Optimal reactive power dispatch using evolutionary computation: Extended algorithms

J.R.Gomes and 0.R.Saavedra

Abstract: Extended evolutionary algorithms for solving the optimal reactive power dispatch are presented. In these approaches, based on evolutionary programming and evolution strategies, mutations in standard deviations have been controlled using a dynamic limits strategy. The proposals have been exhaustively analysed and compared with a state-of-the-art method. Good and reliable performance has been achieved and validation tests using the standard IEEE57 system are reported.

1 Introduction

During the last three decades, there has been a growing interest in evolution-based algorithms for solving different lunds of problems. These algorithms are labelled with the common term ‘evolutionary computation’. There are three main approaches: genetic algorithms (GAS), evolution strategies (ESs), and evolutionary programming (EP).

Each of these mainstream algorithms have clearly dem- onstrated their capability to yield good approximate solu- tions, even in the case of complicated multimodal, discontinuous, non-differentiable, and even noisy or mov- ing response surfaces of optimisation problems [l-51. In these approaches, a population of individuals is initialised and then evolves into a search space throughout a stochas- tic process of selection, mutation, and in some cases, recombination. However, these methods differ in terms of representation, operators and selection process.

Evolution strategies (ESs) were developed in 1960 by Rechenberg and Schwefel in Germany and extended by other authors such as Herdy [6] and Rudolph [7]. The first evolution strategies focused on a single-parent offspring search [8]. In this model, termed (1 + 1)-ES, a single off- spring is created for a single parent and both are placed in competition for survival with selection discarding the poorer solution. In 1973 Rechenberg proposed the use of multiple parents but only a single offspring (p + 1)-ES. More recently, two approaches have been explored, denoted by (p + k)-ES and (p, A)-ES [4]. In the former, p parents generate k offspring and all solutions compete for survival, with the best p individuals being selected as par- ents of the next generation. In the latter, ody A offspring compete for survival and the p parents are completely replaced in each generation. Then, the Me of the individual is h t e d to a single generation.

The original evolutionary programming was introduced by Fogel in 1962 [9] and extended recently by Burgin et al. [lo]. The goal of evolutionary programming is to achieve

G E E , 1999 IEE Proceedmgs online no. 19990683 DOL 10.1O49hpgtd 19990683 Paper fmt received 23rd March and in revised form 14th July 1999 The authors are with the Departamento de Engenharia de Eletricidade, Univer- sidade Federal do MaranhXo, SXo Luis - 65085-580 - MG Brazil

intelligent behaviour through simulated evolution. While the original evolutionary programming was proposed to operate on finite machines and the corresponding discrete representations, most of the present variants are utilised for continuous parameter optimisation problems.

More recently, the technique has been extended and applied to diverse real-valued continuous optimisation problems. Rather than use finite state machines, representa- tions are chosen based on the problem at hand and muta- tion is the main operator used in generating new trials. The last version, called meta-EP incorporates parameter self- adaptation per individual in a similar way to ESs. The various methods can be found in [1&13].

I . 1 Power system applications Despite traditional optimisation techtuques being imple- mented successfully in the existing power systems, there are still dfiiculties. One of these dificulties is the multimodal characteristic of the problems to be handled [14]. In the case of optimal power flow (OPF), due to its non-convex characteristic the majority of techniques converge to a local solution. Other important aspects are the integer variables, because some types of control variables, such as tap posi- tions and capacitor banks, must be integer values. A con- tinuous solution for the problem cannot directly be applied to these controls, because in the case of devices with great discontinuity of values, rounding-off may lead to significant errors. Despite mixed integer programming methods being oriented for this type of problem, the process is more com- plicated than conventional continuous proposals.

The algorithms based on the principles of natural evolu- tion have been applied successfully to a set of problems of numerical optimisation. With a good degree of parallelism and stochastic characteristics, they are adequate for solving complicated problems of optimisation, such as those found in reactive optimisation, distribution systems planning, expansion of transmission systems, etc. [15-191. The litera- ture presents an extensive list of works concerning the application of evolutionary techniques to problems of power systems [20]. In general, these applications concen- trate primarily on power system planning, followed by dis- tribution systems.

