real-parameter quantum evolutionary algorithm for economic load dispatch

Real-parameter quantum evolutionary algorithm foreconomic load dispatch

G.S.Sailesh Babu, D.Bhagwan Das and C. Patvardhan

Abstract: A novel real-parameter optimisation algorithm called the ‘real-parameter quantum evol-utionary algorithm’ is presented. The algorithm pieces together the ideas from evolutionary algor-ithms (EA) and quantum computing to provide a robust optimisation technique that can be utilisedto optimise highly constrained non-linear real-parameter functions. Quantum bits have immenserepresentational power due to their being in superposition of all the basic states at the sametime. New quantum operators designed in this work enable the search to effectively handle thetwin objectives of exploitation and exploration. This enables the search to be pursued with smallpopulation sizes, thereby speeding up the search process and also ensuring that there is noproblem of premature convergence that often plagues pure EA implementations. The power ofthe proposed algorithm is demonstrated by solving the economic load dispatch (ELD) in powersystems. ELD is to find the optimal loadings on the generators so as to achieve minimum operatingcost while satisfying various system and unit-level constraints. The proposed method has beenapplied to standard load dispatch problems reported in the literature including the IEEE 30 bussystem, IEEE 57 bus system and a 110-generator problem, and its performance has been comparedwith the results obtained by other methods. The results adequately demonstrate the enhanced searchpower of the proposed algorithm in terms of obtaining better solutions and provide motivation forits application to other real-parameter optimisation problems in power systems.

1 Introduction

Economic load dispatch (ELD) problem is to obtain optimalloadings on the committed generating units so as to mini-mise the operating cost while satisfying various systemsand unit-level equality and inequality constraints.

Mathematically, ELD can be represented as follows.

Minimise(FPG) (1)

Here, FPG is the total $/h fuel cost and is given by

FPG ¼XNg

i¼1

(aiP2giþ biPgi

þ di þ jei sin (fi(Pgmini� Pgi

))j)

(2)

where Ng is the total number of generating units, ai, bi, di, eiand fi the fuel cost coefficients of ith generator, Pgi

andPgmini

the real power outputs of ith generator in MW andits minimum limit, respectively, and PG is the vector ofreal power outputs of all generators.

A solution of ELD, thus found, must satisfy the equalityand inequality constraints which are as follows.

† Power balance constraint. The total MW generationfrom all the generators must be equal to the sum of powerdemand (PD) and real power loss (Ploss) in the transmission

# The Institution of Engineering and Technology 2008

doi:10.1049/iet-gtd:20060495

Paper first received 14th November 2006 and in revised form 19th July 2007

The authors are with the Department of Electrical Engineering, Faculty ofEngineering, Dayalbagh Educational Institute (Deemed University),Dayalbagh, Agra, Uttar Pradesh, 282005, India

E-mail: [email protected]

22

lines. This is mathematically expressed as

XNg

i¼1

Pgi¼ PD þ Ploss (3)

† Generator capacity constraint. Real power generation ofeach unit should lie between minimum and maximum limit

Pgmini � Pgi � Pgmaxi 8 i ¼ 1, 2, : : : , Ng (4)

In literature, methods based on math programming tech-niques [1, 2] have been reported, which are fast but sufferfrom inferior modelling capabilities and curse of dimen-sionality. Madrigal and Quintana [3] propose an analyticalmethod for solving ELD which is an excellent methodwhen units under consideration have quadratic, continuouscost curves and system losses are neglected. In recentyears, several attempts have been made to solve ELDproblem with stochastic techniques which are inspiredfrom physical world and are highly robust. Simulatedannealing (SA), which is a single-point search technique,and SA-based multi-point search techniques, namely SAwith first move (SAF) and SA with best move (SAB),have been applied for ELD [4]. Genetic algorithms (GAs)and other hybrid methods, which are synergistic combi-nation of more than one stochastic technique to improvethe search capability of the algorithm, have been employedto solve ELD [5, 6]. These methods are robust multi-pointtechniques but are governed by population dynamics andrequire fine-tuning of many parameters.

These iterative search techniques start with an initialpopulation of points in the solution space and move to the

IET Gener. Transm. Distrib., 2008, 2, (1), pp. 22–31

next population according to some operators. These tech-niques differ from each other in details. Heuristics whichemploy history of better solutions obtained in the searchprocess, viz. GA, EA, have been proved to have better con-vergence and quality of solution for ‘difficult’ problemsthan those which employ either no information, viz. blindsearch, or information from last move only, viz. gradientsearch, greedy search and so on. But still, problems ofslow/premature convergence remain and have to betackled with suitable implementation for the particularproblem at hand.

