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Evolutionary Multi-Objective Optimization in Power Systems: State-of-the-Art

F. Rivas-Dávalos, E. Moreno-Goytia, G. Gutiérrez-Alacaraz, J. Tovar-Hernández, Members, IEEE

Abstract— Electric utility industry is currently facing a

market deregulated environment and many technological advances. In this context, the demand for electric power having higher network security, better power quality, improved system reliability, and availability is increasing every day. This complex scenario put the electric utilities under conflicting pressure between meeting the growth demands, reducing its operation cost, keeping maintenance and construction and try to provide lower rates for customers or to improve the company profits. Therefore, solutions for planning, design and operation of power systems involve the simultaneous optimization of multiple objectives, often conflicting between them.

This work presents the state of the art of multi-objective evolutionary algorithms applications to electrical power systems, in order to provide the power system engineering community with the expertise about the development of multi-objective optimization paradigms and trends in the applications of multi-objective evolutionary algorithms, altogether useful for tackling down every-day electrical networks challenges.

Index Terms— Multi-objective Optimization, Evolutionary Algorithms, Electric Power Systems.

I. INTRODUCTION ith the deregulated environment faced by the electric utility industry and recent advances in technology, the

demands for electric power with higher security, quality, reliability, and availability are increasing. This scenario put the electric utilities under conflicting pressure to meeting a growing demand and reducing its cost of operation, maintenance, and construction in order to provide lower rates for customers and to improve profits of companies. Therefore, the power systems planning, control and operation problems involve the simultaneous optimization of multiple objectives where some objectives are often conflicting between them (i.e. simultaneous minimization of designing cost versus maximizing system reliability). Thus, with the purpose of solving these conflicts, between electric utilities and customer demands, power system problems must be formulated as multi-objective optimization problems.

Through the years, several solution methods have been

proposed in the way to crack multi-objective optimization problems down [38]. Most of these methods are based on methods called conventional or classical methods which, as it is known, have many limitations for tackling complex multi-objective optimization problems. On the other hand, methods based on multi-objective evolutionary algorithms have been developed to overcome such limitations [31][41].

Manuscript received April 21, 2007. The authors are with Instituto Tecnológico de Morelia, Av. Tecnológico

No. 1500, Morelia, Mich. México (Phone: 01-55-443-3121570; Fax: 01-55-443-3-17-18; e-mails: frivasd2003@yahoo.co.uk, elmg@ieee.org, ggutier@gmail.com, horaciotovar@mexico.com).

The evolutionary algorithms technique has shown itself as a valuable research tool, gaining world-wide acceptance for finding solutions from many disciplines ranging from bio-cells to aerospace. The welcoming of such technique paved the way for the awake of “evolutionary multi-objective optimization”, an area where investigation efforts are focused to design efficient and effective multi-objective evolutionary algorithms. From this area, several algorithms have been published such as: Non-dominated Sorting Genetic Algorithm (NSGA) [32], Niched-Pareto Genetic Algorithm (NPGA) [33], Multi-Objective Genetic Algorithm (MOGA) [34], Strength Pareto Evolutionary Algorithm 2 (SPEA2) [35], Pareto Archived Evolution Strategy (PAES) [36], Non-dominated Sorting Genetic Algorithm II (NSGA-II) [37], and Micro Genetic Algorithm (µGA) [39]. These algorithms are the foremost representatives of the field.

However, power system engineering has been scarcely touched by evolutionary multi-objective optimization applications. In addition, most of algorithms developed for such area make use of classical methods where multi-objective optimization problems are converted into single-objective optimization problems. This transformation involves the well-know difficulties associated with the classical methods.

In this paper, articles published related to multi-objective evolutionary algorithms applied to finding solutions in power system engineering are surveyed.

The main purpose of this paper is to provide an ample, state-of-the-art, expertise knowledge to the power systems engineering community about the insights and implementation of multi-objective optimization paradigms in electrical networks as well as the trends in the applications of multi-objective evolutionary algorithms in the near future.

W

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II. MULTI-OBJECTIVE OPTIMIZATION PROBLEM

A. Pareto-optimal An optimization problem for more than one objective

function is known as multi-objective optimization if the task of finding involves one or more optimal solutions. In general, a multi-objective optimization problem is mathematically represented by:

)1()(\ xfmMaxMin

subject to

0)(0)(

=≤

≤≤

xhxg

xxx ul

Where

fm(x) is a vector of objective functions x is a vector of independent variables xl is a vector of lower limits xu is a vector of upper limits g(x) is a vector of inequality constraints h(x) is a vector of equality constraints

Each objective function, in the objective function vector,

can be either minimized or maximized. As common rule in multi-objective optimization, the

objective functions represent incommensurable and mutually competing objectives; therefore, there is not a single solution, necessarily, which be the best respective to all objectives. To illustrate this assert, lets consider Figure 1 which shows solutions to a given optimization problem of two objective functions, f1 and f2, being minimized. Also it is possible to take solution 7 as a reference for comparison purposes. So, comparing the universe of solutions versus solution 7 is palpable that solutions 3 and 4 are better solutions than 7 since they have smaller values of objective functions. On the other hand, solutions 8 and 9 are far from being a good solution. In the same context, solutions 1, 2, 5, 6 and 10 are indifferent or incomparable solutions because they are neither better nor worse than solution 7. Consequently, in multi-objective optimization, a case where a single solution simultaneously optimizes all objective functions is extremely rare.