Lai and Ma [15] have presented a modified evolutionary program to solve the reactive power dispatch, obtaining good results. Other authors [17, 181 have applied the same algorithm for other power system problems, reporting

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results using the IEEE30 system. A simpwied evolution strategy has been used in [18] and compared with genetic algorithms and the Lai and Ma algorithm. More recently, a proposal quite similar to [15] has been presented [19]. In spite of these efforts, evolutionary techniques have not yet been explored completely for power system applications.

In this work, in order to obtain new, improved algo- rithms, evolutionary programming and evolution tech- niques have been explored. Firstly, two extensions to solve reactive power dispatch are presented from meta evolution- ary programming proposed by Fogel. Secondly, an evolu- tion strategy based approach has been proposed. Finally, due to the probabilistic nature of the evolutionary algo- rithms, a statistical analysis has been performed. The approaches have been tested using the IEEE57 system, showing good and robust performance.

2 Evolutionary programming (EP)

Evolutionary programming was generalised by Fogel [ 1 I ] to handle numerical optimisation problems. In this approach, each component of a candidate solution is viewed as a behavioural trait rather than a gene. The result- ing change in each behavioural trait will follow a Gaussian distribution with zero mean and some standard deviation. The standard EP algorithm has the following structure (maximisation problem):

t := 0 An initial population P(t) with p individuals is selected at random from a feasible region: P(t) := {xl(z), ..., x,(z)} E I,, where I = R" Calculate the fitness values @(xi) from objective func- tion values (fJ by scaling them to positive values (func- tion S), and possibly by imposing some random alteration K;

where

while termination criterion not fulJiled do

P(0) : {@(Xl(tN, ... 9 @.(x,(t)>>

@(Xk(i)) = mxk(i)) , Ki)

(iii) mutate: xk(t) := m{xk(z) V k E { 1, ..., p; (iv) evaluate P'(t) := {xi (t), ..., x;( t )}:

(v) select: P(t + 1) := S,(P(z) U P'(z)); {@Wl(t))> ...1 @(x;(o>>

z : = t + l end do

On step (iii), an offspring vector x: is created from each parent x, by adding a Gaussian perturbation with mean zero and a standard deviation to each component of vector x,, as follows:

z; = 2, + a,N,(O, 1) vi E 1,. . . ,n (1) where

0 2 = &@(s) + 2% (2) and n is the number of control variables, pi is a mutation scale parameter with 0 < /3 < 1, q is the standard deviation for each individual's mutation, zI represents an offset, and N(0, 1) represents a Gaussian random variable with mean zero and variance one.

In step (v), the population is formed, temporanly, by parents and offspring. The selection mechanism Sq reduces the set of parents and offspring individuals to a set of p parents by performing a tournament as follows.

IEE Proc -Gener Transm Distrih , Vol 146, No 6, November 1999

Each individual xi, i = I, ..., 2p (combined population) must compete with other individuals to get a chance to be transcribed for the next generation (optionally, a subgroup of k individuals can be preserved for the next generation, not participating in the competition). A value wi is associ- ated in accordance with the competition, thus:

P

wi = w; (3) t= l

where q is the number of competitions, and w; is either 0 (loss) or 1 (win) as an individual xi competes with a ran- domly selected individual xr selected from the combined population. Thus, w; is defined as

wt* = { 1 if U, < f r / ( f r + f z )

0 otherwise wheref, is the fitness of a randomly selected individual xr, and j ; is the fitness of x,.

The value U, is determined from a uniformly distributed set U(0, 1). The individuals i = 1, ..., 2p are ranked in descending order of the rank values w,, and the p individu- als having the highest ranks w, are selected to form the next population.

