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Application of Genetic Algorithms to Motor Parameter Determination 2010 CHAPTER 1 1.1 INTRODUCTION The industry is becoming increasingly concerned about the ability of motors to ride through power system disturbances such as voltage dips ((A voltage dip is a short-term reduction in, or complete loss of, RMS voltage. It is specified in terms of duration and retained voltage, usually expressed as the percentage of nominal RMS voltage remaining at the lowest point during the dip. A voltage dip means that the required energy is not being delivered to the load and this can have serious consequences depending on the type of load involved.)) or short duration outages. An improved ridethrough capability improves the reliability of the plant, particularly in the process industry where the failure of a motor can result in considerable downtime. The calculation of reclosing transients after an autoreclose for example, requires knowledge of the motor's electrical and mechanical parameters, which are not always readily available. From the motors' nameplate, the readily available data from the manufacturer are starting torque, breakdown torque, full load torque, full load power factor, full load efficiency etc. It is desirable to be able to extract the motor parameters from such data, which is the purpose of this paper. The Newton-Raphson method has been previously used, but with convergence problems relating to the initial starting points and the requirement for iteration. In this paper, two different techniques, the Newton-Raphson and the genetic algorithms, are used to extract the motor parameters from readily available performance data. Several different induction machines are tested and the results are compared. Many methods can be used to determine motor parameters. Here Newton-Raphson method and advantages of genetic algorithms over this method is discussed. The major drawback of the Newton-Raphson method is that its success depends on the selection of good initial estimates. Although the optimization process may only take a few minutcs, a considerable amount of time and effort can be spent selecting the initial estimates which require familiarity with the particular machine size and parameters. Even when the initial solutions appear reasonable, the optimizer still may not converge to the correct solution. 1 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 1.2 GENETIC ALGORITHM AN OVERVIEW A genetic algorithm (GA) is a search technique used in computing to find exact or approximate solutions to optimization and search problems. Genetic algorithms are categorized as global search heuristics. Genetic algorithms are a particular class of evolutionary algorithms (EA) that use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover. Evolutionary algorithms can be used successfully in many applications requiring the optimization of a certain multi-dimensional function. The population of possible solutions evolves from one generation to the next, ultimately arriving at a satisfactory solution to the problem. These algorithms differ in the way a new population is generated from the present one, and in the way the members are represented within the algorithm. They are part of the derivative- free optimization and search methods that comprise, Simulated annealing (SA) which is a stochastic hill-climbing algorithm based on the analogy with the physical process of annealing. Hill climbing, in essence, finds an optimum by following the local gradient of the function (thus, they are also known as gradient methods). Random Search Algorithms - Random searches simply perform random walks of the problem space, recording the best optimum values found. They do not use any knowledge gained from previous results and are inefficient. Randomized Search Techniques - These algorithms use random choice to travel through the search space using the knowledge gained from previous results in the search. Downhill simplex search 2 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Four types of evolutionary algorithm techniques are presently being used. These are; Genetic Algorithms (GA) Evolutionary Strategies (ES) Genetic Programming (GP) Evolutionary Programming (EP) Search techniques Calculus-based techniques Guided random search techniques Enumerative techniques Direct methods Indirect methods Evolutionary algorithms Simulated annealing Dynamic programming Finonacci Newton Evolutionary strategies Genetic algorithms Parallel Sequential Centralized Distributed Steady-state Generational Fig: 1 Classes of Search Techniques 3 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Genetic algorithms are search methods that employ processes found in natural biological evolution. These algorithms search or operate on a given population of potential solutions to find those that approach some specification or criteria. To do this, the algorithm applies the principle of survival of the fittest to find better and better approximations. At each generation, a new set of approximations is created by the process of selecting individual potential solutions (individuals) according to their level of fitness in the problem domain and breeding them together using operators borrowed from natural genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than the individuals that they were created from, just as in natural adaptation. The GA will generally include the three fundamental genetic operations of selection, crossover and mutation. These operations are used to modify the chosen solutions and select the most appropriate offspring to pass on to succeeding generations. GAs consider many points in the search space simultaneously and have been found to provide a rapid convergence to a near optimum solution in many types of problems; in other words, they usually exhibit a reduced chance of converging to local minima. GAs suffer from the problem of excessive complexity if used on problems that are too large. 