To resolve a multi-objective optimization problem, a different notion of “optimal” is urgently required. The most common notion of optimality in multi-objective optimization is the one called Pareto-optimality [13]. Under this notion, the aim of multi-objective optimization is not focused to find an optimum solution but a set of trade-off optimal solutions (Pareto-optimal solutions). For such solutions, no improvement is possible in any objective function without previously sacrificing at least one objective function.

Applying this concept to Figure 1, then solutions 3 and 4

“dominate” both solution 7 and solutions 8 and 9 are “dominated” by solution 7. The set of non-dominated solutions are those not being dominated by any member of the whole set of solutions P. In Figure 1, it can be seen that from

the total ten solutions of set P, solutions 1, 2, 3, 4 and 5 shapes the non-dominated set. If the whole of solutions in Figure 1 is the total possible space of search, the non-dominated solutions 1, 2, 3, 4 and 5 comprise the Pareto-optimal set. By joining such Pareto-optimal solutions, a curve named Pareto-optimal front is constructed

better

worseindifferent

indifferent

1

2

3

4

6

7

89

10

5

f1

f2 Pareto-optimal dominated

f1

f2

Fig. 1. Illustration of the concept of Pareto optimality.

B. Tow goals in Multi-objective optimization In the presence of intra-conflictive multiple objectives, a

multi-objective optimization problem results in a number of optimal solutions, known as Pareto-optimal solutions. In this context, it is rather difficult choosing one solution over the others if no additional piece of new information about the problem is available for deciding. If further information is ready to plug and play, then is easer to spotlight the search toward one optimal solution. However, if no further information is on hand, all Pareto-optimal solutions are equally relevant. Therefore, it is important to expand the universe of Pareto-optimal solutions as possible in a problem in order to assemble a critical mass of solutions. Thus, it possible to affirm that multi-objective optimization has, actually, two goals [38], this is: a) to find a set of solutions as close as possible to the Pareto-optimal front, and 2) to find a set of solutions as diverse as possible.

III. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHMS Evolutionary algorithms, EAs, have been considered by

many researchers as an umbrella name including four areas: Genetic algorithms, Evolutionary programming, Evolution strategies and Genetic programming. The EAs operation lies on the ground of Darwin’s theory of evolution by natural selection. In such environment, the reproduction, random variation, competition, and selection of contending individuals within some population are common rules. In general, an EA is a set of operators manipulating a population of encoded solutions and evaluating solutions by some fitness function.

Evolutionary algorithms seem particularly suitable to solve

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multi-objective optimization problems, because they deal simultaneously with a set of posible solutions (the so-called population), which allows to find several members of the Pareto optimal set in a single “run”, instead of performing a series of separate runs as is the case when using classical techniques. As an extra, EAs is a better option because they are less susceptible to the shape or continuity of the Pareto front (e.g., they can easily deal with discontinuous or concave Pareto fronts).

Arguably the first multi-objective evolutionary algorithm developed was around mid-1980s and currently is a well established field of research, called “evolutionary multi-objective optimization” [31], where the inherent advantages of the evolutionary techniques are being used in order to design specialized search algorithms to find the Pareto optimal set of a problem.

However, despite the considerable volume of research information in this field, the interest on multi-objective evolutionary algorithms has had a cold start among power systems researchers. An aim for this paper insist on the significance of the field and the many benefits can bring to tackle down challenges in the planning, operation and control of electric distribution and power systems even if renewable energy sources are included.

IV. SURVEY RESULTS All surveyed papers are sorted by methods, application

areas, the year published and problem formulation. The classified results are shown in Table I – Table III. From Table I, it can be observed that in 1994 appeared the first applications of genetic algorithms for a power system problem formulated as multi-objective optimization; however, it was until 2001 that began a real interest for expressing the power systems problems as multi-objective optimization and applying evolutionary algorithms to resolve them. About the applications, environmental/economic dispatch, distribution system planning and reconfiguration are the most popular areas of evolutionary multi-objective optimization methods.

From Table II, it can be observed that, from the 30 papers surveyed, only nine papers present applications of the specialized algorithms developed in the field of “evolutionary multi-objective optimization” (i.e. SPEA, SPEA-II, NPGA, NSGA, NSGA-II, and µGA). There are four papers that present methods based on genetic algorithms with different Pareto sampling techniques from those proposed in the specialized multi-objective evolutionary algorithms. The rest of the papers present evolutionary algorithms that employ conventional methods for multi-objective optimization.

Like conventional algorithms for multi-objective optimization, evolutionary algorithms employing some of those conventional methods are likely to have similar difficulties to find multiple Pareto-optimal solutions. From the surveyed papers, the conventional methods used in evolutionary algorithms are: Linear aggregating, nonlinear

aggregating, goal programming, metric and ε-constrained method.