3 Evolution strategies (ES)

The process of ES is described in [4]. The following pseudocode algorithm summarises the components of the (p + A)-ES evolutionary algorithm, where each individual is characterised by a pair a = (x, q):

z := 0 (i) initialise P(z) := {al(0), ..., ap(0)} E P

where I = Rn+* and ak = (xl, q) Vi E { 1, ..., n }

(ii) evaluate P(z) : {@(a,(t)), ..., @(a,(t))} where @(ak(t)) = f(xk(t));

while termination criterion not fulfilled do (iii) recombine: ah(t) := r(P(t)) V k E {I , ..., A}; (iv) mutate: a'h(t) := m(dk(t) V k E { 1, ..., A}; (v) evaluate P'(t) := {a", (z ) , ..., a'i(t)};

where @(ak(t)) = f(ak(t));

t : = z + l end do

{@(a'\ (41, ..., @(a'i(t)>}

(vi) select: P(t + 1) := SdP(t) U P'(t));

Search points in ESs are n-dimensional vectors x E R", and the fitness value of an individual is identical to its objective function value, i.e. @(a) = Ax), where x is the object variable component of a and each individual includes up to n different variances q (i E { 1, ..., n}).

Different recombination mechanisms are used in ESs either in their usual form, producing one new individual from two randomly selected parent individuals, or in their global form, allowing the taking of components for one new individual from potentially all individuals available in the parent population. Furthermore, recombination is per- formed on strategy parameters as well as on the object var- iables, and the recombination operator may be different for object variables and standard deviations.

The mutation operator m : I +- I (where I = RnCn) yields a mutated individual m(a) = (T, a'), by first mutating the standard deviations and then mutating the object variables as follows:

5x7

a: = a; exp (.r'N(0,1) + .rN;(O,I))

xi = Xi + a p q o , 1)

(4)

( 5 ) The global factor T"(0, 1) allows for any overall change

of the mutability, whereas dV,{O, 1) allows for individual changes of a,. The parameters z and T' are suggested by Schwefel [12] as z = (4/[4n])-' and T' = (d[2n])-'.

In constrast with EP, selection in ES (S,) is completely deterministic, selecting the p best individuals from the union of parents and offspring ((p + A)-selection). The selection is elitist and therefore guarantees a monotonic improving performance.

4 Meta evolutionary programming (MEP)

In [12] Back et al. stated that the standard EP algorithm imposes some parameter tuning difficulties in finding appropriate values for /3, and z,. To overcome these difficul- ties, Fogel developed an extension called meta evolutionary programming (MEP) that self-adapts n variances per indi- vidual in a s d a r way to ES [13]. Then, mutation applied to an individual a = (x, v) produces a = (x', v'), according to

( 6 ) X: = x2 + azNz(O, 1) Vz E 1,. . . ,n

U:. = + J E v Z N ( O , 1 ) (7) where v, = q2 and 5 denotes an exogenous parameter ensuring that v, tends to remain positive. If a variance becomes negative or zero, it is set to a small value E > 0.

An alternative form to eqn. 7 can be obtained by import- ing from ESs the self-adaptation expression for variances given by eqn. 4 [21].

Eqn. 4 is more consistent than eqn. 7, because log- normally distributed alterations generated guarantee positiveness of a,. Thls characteristic is not present in eqn. 7. Furthermore, eqn. 4 provides two search probabil- istic components. The fnst one is responsible for individual local search (for each vector element), while the second one provides perturbations for all the vector components. Thls strategy allows a more uniform search process over the fea- sible region. Therefore, the MEP implemented in this arti- cle utilises eqn. 4.

5 Extensions

Due to the nature of evolutionary algorithms, best solu- tions are expected by increasing the generation number and initial population size. However, in practical applications, solutions in a reasonable time period are required, and in many cases local solutions fulfill these practical require- ments. In the following Sections we present three new extensions of the methods reviewed before. First, the Lai and Ma EP algorithm is reviewed.

5. I Lai and Ma EP algorithm (LM) [I71 An extension of the canonical EP algorithm widely used in the power system literature has been proposed by Lai and Ma [ 11. They have introduced the following modification for the mutation expression (eqn. 1):

V z ~ l , . . . , n (8)

with 0 < pk s 1.

