Advanced engineered-cond to power economic dispatc

Y. H . Song C.S.V.Chou

Indexing terms: Genetic algorithms, Power system optimisation, Economic dhpatch

Abstract: Computational efficiency and reliability are the major concerns in the application of genetic algorithms (GAS) to practical problems. Effort has been made in two directions to improve the performance of GAS: the investigation of advanced genetic operators and the development of genetic algorithm hybrids. In this paper, an advanced engineered-conditioning genetic algorithm hybrid (AEC-CA) is proposed, which is a combination strategy involving local search algorithms and genetic algorithms. Moreover, several advanced techniques which enhance program efficiency and accuracy, such as elite policy, adaptive mutation prediction, nonlinear fitness mapping and different crossover techniques, are explored. Using power economic dispatch problems as a basis for comparisons, the outcome of the study clearly demonstrates the advantages of the AEC-GA.

1 Introduction

Electric power systems and their operation are among the most complex problems in today’s civilisation due to highly nonlinear and computationally difficult environments. The basic requirement of power economic dispatch (ED) is to generate adequate electricity to meet continuously changing customer load demand at the lowest possible cost under a number of constraints. Conventional techniques often use approximations to limit complexity which suffer from myopia for nonlinear, discontinuous search spaces, leading them to less than desirable performance. When search spaces are particularly irregular, algorithms need to be highly robust in order to avoid getting stuck at local optima. Although adding a randomisation step to a hillclimber technique can improve its performance, for many large and complex search spaces this may not be efficient [I, 21.

More recently, attentions have been focused on the applications of modern heuristic techniques, including artificial neural networks [3], simulated annealing [4] and evolutionary computing [5 ] . In particular, based on 0 IEE, 1997

Paper received 16th July 1996 The authors are with the Department of Electrical Engineering and Elec- tronics, Bninel University, Uxbridge UB8 3PH, UK

IEE Proceedings online no. 19970944

tioning genetic approach 1

the principles of genetics and natural selection, evolu- tionary computing is a promising adaptive search tech- nique. As one of such evolutionary models, genetic algorithms (GAS) are being applied to a variety of power system problems where conventional methods have experienced some difficulties. The major applica- tion areas [6] are: (i) unit commitment [7] and genera- tion scheduling [S, 91; (ii) reactive power planning/ dispatch [3]; (iii) distribution network planninghecon- figuration [ll]; (iv) load flow [12]; (v) alarm processing and fault diagnosis [13]; (vi) power system modelling and control [14] etc.

However, it has been realised that simple forms of GA have some drawbacks which prevent the accept- ance of the theoretic performances claimed. Thus vari- ous techniques [ 15-1 81 have been studied to improve genetic search mainly by developing advanced genetic operators and exploring genetic algorithm hybrids. The paper proposes an advanced engineered-conditioning genetic algorithm hybrid (AEC-GA), which is a combi- nation strategy involving local search algorithms and genetic algorithms. Moreover, several advanced tech- niques which enhance program efficiency and accuracy, such as elite policy, adaptive mutation prediction, non- linear fitness mapping, different crossover techniques, are explored. Using power economic dispatch problems as a basis for comparisons, the outcome of the study clearly demonstrates the advantages offered by the advanced engineered-conditioning genetic algorithm hybrid (AEC-GA).

2 Classic economic dispatch problem

The objective of the ED problem is to minimise the total fuel cost at thermal plants:

n

i=l

Subject to the constraint of equality in real power balance

i=l The inequality of real power limits on the generator outputs are

where Fi(Pi) is the individual generation production cost in terms of its real power generation P,. Pi = output generation for unit i n = number of generators in the system

P i m i n 5 Pi I P i m a x (3)

285 IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 3, May 1997

Po = total current system load demand P, = total system transmission losses

The thermal plant can be expressed as input-output models (cost function), where the input is the fuel cost and the output is the power output of each unit. In practice, the cost function could be represented by a quadratic function

