWT Wind turbineFC Fuel cellPCC Point of common couplingSOC State of charge

Complexity 11

S-NSGAII Standard NSGAIIBAT Battery119862grid119905 Cost of the purchased power from

the main grid ($)120590119894120575119894 Hotcold start-up cost of the 119894thcontrollable DG ($)119879off 119894(119905) Time during which the 119894thcontrollable DG has been off at thebeginning of the 119905th schedulingperiod ($)120591119894 Cooling time constant of the 119894thcontrollable DG (s)119906119894(119905) Onoff status of the 119894th controllableDG119864119866119894119905119864119878119895119905 Emission of the 119894th controllableDG119895th DESS at time 119905 (kg)119864grid119905 Emission of the main grid at time 119905(kg)

PWgridminPWgridmax Maximum and minimum powerexchanged with the grid (kW)

PW119866119894119905 Power output of the powergenerators (kW)

PW119866119894minPW119866119894max Maximumminimum outputpower of the 119894th controllable DG(kW)120572119894 120573119894 120574119894 Emission coefficients119864(sdot) Total emission of the MG (kg)119862(sdot) Total operating cost of the MG ($)

X Decision variable vector119879 Number of the time intervals119873119892119873119904 Total amounts of the DGsDESSsCF119866119894119905 Fuel cost of the 119894th controllable DG

at time 119905 ($)V119897119896V119897119896norm Actualnormalized violation values

of the 119896th type of constraint in the119897th individualOM119866119894119905OM119878119895119905 Maintenance cost for the 119894th

DG119895th DESS at 119905 ($)119891119898(x(119894)) Objective function value of the 119894thindividual for the119898th objective119877119898 Penalty coefficient of the fitnessfunction for the119898th objective

V(x(119894)) Overall violation value of the 119894thindividual

PWgrid119905 Power exchanged with the maingrid (kW)119873119871 Number of electricity loaddemands

PW119878119895119905 The 119895th chargeddischarged powerof the DES- Ss at time 119905 (kW)119876ℎ119900119896119905 Quantity of the exhaust heat of the119896th MT at time 119905 (kW)119873119898 Number of the MTs in the MGsystem

SOC119905 Amount of stored energy insidethe BAT at time 119905 (Ah)

SOCminSOCmax Maximumminimum amount ofstored energy inside the BAT (Ah)

119875char(119905)119875dischar(119905) Chargingdischarging power of theBAT at time 119905 (kW)119875charmax119875discharmax Maximumminimumchargingdischarging power of theBAT at time 119905 (kW)120578char120578dischar Chargingdischarging efficienciesof the BATΔ119905 Time interval

PW119866119894ramp Ramp rate of the 119894th controllableDG (kW)

PW1198921(119905) Electricity power output of MT1(kW)119875HL119897(119905) Heat load demand (kW)119901repair(sdot) Repair probability

GENcurGENswi Currentswitch generationSTC119866119894119905 Start-up cost of the 119894th controllable

DG at time 119905 ($)V119896minV119896max Minimummaximum violation

values of the 119896th type of constraintV119897 Overall constraints violation of the119897th individual119908119896 Penalty weight of the 119896th type of

constraint119888 Number of the constraints types119891119898min119891119898max Minimummaximum value for the119898th objective

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 51707071) the NationalPostdoctoral Program for Innovative Talents of China (Grantno BX201600055) and the Postdoctoral Foundation of China(Grant no 2017M610475)

References

[1] C D Korkas S Baldi I Michailidis and E B KosmatopoulosldquoOccupancy-based demand response and thermal comfortoptimization in microgrids with renewable energy sources andenergy storagerdquo Applied Energy vol 163 pp 93ndash104 2016

[2] G Liu Y Xu and K Tomsovic ldquoBidding strategy for microgridin day-ahead market based on hybrid stochasticrobust opti-mizationrdquo IEEE Transactions on Smart Grid vol 7 no 1 pp227ndash237 2016

[3] O Abedinia A Ghasemi and N Ojaroudi ldquoImproved timevarying inertia weight PSO for solved economic load dispatchwith subsidies and wind power effectsrdquo Complexity vol 21 no4 pp 40ndash49 2015