588

Additionally

Pznzt i f k = O

Pk-1

P f z n a l if P k - 1 - P s t e p P f z n a l

Plc = { P k - 1 - P s t e p if fmzn(k) 2 fmzn(k - 1) if fmzn ( k ) < fmzn ( k - 1)

where om,[ is close to 1, /3fina/ - 5.1e3, and bAjstep E [le3,

Thus, the mutation scale is modified during the process and ensures that the search process stops in a local mini- mum. Soon, as the search process starts with high scale values, these will decrease during the process. The speed of scale decreasing of an individual depends on its fitness in such a way that the lower it is, the faster the scale dimin- ishes. Another added modification states that the variables cannot exceed their limits, these being given by the limit value. Good results have been reported by the authors using the IEEE30 bus system.

5.2 Proposed methods In this paper, the standard algorithms MEP and (p + A)- ES have been modified. The main modification is addressed to limit o mutations by introducing dynamic upper and lower bounds. Moreover, a modified ES algo- rithm is performed without recombination.

Dynamic limits allow (3 mutations to fall into an upper and lower limit, both dynamically decreasing exponentially as follows

1 @2] .

g ( t ) m a z = a k a x e x ~ ( - t / T ~ ) (9)

a(t)mzn = a k z m ex~(--t/T2) (10) where oAX and a,& are initial values for each function, t denotes the generation, and TI and T2 are time constants of the exponential functions in eqns. 9 and 10, calculated from final desired values of o,f, and adm, respectively. If any dynamic limit is violated, then o(t) will be given the average of the current values of the functions above.

Eqns. 9 and 10 allow 'large' mutations in the initial gen- erations and 'small' mutations at the end. In other words, in the first iterations diversity is emphasised, while the last generations are dominated by a refined search process (small mutations). Fig. 1 suggests a graphic form of dynamic limits.

01 I I I I 0 50 100 150 200

generation no.

Fig. 1 Dynamic qper b o d and lower bmmd fmtwm (9 U,, (io Umm- . . . . . . . . . . . ......... U

On the other hand, creation of an offspring is performed by talung in account the feasible range of the variable, sim- ilar to the LM proposal, as follows

X:. = 2 2 +a: ( x y z - L C y ) N ( 0 , l ) (11)

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where xim - x,""" are the limits of the control variable x,. If x, exceeds its limit, x, will be given the limit value.

5.2. I Meta bounded evolutionary approaches fMBEP1 - MBEP2): These extended algorithms follow the basic structure of the meta evolutionary algorithm, where an individual is represented by a pair (x, a). The first approach has been named MBEPI, where mutations of a are performed using eqn. 4 and limited by eqns. 9 and 10. The second version, namely MBEP2, is similar to MBEPl but uses deterministic selection rather than the classical EP probabilistic selection.

5.2.2 Bounded evolution strategy approach: In this third proposal, called bounded ES (BES), the canonical (p + d)-ES (Section 3) has been modified by including dynamic limits (eqns. 9 and lo), but without considering recombination. In this way, in all the implemented approaches the main evolutionary operator is mutation. These extended algorithms have been implemented in this work and compared with the LM algorithm.

6 Implementation details

The three approaches proposed have been implemented and compared with the LM algorithm. The application used has been optimal power dispatch.

6.1 Optimal reactive power dispatch The goal of optimal reactive power dispatch is to minimise real power losses and improve the voltage profile by setting generator bus voltages, VAR compensators and trans- former taps. These problem can be in a form penalised as follows:

min f = f e + f, (12) such that

Pf - PZ(V,O) = 0 2 E N B - ~ (13)

QP - QZ(V,O) = 0 i E N ~ Q (14) with

f p = ~ q z ( Q s , - Q f s ) 2 + - K'I2

(15) z E N p v ~ E N P Q

where fe represents the system losses, NB, NS, represent the system nodes set, and the system nodes set excluding the slack bus, respectively, and NpQ, Npv represent the PQ- buses and PV-buses set, respectively. Eqns. 13 and 14 rep- resent the load flow equations. The generator bus voltages and the transformer tap-settings are control variables. pqi and pyi are penalty factors for reactive power violations and voltage violations, respectively. Q,' and represent the violated limits. Pi" - (3: represent the active and reactive power demand at node i, respectively. Penalty parameters are chosen empirically based on experience and the particu- lar application.