4 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 1.3 BIOLOGICAL ASPECT Evolution is a cumulative process. Inheritance is the determinant of almost all of the structure and function of organisms since life began. The amount of variation from one generation to the next is quite small and some molecules, such as those that carry energy or genetic information, have seen very little change since the original common ancestor of several billion of years ago. Inheritance alone does not give rise to evolution because pure inheritance would lead to populations of entirely identical organisms, all exactly like the first one. In order to evolve, there must be something that causes a variation in the structure of the material that an organism inherits material from its parent or parents. In biology, there are several sources of variation. To name a few, mutation, or random changes in inherited material, sexual recombination and various other kinds of genetic rearrangements, even viruses can get into the act, leaving a permanent trace in the genes of their hosts. All of these sources of variation modify the message contained in the material that is passed from parent to offspring. It is an evolutionary truism that almost all variations are neutral or deleterious. Small changes in a complex system often lead to far-reaching and destructive consequences (the butterfly effect in chaos theory) However, given enough time, the search of that space that contains the organisms with their varied inherited material, has produced many viable organisms. Selection is the determining process by which variants are able to persist and therefore also which parts of the space of possible variations will be explored. Natural selection is based on the reproductive fitness of each individual. Reproductive fitness is a measure of how many surviving offspring an organism can produce; the better adapted an organism is to its environment, the more successful offspring it will create. Because of competition for limited resources, only organisms with high fitness survive. Those organisms less well adapted to their environment than competing organisms will simply die out. Evolution can be likened to a search through a very large space of possible organism characteristics. That space can be defined quite precisely. All of an organism‟s inherited characteristics are contained in a single messenger molecule: deoxyribonucleic acid, or DNA. The characteristics are 5 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 represented in a simple, linear, four-element code. The translation of this code into all the inherited characteristics of an organism (e.g. its body plan, or the wiring of its nervous system) is complex. The particular genetic encoding for an organism is called its genotype. The resulting collective physical characteristic of an organism is called its phenotype. In the search space metaphor, every point in the space is a genotype. Evolutionary variation (such as mutation, sexual recombination and genetic rearrangements) identifies the legal moves in this space. Selection is an evaluation function that determines how many other points a point can generate, and how long each point persists. The difference between genotype and phenotype is important because allowable (i.e. small) steps in genotype space can have large consequences in phenotype space. It is also worth noting that search happens in genotype space, but selection occurs on phenotypes. Although it is hard to characterize the size of phenotype space, an organism with a large amount of genetic material (like, e.g., that of the flower Lily) has about 1011 elements taken from a four letter alphabet, meaning that there are roughly 1070,000,000,000 possible genotypes of that size or less. A vast space indeed! Moves (reproductive events) occur asynchronously, both with each other and with the selection process. There are many non-deterministic elements; for example, in which of many possible moves is taken, or in the application of the selection function. Imagine this search process running for billions of iterations, examining trillions of points in this space in parallel at each iteration. Perhaps it is not such a surprise that evolution is responsible for the wondrous abilities of living things, and for their tremendous diversity. All of the genetic material in an organism is called its genome. Genetic material is discrete and hence has a particular size, although the size of the genome is not directly related to the complexity of the organism. 