[4] N Yousefi ldquoSolving nonconvex economic load dispatch prob-lem using particle swarm optimization with time varyingacceleration coefficientsrdquoComplexity vol 21 no 6 pp 299ndash3082016

[5] Y Zhu F Zhuo F Wang B Liu R Gou and Y Zhao ldquoA virtualimpedance optimization method for reactive power sharing in

12 Complexity

networked microgridrdquo IEEE Transactions on Power Electronicsvol 31 no 4 pp 2890ndash2904 2016

[6] A K Basu S P Chowdhury S Chowdhury and S PaulldquoMicrogrids energy management by strategic deployment ofDERsmdasha comprehensive surveyrdquo Renewable amp SustainableEnergy Reviews vol 15 no 9 pp 4348ndash4356 2011

[7] R Morsali M Mohammadi I Maleksaeedi and N GhadimildquoA new multiobjective procedure for solving nonconvex envi-ronmentaleconomic power dispatchrdquo Complexity vol 20 no2 pp 47ndash62 2014

[8] B Zhao Y Yang X Zhang et al ldquoImplementation of a dual-microgrid system with flexible configurations and hierarchicalcontrol in ChinardquoRenewable amp Sustainable Energy Reviews vol65 pp 113ndash123 2016

[9] T Niknam F Golestaneh and A Malekpour ldquoProbabilisticenergy and operation management of a microgrid contain-ing windphotovoltaicfuel cell generation and energy storagedevices based on point estimate method and self-adaptivegravitational search algorithmrdquo Energy vol 43 no 1 pp 427ndash437 2012

[10] A A Moghaddam A Seifi T Niknam and M R AlizadehPahlavani ldquoMulti-objective operation management of arenewable MG (micro-grid) with back-up micro-turbinefuelcellbattery hybrid power sourcerdquo Energy vol 36 no 11 pp6490ndash6507 2011

[11] A G Tsikalakis and N D Hatziargyriou ldquoCentralized controlfor optimizing microgrids operationrdquo IEEE Transactions onEnergy Conversion vol 23 no 1 pp 241ndash248 2008

[12] K Buayai W Ongsakul and N Mithulananthan ldquoMulti-objective micro-grid planning by NSGA-II in primary distri-bution systemrdquo European Transactions on Electrical Power vol22 no 2 pp 170ndash187 2012

[13] E R Sanseverino M L Di Silvestre M G Ippolito A DePaola and G Lo Re ldquoAn execution monitoring and replanningapproach for optimal energy management in microgridsrdquoEnergy vol 36 no 5 pp 3429ndash3436 2011

[14] M Elsied A Oukaour H Gualous and O A Lo BruttoldquoOptimal economic and environment operation of micro-gridpower systemsrdquo Energy Conversion and Management vol 122pp 182ndash194 2016

[15] B Zhao X Zhang J Chen C Wang and L Guo ldquoOperationoptimization of standalone microgrids considering lifetimecharacteristics of battery energy storage systemrdquo IEEE Trans-actions on Sustainable Energy vol 4 no 4 pp 934ndash943 2013

[16] A Soroudi M Aien and M Ehsan ldquoA probabilistic modelingof photo voltaic modules and wind power generation impact ondistribution networksrdquo IEEE Systems Journal vol 6 no 2 pp254ndash259 2012

[17] J Lai H Zhou X Lu X Yu and W Hu ldquoDroop-based dis-tributed cooperative control for microgrids with time-varyingdelaysrdquo IEEE Transactions on Smart Grid vol 7 no 4 pp 1775ndash1789 2016

[18] F J Rooijers and R A M van Amerongen ldquoStatic economicdispatch for co-generation systemsrdquo IEEE Transactions onPower Systems vol 9 no 3 pp 1392ndash1398 1994

[19] F A Mohamed and H N Koivo ldquoOnline management geneticalgorithms of microgrid for residential applicationrdquo EnergyConversion and Management vol 64 pp 562ndash568 2012

