reserve constrained multi-area economic dispatch employing differential evolution with time-varying...

Electrical Power and Energy Systems 33 (2011) 753–766

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Electrical Power and Energy Systems

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Reserve constrained multi-area economic dispatch employing differentialevolution with time-varying mutation

Manisha Sharma, Manjaree Pandit ⇑, Laxmi SrivastavaDepartment of Electrical Engineering, M.I.T.S., Gwalior, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 June 2010Received in revised form 28 October 2010Accepted 9 December 2010Available online 3 February 2011

Keywords:Differential evolutionReserve constrained multi-area economicdispatchMutation strategies

0142-0615/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.12.033

⇑ Corresponding author. Tel.: +91 0751 2665962; faE-mail address: [email protected] (M. Pan

For a power pool that involves several generation areas interconnected by tie-lines, the objective of eco-nomic dispatch (ED) is to determine the most economical generation dispatch strategy that could supplythe area load demands without violating the tie-line capacity constraints. The objective of multi-area eco-nomic dispatch (MAED) is to determine the generation levels and the interchange power between areaswhich would minimize total fuel cost while satisfying power balance constraint, upper/lower generationlimits, ramp rate limits, transmission constraints and other practical constraints. In reserve constrainedMAED (RCMAED) problem inter-area reserve sharing can help in reducing the operational cost whileensuring that spinning reserve requirements in each area are satisfied. The tie-line limits too play a piv-otal role in optimizing the cost of operation. The cost curves of modern generating units are discontinu-ous and non-convex which necessitates the use of powerful heuristic search based methods that arecapable of locating global solutions effectively, with ease. This paper explores and compares the perfor-mance of various differential evolution (DE) strategies enhanced with time-varying mutation to solve thereserve constrained MAED (RCMAED) problem.

The performance is tested on (i) two-area, four generating unit system, (ii) four area, 16-unit systemand (iii) two-area, 40-unit system. The results are found to be superior compared to some recently pub-lished results.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In the power sector, economic dispatch (ED) is used to allocatepower demand among available generators in the most economicalmanner, while satisfying all the physical and operational con-straints. The cost of power generation, particularly in fossil fuelplants, is very high and economic dispatch helps in saving a signif-icant amount of revenue [1]. Although, ED of a single area has beenstudied extensively, multi-area economic dispatch has receivedlimited attention. Many utilities and power pools have limits onpower flow between different areas/regions over tie-lines. Eacharea/region has its own pattern of load variation and generationcharacteristics. They also have separate spinning reserve con-straints. The objective of RCMAED is to determine the generationlevels and the interchange power between areas which would min-imize fuel costs in all areas while satisfying area-wise power bal-ance, upper/lower generation limits, area-wise spinning reserverequirements and transmission constraints. Power utilities try toachieve high operating efficiency to produce cheap electricity. Inthe present competitive power market the operating cost of a

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power pool can be reduced if the areas with more economic unitsgenerate larger power than their load and export the surpluspower to other areas having more expensive units. The benefitsthus gained will depend on several factors like the characteristicsof a pool, the policies adopted by utilities, types of interconnec-tions, tie-line limits, spinning reserve constraints and load distri-bution in different areas. The present paper focuses on theimportance of transmission capacity constraints and area reserveconstraints in the optimal scheduling of generating units in electricpower systems.

Multi-area generation scheduling with import/export con-straints between areas was presented in Ref. [2]. Transmissionconstraints with linear losses were considered in Ref. [3] while solv-ing the multi-area economic dispatch problem by spatial dynamicprogramming. Desell et al. [4] proposed an application of linear pro-gramming to transmission constrained production cost analysis.Farmer et al. [5] presented a probabilistic approach for transmissionconstrained multi-area power systems. MAED was solved with areacontrol error in Ref. [6]. A heuristic multi-area unit commitmentwith ED was proposed in Ref. [7]. Wang and Shahidehpour [8]proposed a decomposition approach using expert systems for non-linear multi-area generation scheduling. The Newton–Rapshon’smethod was applied to solve multi-area economic dispatch problem[9]. An incremental network flow programming algorithm was

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754 M. Sharma et al. / Electrical Power and Energy Systems 33 (2011) 753–766

proposed for the MAED solution with tie-line, area reserve and areademand constraints [10]. Yalcinoz and Short proposed a Hopfieldneural network (HNN) to solve the MAED problem [11]. The MAEDis solved by direct search method, considering transmission con-straints in Ref. [12]. Evolutionary programming is proposed in [13]for multi-area economic dispatch problem. Recently, covariance ma-trix adapted evolutionary strategy has been proposed for MAEDproblems where a Karush Kuhun Tucker (KKT) optimality criterionis applied to guarantee optimal convergence [14].

As compared to the simple ED problem, the RCMAED problem isquite complex due to the stringent area balance constraints, tie-line constraints and area reserve constraints in addition to theother operational constraints. Lee and Breipohl [15] applied adecomposition technique to solve the reserve constrained ED prob-lem with prohibited operating zones (POZ). A hybrid techniquecombining particle swarm optimization (PSO) and sequential qua-dratic programming (SQP) was used in Ref. [16] to model the re-serve constrained dynamic dispatch problem. Wang and Singh[17] have recently proposed a PSO based approach for the reserveconstrained multi-area environmental/economic dispatch prob-lem. A constrained PSO approach is proposed to deal with both en-ergy and reserve allocation in multi-area electricity marketdispatch problem [18]. However, the solution of MAED problemwith valve point loading (VPL) effects, prohibited operating zonesand ramp rate limits has not yet been reported.

Evolutionary optimization approaches are increasingly beingproposed for ED problems having non-differentiable cost functionsdue to their simplicity, the absence of convexity assumptions, andexcellent random parallel search capability. Some of the methodsfound in literature include tabu search, simulating annealing, neu-ral networks [11,19], genetic algorithm [20], particle swarm opti-mization [16–18,21–25], harmony search [26], ant colonyoptimization [27], bacterial foraging [28], artificial immune sys-tem[29] and differential evolution (DE) [30–34]. Ref. [24] providesan extensive review of these heuristic optimization techniques forsolving ED problems. It can be observed that PSO, DE and theirvariants have been more popular due to their reliability, robust-ness, convergence speed, minimum information requirement andease of implementation.

The major challenge faced by evolutionary techniques is tomaintain a proper balance between exploration (global search)and exploitation (local search). Recent research has identified cer-tain issues which need to be tackled such as (i) premature conver-gence and (ii) finding optimal tuning parameters, to improve theperformance of the heuristic search based evolutionary optimiza-tion methods. These limitations are tackled by focusing on (i) var-ious parameter automation strategies and (ii) hybridization ofglobal and local algorithms. Time-varying acceleration coefficients(TVAC) [21] were employed for iteratively controlling the globaland local search components in PSO for solving non-convex EDproblems. Crazy particles were employed in the PSO algorithm[16,22] for stopping premature convergence by increasing popula-tion diversity. In Ref. [25] performance improvement for non-con-vex economic dispatch problem was reported by integrating thePSO with chaotic sequences and crossover operation. The conceptof variable scaling factor based on the one-fifth success rule of evo-lutionary strategies is employed in Ref. [33]. Similarly, hybrid tech-niques have also been successfully employed for maintainingmomentum in the optimization process [30]. In Ref. [31] chaoticDE was combined with quadratic programming to get enhancedperformance. Cultural algorithm has been used to model self adap-tation in DE to get global convergence [34].

This paper focuses on the potential of different DE variants im-proved with time-varying mutation in producing feasible solutionsfor the RCMAED problem formulated with various constraints. Thepaper also compares the solution quality of DE variants with an im-

proved PSO strategy to bring out the similarities and differences inboth these powerful and popular approaches. The results of all DEbased evolutionary strategies are found to be feasible and compa-rable/superior to previously reported results [11,12,14,17].

2. Reserve constrained multi-area economic dispatch

The objective of MAED is to determine the generation levels andthe interchange power between areas to minimize the total fuelcosts in all areas while satisfying power balance, generating limitand transmission capacity constraints. The valve-point effectsintroduce ripples in the heat-rate curves and make the objectivefunction discontinuous, non-convex and with multiple minima.For an accurate modeling of VPL effects, a rectified sinusoidal func-tion [20] is added in the fuel input-power output cost function ofthe ith unit as given below.

FiðPiÞ ¼ aiP2i þ biPi þ ci þ jei � sinðfi � ðPmin

i � PiÞÞj ð1Þ

where Pi is the power generation of ith unit, ai, bi and ci are the fuelcost-coefficients, and ei and fi are the fuel cost-coefficients of the ithunit to model VPL effects. Tie-line power flow between areas plays avery important role in deciding the operating cost in multi-areapower systems. Taking into consideration the cost of transmissionthough each tie-line, the objective function of MAED is given inthe following equation as

Min FT ¼XN

i¼1

FiðPiÞ þXM

j

fjðTjÞ ð2Þ

Here fj is the cost function associated with jth tie-line power flow Tj.There are N number of generating units and M number of tie-lines.