Quantum evolutionary algorithms (QEAs) is a recentbranch of EAs. QEAs have been proved to be effective tooptimise functions with binary parameters [7]. These arecharacterised by population dynamics, individual represen-tation, evaluation function and so on, as in EAs, as well asquantum bit (qubit) representation, superposition of statesand so on, as in quantum computing (QC). The advantageof the QEA is that, unlike the other EAs, it can work withsmall population sizes without getting stuck in localminima and without converging prematurely because ofloss of diversity. In the extreme case, the immense represen-tation power of the qubits enables the use of a single qubitstring to represent the entire search space without compro-mising the quality of solution [8]. This reduces the compu-tational burden and enables the solution of large-sizedproblems. QEAs are very recent and have not yet beenattempted for power system problems. In the same lines,principles of QC have been embedded in various stochasticsearch techniques such as particle-swarm optimisation [9,10] and GAs [11, 12].

Han and Kim [7, 13] have proposed a QEA formulationthat is suitable to optimise functions with binary parameters.However, binary representation is not suitable for many ofthe real-world problems, in general, and power systems pro-blems, in particular. ELD optimises real-parameter func-tions. Using binary numbers to represent the parametersforces a trade-off between accuracy of representation andstring lengths. Hence, in this paper, a novel QEA formu-lation to optimise functions with real parameters called real-parameter QEA (RQEA) is presented which optimise thereal-valued functions. To enable this, new quantum oper-ators have been designed to generate solution strings withreal parameters. In contrast, the method proposed by Hanand Kim [7, 13] performs observation to result in solutionstrings with binary parameters. In this paper, a novelQEA-based technique (RQEA) is presented and applied toELD problem. Results obtained on several examples usingproposed technique are compared with those present inliterature.

The rest of the paper is organised as follows. Section 2describes the preliminaries necessary for understandingQEA. Section 3 describes the proposed technique, RQEA.Section 4 deals with the implementation details for theELD problems. Section 5 presents the computational per-formance of RQEA on ELD problems and comparisonwith other methods. Conclusions are drawn in Section 6.

2 QEA preliminaries

QEA is a population-based probabilistic EA that integratesconcepts from QC for higher representation power androbust search. It maintains a population of individuals inqubits. A qubit-coded individual probabilistically representsall the states in the search space. Thus, it has a better charac-teristic of population diversity than other representations[7]. The preliminary details necessary for understandingQEA are presented in this section.

IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008

2.1 Qubit

QEA uses qubits as the smallest unit of information forrepresenting individuals. Each qubit is represented as

qi ¼ai

bi

� �

State of the qubit (qi) is given by jCil ¼ aij0lþ bij1l, ai

and bi are complex numbers representing probabilistic stateof qubit, that is, jaij

2 is probability of the state being0 and jbij

2 is the probability of the state being 1, such thatjaij

2þ jbij

2 ¼ 1. For the purpose of QEA, nothing is lostby regarding a and b to be real numbers.

In binary-coded QEA, each individual is represented by aset of n qubits in a chromosome string as

Q(t) ¼a1 a2 a3 . . . ai . . . an

b1 b2 b3 . . . bi . . . bn

� �such that

jaij2þ jbij

2¼ 1 where i ¼ 1, 2, 3, . . . , n

For instance, a 3-qubit string with probabilities as

Q(t) ¼

1ffiffiffi2

p�1ffiffiffi

2p

ffiffiffi3

p

2

1ffiffiffi2

p1ffiffiffi2

p1

2

2664

3775

has states that can be represented as

�

ffiffiffi3

p

4j000l�

1

4j001lþ

ffiffiffi3

p

4j010lþ

1

4j011l�

ffiffiffi3

p

4j100l

�1

4j101lþ

ffiffiffi3

p

4j110lþ

1

4j111l

resulting in probabilities of states from ‘000’ to ‘111’ being3/16, 1/16, 3/16, 1/16, 3/16, 1/16, 3/16 and 1/16,respectively. Hence, the information of eight binary states,which requires eight strings in binary representation, is rep-resented by a single string in the qubit system. Thus, a qubitstring with n bits represents a superposition of 2n binarystates and provides an extremely compact representationof the entire search space.