[20] C M Colson M H Nehrir and S A Pourmousavi ldquoTowardsreal-time microgrid power management using computationalintelligence methodsrdquo in Proceedings of the IEEE Power and

Energy Society General Meeting (PES rsquo10) pp 1ndash8 IEEE Min-neapolis Minn USA July 2010

[21] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[22] L Li Study of Economic Operation in Microgrid North ChinaElectric Power University Beijing China 2011

[23] X Li K Deb and Y Fang ldquoA derived heuristics based multi-objective optimization procedure for micro-grid schedulingrdquoEngineering Optimization vol 49 no 6 pp 1078ndash1096 2017

[24] X Li and Y Fang ldquoDynamic EnvironmentalEconomicScheduling for Microgrid Using Improved MOEAD-M2MrdquoMathematical Problems in Engineering vol 2016 Article ID2167153 2016

[25] B Zhao Y Shi X Dong W Luan and J Bornemann ldquoShort-term operation scheduling in renewable-powered microgridsA duality-based approachrdquo IEEE Transactions on SustainableEnergy vol 5 no 1 pp 209ndash217 2014

[26] X Lu X Yu J Lai J M Guerrero and H Zhou ldquoDistributedSecondary Voltage and Frequency Control for Islanded Micro-grids with Uncertain Communication Linksrdquo IEEE Transac-tions on Industrial Informatics vol PP no 99 2016

[27] X Lu X Yu J Lai Y Wang and J M Guerrero ldquoA NovelDistributed Secondary Coordination Control Approach forIslanded Microgridsrdquo IEEE Transactions on Smart Grid no 99pp 1-1

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

multiobjective evolutionary algorithms (MOEAs) have beenintroduced to deal with theMGEEDproblems which ismoreefficient for solving nonconvex objective functions and moreflexible in handling the constraints [10 18 20] Howeverresearchersmainly focused on the accuracy and the efficiencyof algorithms instead of constraints handling strategiesPenalty functionmethod (PFM) which is a common simpleand efficient constraint handling method for constrainedoptimization problems is generally used in MOEAs In[19] the PEM was applied to convert the constrained MGcost optimization problem into an unconstrained one bymodifying the objective function with related penalty itemsThe simulation results showed that PEM could help theoptimization approach to find satisfied feasible solutionsHowever in that study the amount of the constraints wasnot large and the economical and environmental objectiveswere combined as only one optimization task In [21] theauthors introduced a method to handle the constraints inmultiobjective problems taking account of both feasibilityand domination which is called ldquoDebrsquos constraints handlingcriteriardquo It was used in some of the studies [13 22ndash24] andthe results showed that by using thismethod considering bothfeasibility and domination the proportion of the infeasiblesolutions could be reduced evidently [13 23] However thismethod needed to add the overall constraints which wereactually of different types and could not be easily quantifiedIn addition [20] proposed two frameworks based onMOEAsto solve the multiconstrained MGEED problems and bothof the strategies obtained the feasible Pareto sets Howeverthe authors only considered some of the common usedconstraints and the constraint handling approaches werenot customized which may reduce the efficiency when theoptimization frameworks were applied to other MGEEDproblems with different combinations of constraints

From the above researches it is obvious that the con-straints handling problems ofMGEEDhave not been system-atically studied Most of the studies utilized a general methodto deal with the violations However as a practical MOP theMGEED problems have serious challenges to the MOEAs forthe following reasons(1) MGEED problems have various types of constraintssuch as the equality constraints and inequality constraints(2) The dimension of the search space is high Sincealmost every variable in a solution should meet at leastone constraint the amount of the constraints are alwaysvery large which may make the feasible region in unevendistribution(3) When using MOEA as the optimization tool thegeneration of constraints violations may appear in differentsteps of the optimization process(4) As a practical MOP the combinations of the con-straints may change in different scenarios

Considering the above challenges the single and staticconstraint handling strategies may not always adapt to theMGEED problems especially when the constraints are com-plexTherefore it is necessary to study the hybrid approachesin dealing with all kinds of MG constraints