2.1. Area power balance constraints

In MAED problem the power balance constraints need to be sat-isfied for each area. The power balance constraints for area qneglecting losses can be given as

XNq

i¼1

Piq ¼ PDq þXMq

j

Tjq

!¼ 0 ð3Þ

for q = 1,2, . . . , M (areas). For the qth area, PDq is the load demand,Tjq is the tie-line flows from other areas, Nq are number of generat-ing units and Mq represents the number of tie-lines connected tothe qth area.

2.2. Generating limit constraints

The power output of a unit must be allocated within the rangebounded by its lower and upper limits of real power generation asgiven by

Pmini 6 Pi 6 Pmax

i i ¼ 1;2; . . . ;N ð4Þ

2.3. Generator ramp rate limits

When the generator ramp rate limits are considered, the oper-ating limits given in (4) are modified as follows:

MaxðPmini ; Po

i � DRiÞ 6 Pi 6MinðPmaxi ; Po

i þ URiÞ ð5Þ

where Poi indicates the active power output of the ith unit in the

previous hour; DRi and URi are the down and up ramp rate limitsrespectively.

M. Sharma et al. / Electrical Power and Energy Systems 33 (2011) 753–766 755

2.4. Constraints due to prohibited operating zones

Practical generating units have prohibited operating zones dueto physical operational limitations such as faults in the machinesthemselves or the auxiliary equipment, such as boiler and feedpumps [15]. A unit that operates in these zones may experienceamplification of vibrations in its shaft bearing. To avoid operationin these zones, the unit has discontinuous input–output character-istics given by:

Pi 2

Pmini 6 Pi 6 PL

i1

. . .

PUik�1 6 Pi 6 PL

ik

. . .

PUizi 6 Pi 6 Pmax

i

8>>>>>><>>>>>>:

ð6Þ

Here zi are the number of prohibited zones in ith generator curve, kis the index of prohibited zone of ith generator, PL

ik is the lower limitand PU

ik is the upper limit of kth prohibited zone of the ith generator.

2.5. Tie-line limit constraints

The tie-line flows to the qth area should be between the maxi-mum and minimum limits of tie-line flow Tjq.

Tminjq 6 Tjq 6 Tmax

jq j ¼ 1;2; . . . ;Mq ð7Þ

2.6. Area-wise spinning reserve constraints

In MAED problems operating cost can be reduced if the spinningreserve requirement of an area j can be satisfied through multi-area reserve sharing:

XNq

i¼1

Siq P Sq;req þX

k

Reskq ð8Þ

where Nq is the number of generating units in the qth area; Siq is thereserve existing on the ith unit of qth area, and can be calculated as(Pmax

i � Pi), Sq,req is the specified/required spinning reserve in the qtharea and Reskq is the reserve contributed from area k to area q. Wheninter-area aid is not available then Reskq is zero and each area gen-eration has to meet its own reserve. For reserve sharing option thereis no area-wise reserve constraint, only a total reserve requirementwhich is to be met jointly by all areas.

3. PSO and DE based evolutionary optimization strategies

Nature inspired evolutionary soft computing methods areextensively being proposed world-wide for solving complex powersystem optimization problems due to their reliable performanceand ease of implementation. Looking at the number of papers pub-lished during the past few years [16–18,21–25,30–34], it can beconcluded that PSO and DE have so far attracted more attentioncompared to other evolutionary methods due to their parallel ran-dom search techniques suitable for large dimension, heavily con-strained, non-linear, non-convex and multimodal optimizationproblems. As the problem dimension and complexity increases,the classical PSO and DE methods get stuck to some sub-optimalsolution due to their fast convergence property. It becomes essen-tial to allow increased exploration during the initial iterations topermit global search. During the later part of the search however,the population needs to be guided towards the global minima byincreasing its local exploitation capability. This paper aims to pres-ent a brief review and comparison of both PSO and DE variantsusing performance measures such as convergence behavior,

robustness and solution quality for solving the reserve constrainedMAED problem.

3.1. Classical PSO

A PSO proposed by Kennedy and Eberhart [35] is a populationbased modern heuristic search method that operates in parallelwith a group of particles representing the solutions of the problem.The particles update their positions using their own experienceand the experience of their neighbors. The update mode is termedas the velocity of particles. The position and velocity vectors of theith particle of a d-dimensional search space can be represented asXi ¼ ðxi1; xi2; . . . ; xidÞ and Vi ¼ ðv i1;v i2; . . . ;v idÞ respectively. The bestprevious position of a particle is recorded and represented aspbesti ¼ ðpi1; pi2 . . . ; pidÞ. The best particle in the group so far, is rep-resented as pbestg ¼ gbest ¼ ðpg1; pg2; . . . ; pgdÞ. The particle tries tomodify its position using the current velocity and the distancefrom pbest and gbest. The modified velocity and position are givenas follows:

vkþ1id ¼ C½w� vk

id þ c1 � rand1 � ðpbestid � xidÞ þ c2 � rand2

� ðgbestgd � xidÞ� ð9Þ

xkþ1id ¼ xid þ vkþ1

id ð10Þ

Here w is the inertia weight parameter, C is constriction factor, c1, c2

are cognitive and social coefficients, and rand1, rand2 are randomnumbers between 0 and 1. A large inertia weight helps in good glo-bal search while a smaller value facilitates local exploration. There-fore, the time-varying inertial weight is used, given by

w ¼ ðwmax �wminÞ �ðitermax � iterÞ

itermaxþwmin ð11Þ

where iter is the current iteration number while itermax is the max-imum number of iterations. Usually the value of w is varied be-tween 0.9 and 0.4.

3.2. PSO with time-varying acceleration coefficients (PSO_TVAC)

The ability of the PSO technique to fine tune the optimum solu-tion is weak due to the lack of diversity at the end of the search.Therefore, iterative tuning of its coefficients is done in such a man-ner that the cognitive component is reduced while the social com-ponent is increased as the search proceeds. With a large cognitivecomponent and small social component at the beginning, particlesare allowed to roam around the search space instead of moving to-ward the population best during early stages. On the other hand, asmall cognitive component and a large social component allow theparticles to converge to the global optima in the latter part of theoptimization process. The acceleration coefficients are expressedas [36]:

c1 ¼ ðc1f � c1iÞiter

itermaxþ c1i ð12Þ

c2 ¼ ðc2f � c2iÞiter

itermaxþ c2i ð13Þ

where c1i, c1f, c2i and c2f are initial and final values of cognitive andsocial acceleration factors respectively.

3.3. Classical differential evolution

DE is a population based stochastic function optimizer proposedby Storn and Price [37] which works on three basic operations,namely mutation, crossover and selection. DE differs from conven-tional genetic algorithms (i) as mutation is applied first to generate


a trial vector which is then used within the crossover mechanismto produce new offspring and (ii) mutation step sizes are not sam-pled from a known probability distribution function as in GA, butdepend on the difference between individuals of the currentpopulation.

3.3.1. Mutation operationDifferent DE strategies differ in the manner the donor or mutant

vector is formed. In mutation operation a mutant vector is pro-duced for each individual of the current population by mutatinga target vector with a weighted differential. The different variantsof DE are classified using the general notation DE, /a/b/d where arefers to the method for selecting the target individual xi whichwill form the base of the mutated vector, b indicates the numberof difference vectors used to perturb the target vector, and d indi-cates the crossover mechanism used to create the offspring popu-lation [37]. Most papers [34] have explored the variant DE/rand/1/bin, which mean random selection, one difference vector and bino-mial crossover. The best performing variant is found to be problemspecific and needs detailed investigation. The donor or mutant vec-tor, Zi for each target member from the population, is generated fordifferent variants in classic DE as given below [37].