(4) Incremental cost curve data are obtained by taking the derivative of the unit input-output equation resulting in the following equation for each generator:

(5)

Transmission losses are a function of the unit generations and are based on the system topology. Solving the ED equations for a specified system requires an iterative approach since all unit generation allocations are embedded in the equation for each unit. In practice, the loss penalty factors are usually obtained using online power flow software. This information is updated to ensure accuracy. They can also be calculated directly using the B matrix loss formula.

i j i

where B = loss coefficients

3 Overview of genetic algorithms

3. ‘I Basic structure of genetic algorithms Genetic algorithms represent a class of general purpose stochastic search techniques which simulate natural inheritance by genetics and the Darwinian ‘survival of the fittest’ principle. The basic mechanism of a canoni- cal GA consists of the following sequence, which is shown in Fig. 1.

t : = o ‘r’ InitialJse

population P(t)

evaluate Pi t ) 0

mate selection

crossover

mutation

evaluate P(t)

Fig. 1 Basic structure of genetic algorithms

(i) code the problem; (ii) randomly generate initial population strings;

286

(iii) evaluate each string’s fitness; (iv) select highly-fit strings as parents and produce off- springs according to their fitness; (v) create new strings by mating current offsprings, apply crossover and mutation operators to introduce variations and form new strings; (vi) finally, the new strings replace the existing ones. This sequence continues until some termination condi- tion is reached.

3.2 lmplemen tation difficulties GAS are intended to provide a means to attack prob- lems resistant to other known methods. In most cases, although the performance of GAS is comparable to, or better than, conventional optimisation techniques, they still fail to live up to the high expectations engendered by the theory as experienced by other classes of algo- rithms. The main problem of the GA process is prema- ture convergence before the true global optimum solution have been found. During the genetic search, the process converges when the individuals within the population pool are identical, or nearly so. Once this occurs, the crossover operator ceases to produce new individuals, and the algorithms allocates all of its trials in a very small subset of the space. The intuitive reason for premature convergence is that the individuals in the population pool are too alike. However, the mutation operator provides a mechanism for reintroducing lost alleles, but it does so at the cost of slowing down the learning process. To assure that different members of the population pool are different, to minimise the effect of premature convergence, local search algorithm and nonlinear fitness mapping can be introduced. Another problem with genetic algorithms is computation speed, which can be addressed by introducing more advanced genetic operators.

3.3 Improvement approaches There are six ways which can lead to the performance enhancement of GAS: (i) Strings representation: representation of the problem parameter set is important. The encoding must be designed to utilise the algorithm’s ability to transfer information between chromosome strings efficiently and effectively. (ii) Initial string generation: the formation of the initial population is not limited to random generation, but can instead take into account knowledge of the system. (iii) Evaluation function: implementation of an objec- tive function and constraints in a GA are realised within the fitness function. Choosing and formulating an appropriate objective function is crucial to the effi- cient solution of any given GA problem as GAS rely on the evaluation information to guide the search towards a global optimum. (iv) Genetic operators: they form the backbone of the genetic algorithms and they are used to alter the composition of chromosomes. The fundamental genetic operators such as mate selection, crossover, and mutation are used to create children (individuals in the next generation) that differ from their parents (individuals in the previous generation). There are many ways in which these operators can be implemented. (v) Parameter values: including crossover rate, muta-

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tion rate, population size, chromosome length, total generation, can play an important role in performance enhancement. (vi) Hybrid technique: A GA may be crossed with various problem-specific search techniques to form a hybrid that exploits the global perspective of the GA (global search) and the convergence of the problem- specific technique (local search). In this paper, effort has been devoted to developing advanced genetic operators and forming hybrid techniques.