2.2 Observation

The process of generating binary strings from the qubitstring, Q, is called observation. For simplicity, the qubitstring is referred to as Q because it is clear from thecontext whether the first row containing ais is intended orthe reference is to the whole qubits, that is, both ais andbis. In an actual quantum computer, the observation oper-ation results in the collapse of superposition of states toone resultant classical state. However, in QEA simulationon classical computer, collapse of states does not occur.Hence, to observe the qubit string (Q), a string consistingof the same number of random numbers between 0 and 1(R) is generated. The element Pi is set to 0 if Ri is lessthan the square of Qi and 1 otherwise. Table 1 representsthe observation process.

2.3 Updating qubit string

In each of the iterations, several solution strings are gener-ated from Q as given above and their fitness values are com-puted. The solution with best fitness is identified. Theupdating process moves the elements of Q towards thebest solution slightly such that there is a higher probability

23

of generation of solution strings, which are similar to bestsolution, in subsequent iterations. A quantum gate is utilisedfor this purpose so that qubits retain their properties [7, 13].One such gate is rotation gate, which updates the qubits as

atþ1i

btþ1i

" #¼

cos (Dui) � sin (Dui)

sin (Dui) cos (Dui)

� �ati

bti

" #(5)

where aitþ1 and bi

tþ1 denote the probabilities for ith qubit in(tþ 1)th iteration and Dui is equivalent to the step size intypical iterative algorithms in the sense that it defines therate of movement towards the currently perceived optimum.

The above description outlines the basic elements ofQEA. The qubit string, Q, represents probabilistically thesearch space. Observing a qubit string ‘n’ times yields ‘n’different solutions because of the probabilities involved.Fitness of these is computed, and the qubit string, Q, isupdated towards higher probability of producing stringssimilar to the one with highest fitness. The sequence ofsteps continues. The above ideas can be easily generalisedto work with multiple qubit strings that enhance thesearch power further.

3 Real-parameter QEA

The QEA outlined above is good to optimise functions withbinary parameters [7]. However, requirement of handlingreal parameters dictates that RQEA has to be designeddifferently.

The main features of RQEA are as follows.

† RQEA uses qubit representation that has a better charac-teristic of population diversity. This enables the use ofsmaller population with corresponding reduction in compu-tational effort just as in the case of QEA.† Unlike QEA, which uses the concept of observing qubitstrings to generate candidate feasible solutions, RQEA usesspecial quantum evolution operators to generate the candi-date solution strings that comprises real parameters. Theoperator has to strike a balance between adequate explora-tion and full exploitation.† Like QEA, RQEA also uses quantum rotation gate forupdating the qubit strings.† Migration between families of solutions, which are pro-duced from different qubit strings, is employed in RQEAwhich helps in improving the convergence and the qualityof solution strings.

These features along with the inherent representationalpower of qubits and search power of quantum operatorsmake RQEA a powerful, flexible and robust candidatealgorithm for the solution of complicated real-parameterproblem.

One straight forward approach to make RQEA flexibleenough to handle real parameters is to provide multiplequbits for encoding each parameter and then regarding thebinary string generated by collapsing the qubits as repre-senting a real number between 0 and 1. However, for

Table 1: Observation of qubit string

i 1 2 3 4 5 . . . Ng

Q2 0.17 0.78 0.72 0.41 0.89 . . . 0.36

R 0.24 0.07 0.68 0.92 0.15 . . . 0.79

P 1 0 0 1 0 . . . 1

24

functions with steep changes, the level of discretisation isnot sufficient if a small number of qubits encode each par-ameter and allocating more number of qubits per parameterfor a finer level of discretisation would result in unwieldystrings and larger search spaces, making the optimisationeffort cumbersome.

Further, in this representation, changing one bit does notresult in neighbouring solution (consider 101 and 001).Grey code representation can be proposed to mitigate thisproblem but it obscures the relationship between thestring and the objective function value that it yieldsmaking genetically meaningful operators difficult to design.

RQEA has a more sophisticated approach to generatereal-numbered candidate solution strings during the searchprocess. A set of Np qubit strings, Qi

t, i ¼ 1, 2, . . . , Np, ismaintained as in QEA in tth iteration. Correspondingly,another set of Np strings each of Ng real numbers, Pi

t,i ¼ 1, 2, . . . , Np, is maintained. Each Qi also has Ng

qubits, which represent probability amplitudes (ai). Theprobability of generating a real number on the higher(lower) side of current value is given by jaij

2(jbij2). To

have these probabilities equal in the beginning of search,ai and bi of Qi are initialised with 0.707. The value ofeach element of Pi, i ¼ 1, 2, . . . , Np, is initialised with arandom number between the minimum and maximumvalues allowed for the parameter. Each pair consisting Qi

t

and Pit represent the ith family in tth iteration.