(1) Strategy I: DE/best/1:

Ziðt þ 1Þ ¼ xi;r1ðtÞ þ fm½xi;r2ðtÞ � xi;r3ðtÞ�

(2) Strategy II: DE/rand/1:

Ziðt þ 1Þ ¼ xi;bestðtÞ þ fm½xi;r2ðtÞ � xi;r3ðtÞ�

(3) Strategy III:DE/rand-to-best/1:

Ziðt þ 1Þ ¼ xiðtÞ þ fm½xi;bestðtÞ � xiðtÞ� þ fm½xir1ðtÞ � xir2ðtÞ�

(4) Strategy IV:DE/best/2:

Ziðt þ 1Þ ¼ xibestðtÞ þ fm½xi;r1ðtÞ � xir2ðtÞ� þ fm½xir3ðtÞ � xir4ðtÞ�

(5) Strategy V: DE/rand/2:

Ziðt þ 1Þ ¼ xir5ðtÞ þ fm½xi;r1ðtÞ � xir2ðtÞ� þ fm½xir3ðtÞ � xir4ðtÞ� ð14Þ

where i = 1, 2, . . . , R is the individual’s index of population; t repre-sents iteration count; the target vector is denoted by xi while r1, r2,r3, r4 and r5 are mutually different integers and also different fromthe running index i. These integers are randomly selected with uni-form distribution from the population set and fm is a real parametercalled mutation factor, which controls the amplification of the dif-ferential variation between two individuals to avoid search satura-tion. The mutation factor is usually taken between (0–2). In thispaper, a detailed study of mutation strategy I to strategy V is carriedout to find the best strategy for a given MAED problem with reserveconstraints.

3.3.2. Crossover or recombination operationFor increasing population diversity of the perturbed parameter

vector, crossover operation is introduced after the mutation oper-ation. Recombination is employed to generate a trial vector Ui byreplacing certain parameters of the target vector (xi) with the cor-responding parameters of the randomly generated mutant vector(Zi).

Uijðt þ 1Þ ¼Zijðt þ 1Þ if ðrandðjÞ 6 CRÞ or ðj ¼ rand intðiÞÞxijðtÞ if ðrandðjÞ > CRÞ or ðj – rand intðiÞÞ

�ð15Þ

Here, j = 1, 2, . . . , N is the position in n-dimensional individual;rand(j) is a uniform random number within range [0, 1] for the jthdimension variable, and CR is a crossover or recombination rate in

the range [0, 1]. The performance of a DE algorithm usually dependson three variables; the population size N, the mutation factor fm andthe recombination rate CR.

3.3.3. SelectionSelection is the procedure of producing better offspring. To de-

cide whether the trial vector (Ui) should be included in the popu-lation in the next generation, it is compared with the corresponding target vector xi using the greedy criterion. The evaluationfunction f(Ui(t + 1)) of each trial vector is compared with the eval-uation function f(xi(t) of its parent target vector xi(t) and selectionfor a minimization problem can be carried out using the followingcriterion:

xiðt þ 1Þ ¼Uiðt þ 1Þ if ðf ðUiðt þ 1ÞÞ < f ðxiðtÞÞxiðtÞ otherwise

�ð16Þ

3.4. Time-varying chaotic differential evolution

Chaos describes the complex behavior of a non-linear, deter-ministic, dynamical system which is highly sensitive to initial con-ditions [38]. The application of chaotic sequences in place ofrandom sequences in evolutionary optimization methods hasproved to be a promising strategy to diversify population and im-prove performance by preventing premature convergence to localminima [31,39,40]. Coelho and Mariani [31] integrated chaotic se-quences with the mutation factor in differential evolution to im-prove solution quality. Caponetto et al. [39] proposed variouschaotic sequences in evolutionary algorithms (EAs) in place ofthe random numbers. Shengsong et al. [40] hybridized chaoticoptimization with linear programming (LP) to solve the complexoptimal power flow problems.

In the DE algorithm, mutation rate fm and crossover rate CR sig-nificantly affect the performance of the algorithm. The smaller themutation rate fm, longer time will be required for convergence. Lar-ger values of fm facilitate exploration, but may cause the algorithmto overshoot good optimal solution. Actually, the value of fm shouldbe small enough to enable the algorithm to explore tight valleysand large enough to allow global exploration in order to maintaindiversity. Similarly, a higher CR creates more diversity and betterexploration in the new population. In classical DE both fm and CRare fixed, so a lot of parameter tuning is required to achieve globalbest results. More so, because the selection of optimal tuningparameters is found to depend on population size too [32,41]. Thisproblem can be solved by employing time-varying mutation andcrossover rates. The parameters can be varied with time linearlyor in a random manner. The use of chaotic sequences in DE is use-ful to escape easily from local minima. One of the simplest dy-namic systems evidencing chaotic behavior is the iterator calledthe logistic map [39], whose equation is given by

yðtÞ ¼ l� yðt � 1Þ � ½1� yðt � 1Þ� ð17Þ

where t is the iteration count and l is a control parameter, 0 6 l 6 4.The behavior of the system represented by Eq. (17) significantlychanges with the variation in l. The value of l controls the variationof the chaotic sequence. A number of DE variants with time-varying fm

and CR are proposed here. The combination of DE with chaotic se-quences (DEC) based on logistic map can be modeled as below.

Variant 1/DEC(1): The parameter fm is varied as per equation(18) where fm(0) lies between [0, 1]. The index ‘t’ is the current iter-ation and fm(t) is the new mutation factor based on the logisticmap.

fmðtÞ ¼ l� fmðt � 1Þ � ½1� fmðt � 1Þ� ð18Þ

The parameter l decides whether the mutation rate fm oscillates be-tween a limited sequence, varies chaotically or stabilizes to a con-


stant value. A very small difference in fm(0) causes significant differ-ence in its variation pattern. The system at (18) is deterministic anddisplays chaotic behavior when l = 4 and fm(0) R {0, 0.25, 0.5,0.75, 1.0}. The variation of fm has been shown for l = 3 and l = 4in Fig. 1a and b, respectively.

Variant 2/DEC(2): The parameter fm is decreased from an initialvalue f2i to a final value f2f with the progress of the optimizationalgorithm, as per the dynamics given below:

f1ðtÞ ¼ l� f1ðt � 1Þ � ½1� f1ðt � 1Þ� ð19Þ

fmðtÞ ¼ ðf2f � f2iÞiter

itermaxþ f2i

� �f1ðtÞ ð20Þ

Variant 3/DEC(3): The parameter fm is increased from an initialvalue f2i to a final value f2f with the progress of the optimization

0 5 10 15 20

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

gen

Mut

atio

n fa

ctor

Fig. 1a. Variation of fm for DEC1 s

0 5 10 15 20 20

0.5

1

1.5

gene

Mut

atio

n fa

ctor

Fig. 1c. Variation of fm for DEC2 s

0 5 10 15 2050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

gen

Mut

atio

n fa

ctor

Fig. 1b. Variation of fm for DEC1 s

algorithm, as per the dynamics given by (19) and (20). The varia-tion of fm for variants DEC2 (l = 4) and DEC3 (l = 3) is plotted inFig. 1c and d respectively.

Variant4/TVDE: A time-varying DE (TVDE) is proposed here whereboth mutation rate fm and crossover rate CR, are varied iteratively;higher value (parmax) at the beginning and then linearly decreasingto smaller value (parmin), to get better exploration and exploitation.Both fm and CR parameters are represented here with par.

parðiterÞ ¼ ðparmax � parminÞ �ðitermax � iterÞ

itermaxþ parmin ð21Þ

Variant 5/TVM-DE: Time-varying mutation (TVM) factor using Eq.(21) is applied with fixed CR.

Variant 6/TVCR1-TVCR3: Linearly time-varying CR is appliedusing Eq. (21) in combination with variant 1(DEC1), variant 2

25 30 35 40 45 50eration

trategy; l = 3, f1(t = 0) = 0.48.

5 30 35 40 45 50ration

trategy; l = 4, f1(t = 0) = 0.48.

25 30 35 40 45 50eration

trategy; l = 4, f1(t = 0) = 0.48.

Mut

atio

n fa

ctor

0 5 10 15 20 25 30 35 40 45 50

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

generation

Fig. 1d. Variation of fm for DEC 3 strategy; l = 3, f1(t = 0) = 0.48.


(DEC2), and variant 3 (DEC3) and referred to as time-varying cross-over rate (TVCR), i.e. TVCR1–TVCR3 respectively.

4. Solution of reserve constrained MAED Problem using PSO andDE variants

The paper presents solution of MAED problem with reserve con-straints employing PSO and DE variants and critically comparestheir features for practical power system operation.

Step (1) Parameter setup

The PSO and DE parameters such as population size, the bound-ary constraints of optimization variables, cognitive and socialacceleration coefficients, the mutation factor (fm), the crossoverrate (CR), and the stopping criterion of maximum number of gen-erations are selected.

Step (2) Initialization of an individual population

For a population size R, the particles are randomly generatedand normalized between the maximum and the minimum operat-ing limits of the N generator units and M number of tie-lines. Theith particle is represented as

Pi ¼ ðPni1; P

ni2; . . . ; Pn

iN; Tni1; T

ni2; . . . ; Tn

iMÞ; where i ¼ 1;2 . . . ;R ð22Þ

The jth dimension of the ith particle is normalized as given below tosatisfy the generation limit constraints given by (4) and tie-line lim-its given by (7). For generators with ramp rate constraints, the unitgeneration limits are modified using (5). Here, r is a uniformly dis-tributed random number between 0 and 1.

