SELECTED TOPICS in MATHEMATICAL METHODS and COMPUTATIONAL TECHNIQUES in ELECTRICAL ENGINEERING. Heuristics were first introduced by G Polya in 1945 3 Metaheuristics. 2 and were developed later in the 70 s when various The new paradigms were called metaheuristics and were. heuristics were also introduced for specific purpose first introduced at mid 80s as a family of searching. problems in different fields of science and technique algorithms able to approach and solve complex. including electrical engineering 3 Basically a optimization problems using a set of several general. heuristic is designed to provide better computational heuristics The term metaheuristic was proposed in 18. performance as compared to conventional optimization to define a high level heuristic used to guide other. techniques at the expense of lower accuracy However heuristics for a better evolution in the search space. the rules of thumb underlying a heuristic are often Although traditional stochastic search methods are. very specific to the problem under consideration mainly guided by chance solutions change randomly. Moreover since heuristics are problem solving from one step to another they can be used in. techniques based on solvers expertise they use domain combination with metaheuristic algorithms to guide the. specific representations and hence general heuristics search process and to accelerate the convergence. can be successfully defined only for fundamental Most metaheuristics algorithms are only. problem solving methods such as search methods 6 approximation algorithms because they cannot always. A wide range of heuristic search strategies can be find the global optimal solution But the most attractive. cited from the literature from uninformed approaches feature of a metaheuristic is that its application requires. basically non heuristic like Depth First Search DeFS no special knowledge on the optimization problem to be. Breadth First Search BrFS or Uniform Cost Search solved hence it can be used to define the concept of. UnCS to informed approaches mainly heuristic like general problem solving model for optimization. Best First Search BeFS Beam Search BeS or the A problems or other related problems 4 12. Search A S 5 6 7 8 10 Uninformed or blind search Since their introduction in the mid 80s till now. strategies are applied with no information about the metaheuristic methods for solving optimization. search space other than the ability to distinguish problems have been continuously developed allowing. between an intermediate state and a goal state addressing and solving a growing number of such. Informed search strategies use problem specific problems previously considered difficult or even. knowledge Usually this knowledge is represented using impossible to solve These methods include simulated. an evaluation function that assesses either the quality of annealing tabu search evolutionary computation. each state in the search space or the cost of moving techniques artificial immune systems memetic. from the current state to a goal state using various algorithms particle swarm optimization ant colony. possible paths In case of BeFS among all possible algorithm differential evolution harmony search. states at one level the algorithm chooses to expand the honey bee colony optimization etc. most promising one in terms of a specified rule 9 The next section presents a brief review of basic. BeS is an enhanced version of the BeFS algorithm issues for the most commonly used metaheuristics cited. The improvements consist in reducing the memory above Several applications of these methods in the field. requirements 11 With this aim in view BeS is defined of power systems will be presented in section 5. based on BrFS which is used to build the search tree At. each level all new states are generated and the heuristic. function is computed for each state that a inserted in a 4 Metaheuristic methods. list ordered by heuristic function values The list is of. limited length equal to the so called beam width This 4 1 Simulated Annealing. limits the memory requirements but the compromise Studies on Simulated Annealing SA were developed in. risk to pruning out the path to the goal state the 1980s based on the Metropolis algorithm 17 which. The A search algorithms uses a BeFS strategy and a was inspired by statistical thermodynamics where the. heuristic function that combines two metrics the cost relationship between the probabilities of two states A. from the origin to the current state or the cost so far and B with energies EA and EB at a common tempera. and an estimation of the cost from the current state to a ture T suggests that states with higher energies are less. goal state or the cost to goal The A algorithm is probable in thermodynamic systems Thus if the system. considered to be very efficient 7 is in state A with energy EA another state B of lower. One of the shortcomings of the search strategies energy EB EA is always possible Conversely a state. presented above is the numerical inefficiency of the B of higher energy EB EA will not be excluded but it. search process especially for high dimensional will be considered with probability exp EB EA T. problems Thus significant efforts were devoted to Application of Metropolis algorithm in search. identify new heuristics able to successfully cope with the problems is known as SA and is based on the possibility. above problems of moving in the search space towards states with poorer. ISSN 1792 5967 96 ISBN 978 960 474 238 7, SELECTED TOPICS in MATHEMATICAL METHODS and COMPUTATIONAL TECHNIQUES in ELECTRICAL ENGINEERING. Table 1 Pseudocode for Simulated Annealing Table 2 Pseudocode for Tabu Search. Data initial approximation X0 initial temperature T Data length of the Tabu list LT number of. number of iteration for a given temperature nT intermediate solutions N. Optimal solution Xbest X0 Initialization approximation X Tabu list TABU X. WHILE stopping criterion not met Optimal solution Xbest X. n 0 i 0 WHILE stopping criterion not met, WHILE n nT DO prepare the Tabu list if Length TABU LT. choose a new approximation Y then delete the oldest item from the list. accept or reject the new approximation based generate N new aproximations in the. on the Metropolis rule Xi 1 G Xi Y T neighborhood of X and select the best candidate. update optimal solution if Xi 1 is better than solution Y which is not TABU. Xbest then Xbest Xi 1 update current approximation X Y and add it. next n iteration n n 1 in the Tabu list Add TABU X. update temperature T update optimal solution if X is better than Xbest. next T iteration i i 1 then Xbest X, values of the fitness function Starting from a Table 3 Pseudocode for Evolution Strategy. temperature T and an initial approximation Xi with a Data number of parents and offsprings k. fitness function Fitness Xi a perturbation is applied to Initialization create initial population P Pi i 1. Xi to generate a new approximation Xi 1 with and initialize the best solution Best void. Fitness Xi 1 If Xi 1 is a better solution then Xi i e WHILE stopping criterion not met. Fitness Xi 1 Fitness Xi the new approximation will evaluate P and update the best solution Best. replace the old one Otherwise when Fitness Xi 1 reproduction stage select fittest individuals from. P and create parent population R Rj j 1, Fitness Xi the new approximation will be considered. mutation stage apply stochastic changes to parents. with probability pi exp Fitness Xi Fitness Xi 1 T, and create k offsprings for each parrent. The above steps are repeated for a given number of R Rj j 1 Q Qi i 1. times for a constant value of temperature then mutation stage replace the current population with. temperature is updated by decreasing its value and the the mutated one. iterative process continues until a stopping criterion is Pi Qi i 1. met The pseudocode for SA is shown in Table 1, common notation is ES The main genetic operator. 