For ith family, Nc solution strings, pijt , j ¼ 1, 2, . . . , Nc,

are generated using Qit, Pi

t and PBESTt , where PBEST

t is thesolution string with best fitness found so far. This processis described later in this section. The fitness of each of thestrings, pij

t is evaluated after ensuring constraint compliance.Two neighbourhood operators, neighbourhood operator 1

(NO1) and neighbourhood operator 2 (NO2) are used togenerate the Nc neighbourhood solution strings, and thebest out of these, Ci

t, is determined for each family i. If Cit

is better than Pit, it replaces Pi

t to become Pitþ1. The best

out of all Pitþ1s, i ¼ 1, 2, . . . , Np, replaces PBEST

t tobecome PBEST

tþ1 if it is found better.

3.1 Evolving qubit

Unlike QEA, where observation of each qubit results ineither of the classical states 0 or 1, in RQEA, the qubit isevolved. Here, evolving a qubit implies that state of thequbit, which is a superposition of state 0 and 1, is shiftedto a new superposition state. This change in probabilitymagnitudes jaj2 and jbj2 with change in state is transformedinto real-valued parameters in the problem space by twoneighbourhood operators, NO1 and NO2.

3.1.1 Neighbourhood operator 1: Given in the tth iter-ation, Np qubit strings Qi (8i ¼ 1, 2, 3, . . . , Np), each havingNg elements, NO1 generates solution strings pij

t (8j ¼ 1, 2,3, . . . , Nc), each having Ng elements. This is done for a par-ticular i, j as follows.

An array Rij is created with Ng elements generated atrandom such that every element in Rij is either þ1 or 21.Let rijk be the kth element in Rij. Then uijk

t is given by

u tijk ¼ u t�1

ijk þ rijkd00 (6)

where d is the alteration in angle and uijkt is the rotated angle

which is given as tan21(bijk/aijk). d is randomly chosen inthe range [0, uijk

t21] if rijk ¼ 21 and in the range [uijkt21,

p/2] if rijk ¼ þ1.


The new probability amplitudes, aijkt and bijk

t , are calcu-lated using rotation gate as

atijk

btijk

" #¼

cos d sin d

� sin d cos d

� �at�1ijk

bt�1ijk

" #(7)

These probabilities are then transformed to solution spaceto determine individual element as

Ptijk ¼ (at

ijk)2(Pkmax � Pkmin) þ Pkmin (8)

where Pkmax and Pkmin are the maximum and minimumallowable values for Pk 8k.

Pseudo-code of NO1 is presented in Fig. 1. Fig. 2 illus-trates the functioning of NO1. For the sake of simplicity,family identifier and child number identifier have not beenused in the illustration and uijk

t implies ukt . Fig. 2 shows

the mechanism for case R ¼ þ1 on the left and R ¼ 21on the right.

3.1.2 Neighbourhood operator 2: NO2 works justas NO1 except that it generates a point between Pb

and PBEST and is primarily utilised to exploit searchspace. The NO2 determines the Pijk

t using (8) suchthat Pijk

t ¼ (aijkt )2 (Pkmax 2 Pkmin) þ Pkmin, where

Pkmax ¼max(PBEST ik, Cit) and Pkmin ¼ min(PBEST ik, Ci

t)Here, Ci

t is the best child of ith family in tth iteration.The rationale for two neighbourhood operators is as

follows. NO1 has a greater tendency for exploration in thesense that solutions generated for a given string could bequite different from the given string. NO2 has a greater ten-dency towards exploitation because, as the algorithm pro-gresses, the values of Pj would converge towards PBESTj.The desired property of search by any evolutionary algor-ithm is that exploration should gradually yield to

Fig. 1 Pseudo-code for NO1 for kth element of ith family’s jthchild in tth iteration


exploitation. Thus, the two operators are used with the fre-quency of NO1 taken high to start with and increasing theproportion of NO2 as the search progresses. Table 2shows the frequency of the use of NO1 and NO2.

The neighbourhood operators thus generate new feasiblesolution strings directly from the existing strings. Thisapproach removes all disadvantages of binary represen-tation of real numbers and, at the same time, balancesexploration and exploitation in the sense that it adopts the‘step-size’ from large initially to progressively smaller size.