In remainder of this paper the system models andobjective functions are established in Section 2 The hybrid

constraints handling strategy (HCHS) based on nondom-inated sorting genetic algorithm II (NSGAII) is designedin Section 3 Thereafter the HCHS-NSGAII is applied toseveral practical MGEED problems with different constraintscombinations to study the efficiency of the proposed hybridconstraints handling strategy in Section 4 Finally the con-clusions of the study are presented

2 Modeling of MGEED

21 MG System Description TheMGEED problem is to allo-cate the output of every distributed component to meet thepredicted electricity and heat load demand without prejudiceto any of the constraints through the whole 24-hour processwhile maximizing the financial and ecological benefit of theMG system In this section the proposed objective functionsand constraints are discussedThe structure of the typicalMGused in this paper is shown in Figure 1 It can be seen thattwo kinds of uncontrollable DGs namely photovoltaic cells(PVs) and wind turbines (WTs) are considered of which themathematical models and related parameters are describedin [22 24] Two microturbines (MTs) and two fuel cells(FCs) with different parameter settings are considered as thecontrollableDGs to supply electricity powerAbattery bank isalso included as theDESS Besides the power exchangedwiththemain grid is consideredwhen theMG system turns to gridconnected mode by the PCC (point of common coupling)

22 Objective Functions In this paper the total operatingcost is considered as one of the objectives to be minimizedwhich contains the fuel cost the start-up cost the mainte-nance cost and the outcome by introducing the power fromthemain gridThus the objective function forminimum costcan be described below

min119862 (X)= 119879sum119905=1

119873119892sum119894=1

(CF119866119894119905 (X) + STC119866119894119905 (X) +OM119866119894119905 (X))

+ 119873119904sum119895=1

(OM119878119895119905 (X)) + 119862grid119905 (X)

(1)

where X is the decision variable vector 119879 is the number ofthe time intervals 119873119892 and 119873119904 are the total amounts of theDGs and DESSs respectively CF119866119894119905 and STC119866119894119905 are the fuelcost and the start-up cost of the 119894th controllable DG at time119905 respectively OM119866119894119905 and OM119878119895119905 are the maintenance costsfor the 119894th controllable DG and the 119895thDESS at time 119905 respec-tively119862grid119905 is the cost of the purchased power from themaingrid The model functions of CF119866119894119905 OM119866119894119905 OM119878119895119905 119862grid119905can be found in [22 23] and the start-up cost function isdescribed below

STC119866119894119905 = 120590119894 + 120575119894 [1 minus exp(minus119879off 119894119905120591119894 )] (1 minus 119906119894 (119905)) (2)

where 120590119894 and 120575119894 are the hot start-up cost and cold start-upcost of the 119894th controllable DG respectively 119879off 119894(119905) is the

Complexity 3

Circuit breaker

Circuit breaker

PV

MT

Circuit breaker

Circuit breaker

BAT

MT

WT

Circuit breaker

Sensitive load

Interruptible load

Adjustable load

Heat load

PCC

Feeder A

Feeder B

Feeder C

FC

Main grid




min119864 (X) = 119879sum119905=1

119873119892sum119894=1

119864119866119894119905 +119873119904sum119895=1

119864119878119895119905 + 119864grid119905 (3)


119864119866119894119905 = 120572119894 sdot PW1198661198941199052 + 120573119894 sdot PW119866119894119905 + 120574119894 (4)










119873119892sum119894=1

PW119866119894119905 +119873119904sum119895=1

PW119878119895119905 + 119875grid (119905) minus119873119871sum119897=1

119875119864119871119897 (119905) = 0 (7)


119873119898sum119896=1

119876ℎ119900119896119905 + 119875HL119897 (119905) = 0 (8)





119875char (119905) le 119875charmax


4 Complexity






3 HCHS-NSGAII








119891119896 (PW119892119896 (119905))) (13)


119875grid (119905)= 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905

(14)


PWgridmin

le 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905 le PWgridmax

(15)




Complexity 5





endendreturn PW1015840119866119894119905





= (GENcurGENswi

)2 GENcur le GENswi











6 Complexity















119865119898 (x(119894)) = 119891119898 (x(119894)) + 119866119898 (x(119894)) (19)