4 Adwanced implementation of genetic algorithms

4. I Encoding and decoding Encoding is the process of coding a problem as a number of finite strings. It typically utilises the binary alphabet (0, 1). Two types of encoding schemes have been developed by researchers which are called series encoding and embedded encoding [15]. The series encoding simply stacks each unit’s output value struc- ture in series with each other in the string. Each unit’s output gene structure is assigned the same number of loci within the string. The embedded encoding scheme uses the same system for representation and decoding as the first, except the assigned gene structures are embedded within each other throughout the string. The string is made up of a series of smaller gene structures, each containing one gene locus for each unit. It has been reported that series encoding can provide a better ED solution [15]. In this paper, a binary series coding is used throughout all the GAS.

Decoding a binary string into an unsigned integer can play very important roles in GA implementation. The inequality power limit constraint is performed in such a way that the individual string is normalised over the unit’s operating region. The inequality constraints are handled in this manner, which efficiently reduces the searching space, and thus enhances the perform- ance of the system.

The decoding method is formulated in eqn. 7:

value =bit0 * 2’ + bit1 * 2l + . . - + bit, * 2’

(7 ) * 2chrom-length ’ ‘ f bitchromdength

If the optimised parameter belongs to (P,,,,, P,,,,), the decoding value of the parameter is computed by eqn. 8.

(8) Pz min + value * (e max - pz min)

2chrom-length-1 P, =

4.2 Objective and fitness function form dation As the fitness value is the only information GAS used to guide the search towards the optimal points, the fit- ness function must be formulated effectively. The major difficulty in fitness formulation is in handling constraints. The LaGrange function is used so that constraints are implemented by imposing penalties in individuals that violate them.

Since a chromosome string’s fitness will be compared with all other strings within the same population, an absolute measure of optimality is not required. Likewise, the constraint equation error can be calculated as a comparison within the same population. The result is a fitness function based on a percenage rating. It can not only treat equality constraints and

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objective function effectively, but also can consider additional constraints very easily.

4.2. I Power balance constraht handling: The power balance constraint equation can be rewritten as:

i=l (9)

where P, = output generatien for unit i n = number of generatsrs in the system Po = total current system load demand PL = total system transmission losses In addition, the inequality constraints are taken into consideration during the optimisation process. If the output of a generator unit is over its upper limit, a pen- alty factor p2 as shown in eqn. 10 is added into the power balance equation.

Otherwise, if the output of a generator is less than its lower limit then the penalty factor is computed by eqn. 11

~z = [pzmin/PzI2 (11) where p, = power balance penalty factor for unit i.

The power balance equation can be rewritten as I n I n

Any power balance error can be further eliminated by redistributing the portion of error iato each unit

D

4.2.2 Fitness function formulation: Based on the lambda-iteration method [3, 151, the lowest cost operating point can be found by plotting the incremental cost characteristics for each of these generating units on the same graph. The solution is the point at which the minimum incremental cost rate (A) is equal and simultaneously satisfies the specified demand.

An error term is then introduced and a measure of this error is calculated. The constraint is met if each term in the equality is equal to the average incremental cost rate for the string, the incremental cost error term is then

n

z = 1

where A,,, = incremental cost error term A, = incremental cost for unit i

davg = average incremental cost of individual string. The result is a fitness function based on a percentage rating as follows.

(16) stringerr - minerr maxerr - minerr

%err =

where %err = percentage of striag’s constraint error

287

stringerr = string’s error in meeting constraint equation minerr = minimum constraint error within population maxerr = maximum constraint error within population By combining the two errors, the fitness function objective becomes:

O B J = min[%)\err + %Zerr] (17) where Aerr is the percentage of incremental cost error and %lerr is the percentage of power balance error By converting the minimum problem into a maximum problem, the final form of the fitness function becomes:

f = sf1 x [(l - %Xerr)“pl] + sf2 x [(l - %lerr)”p2] (18)

where sf = scaling factor for emphasis within function sp = scaling factor for emphasis over entire population.