3.2 Updating qubit string

In the updating process, the individual states of all the qubitsin Q are modified so that the probability of generating a sol-ution string which is similar to the current best solution isincreased in each of the subsequent iterations. The amountof change in these probabilities is decided by learningrate, Du. Updating process is done as explained in Section2 using (5) for a minimisation problem. Table 3 presentsthe choice of Du for various conditions of objective functionvalues and kth element of Pi and PBEST in tth iteration.

The updating process is illustrated in Fig. 3 for kthelement for tth iteration, that is, changes in state of kthqubit and corresponding change in probability amplitudes.

Premature convergence indicates the identicalness of sol-utions in population and is prevalent in most of thepopulation-based search techniques. In QEA, the diversityin population is decided by the qubit string. When a qubithave a as 0.707, diversity is highest. The diversity is leastwhen the value of a is near the extremes, that is, 0 or1. In QEA, learning rate (Du) decides the speed at whichthe qubits move from the initial value of 0.707 to a finalvalue of 0 or 1 in the updating process. This learning rateis kept small enough to ensure that shift of a from 0.707to 0 (or 1) takes a large number of iterations. Hence, theprobability of generating solution strings that are extremelysimilar to the ‘best solution found so far’ is high only whenmost of the qubits have converged very close to either 0or 1.

In contrast, too small a learning rate will result in slowconvergence speeds. Hence, proper choice of Du is necess-ary. In this work, learning rate has been chosen as 0.001pwhich, on experimentation, was found to be working

Fig. 2 Graphical illustration of NO1

25

satisfactorily for the test cases considered. Similar value hasbeen selected in [7].

Moreover, the updating process alters the values ofelements of qubit string selectively as explained inTable 3. Hence, shifting of one qubit to its extreme valuedoes not force other qubits to generate similar realnumbers. Hence, qubit representation will provide betterdiversity even if a part of the solution string remains samefor a long time.

3.3 Migration

In the proposed technique, two levels of migration areincorporated, local and global. In local migration, one ran-domly chosen Pt is used to update some Qi

t. In globalmigration, PBEST is used to update Qi

t8i ¼ 1, 2, . . . , Np.

Apart from the algorithmic details discussed in thissection, some implementation details have to be takencare of for solution of ELD using RQEA.

4 RQEA for ELD

The ideas mentioned above provide an excellent frameworkfor the design of an iterative improvement optimisationalgorithm. Its simplicity implies that it can be adopted tosolve a variety of problems. But it also implies that forthe solution of any reasonable-sized problem, severaldetails are to be appropriately designed. Also, in heavily

Fig. 3 Updating kth element of qubit string Q

Table 2: Frequency of the use of NO1 and NO2

Stage of search Proportion

of NO1 (%)

Proportion

of NO2 (%)

first one-fifth iterations 90 10

second one-fifth iterations 70 30

third one-fifth iterations 50 50

fourth one-fifth iterations 30 70

last one-fifth iterations 10 90

Table 3: Calculation of Du for tth iteration

Fitness Elemental values Du

X Cij ¼ PBEST j 0

F(P) . F(PBEST) Cij . PBEST j 0.001p

Cij , PBEST j 20.001p

F(P) , F(PBEST) Cij . PBEST j 20.001p

Cij , PBEST j 0.001p

F(P) ¼ F(PBEST) Cij . PBEST j 0

Cij , PBEST j 0

26

constrained scenarios, effort is needed to handle the con-straints and ensure the feasibility of solutions obtained.Furthermore, convergence can be improved by incorporat-ing special domain-specific knowledge particular to theproblem at hand. The implementation details of RQEAare presented here.

4.1 Problem representation

Since the task at hand in any ELD problem is to determinethe loading of each generator, the most obvious represen-tation scheme for representing the complete search spaceis to use real numbers between 0 and 1 with 0 representingPgmin and 1 representing Pgmax and any other numberbetween 0 and 1 to be scaled between Pgmin and Pgmax todetermine the actual loading of the generator. Example ofsuch representation is found in [4, 6, 14].

However, with such a representation, the search stringcan only represent the point it describes in the solutionspace. It does not have the power to represent the historyof the previous moves in terms of direction of change inloading of generators. However, in RQEA, the searchstring composed of qubits stores information regarding thedirection of change and in the loadings for each generatoras evidenced by the history of previous moves. The ideais as follows.