6.2 Algorithm implementation The implemented algorithm basically follows the sequence presented in Sections 2 4 . The fitness function Q, corre- sponds to the objective function f (eqn. 12). The control vector x is formed by the generator bus voltage and trans- former tap-setting. The initial population of p candidate solutions satisfying eqns. 13 and 14 is generated at random (i.e. candidates must satisfy the load flow equations).

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6.2.1 Parameters used in practical implementa- tion: The parameters of dynamic limits (eqns. 9 and 10) have been assumed as aiin = I t 2 , a& = 1, aAn = 10-4 and aAax = IC2, where T,, represents the amount of gen- erations used. Other specific parameters of implemented algorithms are presented in Table 1. In the current imple- mentation, penalty parameters remain constant during the process; the values considered in tests are pqi = IO4 and pvi = 105.

Table 1: Parameters of implemented algorithms

Parameter LM MBEPl MBEP2 BESloo BESZOO

Y 60 60 60 30 30 A. 60 60 60 60 60

Tnl, 200 200 200 100 200 Po 0.10 -

Ps;ep 0.10 -

- - - - - - Pf 5E-4 - - - -

Selection Sq Sq s d s d s d - - - 9 60 60

*If in two consecutive generations fitnesses do not diminish, then p is reduced by f i f i It is a small modification introduced in the LM algorithm that improves the performance

The performance of algorithms is affected by the choice of parameters. Thus, after several tests, these parameters have been selected in order to get the best performance of each implemented algorithm.

6.2.2 Competition parameter: Parameter q defines the number of competitions to which each indwidual of combined population is submitted. A small value for q leads to random behaviour of wi (eqn. 3). On the other hand, a very high q compared with population tends to deterministic behaviour [I]. In this work it has been assumed that q = p.

Notice that the MBEPl and MBEP2 algorithms are dif- ferent in the type of selection used. While MBEPl uses probabilistic selection (Fogel's q-tournament selection), MBEP2 uses deterministic selection similar to classical ES. The parent and offspring populations are 60, similar to parameter q.

The modified algorithm BES is based on the (p + d)-ES without recombination. Two cases 100 and 200 genera- tions, respectively, have been simulated. The parameters used in the implementation are given in columns 5 and 6 of Table 1.

7 Testresults

The EP approach proposed by Lai and Ma [ lq , and the three modified algorithms based on EP and ES have been implemented. Tests have been performed using the IEEE57 standard system. The network consists of 7 generator- buses, 50 load-buses and 80 branches, of whch 17 branches are under load-tap-setting transformer branches. All power and voltage quantities are per-unit values. The platform used was a Pentium 133MHz.

Tables 2 and 3 show the main characteristics of these systems. Voltage h u t s have been considered 0.9-1.10p.u. for PV-buses and 0.95-1.05 for PQ-buses. Taps limits have been assumed to be 0.9&1.10. Total losses in the base-case have been 0.2793p.u. (base-case corresponds to the opera- tion point given in the standard IEEE57 data file).

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Table 2: Reactive power generation limits

Bus 1 2 3 6 8 9 12

OF"' -1.40 -0.17 -0.10 -0.08 -1.40 -0.03 -0.3

OFax 2.00 0.50 0.60 0.25 2.00 0.09 1.55

Table 3: Tap-setting and vokage limits

PV buses PQbuses Taps

vgmh vgmax vmin vmax amin amax

0.9 1.1 0.95 1.05 0.90 1.10

Due to the probabilistic characteristic of evolutionary algorithms, results reported here correspond to an average from 20 trials. From a practical point of view, we are inter- ested in reliable software tools that supply good solutions every time. Thus, to evaluate the quality of the proposals, dispersion measures are fundamental.

Fig. 2 shows the performance of four algorithms, say LM, MBEPl, MBEP2 and BES. Tests have been per- formed for 200 generations. Fig. 3 shows a zoom of the last 100 generations. Each point corresponds to the average (over 20 trials) of the best fitness on the current generation.