Pnij ¼ Pij min þ rðPij max � Pij minÞ ð23Þ

Step (3) Implementation of area-wise reserve constraintsTo implement the area-wise reserve constraints specified by (8),the following steps are employed. If for the ith population particle,the existing reserve in the mth area is less than Sm,req, (the requiredreserve in area m) then Pij,m, the generation of the jth unit of area mis modified as given below.

Pij;m ¼ Pij;m � Diffm �Pmax;jP

jPmax;jwhere Diffm

¼ Sm;req �X

j

ðPmax;j � PijmÞ ð24Þ

The area-wise reserve requirement specified by Eq. (8) ischecked for the generated initial population. If the reserve of mtharea is found to be less than specified for any population, thenthe shortfall in reserve (Diffm) is deducted in ratio of the ratings

of generators in that area, from each unit allocation of the area,to satisfy the reserve requirement. To satisfy area reserve require-ments, after every population update also, the existing state of thepopulation variables is modified in the above manner.

Step (4) Implementation of constraints due to prohibited oper-ating zones

If any element Pij of the initial population (or updated popula-tion) lies within the kth prohibited operating zone, it is again mod-ified and assigned the generation value corresponding to the lowerboundary (PL

ijk) or the upper boundary ðPUijkÞ of the zone, as per the

given logic. Here, Pmid,k is the mid point of the kth prohibited zone.

Pij ¼PL

ijk if PLijk 6 Pij < Pmid;k

PUijk if Pmid;k 6 Pij 6 PU

ijk

(ð25Þ

Step (5) Evaluation of the individual populationThe strength of each individual particle in the swarm is evaluatedto judge its merit using a fitness function called evaluation func-tion. The evaluation function should be such that cost is minimizedwhile constraints are satisfied. One of the methods for this is thepopular penalty function method. In this method, the penalty func-tions composed of squared or absolute violations are incorporatedin the fitness function, and are set to reduce the fitness of theparticle according to the magnitude of the violation. The penaltyfunction approach, thus, converts a constrained optimization prob-lem into an unconstrained optimization problem. The fitness func-tion values need to be calculated for each particle in order to findits merit. The evaluation function used here is computed from thecost Eq. (2) and area balance Eq. (3), and can be written as

MinXN

i¼1

FiðPiÞ þXM

q¼1

aq

XNq

i¼1

Piq � PDq þXMq

j

Tjq

!" #2

ð26Þ

For the qth area, aq is the penalty parameter, PDq is the load demand,Tjq is the tie-line flows from other areas, Nq are number of generat-ing units and Mq represents the number of tie-lines connected tothe qth area.

The second term imposes a penalty on the particle in terms ofincreased cost if power balance constraints of all the areas arenot satisfied. Transmission losses are neglected here for the sakeof simplicity and comparison with previous results.

Step (6) Mutation operation to generate donor vector

The parameters are iteratively updated to improve the fitness.In PSO the parameters are updated using Eqs. (9)–(13) while inDE mutation adds a vector differential to a population vector ofindividuals; the donor or mutant vector is generated by using Eq.


(14) corresponding to the chosen DE variant. For implementing DEwith time-varying mutation Eqs. (18)–(21) are applied for itera-tively changing fm in a predefined manner.

Step (7) Recombination operation

Recombination is applied in DE using Eq. (15) to generate a trialvector by probabilistically replacing certain parameters of the tar-get vector with the corresponding parameters of the randomlygenerated donor or mutant vector in step 5.

Step (8) Selection operation

The values of the evaluation function are calculated for the up-dated positions of the particles. In PSO if the new value is betterthan the previous pbest, the new value is set to pbest. Similarly, va-lue of gbest is also updated as the best vector among pbest. In DEthe trial vector Ui(t + 1) replaces its parent target vector xi(t) if itscost is found to be better otherwise the target vector is allowedto proceed to the next generation.

Step (9) Stopping criterion

A stochastic optimization algorithm is stopped either based onthe tolerance limit or maximum number of iterations. For compar-ison purpose, maximum iteration is adopted as the stopping crite-rion here.

5. Results and discussion

The tie-line constraints, area-wise spinning reserve constraintsand area power balance constraints make the reserve constrainedMAED problem much more complex and difficult to solve as com-pared to the classical ED problem. Different PSO and DE variantsare tested for the proposed RCMAED problem on three systemshaving different sizes and complexities. The performance of bothPSO and DE variants is compared with previously published results[11,12,14,17] and is found to be better. The paper (i) compares dif-ferent DE variants for reserve constrained MAED problem, (ii) com-pares the best DE variant with its close competitor PSO and itseffective variant PSO_TVAC, (iii) investigates the influence of re-serve constraints and inter-area aid on the total fuel cost and (iv)compares the cost for tie-line capacity variation.

Table 1Comparison of best results of DE and PSO variants for test system I.

Strategy P1 P2 P3 P4

CMAES [17] 560.9383 168.9300 99.9890 290HNN [11] Not available (NA)DSM [12] Not available (NA)All DE variants 445.1223 138.8777 212.0427 323PSO_TVAC 444.8047 139.1953 211.0609 324

a V1 and V2 are area power balance violations for the two areas.

Fig. 2. Two-area, four-unit system.

5.1. Description of the test systems

5.1.1. Test system IThe first test system consists of a two-area system with four

generating units [11,12] as shown in Fig. 2. This system is consid-ered here for the purpose of comparison with previous results. Thepercentage of the total load demand in area 1 is 70% and 30% inarea 2. The load demand (PD) and tie-line flow limit are set at1120 MW and 200 MW respectively for solving MAED. The globalbest for this system has been reported at $10,605 [11,12]. Ref.[14] has reported $10,574 but the reported results are infeasibleas they do not satisfy the area power balance constraints. Table 1gives the comparison of reported results with already published re-sults. For this system, all the DE variants converged to the globalbest solution. The superiority of the reported results is evidentfrom its ability to satisfy all constraints and produce feasible re-sults. For solving RCMAED the area reserves are taken as 40%(313.6 MW) and 30% (100.8 MW) of area load demand respec-tively, and the tie-line limit is assumed to be 300 MW.

5.1.2. Test system IIThe second system [17] has four areas and four generating units

in each area as shown in Fig. 3. The fuel characteristic data, unitoperating limits and tie-line limits are taken from [17]. The arealoads are 30 MW, 50 MW, 40 MW and 60 MW respectively. Thearea spinning reserve requirement is 30% of area load demand ineach area, i.e. 9 MW, 15 MW, 12 MW and 18 MW respectively forthe four areas. The minimum cost reported for this system usingPSO with local search [17] is $2166.82/h with inter-area aid and$2191.14/h without inter-area aid. The DE variants and PSO_TVACalgorithms used in this paper have achieved better and feasible re-sults for this system. For the sake of comparison with Ref. [17], theemission content corresponding to the best cost solution has beencalculated and shown here.

P12 Cost ($/H) V1a V2a

.1427 �194.39 10574.00 140.00 �140.010605.00 NA NA10605.00 NA NA

.9573 �200.0000 10604.6740 0.0 0.0

.9391 �200.0000 10604.6781 0.0 0.0

Fig. 3. Four area 16 generating unit system.

Fig. 4c. Effect of variation of CR on DE performance; strategy III: DE/rand-to-best/1.

Fig. 4d. Effect of variation of CR on DE performance; strategy IV: DE/best/2.


5.1.3. Test system IIIThe third system is a large system from [30] which has units

with valve point loading effects, ramp rate limits and some unitshaving prohibited operating zones. Multi-area economic dispatchfor such a large and complex system has not yet been reported.The system is assumed to be divided into two areas; the firsttwenty units are assumed to be in area one while the remaining20 units are in area two. The total load is 10,500 MW out of whicharea one load is taken as 7500 MW and load at area two is3000 MW. The tie-line capacity is taken as (i) 1500 MW and (ii)2000 MW.

In classical PSO both the acceleration coefficients are takenequal to 1.5 for all the three test systems. For PSO_TVAC the cogni-tive and social acceleration factors were varied between 2.2 and1.6. Simulations were carried out using MATLAB 7.0.1 on a PentiumIV processor, 2.8 GHz. with 1 GB RAM.