All of these evolutionary algorithms hybrid with conventional methods suggest a way to convert a multi-objective optimization problem into a single-objective optimization problem. For instance, a linear aggregating method suggests minimizing a weighted sum of multiple objectives, the ε-constrained method suggest optimizing one objective function and use all other objectives as constraints, and goal programming methods suggest minimizing a sum of deviation of objectives from user-specified targets.

All of these methods require some knowledge about the problem in order to define the parameters used to convert a multi-objective optimization problem into a single-objective optimization problem. These parameters largely affect which Pareto-optimal solution will result from the method. Therefore, these methods are recommended only if enough knowledge about the problem is available.

Another disadvantage is that some of the conventional methods are susceptible to the shape of the Pareto front. For instance, the linear weighted sum of multiple objectives and the metric methods have the disadvantage of missing concave portions of the Pareto-optimal front, which is a serious drawback in most real-world problems.

The main disadvantage of goal programming methods is that they yield a Pareto-optimal solution only if the targets are chosen in the feasible solutions region. Furthermore, specifying targets before search may limit the search space and therefore “miss” desirable solutions.

Despite these disadvantages, the conventional methods can be efficient and effective if enough information about the problem is available and if the Pareto front is not difficult to approach.

In Table III it is shown that most of the problems are formulated with two objective functions to optimize. Furthermore, the majority of the problems formulated with more than two objective functions are converted into single-objective optimization problems.

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TABLE I APPLICATIONS OF EVOLUTIONARY ALGORITHMS FOR MULTI-OBJECTIVE POWER SYSTEM PROBLEMS

Application Year a b c d e f g h i j k l m 1994 [1] 1995 1996 1997 [2] 1998 1999 2000 [24] 2001 [3] [21] [28] 2002 [30] 2003 [6,

17,19] [4] [5] [25

]

2004 [18] [7] [8] [9] [16] [26] 2005 [23] [12] [10,27] [11] 2006 [14,20] [15,29] [13] 2007 [22]

The following labels are used: a = POWER DISTRIBUTION NETWORK PLANNING, b = ENVIRONMENTAL/ECONOMIC DISPATCH, c =

PASSIVE FILTER PLANNING, d = DESIGN OF POWER STABILIZERS, e = POWER DISTRIBUTION NETWORK RECONFIGURATION, f = DEMAND MANAGEMENT, g = VOLT/VAR CONTROL AND NETWORK RECONFIGURATION, h = REACTIVE POWER COMPENSATION PLANNING, i = DISTRIBUTED GENERATION PLANNING, j = PARALLEL POWER FLOW CONTROL, k = TRANSMISSION NETWORK PLANNING, l = OPTIMAL CAPACITOR PLACEMENT IN POWER DISTRIBUTION SYSTEMS, m = LOCATION OF AUTOMATIC VOLTAGE REGULATORS

TEVOLUTIONARY ALGORITHMS APPLIED O

Approach SPEA SPEA-II NSGA NSGA-II NPGA MICRO-GA GENETIC ALGORITHMS with Other Paresampling techniques GENETIC ALGORITHMS with Linear aggregating methods GENETIC ALGORITHMS with Goal programming methods GENETIC ALGORITHMS with ε-constrainmethod EVOLUTIONARY PROGRAMMING withGoal programming methods EVOLUTIONARY PROGRAMMING withLinear aggregating method EVOLUTIONARY PROGRAMMING withMetric method (Global criterion method) EVOLUTIONARY STRATEGY with a nonlinear aggregating method (Multiplicativmethod)

ABLE II N MULTI-OBJECTIVE POWER SYSTEMS PROBLEMS

References [6, 27, 20] [23] [21, 20] [14, 29] [17] [22]

to [2, 3, 8, 10]

[1, 5, 13, 15, 16, 30]

[4, 12, 26, 25]

ed [11]

[7, 18]

[19]

[28]

e [9, 24]

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V. CONCLUSIONS The academic interest over multi-objective evolutionary

algorithms has been slowly rising among power systems researchers. Most of the applications have been implemented based on evolutionary algorithms with a mixture taste of conventional multi-objective optimization methods. The resulting algorithms are designs focus on privileging the system preferences of users to facilitate finding a unique solution. However, it seems that at the moment, the power system engineering community has not seduced by the usefulness of a systematic framework for directing the search toward Pareto-optimal front for finding the maximum number of non-dominated solutions in one-single simulation run. Therefore, this paper also is an effort to bring attention from power engineering community about the outmost relevance for applying evolutionary multi-objective optimization field and algorithms because those algorithms have been designed to overcome the limitations that conventional methods have. In addition, in the view of the authors, it is fundamental for the power engineering community to know how is the level of applicability to power system problems of multi-objective evolutionary algorithms to define theirs future research trends and applications.

In this way, another contribution of this paper is a list of bibliographic references to be used as a source for students and researchers interested in solving power system problems using multi-objective evolutionary algorithms.

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