6 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 CHAPTER 2 2.1 METHODOLOGY In a genetic algorithm, a population of strings (called chromosomes or the genotype of the genome), which encode candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem, evolves toward better solutions. Traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible. The evolution usually starts from a population of randomly generated individuals and happens in generations. In each generation, the fitness of every individual in the population is evaluated, multiple individuals are stochastically selected from the current population (based on their fitness), and modified (recombined and possibly randomly mutated) to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population. If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached. Genetic algorithms find application in bioinformatics, phylogenetics, computational science, engineering, economics, chemistry, manufacturing, mathematics, physics and other fields. A typical genetic algorithm requires: 1. a genetic representation of the solution domain, 2. a fitness function to evaluate the solution domain. A standard representation of the solution is as an array of bits. Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size, which facilitates simple crossover operations. Variable length representations may also be used, but crossover implementation is more complex in this case. Tree-like representations are explored in genetic programming and graph-form representations are explored in evolutionary. 7 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 The fitness function is defined over the genetic representation and measures the quality of the represented solution. The fitness function is always problem dependent. For instance, in the knapsack one wants to maximize the total value of objects that can be put in a knapsack of some fixed capacity. A representation of a solution might be an array of bits, where each bit represents a different object, and the value of the bit (0 or 1) represents whether or not the object is in the knapsack. Not every such representation is valid, as the size of objects may exceed the capacity of the knapsack. The fitness of the solution is the sum of values of all objects in the knapsack if the representation is valid or 0 otherwise. In some problems, it is hard or even impossible to define the fitness expression; in these cases, interactive genetic algorithms are used. Once we have the genetic representation and the fitness function defined, GA proceeds to initialize a population of solutions randomly, and then improve it through repetitive application of mutation, crossover, and inversion and selection operators. 2.2 GA OPERATORS A basic genetic algorithm comprises three genetic operators. Initially many individual solutions are randomly generated to form an initial population. The population size depends on the nature of the problem, but typically contains several hundreds or thousands of possible solutions. Traditionally, the population is generated randomly, covering the entire range of possible solutions (the search space). Occasionally, the solutions may be "seeded" in areas where optimal solutions are likely to be found. • Selection • Crossover • Mutation Starting from an initial population of strings (representing possible solutions), the GA uses these operators to calculate successive generations. First, pairs of individuals of the current population are selected to mate with each other to form the offspring, which then form the next generation. 8 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 2.2.1 Selection This operator selects the chromosome in the population for reproduction. The more fit the chromosome, the higher its probability of being selected for reproduction. Thus, selection is based on the survival-of-the-fittest strategy, but the key idea is to select the better individuals of the population, as in tournament selection, where the participants compete with each other to remain in the population. The most commonly used strategy to select pairs of individuals is the method of roulette-wheel selection, in which every string is assigned a slot in a simulated wheel sized in proportion to the string‟s relative fitness. This ensures that highly fit strings have a greater probability to be selected to form the next generation through crossover and mutation. After selection of the pairs of parent strings, the crossover operator is applied to each of these pairs. 2.2.2 Crossover The crossover operator involves the swapping of genetic material (bit-values) between the two parent strings. This operator randomly chooses a locus (a bit position along the two chromosomes) and exchanges the sub-sequences before and after that locus between two chromosomes to create two offspring. For example, the strings 1110 0001 0011 and 1000 0110 0111 could be crossed over after the fourth locus in each to produce the two offspring: 1110 0110 0111 and 1000 0001 0011. The crossover operator roughly imitates biological recombination between two haploid (single chromosome) organisms. An alternative example is where non-binary chromosomes are used Parent A = a1 a2 a3 a4 | a5 a6 Parent B = b1 b2 b3 b4 | b5 b6 The swapping of genetic material between the two parents on either side of the selected crossover point, represented by “|”, produces the following offspring: Offspring A‟ = a1 a2 a3 a4 | b5 b6 Offspring B‟ = b1 b2 b3 b4 | a5 a6 9 