5.2. Mutation strategies

Most papers [30,32–34] have not mentioned the strategyadopted for mutation while some have used DE/rand/1/bin [31].In the present paper, a detailed comparison is carried out for find-ing out the best mutation strategy from Eq. (14) using minimumcost and its standard deviation as the performance measures. Theresults for the first test system are plotted here (for a populationsize of 25) in Fig. 4a–e for all the five DE variants discussed in Sec-tion 3.4. The idea is to present an extensive comparison of DE andits hybrid variants with PSO and its variant for reserve constrainedMAED problem with complex constraints. It is observed that DEmutation strategies I, II and IV perform better for lower values ofCR. In the case of strategy III and strategy V the performance im-proves as CR is increased. All DE strategies are found to be highlysensitive to CR, as the standard deviation (SD) drastically increasesif appropriate value of CR is not employed. On the whole, the per-formance of strategy III (DE/rand-to-best/1) shown in Fig. 4c is

Fig. 4a. Effect of variation of CR on DE performance; strategy I: DE/best/1.

Fig. 4b. Effect of variation of CR on DE performance; strategy II: DE/rand/1.

Fig. 4e. Effect of variation of CR on DE performance; strategy V: DE/rand/2.

found to be the best with least standard deviation. Therefore,throughout this paper mutation strategy III has been employedfor all five DE variants. For the time-varying strategies (TVCR1–TVCR3 and TVM-DE) if the value of fm and CR are varied over thecomplete range, i.e. (0–1) for CR and (0–2) for fm then good resultsare not obtained for the systems tested here. Variation of boththese parameters in a narrow critical range from 0.9 to 0.8 wasfound to give satisfactory performance. For classical DE case,fm = CR = 0.9 is found to give the best results. Similar performancewas observed for other test systems also.

5.3. Comparison of DE and PSO variants

Tables 2 and 3 give a detailed study of DE variants, PSO andPSO_TVAC for RCMAED problems out of 50 trials for the first twotest systems. Later, Table 9 compares these variants for a large sys-tem with complex constraints like VPL, POZ, ramp rate limits formulti-area operation. The PSO and DE strategies are compared herein-depth, to focus on their individual similarities and differences;merits and shortcomings; solution quality and consistency anddependence on tuning parameters. An insight into these strategieswould result in saving a lot of time and effort which is otherwisespent in tuning the different parameters of these algorithms.

Table 3Comparison of DE variants with PSO and PSO_TVAC for RCMAED (test system II; 50trials).

Variant Minimum cost($/H)

Mean cost($/H)

Maximum cost($/H)

SD

DEC1 2127.79244 2128.06894 2133.76267 1.2084DEC2 2127.67411 2127.65387 2130.0064 0.2336DEC3 2127.59244 2128.0594 2135.36251 1.6079TVM-DE 2131.07449 2138.4534 2436.4298 2.0122DE 2131.74232 2134.1831 2140.5474 1.8408TVCR1 2129.31486 2132.62958 2138.33964 2.9350TVCR2 2129.31803 2133.62897 2137.39782 1.6355TVCR3 2142.24404 2165.66144 2170.76476 6.0172TVDE 2127.93597 2133.13897 2138.35842 2.6474PSO_TVAC 2147.00477 2194.11962 2225.70475 11.9431PSO 2153.75742 2770.9073 30984.44153 1128.5403

Table 2Comparison of DE variants with PSO and PSO_TVAC for RCMAED (test system I; 50 trials).

Variant P1 P2 P3 P4 Pt (tie-line) Minimum cost ($/H) Maximum cost ($/H) SD

DEC1 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10567.1873 0.0076DEC2 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10566.9942 0.0DEC3 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10567.1913 0.0382TVM-DE 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10566.9942 0.0DE 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10566.9942 0.0TVCR1 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10597.9979 6.8545TVCR2 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10566.9942 0.0352TVCR3 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10664.0845 36.3105TVDE 369.5736 114.4263 296.00 340.00 �300.00 10566.9942 10566.9942 0.0PSO_TVAC 370.9293 113.0706 296.00 340.00 �300.00 10567.0059 10613.7863 11.0512PSO 383.7126 100.2874 296.00 340.00 �300.00 10568.2696 10641.5278 91.4719


Some DE strategies converged with zero/near zero standarddeviation (SD) while some (TVCR1 and TVCR3) produced higherSD. The PSO_TVAC performed better than PSO but inferior to mostDE variants. The PSO_TVAC could not achieve the minimum cost;instead converged to near global best value for all tested values ofcognitive and social acceleration factors. The interesting pointabout PSO_TVAC is that it gives consistent performance for allvalues of acceleration coefficients, while DE variants are highlysensitive to CR and fm (or their upper and lower variation ranges).It can be seen from Table 4 that for some values of CR the perfor-mance is very poor. Another significant difference is that DErequires less population but takes larger number of iterations toconverge while PSO_TVAC works with larger populations and les-ser iterations.

The time-varying DE strategies were found to be sensitive to theminimum and maximum parameter variation limits parmin andparmax in Eq. (21). Best performance was recorded for parmin = 0.8and parmax = 0.9 for both CR and fm. The performance of DEC1–DEC3 was found to be quite comparable and better than time-vary-ing DE in terms of minima as well as SD. The performance of DEC2was the best; therefore detailed results of the DEC2 are presentedhere for the studied cases.

Table 4Effect of CR and NP on mean value and standard deviation of DEC2 (test system I strategy

NP CR = 0.2 CR = 0.4 CR = 0.6

5 10596.9979 (4211.3603)a 10627.1877 (73463.5407) 10606.9942 (10 10576.9964 (5.2943) 10579.9942 (5.5605) 10586.9942 (15 10573.5439 (4.7747) 10569.38954 (3.0557) 10567.1624 (20 10571.6504 3.2790) 10567.4959 (0.5543) 10567.4113 (25 10570.4984 (2.5239) 10567.1256 (0.1716) 10567.0291 (

NP: population size.a The value in bracket is the standard deviation.

5.4. Effect of population size and crossover rate

The population size is another important factor for achievinggood results in evolutionary optimization methods. It has been re-ported that increasing population improved the performance ofPSO algorithm [23,43]. The optimal population size is found to de-pend on problem complexity and dimension. Storn and Price sug-gested a population size of 5–10 times that of problem dimension[42]. In case of PSO, Ref. [23] clearly links population to perfor-mance while similar observations are reported for DE in Ref. [32]for various ED problems. Both references show that larger thedimension, larger is the population size required to achieve goodresults. A lower population can achieve the minima but the stan-dard deviation is higher.

Different population sizes were tested and results of mean andSD are given in Tables 4 and 5 for reserve constrained MAED prob-lem for test system I and test system II for the DEC2 algorithm. Glo-bal best results ($10566.9942/h) were obtained for a populationsize of 25 for the first system with zero SD. The global best results($2127.81194/h) for the second system were found for a popula-tion size of 250. For both the systems, as the population size wasincreased, the standard deviation reduced and finally becamezero/near zero, indicating global convergence of the algorithm.For lower population size, near global results were obtained atlower value of CR. A high CR with low population results in veryhigh SD indicating instability of the solution. The SD for the largersystem is found to be higher than for the smaller system.

5.5. Effect of tie-line and reserve constraints

The results of RCMAED solution for test system I using DEC2with area-wise reserve constraints specified in Section 5.1 are tab-ulated with different loads (Case I: 800 MW and Case II: 1120 MW)and tie-line capacities in Table 6. The performance is comparedwith (i) change in load, (ii) change in tie-line capacity, (iii) changein area reserve and (iv) reserve sharing between areas. For all casesthe DE variants were found to perform well, but the DEC2 con-verged to the global best solution with zero/near zero SD. It canbe seen that with increase in tie-line capacity from 90 MW to

= 3, Mu = 4).

CR = 0.8 CR = 0.9

148994.7855) 10657.1784 (157434.9808) 10659.4754 (330005.8708)6.4479) 10572.9942 (3.5537) 10595.9878 (7.3727)0.0738) 10567.0082 (0.0480) 10567.0044 (0.0365)0.5318) 10566.9947 (0.00) 10566.9945 (0.0087)0.0338) 10566.9942 (0.00) 10566.9942 (0.00)

Table 5Effect of CR and NP on mean value and standard deviation of DEC2 (test system II strategy = 3, Mu = 4).