4.2.3 Nonlinear fitness mapping: In the genetic search, the process usually quickly converges in the early stages. However, in the mature run, it often strug- gles between the global optimal solution or may be trapped in the suboptimal point in the space being searched. One of the causes which hinders the search towards the solution point is the quantisation error of decoding the string. One way to overcome this problem is to use an infinite length of bit string representation, which is not possible in a real application. The other way is to use nonlinear fitness mapping which defines a metric over the space of structures and assures that, at each point, the distance between any two structures is greater than some minimum distances. Thus, at the start of the search, the space is sampled over a rela- tively coarse grid, and as the search progresses, the grid size is gradually reduced.

The idea of nonlinear fitness mapping comes from the bilinear transformation method of filter design. The following form is used in the paper:

tan-l(?r x %err) 0.402 x ?r

%err0 =

where %err is the original percentage of string’s con- straint error. %erro is the new percentage of string’s constraint error.

Therefore, the improved fitness function becomes:

f = s f i x [(1-%Xerr0)”p1]+sf~ x [ ( ~ - % ‘ o l e r r ~ ) ~ p ~ ] (20)

4.2.4 Sigma truncation scaling method: At the start of GA runs, it is common to have a few extraordi- nary individuals in a population of mediocre col- leagues. During the selection process, the extraordinary individuals would take over a significant proportion of the finite population in a single generation, and this is undesirable. In fact, this is a leading cause of prema- ture convergence. As GA searching process bases on the fitness information of the population, the art lies in how distinguishable the potential solutions can be selected and recombined. Regulation of the number of copies is especially important in small population genetic algorithms. One of the useful scaling procedure is sigma truncation. This method was designed as an improvement of linear scaling both to deal with nega- tive evaluation values and to incorporate problem dependent information into mapping itself. The new fitness is calculated according to:

f i s c = f z + ( f a v s - c x 4 (21) 288

whereJ; = raw fitness of the individual string = scaled fitness of the individual string

fa, = average fitness of the population c = number of expected copies desired for the best pop- ulation member CT = the standard deviation of the population

4.3 Advanced genetic operators

4.3. ’i Selection strategy including elite policy: Although there are a number of selection criteria, reproduction can be typically implemented in the func- tion selection as a linear search through a roulette wheel with slots weighted in proportion to string fitness values. This roulette wheel selection gives a straightfor- ward way of choosing offspring for the next genera- tion. The selected candidates are gathered in a temporary mating pool and are ready for further processing.

Elitism is a technique used to save early solu ensuring the survival of the most fit strings in rent population to be carried over to the next tion. Elitism compares the results of the mo population to the elite population. It then combines the two populations and determines the best results from both population in order of decreasing fitness value. If a duplication is found, elitism eliminates this duplica- tion. This combination of the most fit strings becomes the elite population. The process continues for each generation so that accuracy and convergence capability can be maintained in this algorithm.

4-32 Various crossuver techniques: Crossover exchanges the corresponding genetic material from two parent chromosomes, allowing beneficial genes on dif- ferent parents to be combined in their offspring. There

ossover techniques called one- point crossover and uniform , GAS have relied upon one-

and two-point crossover operators. But there are many situations in which having a higher number of crosso- ver points is beneficial. Considerable effort has been directed towards theoretical comparisons between dif- ferent crossover operators by a number of researchers. With smaller populations more disruptive crossover operators such as uniform crossover are likely to yield better results because they help to overcome the limited information capacity of a smaller population and the tendency for more homogeneity. However, with larger populations providing sufficient sampling accuracy, less disruptive crossover operators (i.e. two-point) are more likely to work better.

But these theories are not sufficiently general to predict when to use crossover, or what form of crossover to use. For example, the theories do not consider that the coding scheme, selection scheme and the fitness function itself may all have an effect on the relative utility of crossover. Current genetic algorithm theory is inadequate for selecting which operators to use before the GA is initiated. To make the GA adapted, there have been two approaches to coupling adaptive mechanisms, known as nonself-adaptive and self-adaptive. The nonself-adaptive approach adapts the GA over the course of many runs. In contrast, the self-adaptive approach adapts the GA as it solves a single specific problem once. This approach allows for

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the simultaneous exploration of both the problem space and some space of different GAS.