Fig. 4 Structure of RQEA

Fig. 5 Pseudo-code for RQEA


Table 4: Operating costs for IEEE 30 Bus System with and without considering losses

Losses neglected Losses considered

[16] [17] [18] RQEA [16] [17] [18] RQEA

Pg1 p.u. 0.10954 0.10954 0.1095 0.1097 0.11516 0.11516 0.1152 0.1250

Pg2 p.u. 0.29967 0.29967 0.2997 0.2998 0.30552 0.30552 0.3055 0.3150

Pg3 p.u. 0.52447 0.52447 0.5245 0.5242 0.59724 0.59724 0.5972 0.5350

Pg4 p.u. 1.01601 1.01601 1.0160 1.0161 0.98088 0.98088 0.9809 1.0150

Pg5 p.u. 0.52469 0.52469 0.5247 0.5242 0.51421 0.51421 0.5142 0.5150

Pg6 p.u. 0.35963 0.35963 0.3596 0.3599 0.35417 0.35417 0.3542 0.3554

cost ($/MW h) 600.114 600.114 600.11 600.0892 607.777 607.777 607.78 606.0297

loss p.u. 0.03318 0.03318 0.0332 0.0264

Since there is one dependent and n2 1 independent gen-erators, any change at a particular point in search space canbe in one of the 2n21 directions with two possibilities, thatis, an increase or decrease for each generator. Since theapplication of NO1 or NO2 on the qubit string can resultin any one of these 2n21 possibilities as described above,the n2 1 qubit string gives a complete representation ofthe space of possible changes.

4.2 Population size and number of children

The selection of population size and number of children tobe produced in each generation is a compromise between awider exploration of the search space and increased compu-tational burden. The choice is somewhat restricted if theevaluation of the optimisation goal requires a substantialamount of computational resources. In contrast, if the evalu-ation of the objective function is computationally fast,larger number of children can be produced which willresult in better exploration of search space at each step. Inthe proposed technique, a total of Np

�Nc solution stringsgenerated in each iteration, where Np is the number ofqubit strings and Nc the number of solutions generatedfrom each qubit string in one iteration. In this implemen-tation, suitable values of Np and Nc have been found exper-imentally to be 5 and 10, respectively.

Fig. 6 Convergence of RQEA for example 1

Table 5: Best costs obtained from 30 runs of RQEA forIEEE 30 bus system with losses being neglected

Best run Worst run Average Standard deviation

600.0892 600.1116 600.0995 0.00748


4.3 Constraint handling

In this formulation, all the constraints are strictly adhered tothroughout the search. Hence, imposing penalty on infeas-ible solutions is not necessary. This is achieved as follows:

† Inequality constraint: given by (4).The two quantum operators, NO1 and NO2, are designed sothat the mapping of qubits from u-space to generator load-ings in solution space does not violate generator capacityconstraints.† Equality constraint: given by (3).The equality constraint of power balance is taken care of byconsidering Ng2 1 generations as part of the solution stringand calculating Ngth generation from load flow analysis.

4.4 Stopping criterion

The stopping criterion can be based on the number of iter-ations, number of fitness evaluations and convergence ofsearch or a combination of them. The diversity in popu-lation is decided by the real-valued qubit string. When thequbits have a value of a as 0.707, diversity is highest.The diversity is least when the value of a is near theextremes, that is, 0 or 1. Hence, the level of convergencecan be termed as the number of qubits which havereached very close to 0 or 1.When this number is equal tothe number of elements in Q, the chances of the observationprocess generating diverse solutions becomes very low andone can say that the search has converged. In this work,maximum number of iterations is 1000 for first two testcases and 10 000 for the third test system.

Various details and structural improvements discussed inprevious sections have been incorporated to develop RQEA.The structure is presented in Fig. 4 and the pseudo-code isgiven in Fig. 5.

5 Test results

RQEA has been applied on various standard test systemsand results have been compared with those of SA variants[4], HSS [6], PSO-based method [15] and RGA from[16–18]. Solution of analytical ELD [3] has been presentedwherever it is applicable, that is, lossless system with con-tinuous and quadratic cost curves. As analytical ELD pro-vides global optimum for these cases, more explicitcomparison of these methods may be presented. The plat-form used for implementing RQEA is a Pentium IVmachine with 1 GB RAM and Matlab is used as program-ming language. However, the fact that CPU times do varywith many parameters such as RAM, speed of RAM,