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Fig: 2 Crossovers 10 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 2.2.3 Mutation The two individuals (children) resulting from each crossover operation will now be subjected to the mutation operator in the final step to forming the new generation. This operator randomly flips or alters one or more bit values at randomly selected locations in a chromosome. For example, the string 1000 0001 0011 might be mutated in its second position to yield 1100 0001 0011. Mutation can occur at each bit position in a string with some probability and in accordance with its biological equivalent, usually this is very small, for example, 0.001. If 100% mutation occurs, then all of the bits in the chromosome have been inverted. The mutation operator enhances the ability of the GA to find a near optimal solution to a given problem by maintaining a sufficient level of genetic variety in the population, which is needed to make sure that the entire solution space is used in the search for the best solution. In a sense, it serves as an insurance policy; it helps prevent the loss of genetic material. 2.3 FITNESS ASSESSMENT A fitness function must be devised for each problem to be solved. The fitness function and the coding scheme are the most crucial aspects of any GA. They are its core and determine its performance. The fitness function must be maximized. In most forms of evolutionary computation, the fitness function returns an individual‟s assessed fitness as a single real-valued parameter that reflects its success at solving the problem at hand. That is, it is a measure of fitness or a figure-of-merit that is proportional to the "utility" or "ability" of that individual represented by that chromosome. This is an entirely user-determined value. The general rule to follow when constructing a fitness function is that it should reflect the value of the chromosome in some "real" way. If we are trying to design a power amplifier that produces an output power over a specific audio range, then the fitness function should be the amount of audio power per Hertz, that this amplifier produces, over this specific range. 11 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 2.4 SOME PROPERTIES OF GA generally good at finding acceptable solutions to a problem reasonably quickly free of mathematical derivatives no gradient information is required free of restrictions on the structure of the evaluation function fairly simple to develop do not require complex mathematics to execute able to vary not only the values, but also the structure of the solution get a good set of answers, as opposed to a single optimal answer make no assumptions about the problem space blind without the fitness function. The fitness function drives the population toward better solutions and is the most important part of the algorithm. not guaranteed to find the global optimum solutions probability and randomness are essential parts of GA can by hybridized with conventional optimization methods potential for executing many potential solutions in parallel 12 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 2.5 DIFFERENCES BETWEEN SEARCH METHODS The most significant differences between the more traditional search and optimization methods and those of evolutionary and genetic algorithms are that these algorithms, work with a coded form of the function values (parameter set), rather than with the actual parameters themselves. So, for example, if we want to find the minimum of the objective function f (x) = x2 + 5x + 6 , the GA would not deal directly with x or f(x) values, but would work with strings that encode these values. In this case, strings representing the binary x values would be used, use a set, or population, of points spread over the search space to conduct a search, not just a single point on the space. This provides GAs with the power to search spaces that contain many local optimum points without being locked into any one of them, in the belief that it is the global optimum point, the only information a GA requires is the objective function and corresponding fitness levels to influence the directions of search. Once the GA knows the current measure of "goodness" about a point, it can use this to continue searching for the optimum, use probabilistic transition rules, not deterministic ones, are generally more straightforward to apply, can provide a number of potential solutions to a given problem, the application of GA operators causes information from the previous generation to be carried over to the next. CHAPTER 3 13 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 3.1 A SIMPLE GA BY HAND Let f(x) = x²; We have to maximize this function on the integer interval [0-31]. For this we will initially create a population of strength 4(n), fitness function is calculated string wise and next generation is created by using a roulette wheel. Strings with high fitness value will be transmitted to next generation and others die off. 14 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Table above clearly indicates how Genetic algorithm work when maximizing a function. Roulette Wheel is a way of choosing members from the population of chromosomes in a way that is proportional to their fitness. It does not guarantee that the fittest member goes through to the next generation merely