Empirical comparisons among one-point, two-point, uniform crossovers have been carried out in Section 5 which shows that the two-point crossover yields a better result in ED applications.

4.3.3 Adaptive mutation prediction: It has been reported the mutation prediction [ 151 resembles the results found using normal mutation. The intention using mutation prediction rather than normal mutation is to reduce the number of times that the random number generator is called. By use of geometric mean probability theory to work out the probability of muta- tion, much of the computation effort can be saved without sacrificing accuracy. The theory states that, given n independent tasks, with equal probability of occurrence, the task will occur at [ Uprobability] posi- tion and every bit thereafter that is a multiple of that number.

online average cost

F(t)

from last generation F(t-1)

Fig.2 Mutation prediction

The adaptive mutation prediction, illustrated in Fig. 2, is proposed to automatically adjust the muta- tion probability during the optimisation process based on the cost information. The difference between the current online average cost and the past online average cost gives the mutation controller a clue for estimating the probability of mutation. According to the look-up table of Table 1, the control action can be determined. Hence, the probability of mutation is

(22) where B is a problem dependent coefficient which is chosen as 0.001 in the ED application.

Table 1: Decision Table for adaptive mutation prediction

pmutation(t) = pmutation(t - 1) + y * 0

AHt) -20 -15 -10 -5 -1 0 1 5 10 15 20

Y -5 -4 -3 -2 -1 0.5 1 2 3 4 5

The heuristic information for adjusting the mutation probability rate is, if the changes in average cost are very small, increase the mutation probability rate until the average cost begins to reduce in the next genera- tions and vice versa.

4.4 Engineered-conditioning operator (ECO) An engineered-conditioning operator based on the first- order gradient method can be seen as a DNA repair system found in natural genetic mechanisms [20]. It is visualised that the GA-hybrid exhibits more reliablity than the pure GA as they complement each other. The local search algorithm will try to optimise locally, while the genetic algorithm will try to optimise globally. The EC-GA hybrid has been used in the past to achieve higher reliability in solving MFD problems [17, 181. Miller, Potter, Gandham and Lapena [lX] used three local search or 'conditioning' tests in the ECO. These are, in order of application, the superset, substitute, and subset evaluation tests. In terms of the running

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time, the complexity of this test is O(n2). To overcome the time-consuming trial and error process, this ECO is incorporated with problem specific knowledge to ensure that it works well with GA on power economic dispatch. The new ECO proposed is more accurate, and the complexity of the test is only O(n).

The engineered-conditioning operator for power eco- nomic dispatch works as follows: In the end of the run of each generation, the fittest individual is copied into a temporary population waiting for the next process. The ECO will try to compare each incremental cost curve (X) of this individual and select a particular pair of units to move. In this case two units with extreme X are selected. It would be possible to move the power of these two units by flipping the bit on and off such that the average incremental costs will be minimised while maintaining no power balance error on the system. The individual will be introduced back into the current pop- ulation if any further improvement can be made. The implementation of the engineered-conditioning opera- tor is shown in Fig. 3.

I pick fittest individual 1 in population

I

cost curve h,,,,~~, cost curve hMIN, if hi ' ~ M A X ; ~ M A X = ~ I i f ~ ~ - = ~ M I N , A M I N =hi

move cursor to move cursor t o L.B.S. of Pi

I I I c evaluate new fitness

individual in current population

individual in population

Fig. 3 Engineered-conditioning operator

4.5 Advanced engineered-conditioning genetic algorithms (AEC-GAS) By coupling the advanced genetic operators with engi- neered-conditioning operator, an advanced engineered- conditioning genetic algorithm (AEC-GA) is proposed. The local improvement offered by the ECO is used to improve the diagnosis obtained from the global search- ing of the genetic algorithm. The AEC-GA hybrid works as follows: run the genetic algorithm, but after each generation, remove the fittest individual, apply the ECO to this individual, and introduce the hopefully improved individual back into the population. The hybridisation techniques are very much problem spe- cific and care should be taken when enhancing genetic algorithms so as not to harm the benefits offered by genetic algorithm. The overuse of local improvement