5

lo r

10 I.$ 3 -

-1 \ lo 0 50 100 150 200

generation no. Fig. 2 Best average Jhess per generution - LM

~ BES ......... MBEPl . . . . . . . . . . . MBEP2

The three algorithms proposed outperform [Note I] the algorithm of Lai and Ma. On the other hand, probabilistic and deterministic selection (MBEP1 and MBEP2, respec- tively), show a similar performance. However, from the point of view of a local search, the performance of MBEP2 is better than MBEP1. MBEP2 and BES differ basically in the mutation mechanism. MBEP2 uses ,U = d and a parent generates a unique offspring, while BES uses ,U < d and a parent can be selected more than once for mutation. The

Note 1: Although great effort has been made to tune the parameters to obtain the best performance of each algorithm, it is not guaranteed that the parame- ters selected are the best. Of cow, if other parameters are used, this relative performance au ld be different

590

impact of this difference in the performance of the algo- rithm has been shown to be dramatic.

2.0r 1,8i 1.6

1 . 4 1 *: ... L-, . . I . . I

0.6

0.4

.

o,2i 100 0 120 140 160 180 200

generation no.

In the following, in order to validate the proposed approaches from a robustness point of view, a statistical analysis is presented.

Table 4 presents a comparative analysis of implemented algorithms relative to loss minimisation. Tests have been performed over 20 trials discarding the 2 worst cases. Columns 2 4 show the minimum, maximum, average and standard deviation of losses, respectively. Column 5 gives average losses relative to base-case losses. Column 6 gives the total CPU time, in minutes, excluding the data reading. These times correspond to the average time calculated over 10 trials. Around 80% of time spent is due to the evalua- tion process (loadflow, checking limits, objective function evaluation, etc.), while the remaining time is due to the evo- lutionary process. MBEP2 and BES algorithms are less- time consumers because they use deterministic selection. This Table is important because it shows the robustness and reliability of proposals. We have in mind an algorithm that supplies optimal and feasible solutions with minimal dispersion. Despite the fact that all proposals show good performance, the best result has been obtained with BES for 200 generations, i.e. the same used on the other meth- ods. Notice that good results have already been obtained for 100 generations.

Table 4: Statistic performance of implemented algorithms

Global performance - -

Method Gmin Gmax 4 Oe% G'% CPU

LM 0.2484 0.2922 0.2641 4.89 94.56 1.56

MBEPl 0.2474 0.2848 0.2643 3.68 94.62 1.55

MBEPZ 0.2482 0.2830 0.2592 3.38 92.79 1.36

BES,oo 0.2438 0.2630 0.2541 2.30 90.99 1.34

BE&oo 0.3417 0.2486 0.2443 0.82 87.47 0.69

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Until now, results of algorithms were analysed in terms of optimality and dispersion measures. Next, algorithm performances are analysed in terms of feasibility.

Tables 5-7 show test results distributed in deciles. felf represents a measure of feasibility of solutions. A value equal to 100% means that the feasibility is a maximum and no violations are registered. flfe' represents the ratio between the objective function and base-case losses. It indi- cates, when the feasibility is loo%, the reduction of losses in relation to the base-case. In the case of unfeasibility, this column has no meaning. V,,% and N , represent the maxi- mum voltage violation observed and the number of buses with violations, respectively. Notice that no reactive viola- tions have been registered in all tests.

Table 5: Deciles distribution for LMalgorithm

Table 5 shows the deciles distribution for the LM algo- rithm. Over 20 trials, 10% of cases have presented viola- tions. A similar performance is observed for MBEPl and MBEP2 in Table 6. On the other hand, the BES algorithm gives the best performance, as shown in Table 7. No viola- tion is observed, and in all cases loss reduction has been achieved. Furthermore, solutions obtained over 100 genera- tions fulfd practical requirements, i.e. reasonable loss reduction and complete feasibility are achieved.

Tables 8 and 9 show the best solutions of the control variables obtained by the LM, MBEP2 and BES methods, respectively. Base-case (BC) values are included. The best solution has been obtained with the BES method, achieving loss reduction of 13.48%.

Table 8 Generator bus voltages

LM Generator bus voltages

Decil felt % fife: Yo V, % Nv - MIN 100 88.93 -

4 100 89.79 - 4 100 90.86 - 4 100 91.51 -

- - - - 0 4 100 91.90 - - 4 100 93.91 -

D6 100 96.34 - 4 100 98.74 -

- - - 5 100 101.2 -

a, 99.52 105.4 0.05 2 MAX 90.13 112.7 0.05 2

Table 6 Deciles distribution for MBWl and MB€R algo- rithms

MBEPl MBEP;!