NP CR = 0.2 CR = 0.4 CR = 0.6 CR = 0.8 CR = 0.9 CR = 1

5 2187.7343(7.2376)a

2194.1003(10.4663)

2187.2849(16.2459)

2658405.8748(456345.9356)

1059506.9564(1528806.7854)

4488078.9817(38804007.3675)

10 2197.3893 (6.7978) 2181.6983 (8.2870) 2179.8036(11.1292)

2190.5734 (15.7954) 2183.6803 (16.0905) 753898.8750 (9099278.9834)

25 2181.6873(11.4753)

2195.1709 (7.8607) 2186.5904 (8.6001) 2186.8398 (10.2402) 2186.7603 (5.5643) 2173.6643 (9.1503)

50 2179.1703 (6.8524) 2187.2603 (6.8527) 2188.7783 (7.0754) 2187.3703 (5.3788) 2164.2003 (7.9661) 2177.2803 (11.0206)75 2186.3783 (6.8896) 2194.6673 (5.5072) 2191.8394 (7.3020) 2179.6973 (9.4123) 2162.5453 (8.6475) 2155.3389 (7.3961)

100 2180.1673 (3.9269) 2188.2563 (6.9154) 2187.2043 (9.4423) 2178.8193 (8.0703) 2158.7603 (5.2833) 2153.9673 (7.7729)250 2169.1673 (5.7188) 2181.2503 (6.2822) 2162.2783 (9.8849) 2156.8403 (9.2888) 2127.65387 (0.7146) 2127.85387 (0.02336)

NP: population size.a The value in bracket is the standard deviation.

Table 6Effect of load, tie-line flow and area-wise reserve on operating cost of MAED (test system I, DEC2).

Area 1 reserve (MW) Area 2 reserve (MW) Tie-line capacity P1 (MW) P2 (MW) P3 (MW) P4 (MW) Pt (MW) Cost ($/H)

Case I: Total load (Pd = 800; PD1 = 560 MW, PD2 = 240 MW)313.6 100.8 90 358.9968 111.0031 112.6133 217.3866 �90.00 7811.9436

120 336.3323 103.6677 127.0933 232.9066 �120.00 7791.2531200 275.8934 84.1066 165.7066 274.2933 �200.00 7754.6899

With reserve sharing between areas (total reserve = 414.4 MW) 00 426.9906 133.0094 100.0000 140.0000 0.0 7900.420210 419.4357 130.5643 100.0000 150.0000 �10.00 7888.265025 408.1034 26.8966 100.0000 165.0000 �25.00 7871.152850 389.2163 120.7837 100.0000 190.0000 �50.00 7845.6211

Case II: Total load (Pd = 1120; Pd1 = 784, Pd2 = 336)265 130 250 407.3479 126.6520 246.0000 340.0000 �250.00 10578.5201

300 389.2162 120.7837 270.0000 340.0000 �274.00 10571.0417274.4 134.4 260 399.7931 124.2069 256.0000 340.0000 �260.00 10574.9675

270 392.5404 121.8595 265.6000 340.0000 �269.60 10572.1438313.6 100.8 300 369.5736 114.4263 296.0000 340.0000 �300.00 10566.9942With reserve sharing between areas (total reserve = 414.4 MW) 00 596.2194 187.7806 115.5093 220.4907 0.0 10816.4825

10 588.6646 185.3354 120.3360 225.6640 �10.00 10801.873725 577.3323 181.6677 127.5760 233.4240 �25.00 10780.753650 558.4452 175.5548 139.6427 246.3573 �50.00 10747.668490 528.2257 165.7743 158.9494 267.0506 �90.00 10700.2308

120 505.5611 158.4389 173.4293 282.5707 �120.00 10669.0941200 456.3699 133.9527 206.2385 323.4389 �193.68 10609.0757250 407.2359 126.7640 246.2669 339.7331 �250.00 10578.5405


200 MW, the cost decreased from $7811.9436/h to $7754.6899/hfor Case I. No feasible solution was available for tie-line capacityless than 90 MW. But when the same reserve was shared betweenareas, solutions were available for lesser tie-line capacities. In thisformulation when both areas operated separately without tie-lineflow, the cost was $7900.4202/h, which reduced to $7845.6211/hfor a tie-line capacity of 50 MW. Similar results were found forCase II. When reserve requirement was increased, the tie-linecapacity had to be increased to get a feasible solution. Tie-linecapacity had to be increased to 300 MW for area-wise reserverequirement of 313.6 MW and 100.8 MW respectively for Case II.When reserve sharing between areas was permitted by relaxingthe area-wise reserve constraint, operation was possible with les-ser tie-line capacities. Table 6 presents the complex relationshipbetween reserve requirements, tie-line capacities and operatingcost. The DE variants are able to achieve best cost solutions forall these cases.

For the test system II the effect of area-wise reserve constraints,on cost of operation was studied (i) with inter-area aid, (ii) withoutinter-area aid and (iii) with inter-area reserve sharing. In the firsttwo cases the two areas have separate reserve requirements tobe satisfied. In the third case there is a common reserve require-ment which is to be met by the different areas together. The resultsobtained by DEC2 are compared with already published results[17] for the first two cases in Table 7. When area-wise reserveconstraints were imposed the best cost achieved by DE was$2152.9646/h against the already published $2166.82/h [17]. The

corresponding emission value reported in [17] was 3.3152 ton/hagainst 2.87017 ton/h by DEC2. In [17] the area reserve constraintof area 1 was violated and area balance constraints were also notfully satisfied. It can be seen from Table 7 that the DEC2 methodgives full satisfaction of area-wise power balance and reserveconstraints.

The cost increased to $2181.26094/h (emission: 3.01119 ton/h)when no inter-area aid was available. The results achieved by DEwere superior to PSO [17] $2191.14/h (emission: 3.7493 ton/h) inthis case too, as the cost achieved was less and constraints (areabalance as well as area reserve) were fully satisfied. The cost wasfurther reduced to $2127.81194/h when area-wise reserve con-straints were relaxed to total reserve constraint, i.e. reserve sharingbetween areas. The solution indicates that area 1 no longer has therequired reserve of 9 MW as other areas are sharing its reserverequirement. Table 8 shows the effect of tie-line limits on reservesharing and cost of operation for (i) area-wise reserve constraintand (ii) reserve sharing between areas. Operating cost reduceswhen reserve requirement is shared between areas. For both casesit can be observed that the operating cost reduced when tie-linelimit constraints were not imposed.

5.6. Performance on large system

To show their effectiveness for practical MAED cases, the per-formance of the various DE variants is also tested on a large systemhaving units with ramp rate constraints, prohibited operating

Table 7Comparison of results with and without inter-area aid (test system II).

Power output (MW) With inter-area aid [17] With inter-area aid (DEC2) W/O inter-area aid [17] W/O inter-area aid (DEC2) Reserve sharing (DEC2)

P1 13.20 13.99999 10.74 10.89434 13.99882P2 6.49 9.99364 9.43 9.99990 9.99987P3 12.01 6.57665 5.03 3.34036 5.69721P4 11.28 9.42952 5.33 5.76539 11.98285P5 20.47 18.05712 25.07 15.59196 24.99881P6 6.57 7.32879 6.71 9.62606 11.99570P7 13.16 13.66129 9.80 12.46762 10.94960P8 15.03 13.91045 8.46 12.31438 8.36306P9 5.72 9.81839 10.83 10.08992 0.07649P10 9.71 8.45018 16.46 9.05573 0.05389P11 6.63 8.01054 6.62 9.68112 18.17293P12 22.78 20.85971 6.22 11.17323 29.99971P13 7.59 10.99580 7.54 10.99854 10.99540P14 11.23 10.01365 15.87 15.11019 0.05133P15 5.20 7.87430 10.71 14.32973 22.60373P16 14.02 11.02001 26.04 19.56156 0.06055T1 �3.16 �5.97795 0.00000 0.00000 �4.31812T2 �0.88 �3.99997 0.00000 0.00000 �3.99921T3 16.99 19.97772 0.00000 0.00000 19.99604T4 �3.20 �3.49995 0.00000 0.00000 �3.49951T5 5.16 0.47963 0.00000 0.00000 5.48859T6 0.48 �0.36110 0.00000 0.00000 0.80430Reserve area 1 6.02 9.00020 18.47 19.00001 7.32125Reserve area 2 19.77 22.04235 24.96 24.99998 18.69282Reserve area 3 75.16 72.86118 79.87 80.00000 71.69698Reserve area 4 52.96 51.09624 30.84 30.99997 57.28899Total reserve 153.91 154.99998 154.1400 154.99997 155.00004Violation (area 1) 0.03 0.00000 0.53 0.0000 0.0000Violation (area 2) 0.11 0.00000 0.04 0.0000 0.0000Violation (area 3) 0.28 0.00000 0.13 0.0000 0.0000Violation (area 4) 0.67 0.00000 0.16 0.0000 0.0001Cost ($/H) 2166.82 2152.9646 2191.14 2181.26094 2127.81194Emission (ton/h) 3.3152 2.87017 3.7493 3.01119 5.7614

Table 8Effect of tie-line limits on reserve sharing and cost of operation (test system II).