289

operators in the early stages of GA evolution can destroy the randomness of the population. This early reduction in diversity can possibly lead to premature convergence and result in poor overall performance. Therefore, a phased-in AEC-GA has been adopted to overcome this pitfall. The ECO is applied only when the following conditisn is satisfied: generation > threshold. The flmchart of the AEC-GA is illustrated in Fig. 4.

evaluate P(t)

elitist policy

i f generation 2, threshold -

I I initialise population P i t ) z--I evaluate Pi t )

condition?

Fig.4 Flowchart ofproposed AEC-GAS

5 Simulation re*suRs and performance

5. I Six-unit te& power system To focus on the evaluation of the proposed AEC-GA, a six-unit power system is used [3]. The data used in this paper are obtained from 1131. For simplicity, trans- mission losses are ignored ia the tests. A large number of simulations have been conducted to evaluate and compare both online ptxformance (average perform- ance at each generathn) and off-line performance (best performance at each generation) of AEC-GAS and Conventional GAS. The parameters used in AEC-GA and CGA are as follows: population size = 100; subchromosome length = 11; chromosome length = 66; optimised parameter = 6, crossover rate = 0.98; initial mutation rate = 0.01.

Table 2: Effect of nonlinear fitness mapping

Algorithms With nonlinear mapping Without mapping

Error rate of CGA 3.71 x IO-2% 2.6 x

5.2 Effect of nonlinear fitness mapping Table 2 compares the effect of nonlinear fitness mapping on relative performance when the load demand is 1800.00MW. The error rate is the percentage error of the average result of each GA from the optimal cost which in this case is $16579.33. From the results, it is clear that the performance is significantly improved when the nonlinear fitness mapping is applied.

5.3 Comparison of crossover techniques Although many different forms of crossover exist, this work considers only three forms of crossover; one- point, two-point and uniform crossover. Although these three forms are very commonly used, they repre- sent extremes. One-point and two-point are the least disruptive of material, while uniform crossover is the most disruptive. Also Booker has noted that, in terms of positional and distributional bias, both two-point and uniform crossover are considerably different. Despite theoretical analysis, it appears difficult to decide in advance which form of crossover to use. Thus, it is natural to allow the GA to explore a relative mixture of these operators as the motivation, since dif- ferent mixtures will represent different intermediate search characteristics between the two extremes.

Table 3: Effect of different crossover techniques

Algorithms One-point Two-point Uniform

Error rate of CGA 5.28 x 4.55 x IO”% 8.47 x IO-’%

Error rate of ACE-GA 4.37 x 2.29 x 2.01 x IO-’%

Table 3 compares the relative performance of one- point, two-point and uniform crossover again with a load demand of 1800.00MW. The results clearly show that two-point crossover leads to the best performance. This effectively suggests for this 6-unit ED problem, the population size of 100 provides enough sampling in which the less disruptive crossover technique such as two-point crossover works better.

5.4 Comparison of mutation techniques In this Section, different mutation techniques are com- pared, such as normal mutation, mutation prediction aud adaptive mutation prediction. Table 4 compares the relative performance of the different mutation tech- niques. The results using mutation prediction resemble the results found using normal mutation. The only observable difference is the amount of time it takes to converge to a solution. The mutation prediction mini- mises computer time without sacrificing accuracy. The adaptive mutation prediction not only reduces the com- putational time, but also improves both online and off- line performances by regulating the degree of disrup- tion in the population providing the necessary sampling accuracy for complex search spaces.