Deci I

MIN

4 4 4 0 4

4

4 a, a, MAX

D6

Glf %

100 100 100 100 100 100 100 100 100 99.8 14.4

fl&: Yo

88.57 91.10 91.94 92.56 93.32 94.51 95.81 97.29 99.81 103.4 687

_ _ 0.37 4 0.37 4

felt % @:%

100 88.89 100 89.47 100 90.13 100 90.69 100 91.76 100 92.87 100 93.54 100 94.84 100 96.73 61.7 258 9.09 1213

-

0.46 0.46

-

5 5

Table 7: Deciles distribution for BES algorithm

BESIOO BESZOO

Decil felt YO 176: % f& YO ffe: Yo MIN 100 87.31 100 86.52 D1 100 88.42 100 86.69 4 100 89.05 100 86.86 4 100 89.83 100 87.02 0 4 100 90.11 100 87.23 4 100 91.16 100 87.47 D 6 100 92.85 100 87.67 4 100 93.33 100 87.88 5 100 93.44 100 88.67 a, 100 94.64 100 89.03 MAX 100 99.71 100 90.45

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Bus BC

1 1.0400 2 1.0100 3 0.9850 6 0.9800 8 1.0050 9 0.9800

1.0150 fG :% 100

LM

1.0745 1.0618 1.0510 1.0515 1.0692 1.0335 1.0401 88.93

MBEP;!

1.0730 1.0529 1.0505 1.0449 1.0729 1.0382 1.0422 88.89

BESZOO

1.0725 1.0596 1.0484 1.0424 1.0662 1.0344 1.0440 86.52

Table 9: Transformer tap-settings

Transformer tap-settings

Branch BC LM MBEPZ BESzOo

(4, 18) 0.9700 1.0190 1.0276 1.0443 (4, 18) 0.9780 1.0190 1.0276 1.0443 (20,21) 1.0430 0.9248 0.9997 1.0076 (24,25) 1.000 0.9252 0.9491 1.0097 (24,25) 1.000 0.9252 0.9491 1.0097 (24,26) 1.0430 1.0416 1.0103 1.0097 (7,29) 0.9670 1.0136 0.9916 1.0435 (32,34) 0.9750 0.9890 1.0395 0.9803 (11,41) 0.9550 0.9003 0.9369 1.0221 (1 5, 45) 0.9550 1.0378 0.9849 1.0381 (14,46) 0.900 0.9751 0.9970 1.0220 (IO, 51) 0.9300 0.9707 1.0364 1.0296 (13, 49) 0.8950 0.9272 0.9463 1.0248 (11,43) 0.9580 1.0430 1.0034 1.0221 (40,56) 0.9580 1.0884 0.9735 0.9835 (39.57) 0.9800 0.9737 0.9672 0.9899 (9.55) 0.9400 1.0467 1.0058 1.0344

8 Conclusions

In this paper we presented three extended algorithms based on meta evolutionary programming and evolution strate- gies. In the three proposals, mutations in standard devia- tions have been controlled using dynamic limits. A comparative study between these approaches and the Lai and Ma evolutionary algorithm has been performed. In order to validate it, due to the probabilistic nature of algo- rithms, a statistical analysis has been presented. The three proposals outperform the state-of-the-art algorithm. The inclusion of dynamic limits for the standard deviation has

59 1

been shown to be fundamental in the performance of pro- posals.

The comparative study has shown that the BES algo- rithm performs better than the other approaches. In 100% of tests, feasible solutions with loss reduction have been achieved. Exhaustive tests were performed and reported using the standard IEEE57 system in this work.

8 Acknowledgments

The authors wish to acknowledge the support from Con- selho Nacional de Desenvolvimento Cientifico e Tecnolog- ico-CNpq and Superintendencia de Desenvolvimento da AmazGnia-SUDAM, Brazil. Also, the authors would like to thank the anonymous referees for their help and useful suggestions.

9 References

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