Power output(MW)

Area-wise reserve (with tie-linelimit)

Area-wise reserve (without tie-linelimit)

Total area reserve (with tie-linelimit)

Total area reserve (without tie-linelimit

P1 14.00000 14.00000 13.99994 14.00000P2 9.99929 10.00000 10.00000 9.99937P3 3.97572 0.05000 4.88896 0.05002P4 11.99877 10.64995 12.00000 0.05112P5 24.97140 25.00000 25.00000 23.60446P6 11.99991 12.00000 11.99901 11.99475P7 6.06372 0.05002 19.56806 0.05000P8 14.94026 0.05000 0.54377 0.05000P9 0.08507 22.61261 0.05000 29.99979P10 0.05029 25.38747 0.05319 30.00000P11 18.24800 29.99992 18.29668 30.00000P12 30.00000 30.00000 30.00000 29.99996P13 10.68162 0.05000 10.99998 0.05000P14 0.06735 0.05000 0.05000 0.05032P15 22.86856 0.05000 22.47434 0.05019P16 0.05000 0.05000 0.07604 0.05000T17 �5.99546 18.06411 �5.11105 32.75988T18 �3.98338 �29.16844 �4.00000 �39.66324T19 19.95265 15.80428 19.99996 1.00387T20 �3.50000 �34.99849 �3.49988 �34.99996T21 5.47985 40.16263 5.49967 53.45905T22 0.90000 3.83308 0.90000 5.33656Reserve (area 1) 9.02623 14.30006 8.11111 24.89950Reserve (area 2) 17.02472 37.89998 17.88916 39.30079Reserve (area 3) 71.61663 12.00000 71.60012 0.00025Reserve (area 4) 57.33247 90.80000 57.39964 90.79949Total reserve 155.00005 155.00004 155.00004 155.00004Cost ($/H) 2127.67411 2064.81707 2127.65387 2056.58311Emission (ton/h) 5.8993 7.4663 6.2056 8.5841


zones and valve point loading effects along with tie-line con-straints. The problem is very complex due to multiple non-linear,

non-convex constraints. To the best of authors’ knowledge all theseconstraints have not yet been modeled in MAED. The overall cost is

Table 9Comparison of DE variants with PSO variants for MAED problem (test system III; 50 trials).

Variant Minimum cost ($/H) Mean cost ($/H) Maximum cost ($/H) SD cpu time/trial

DEC1 127815.1341 127895.1163 12796.1329 1.7404 4.078177DEC2 127745.5241 127751.4067 127785.2131 0.3842 4.101123DEC3 127815.2058 127851.8429 127863.6224 1.7012 3.993912TVM-DE 127964.2408 127968.4991 127970.0192 0.5973 4.125524DE 127881.0959 127881.0959 127881.0959 0.0000 3.891448TVCR1 127825.2058 127851.8429 127863.6224 1.5488 4.210325TVCR2 127949.1199 127949.9739 127950.1959 0.8126 4.422722TVCR3 127744.6433 127746.6392 127749.3793 2.4838 4.647221TVDE 127746.1545 127748.5857 127750.1912 1.3893 4.772108PSO_TVAC 127744.5609 127747.1205 127750.1545 5.0418 0.674349PSO 129045.1049 140921.7508 2130278.8155 80.3348 0.641703

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10 x 107

generation

cost

DEDEC1DEC2DEC3PSO TVAC

Fig. 5. Convergence characteristics of the PSO and DE strategies for reserveconstrained system (two-area, four-unit system).


heavily influenced by tie-line limits, area loads and cost curves. Re-sults are compiled in Table 9 from where it is clear that for thislarge and complex system all DE variants efficiently converged inthe vicinity of the best solution with full constraint satisfaction.The PSO approach produced much higher SD than PSO_TVAC ap-proach. Thus PSO_TVAC performed better than PSO but inferiorto most DE variants.

The time-varying DE strategies were found to be sensitive to theminimum and maximum parameter variation limits parmin andparmax in Eq. (21). Best performance was recorded for parmin = 0.8and parmax = 0.9 for both CR and fm for mutation strategy III. Clas-sical DE converged to a sub-optimal solution with zero SD indicat-ing premature convergence for larger dimension problem while allthree DEC strategies performed better that classical DE. The perfor-mance of DEC1–DEC3 was found to be quite comparable and betterthan TVCR1–TVCR3 in terms of minima as well as SD. The perfor-mance of DEC2 was the best with least cost and least SD. Thereforedetailed results of the DEC2 are presented here for the studiedcases. Table 10 illustrates the effect of tie-line capacity variationon optimal cost of operation. When the tie-line capacity was raisedfrom 1500 MW to 2000 MW the cost of operation reduced from$131549.6080/h to $127745.5241/h. The detailed results show that

Table 10Best results for test system III (DEC2).

Tie line limit = 2000

P1 113.6880 P22 549.9712P2 107.9724 P23 548.8130P3 79.4325 P24 549.0277P4 169.5169 P25 548.7569P5 96.4316 P26 549.8162P6 139.9064 P27 10.1824P7 298.4421 P28 18.3553P8 297.8600 P29 10.8439P9 243.3245 P30 96.8428P10 300.0000 P31 180.6885P11 311.5121 P32 160.7230P12 98.2914 P33 80.6223P13 125.0750 P34 107.7962P14 497.5848 P35 199.7514P15 191.7050 P36 90.5810P16 360.1095 P37 69.9961P17 496.6322 P38 106.7186P18 498.0519 P39 29.9162P19 525.6573 P40 548.7375P20 548.8070 T12 (T-line) �1999.9993P21 541.8591 – –SD 1.7012Cost ($/H) 127745.5241V1a 0.0000V2b 0.0000

a Area 1 power balance violation.b Area 2 power balance violation.

the solution is feasible with zero area power balance violation. Thetie-line and generating unit limits, ramp rate limits and prohibitedoperating zone constraints too are fully satisfied.

Tie line limit = 1500

P1 114.0000 P22 550.0000P2 41.2339 P23 254.0000P3 72.2040 P24 467.5633P4 190.0000 P25 488.3421P5 97.0000 P26 548.5202P6 140.0000 P27 10.0000P7 300.0000 P28 10.0000P8 300.0000 P29 11.0080P9 169.7833 P30 97.0000P10 300.0000 P31 190.0000P11 375.0000 P32 164.5855P12 375.0000 P33 163.7269P13 500.0000 P34 199.3732P14 500.0000 P35 90.0000P15 191.7132 P36 166.5709P16 500.0000 P37 110.0000P17 500.0000 P38 110.0000P18 500.0000 P39 77.3028P19 517.7832 P40 242.0000P20 316.2894 T12 (T-line) �1499.9929P21 550.0000 – –SD 1.7634Cost ($/H) 131549.6080V1a 0.0000V2b 0.0000


5.7. Convergence characteristics

The convergence behavior of the DE and PSO variants was com-pared. The results of classical DE, all three DEC strategies andPSO_TVAC were plotted for one trial of 100 iterations as shownin Fig. 5 for the four-unit system. PSO_TVAC shows the best con-vergence characteristics due to the proper tuning of social and cog-nitive coefficients during the search. Convergence behavior of DEstrategies with chaotic/or time-varying mutation was found to besimilar but slower compared to PSO_TVAC. It is seen that DE vari-ants take longer to converge than PSO_TVAC. Similar results wereobserved for the other systems too. Tables 2, 3 and 9 show thecpu time required by all the DE and PSO variants discussed in Sec-tion 3.4 for one trial.

6. Conclusion

The paper presents a close comparison of classic PSO and DEstrategies and their variants for solving the RCMAED problem withcomplex constraints. An in-depth analysis of various classic DEstrategies is carried out which clearly highlights the dependenceof all DE variants on fm and CR. A comparison of PSO_TVAC andclassic DE reveals that the former converges to near global solu-tions in all trial runs, for all tested values of tuning parametersand population sizes. DE on the other hand operates well only ina very narrow range of tuning parameters; otherwise it does notconverge at all or produces very high SD. To combat this depen-dence on tuning parameters, some time-varying mutation andcrossover strategies are proposed here which are found to producefeasible and robust solutions for large multi-area ED problem withreserve constraints. The time-varying DE algorithms are found tobe capable of finding better solutions than previously reported re-sults under varying loads, tie-line limits and reserve requirements,with/without inter-area aid, with full constraint satisfaction. Forlarge systems classic DE suffers from premature convergence whilethe time-varying DE variants are capable of achieving global bestsolutions efficiently for non-convex problems.