that it has a very good chance of doing so. It works like this: Imagine that the population‟s total fitness score is represented by a pie chart, or roulette wheel. Now you assign a slice of the wheel to each member of the population. The size of the slice is proportional to that chromosomes fitness score. i.e. the fitter a member is the bigger the slice of pie it gets. Now, to choose a chromosome all you have to do is spin the ball and grab the chromosome at the point it stops. Fig3: Roulette Wheel 15 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 3.2 IMPLIMENTATION OF GENETIC ALGORITHM The genetic algorithm can be used to calculate the equivalent circuit parameters of an induction machine as shown in figure 4. The locked rotor, breakdown, and full load torque equations form a multiobjective optimization problem, where each equation is a function of three or more machine parameters. The three torque functions can be written as follows. Fig 4: equivalent circuit of an induction motor. 16 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 where F1 is the error in the full load torque, F2 is the error in the locked rotor torque, F3 is the error in the breakdown torque, R2 is the rotor resistance, R1 is the stator resistance, X2 is the rotor reactance, X1 is the stator reactance, and Xm is the magnetizing reactance. For simplicity, the stator and rotor leakage reactances are combined into one leakage reactance (XI). The stator and rotor reactances can be extracted after the optimization by knowing the design class of the machine. The magnetizing reactance (Xm) can be calculated using the full load power factor equation after R1, R2, and X1 have been calculated, using the genetic algorithm. Each parameter is coded as a 14 bit unsigned binary number, and together they form one 42 bit string as shown in figure 5. The maximum value each parameter can have, based on an accuracy of three decimal places, is 16.384 ohms. In this case, the error function is chosen as the sum of the squares of the toque error functions, while the fitness function is the inverse of the error. The aim of the genetic algorithm is to minimize the error or to maximize the fitness. Fig 5: The genetic algorithm string 17 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 3.3 Steady state parameter and torque results Three induction motors (table) with known equivalent circuit parameters were used to test four different versions of the genetic algorithm. Table1: Actual Machine Parameters Each version uses the same population size, random number generator, string size, objective functions, and fitness function, yet each is slightly different in its approach. A description of each version follows. V1: Version 1 uses stochastic sampling with replacement (weighted roulette wheel) as described earlier for reproduction. Simple crossover and mutation are also used, that is, one randomly selected crossover point and one bit change per thousand bit transfers for each string. This version is the simple genetic algorithm. V2: Version 2 is identical to version 1 except that deterministic sampling is used instead of stochastic sampling for its reproduction scheme. The deterministic sampling scheme calculates the probabilities of selection as usual, the string fitness divided by the total fitness. Then each string is assigned an expected number based on its probability of selection. The actual number of times a string is copied into the mating pool is found from the integer part of the expected number. If additional strings are needed to fill the new population then the fractional parts of the expected number are sorted and the strings are selected from the top of the sorted list. This selection scheme has proved superior to straight roulette wheel selection since it guarantees that tit strings will be copied into the mating pool. 18 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 V3: Version 3 uses the deterministic sampling scheme with two-point crossover. The two-point crossover operator swaps all binary digits between two randomly selected points along the string. V4: Version 4 uses the deterministic sampling selection scheme with a crossover operator for each parameter. This means that there is one randomly selected crossover point for each parameter or three crossover points along the entire string. This algorithm is superior to the other three since it produces consistently good results. The results of each version of the genetic algorithm are given in the tables below. Comparisons can be made with table 1 which shows the actual equivalent circuit parameters for each induction machine. In addition, results from Quattro Pro's Newton- Raphson search routine is provided. The results of each version of the genetic algorithm are given in the tables below. Comparisons can be made with table 1 which shows the actual equivalent circuit parameters for each induction machine. In addition, results from Quattro Pro‟s Newton-Raphson search routine are provided. The results in using version 1 of the genetic algorithm are shown in table 4.2 for 3 different horsepower sizes. The estimated parameters are compared with the actual parameters and the errors calculated. Considerable errors are produced for some parameters, for example 85% error in R1 and even larger errors in R2. This version would be unacceptable. Table 