27

Table 6: Comparison of solutions and operating costs for example 2

Units SA [4] SAF [4] SAB [4] HSS [6] EP-SQP [15] PSO-SQP [15] RQEA

Z1 629.78 627.86 628.06 628.23 628.3136 628.3205 628.3170

Z2 300.66 299.17 298.65 299.22 299.1715 299.0524 299.1991

Z3 301.33 298.84 298.48 299.17 299.0474 298.9681 299.1990

A1 158.58 159.16 159.44 159.12 159.6399 159.4680 159.7334

A2 163.50 159.66 159.71 159.95 159.6560 159.1429 159.7331

A3 163.03 159.27 159.44 158.85 158.4831 159.2724 159.7330

A4 161.08 159.19 159.62 157.26 159.6749 159.5371 159.7324

B1 154.61 159.73 159.73 159.93 159.7265 159.8522 159.7329

B2 159.58 159.64 159.74 159.86 159.6653 159.7845 159.7331

B3 104.49 111.47 111.87 110.78 114.0334 110.9618 107.4875

C1 75.00 75.00 75.00 75.00 75.00 75.00 75.0000

C2 60.00 60.00 60.00 60.00 60.00 60.00 60.0000

C3 88.40 91.01 90.25 92.62 87.5884 91.6401 92.3994

($/MW h) 24 484.89 24 260.72 24 259.56 24 275.71 24 266.44 24 261.05 24 252.95

Operating cost of each generator is given as Fi(Pgi) ¼ aiPgi

2þ biPgi

þ diþ jei sin( fi (Pgmini2 Pgi

))j

CPU, Cache, OS, Programming Environment and so on, andit is meaningless to compare the results unless all these par-ameters are matched, prompted us to provide the perform-ance of RQEA using number of function evaluations asthe criterion rather than CPU times.

28

5.1 Test case 1

For IEEE30 bus system, two cases were considered, that is,with and without transmission losses. The generator fuelcost coefficients are provided in [18]. The results obtained

Fig. 7 Convergence of RQEA for example 2

Fig. 8 Best costs obtained by RQEA in 30 different runs forexample 2

have been compared with those presented in Table 2 of[16–18] with cost minimisation as the sole objective.Table 4 presents the operation costs of various techniquesand the p.u. loadings on the generators for transmissionlosses neglected and the results of the same system whentransmission losses were considered. Fig. 6 presents theconvergence curve of RQEA for example 1. Table 5 pre-sents the best, worst and average values of best costsobtained by RQEA in 30 runs of example 1 without consid-ering losses.

It is evident that RQEA has converged to better solutionswhen losses are not considered and when losses wereincluded. The difference in the results of the worst andthe best runs out of 30 different runs is only 0.012. This indi-cates the consistency of the proposed technique in reachingthe region of better solutions.


5.2 Test case 2

In this example, a 13-generator system is considered.Generator cost curves includes valve point discontinuitiesand the test data are provided in [6]. Results obtained arecompared with those obtained with SA variants [4], HSS[6] and SQP integrated with EP (EP-SQP) and particleswarm optimisation (PSO-SQP) as presented in Table 6 of[15]. Table 6 presents the results obtained. Fig. 7 illustratesthe convergence of RQEA and Fig. 8 presents the best costsobtained by RQEA in 30 different runs for example 2.

The solution quality of RQEA is better than thoseobtained by other methods presented in Table 6 and pro-vides an improvement of $6.61 over the next best resultobtained by SAB [4]. The horizontal axis in Fig. 7 showsthe number of evaluations. However, the flat portion inthe curve, that is, from 5000 evaluations to 18 000 evalu-ations also contains some improvements shown by points.Discontinuous cost curves and the nonlinear constraints ofthe test case make it difficult to solve. However, RQEA isable to reach the global minimum eventually. In this diffi-cult problem with discontinuous cost curves, RQEA has

Table 7: Comparison of solutions obtained by TSA, SDEand RQEA

Unit TSA [19] SDE [20] RQEA

Z1 628.319 628.3185 628.3185

Z2 299.1993 299.1993 299.1993

Z3 331.8975 294.4818 294.4835

A1 159.7305 159.7331 159.7331

A2 159.7331 159.7331 159.7331

A3 159.7306 159.7331 159.7330

A4 159.7334 159.7331 159.7330

B1 159.7308 159.7331 159.7331

B2 159.7316 159.7331 159.7331

B3 40.0028 77.3999 77.3997

C1 77.3994 77.3999 77.3997

C2 92.3932 92.3999 92.3996

C3 92.3986 92.3999 92.3991

($/MW h) 24 312.78 24 173.888 24 164.048


exhibited excellent consistency with only 0.16% differencebetween best and worst solutions and a standard deviationof 6.77.