Acknowledgements

The authors sincerely acknowledge the financial support pro-vided by UGC under major research project entitled Power SystemOptimization and Security Assessment Using Soft Computing Tech-niques, vide F No. 34-399/2008 (SR) dated, 24th December 2008and AICTE New Delhi for financial assistance under RPS Project FNo. 8023/RID/BOR/RPS-45/2005-06 dated 10/03/2006. The authorsalso thank the Director, M.I.T.S., Gwalior for providing facilities forcarrying out this work.

References

[1] Wood AJ, Wollenberg BF. Power generation, operation and control. John Wiley& Sons; 1984.

[2] Shoults RR, Chang SK, Helmick S, Grady WM. A practical approach to unitcommitment, economic dispatch and savings allocation for multiple area pooloperation with import/export constraints. IEEE Trans Power Apparatus Syst1980;99(2):625–35.

[3] Doty KW, McEntire PL. An analysis of electrical power brokerage systems. IEEETrans Power Apparatus Syst 1982;101(2):389–96.

[4] Desell AL, Tammar K, McClelland EC, Van Home PR. Transmission constrainedproduction cost analysis in power system planning. IEEE Trans PowerApparatus Syst 1984;PAS-103(8):2192–8.

[5] Farmer ED, Grubb MJ, Vlahos K. Probabilistic production costing oftransmission-constrained power systems. In: 10th PSCC power systemcomputation conference; 1990. p. 663–9.

[6] Helmick SD, Shoults RR. A practical approach to an interim multi-areaeconomic dispatch using limited computer resources. IEEE Trans PowerApparatus Syst 1985;104(6):1400–4.

[7] Ouyang Z, Shahidehpour SM. Heuristic multi-area unit commitment witheconomic dispatch. IEE Proc C 1991;138(3):242–52.

[8] Wang C, Shahidehpour SM. A decomposition approach to non-linear multi-area generation scheduling with tie-line constraints using expert systems. IEEETrans Power Syst 1992;7(4):1409–18.

[9] Wernerus J, Soder L. Area price based multi-area economic dispatch with tie-line losses and constraints. In: IEEE/KTH Stockholm power tech conference,Sweden; 1995. p. 710–15.

[10] Streifferet Dan. Multi-area economic dispatch with tie-line constraints. IEEETrans Power Syst 1995;10(4):1946–51.

[11] Yalcinoz T, Short MJ. Neural networks approach for solving economic dispatchproblem with transmission capacity constraints. IEEE Trans Power Syst1998;13(2):307–13.

[12] Chen CL, Chen N. Direct search method for solving economic dispatch problemconsidering transmission capacity constraints. IEEE Trans Power Syst2001;16(4):764–9.

[13] Jayabarathi T, Sadasivam G, Ramachandran V. Evolutionary programmingbased multi-area economic dispatch with tie-line constraints. Electr MachPower Syst 2000;28:1165–76.

[14] Manoharan PS, Kannan PS, Baskar S, Iruthayarajan M Willjuice. Evolutionaryalgorithm solution and KKT based optimality verification to multi-areaeconomic dispatch. Electr Power Energy Syst 2009;31:365–73.

[15] Lee FN, Breipohl AM. Reserve constrained economic load dispatch withprohibited operating zones. IEEE Trans Power Syst 1993;8(1):246–54.

[16] Aruldoss T, Victoire Albert, Jeyakumar A Ebenezer. Reserve constraineddynamic dispatch of units with valve point effects. IEEE Trans Power Syst2005;20(3):1273–82.

[17] Wang Lingfeng, Singh Chanan. Reserve constrained multi-area environmental/economic dispatch based on particle swarm optimization with local search.Eng Appl Artif Intel 2009;22:298–307.

[18] Azadani E Nasr, Hosseinian SH, Moradzadeh B. Generation and reservedispatch in a competitive market using constrained particle swarmoptimization. Electr Power Energy Syst 2010;32:79–86.

[19] Park JH, Kim YS, Eom IK, Lee KY. Economic load dispatch for piecewisequadratic cost function using Hopfield neural network. IEEE Trans Power Syst1993;8:1030–8.

[20] Walter DC, Sheble GB. Genetic algorithm solution of economic load dispatchwith valve point loading. IEEE Trans Power Syst 1993;8(3):1325–32.

[21] Chaturvedi Krishna Teerth, Pandit Manjaree, Srivastava Laxmi. Particle swarmoptimization with time varying acceleration coefficients for nonconvexeconomic power dispatch. Electr Power Energy Syst 2009;31(6):249–57.

[22] Chaturvedi Krishna Teerth, Pandit Manjaree, Srivastava Laxmi. Particle swarmoptimization with crazy particles for nonconvex economic dispatch. Appl SoftComput 2009;9(3):962–9.

[23] Chaturvedi Krishna Teerth, Pandit Manjaree, Srivastava Laxmi. Self-organizinghierarchical particle swarm optimization for nonconvex economic dispatch.IEEE Trans Power Syst 2008;23(3):1079–87.

[24] Vlachogiannis John G, Lee Kwang Y. Economic load dispatch – a comparativestudy on heuristic optimization techniques with an improved coordinatedaggregation-based PSO. IEEE Trans Power Syst 2009;24(2):991–1001.

[25] Park JB, Jeong YW, Shin JR. An improved particle swarm optimization fornonconvex economic dispatch problems. IEEE Trans Power Syst2010;25(1):156–66.

[26] Vasebi A, Fesanghary M, Bathaee SMT. Combined heat and power economicdispatch by harmony search algorithm. Electr Power Energy Syst2007;29:713–9.

[27] Song YS, Chou CS, Stoham TJ. Combined heat and power economic dispatch byimproved ant colony search algorithm. Electr Power Syst Res1999;52(2):115–21.

[28] Panigrahi BK, Pandi V Ravikumar. Bacterial foraging optimisation: Nelder–Mead hybrid algorithm for economic load dispatch. IET Gen Transm Distrib2009;2(4):556–65.

[29] Vanaja B, Hemamalini S, Simon SP. Artificial immune based economic loaddispatch with valve-point effect. In: IEEE region 10 conference, TENCON 2008;19–21 November, 2008. p. 1–5.

[30] Wang SK, Chiou J-P, Liu CW. Non-smooth/non-convex economic dispatch by anovel hybrid differential evolution algorithm. IET Gen Transm Distrib2007;1(5):793–803.

[31] Coelho LDS, Mariani VC. Combining of chaotic differential evolution andquadratic programming for economic dispatch optimization with valve-pointeffect. IEEE Trans Power Syst 2006;21(2):989–96.

[32] Noman Nasimul, Iba Hitoshi. Differential evolution for economic load dispatchproblems. Electric Power Syst Res 2008;78:1322–31.

[33] Chiou J-P. A variable scaling hybrid differential evolution for solving large-scale power dispatch problems. IET Gen Transm Distrib 2009;3(2):154–63.

[34] Coelho LDS, Souza RCT, Mariani VC. Improved differential evolution approachbased on cultural algorithm and diversity measure applied to solve economicdispatch problems. Math Comput Simulat 2009;79:3136–47.

[35] Kennedy J, Eberhart R. Particle swarm optimization. In: Proc IEEE conf onneural networks (ICNN’95), vol. IV, Perth, Australia; 1995. p. 1942–8.

[36] Ratnaweera A, Halgamuge SK, Watson HC. Self-organizing hierarchical particleswarm optimizer with time varying acceleration coefficients. IEEE Trans EvolComput 2004;8(3):240–55.

[37] Storn R, Price K. Differential evolution: a simple and efficient adaptive schemefor global optimization over continuous spaces, Tech. rep. TR-95-012.International Computer Science Institute, Berkeley, CA; 1995.

[38] Li B, Jiang W. Optimizing complex functions by chaos search. Cybern Syst1998;29(4):409–19.


[39] Caponetto R, Fortuna L, Fazzino S, Xibilia MG. Chaotic sequences to improvethe performance of evolutionary algorithms. IEEE Trans Evol Computat2003;7(3):289–304.

[40] Shengsong L, Min W, Zhijian H. Hybrid algorithm of chaos optimization and LPfor optimal power flow problems with multimodal characteristic. Proc InstElectr Eng Gen Transm Distrib 2003;150(5):543–7.

[41] Sharma Manisha, Pandit Manjaree, Srivastava Laxmi. Multi-area economicdispatch with tie-line constraints employing evolutionary approach. Int J Eng

Sci Technol (IJEST) 2010;2(3):133–50 [Special Issue Application Comput IntelEmerging Power Syst].

[42] Storn R. Differential evolution – a simple and efficient heuristic forglobal optimization over continuous spaces. J Global Optim 1997;11(4):341–59.

[43] Alrashidi MR, El-Hawary ME. Hybrid particle swarm optimization approach forsolving the discrete OPF problem considering the valve loading effects. IEEETrans Power Syst 2007;22(4):2030–8.

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