4 2 Tabu Search that controls the evolution from one generation to. Tabu Search TS was introduced by 18 as a search another is mutation. strategy that avoids returning to solutions already visited In a general ES model where k each. by maintaining a Tabu list which stores successive generation starts with a population of individuals The. approximations Since the Tabu list is finite in length at fitness of each individual is computed to rank them in. some point after a number of steps some solutions can descending order of their fitness Amongst the current. be revisited Adding a new solution to a complete Tabu population only the first fittest individuals are. list is done by removing the oldest one from the list selected to create the parent population this selection. based o a FIFO principle First In First Out phase is sometimes called truncation Next each of the. New approximations can be generated in different parents will create by repeated mutation k. ways The pseudocode presented in Table 2 uses the offsprings Eventually the new mutated population will. following procedure at each step a given number of new replace the old one and the algorithm reiterates The. approximations are generated in the neighborhood of the pseudocode for the ES model is shown in Table 3. current solution X but considering as feasible only the. ones which are not in the Tabu list Amongst the new 4 4 Genetic Algorithms. approximations the best one is chosen to replace the The Genetic Algorithm GA is a step forward of ES in. current solution being also introduced in the Tabu list the general framework of Evolutionary Computation. GAs have been designed and developed by Holland 15. 4 3 Evolution Strategy and later by Goldberg 26 and De Jong 25. Evolution Strategy ES was first proposed in 13 as a GAs are search strategies that are based on specific. branch of evolutionary computation It was further mechanisms of genetics and natural selection using. developed after 1970 s An ES may be described by two three basic operators selection crossover and mutation. main parameters number of parents in a generation For each generation selection is used to choose parent. and number of offsprings created in a generation A individuals based on their fitness function After. ISSN 1792 5967 97 ISBN 978 960 474 238 7, SELECTED TOPICS in MATHEMATICAL METHODS and COMPUTATIONAL TECHNIQUES in ELECTRICAL ENGINEERING. Table 4 Pseudocode for Genetic Algorithm Table 5 Pseudocode for Differential Evolution. Data population size N crossover rate c and mutation Data population size parents N weighting factors. rate m Initialization create initial population P Pi i 1 N. Initialization create initial population P Pi i 1 N Evaluate current population P and store the fittest. and initialize the best solution Best void individual as Best. WHILE stopping criterion not met WHILE stopping criterion not met. evaluate P and update the best solution Best FOR i 1 TO N DO. initialize offspring population R void select two different individuals Xr1 and Xr2. create offsprings other than Xi, FOR k 1 TO N 2 DO apply mutation Xi Xi Xr1 Xr2. selection stage select parents Q1 and Q2 from apply crossover Xi Xi Xi Xi. P based on fitness values evaluate Xi and replace Xi with Xi anytime. crossover stage use crossover rate c and when Fitness Xi Fitness Xi. parents Q1 Q2 to create offsprings S1 S2 update the best solution Best. mutation stage use mutation rate m to apply, stochastic changes to S1 and S2 and create. mutated offsprings T1 and T2 solutions to which the mechanisms of DE are applied. add T1 and T2 to offspring population Thus for a reference individual Xi two different. R R T1 and T2 individuals Xr1 and Xr2 other than Xi are randomly. replace current population P with offspring selected and an arithmetic mutation is applied to Xi. population R P R based on the difference between Xr1 and Xr2 to produce. elitism replace the poorest solution in P with the a mutant Xi Then an arithmetic crossover based on the. best solution stored in Best, difference between current and mutated solutions is. applied to generate the new estimation Xi Xi will, selecting a pair of parent chromosomes they enter the replace the reference solution and the best one anytime. crossover stage to generate two offsprings Crossover is when its fitness function is better The pseudocode for. useful to create new individuals or solutions that inherit the DE algorithm is shown in Table 5. good characteristics from both parents Newly created. individuals will be altered by small scale changes in the 4 6 Immune Algorithms. genes applying mutation operator Mutations ensure the The Immune Algorithm IA was proposed first be 19. introduction of novelty in the genetic material After to simulate the learning and memory abilities of. completing the offspring population this will replace immune systems The IA is a search strategy based on. the parents from the previous generation and the genetic algorithm principles and inspired by protection. selection crossover mutation process will be resumed mechanisms of living organisms against bacteria and. for a next generation To avoid losing the best solution viruses The problem coding is similar for both GA and. due to the stochastic character of the search procedure IA except that chromosomes in GA are called. described above 25 has proposed to apply a special antibodies in IA and problem formulation i e objective. replacement procedure called elitism that makes a or fitness functions are coded as antigens in IA. optimization techniques were proposed to solve such problems This paper provides basic knowledge about most widely used meta heuristic optimization techniques and their application in optimization problems in power systems Key Words Heuristic optimization Metaheuristic optimization Power systems Efficiency 1 Introduction

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