Complexity 7


Start



GENcur = 0





End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi


No

Yes


8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3

Circuit breaker

Circuit breaker

PV

MT

Circuit breaker

Circuit breaker

BAT

MT

WT

Circuit breaker

Sensitive load

Interruptible load

Adjustable load

Heat load

PCC

Feeder A

Feeder B

Feeder C

FC

Main grid




min119864 (X) = 119879sum119905=1

119873119892sum119894=1

119864119866119894119905 +119873119904sum119895=1

119864119878119895119905 + 119864grid119905 (3)


119864119866119894119905 = 120572119894 sdot PW1198661198941199052 + 120573119894 sdot PW119866119894119905 + 120574119894 (4)










119873119892sum119894=1

PW119866119894119905 +119873119904sum119895=1

PW119878119895119905 + 119875grid (119905) minus119873119871sum119897=1

119875119864119871119897 (119905) = 0 (7)


119873119898sum119896=1

119876ℎ119900119896119905 + 119875HL119897 (119905) = 0 (8)





119875char (119905) le 119875charmax


4 Complexity






3 HCHS-NSGAII








119891119896 (PW119892119896 (119905))) (13)


119875grid (119905)= 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905

(14)


PWgridmin

le 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905 le PWgridmax

(15)




Complexity 5





endendreturn PW1015840119866119894119905





= (GENcurGENswi

)2 GENcur le GENswi











6 Complexity















119865119898 (x(119894)) = 119891119898 (x(119894)) + 119866119898 (x(119894)) (19)




Complexity 7


Start



GENcur = 0





End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi


No

Yes


8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity






3 HCHS-NSGAII








119891119896 (PW119892119896 (119905))) (13)


119875grid (119905)= 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905

(14)


PWgridmin

le 119875119864119871 (119905)minus 119873119892sum119894=2


119891119896 (PW119892119896 (119905)))

+ 119873119904sum119895=1

PW119878119895119905 le PWgridmax

(15)




Complexity 5





endendreturn PW1015840119866119894119905





= (GENcurGENswi

)2 GENcur le GENswi











6 Complexity















119865119898 (x(119894)) = 119891119898 (x(119894)) + 119866119898 (x(119894)) (19)




Complexity 7


Start



GENcur = 0





End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi


No

Yes


8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5





endendreturn PW1015840119866119894119905





= (GENcurGENswi

)2 GENcur le GENswi











6 Complexity















119865119898 (x(119894)) = 119891119898 (x(119894)) + 119866119898 (x(119894)) (19)




Complexity 7


Start



GENcur = 0





End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi


No

Yes


8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity















119865119898 (x(119894)) = 119891119898 (x(119894)) + 119866119898 (x(119894)) (19)




Complexity 7


Start



GENcur = 0





End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi


No

Yes


8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7


Start



GENcur = 0





End

GENcur = GENcur + 1

Yes

No

GENcur gt GENswi


No

Yes


8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity





406080

100120140160180

Load

(kW

)

6 12 18 240 Time (h)







5 Conclusions


Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9


0

50

100

150

200

250

300Em

issio

n (k

g)

30 40 50 60 70 80 90 10020Overall cost ($)



30 40 50 60 70 80 90 10020Overall cost ($)

0

50

100

150

200

250

300

Emiss

ion

(kg)




80 100 120 140 160 180 20060Overall cost ($)

250

300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)



300

350

400

450

500

550

600

650

700

Emiss

ion

(kg)

100 120 140 160 180 20080Overall cost ($)





10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity


720

740

760

780

800

820

840

860

880Em

issio

n (k

g)

120 140 160 180 200 220100Overall cost ($)



720

740

760

780

800

820

840

860

880

Emiss

ion

(kg)

130 140 150 160 170 180 190 200 210120Overall cost ($)




880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

180 190 200 210 220 230 240 250170Overall cost ($)



880

900

920

940

960

980

1000

1020

Emiss

ion

(kg)

190 200 210 220 230 240 250180Overall cost ($)




Nomenclature






Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11
























Acknowledgments


References






12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




12 Complexity
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




a hybrid constraints handling strategy for...

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