Table 4: Effect of different mutation techniques

Adaptive Algorithms mutation

prediction

Normal Mutation mutation prediction

~~

Error rate of CGA- 5.60 x IO-’% 4.58 x 3.50 x

Error rate of ACE-GA 3.41 x 3.13 x 2.19 x

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Error rate of ACE-GA 7.61 x 2.29 x 10-4%

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Table 5: Results obtained by traditional methods, CGAs and AEC-GAS with load demand of 1800 MW

Load Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 demand cost ($) (MW) (MW) (MW) (MW) (MW) (MW) (MW)

Algorithm

Ref [31 184.00 166.20 54.40 590.00 402.70 402.70 1800.00 16609.57

CGA 254.06 244.51 77.07 561.91 329.57 331.98 1800.00 16582.03

AEC-GA 248.07 217.73 75.30 587 70 335.60 335.60 1800.00 16579.33

Table 6: Results obtained by CGAs and A

Load Unit 1 Un i t2 Un i t3 Unit 4 Unit 5 Unit 6 demand cost ($) (s) (MW) (MW) (MW) (MW) (MW) (MW) (MW) Algorithm

CGA 194.13 153.04 56.46 518.74 296.71 300.92 1520.00 14170.79 29.12

AEC-GA 191.17 172.31 56.93 524.67 287.43 287.59 1520.00 14169.54 5.78

CGA 450.37 335.72 97.86 541 -55 407.46 404.04 2238.00 20502.72 44.97

AEC-GA 365.40 301.92 11 1.96 590.00 429.20 439.52 2238.00 20470.48 9.88

5.5 Comparison of accuracy and computation time Table 5 lists results obtained by traditional methods [3], CGAs and AEC-GA, respectively, when the load demand is 1800MW. The results obtained from GAS are better than the traditional method. When compar- ing AEC-GA and CGA, AEC-GA only takes 7.1 s to reach the optimum ($16579.33) with a tremendous reduction on iterations and computational effort as CGA can only reach suboptimum ($16582.03) but takes 36.7s.

In reality, the load demand will be unpredictable, and it varies throughout the day. Table 6 lists results of another two loading conditions (1 520 MW and 2238MW) for CGA and AEC-GA. The results again show that AEC-GA outperforms CGA in both accu- racy and computation time.

loor

2ot

16710 167501 \ 16630 ... ..... .. .................

0 10 20 30 LO 5 generations

0

0 ; L

O: 0 0 -

20

- on-line cost ---off-l ine cost .... off-line loaderr Fig.5

To appreciate the evolutionary process, Figs. 5 and 6 illustrate the optimisation progress during each run for both online and off-line performance of CGA and AEC-GA with a load demand of 1800MW.

6 Conclusions

The biological mechanisms of bacteria provide additional insight into the effective implementation of advanced genetic operators. The proofreading system found in the high reproductive rate of bacteria has to be robust enough to remove replication errors. The artificial model is analogous to this. The ECO can be seen as one of the proofreading systems. This uncovers the reason why AEC-GA hybrid is significantly better than the CGA as demonstrated in the ED applications. Clearly, the local improvement nature of the ECO certainly plays an important role in improving the dominant individual of the GA populations.

2ot

0 10 20 30 LO I

generations

- on-Line cost ---off- l ine cost .... off-line loaderr Fig.6

The adaptive mutation prediction proposed in this paper not only reduces the computational time, but also improves both online and off-line performances by regulating the degree of disruption. It is believed that there are two important situations in which disruption is advantageous: (i) late in the evolutionary search process when the population is rather homogeneous; (ii) when the population size is too small to provide the

29 1 IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 3, May 1997

necessary sampling accuracy for complex search spaces. However, excessive disruption of the sample distribution greater than the genetic operators can handle would destroy the potential genes and lead to a poor online and off-line performance. Therefore, care should be taken while maintaining genetic diversity not to upset the genetic equilibrium. Consequently, the less disruptive two-point crossover yields better results in this case.

The combination of the nonlinear fitness mapping and the sigma truncation scaling is highly beneficial. The quality of recombining the potential genes has been uplifted by these methods. Overall, the improved efficiency, accuracy and reliability achieved by the pro- posed AEC-GA hybrid demonstrates its advantages in power system optimisations. Work is continuing on the development of other types of hybrid techniques [22] and in the evaluation of their performance over large- scale systems.

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