Application of Genetic Algorithms to Motor Parameter Determination

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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.


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       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


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 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




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     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.




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     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


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     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.




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     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.



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     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.




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     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




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     Application of Genetic Algorithms to Motor Parameter Determination       2010




Fig: 2 Crossovers




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     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.
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     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




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      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
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     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.




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     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




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     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.




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     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




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     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.

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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


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     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




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    Application of Genetic Algorithms to Motor Parameter Determination       2010




Table 3: Results of genetic algorithm V2




Table 4: Results of genetic algorithm V3




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     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.




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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.




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Fig 6: Effect of bit length and population size on generation number




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                                        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.




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                                     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




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