The generators C1 and C2 are in ramping and, hence,load shared by them is fixed at 75 and 60 MW, respectively,in the above example [4, 6, 15]. However, some authorsconsider a variation of the same by relaxing restriction ongenerations of C1 and C2 [19, 20]. Since the constraint isrelaxed, it is possible to obtain better solutions. RQEAhas been used to solve this variation and the generationsobtained have been compared with those presented inTable 1 of [20]. Table 7 presents the generations obtainedby a Tabu search-based method [19], Self-adaptive differen-tial evolution technique [20] and RQEA. The fitness of thesestrings has been computed using (2). Quality of solutionobtained by RQEA is superior.

5.3 Test case 3

A large system with 110 generators has been considered andthe performance of RQEA has been observed for light(10 000 MW), medium (15 000 MW) and heavy(20 000 MW) loads. Data for generator cost curves can beobtained from [21]. The operating costs obtained havebeen compared with those obtained by SA variants andanalytical ELD. As the analytical ELD provides a globaloptimum for these cases, the performance of techniques inreaching global minimum can be compared. The compari-son of results obtained in 30 different runs is presented inTable 8.

The consistency of RQEA is evident from the fact that the% distance of worst of the solutions is of the order of 1026

in the case of light load and even smaller, that is, of theorder of 1027 for other two load conditions. This alsoemphasises the scalability of the proposed technique.Fig. 9 shows the convergence of RQEA for different loadconditions considered. In this, y-axis consists of costs nor-malised between 0 and 1with 0 indicating the minimumcost of the run and 1 the maximum cost.

6 Discussion

In RQEA, NO1 and NO2 are responsible for searching thesolution space. A fine balance between exploration andexploitation of the search space is essential for any algor-ithm to be effective. To evaluate and illustrate the effect

Table 8: Comparison of solutions obtained from 30 different runs for example 3

Loading condition Analytical [3] SA [4] SAB [4] SAF [4] RQEA % Distance,

from global min

Light

best 131 941.8838 145 550.4412 140 385.7586 141 107.8541 131 941.8851 9.853 � 1029

average 146 757.7060 141 213.4207 141 215.1159 131 942.0439 1.213 � 1026

worst 147 476.4295 141 900.2431 141 398.0923 131 942.4931 4.618 � 1026

best 197 988.1775 216 100.5475 206 921.9057 207 380.5164 197 988.1793 9.091 � 1029

Medium

average 216 365.7269 207 764.7398 207 813.3717 197 988.1835 3.030 � 1028

worst 216 823.5408 208 197.0059 208 012.6248 197 988.2006 1.167 � 1027

best 313 211.5688 314 647.0416 313 279.8825 314 532.8747 313 211.5688 0.00

Heavy

average 315 659.1453 314 271.7484 314 635.3244 313 211.5983 9.419 � 1028

worst 317 385.2167 314 723.8825 314 783.5061 313 211.8189 7.985 � 1027

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Fig. 9 Convergence of RQEA for example 3, for low, medium and heavy-load condition

Fig. 10 Effect of neighbourhood operators on convergence of RQEA

of the neighbourhood operators NO1 and NO2, test case 2has been evaluated with different combinations of NO1and NO2 for 1000 iterations flat and random initialisation.These cases are as follows:

1. NO1 only;2. NO2 only;3. either NO1 or NO2 is applied with equal probabilitythroughout the run;4. frequency of application of NO1 and NO2 in differentsearch stages as per Table 2.

Fig. 10 illustrates the typical convergence of all theabove-mentioned cases. Application of NO1 alone resultedin quick and large jumps initially and the search gets stuck.On the contrary, NO2 provides finer search. However, con-vergence is slow and also converges to suboptimal solution.Simultaneous application of both the operators has beenexplored and was found better than the application of anyone of the operators. For the problem chosen, case 4 hasbeen found to provide better quality of solutions andhence used.

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7 Conclusions

ELD is an important problem in power systems. The objec-tive is to find the optimal loadings on the generators so asto achieve minimum operating cost. The proposed technique,RQEA, is equipped with qubit representation, quantum oper-ators. A novel technique of evolving qubit to enable RQEA tohandle real parameters directly has also been developed.RQEA has been applied to standard ELD problems and com-pared with other methods. The performance of RQEA isbetter in the sense that it has provided better quality solutionsand has exhibited excellent consistency in reaching globalminima. RQEA is computationally inexpensive as it requirescomparatively smaller population sizes. The method requireslesser number of parameters to be tuned than EA, and henceits implementation is user-friendly. The method developed isquite general. Results obtained are promising and theapproach can be applied to other complex problems andcan be extended to multi-objective problems.

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real-parameter quantum evolutionary algorithm for economic load dispatch

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