4.3 has the parameter results using version 2. Although the parameter errors are not as large as for version 1, they are still considerable. Version 3 does not produce substantially better results than version 2 as shown in the parameter errors of table 4.4. Version 4 has the best results with acceptable errors in R2, X1, and Xm as shown in table 4.5. Larger errors were produced in R1 for the small and large motors. However, table 4.6 shows that errors in R1 do not affect the torque calculations significantly. For example, while there were 15% and 24% errors in R1, for the 5 hp and 500 hp motors, the maximum error produced in the estimation of any toque is 2%. This is not unexpected since the error function was defined to minimize the torques, not the electrical 19 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 parameters. Tables 4.7 and 4.8 show that acceptable results are obtainable using Newton- Raphson techniques provided good initial estimates of parameters are used. However table 4.9 shows that a slight change in the initial estimate of a parameter can cause the Newton-Raphson to converge to an entirely wrong solution as shown for the leakage and magnetizing reactances. The genetic algorithm is more robust in this regard. Table2: Results of genetic algorithm VI 20 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Table 3: Results of genetic algorithm V2 Table 4: Results of genetic algorithm V3 21 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Table 5: Results of genetic algorithm V4 The performance of the error function when each version of the genetic algorithm is used is compared in figurt 4.1. The results show that all versions eventually converge, thus producing low errors in the torques, but not necessarily low errors in the parameters. Version 4 however converges fastest with acceptable errors in the parameters as well as the torquea. Figure 4.2 shows the convergence of machine paramders using the Newton- Raphson method. Two cases are used to test the convergence of the Newton-Raphson method. In case 1 the initial guess of the machine parameters was good, and the optimizer converged to the c o m t machine parameters. In case 2 the paramder xl was changed from 1.05 to 1.00 while all other parameters remained the same, and the optimizer failed to converge. 22 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 3.4 Effect of population size and bit length on the algorithm Performance While 14 bits were used for each parameter and a population size of 250 strings was used in this paper, it is of interest to examine the effect of different bit lengths and population numbers on the ability of the algorithm to converge. The results are presented in figure showing a bit length variation from 8 to 18 and the number of strings or population size from 100 to 500 for version V4. The number of generations is basically an indication of the time to converge. If the number of bits is less than 10 then poor results are obtained regardless of the population size. If the population size is less than 150 strings then poor results are obtained, regardless of the bit length. Also if the population is too large, greater than say 450 strings, the performance deteriorates regardless of bit size, indicating that perhaps other advanced operators like dominance and segregation may have to be used to produce reasonable results. The graph shows that bit sizes between 10 and 18 and population sizes between 150 to 400 would give consistently good results. A smaller bit size and population size has advantages in computer space requirements and processing time. 23 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 Fig 6: Effect of bit length and population size on generation number 24 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 CONCLUSION This paper has applied the genetic algorithm to the problem of motor parameter determination. Several different versions were examined by calculating the parameters for a small (5 hp), medium (50 hp), and large (500 hp) induction motor. Version 4 produced extremely good results when the torques was generated from the equivalent circuit with known parameters. Larger errors were produced when using actual data from several manufacturers due to the neglect of parameter variations and deep bar effects in the model. The results were still acceptable for 5 of the 7 manufacturers. The use of the Newton-Raphson method was also demonstrated and its sensitivity to the initial starting values highlighted. 25 Department of Electrical and Electronics Application of Genetic Algorithms to Motor Parameter Determination 2010 REFERENCES [l] T.A Higgins, W.L. Snider, P.L. Young, and H.J. Holley, "Report on Bus Transfer: Part I – Assessment Application", IEEE Transactions on Energy Conversion, vol. 5 , no. 3, September 1990. [2] B.K. Johnson and J.R. Wfis, "Tailoring Induction Motor Analytical Models to Fit Known Motor Performance Characteristics and Satisfy Particular Study Needs", IEEE Transactions on Power Sytems, vol. 6, no. 3, August 1991. [3]Theory and performance of electrical machines- J.B Gupta(Part III, AC machines, page:440-) [4] „Genetic Algorithms in search optimization and machine learning‟-David E. Goldberg [5]Wikipedia encyclopedia 26 Department of Electrical and Electronics

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