Comparison of Differential Evolution and Genetic Algorithm in the Design of a 2MW Permanent Magnet Wind Generator A.D.Lilla, M.A.Khan, P.Barendse Department of Electrical Engineering, University of Cape Town Energy Postgraduate Conference 2013

INTRODUCTION  The inherent complex structure of electrical machines makes an optimum design a challenging task.  The Genetic Algorithm (GA) is a benchmark in machine design optimization due to its gradient-free nature and its ability to efficiently find global optima.  Differential evolution (DE) is a metaheuristic optimization routine, and has recently been applied to many optimization problems with much success.  This work uses a RFPMG analytical model, to design a 2MW direct drive permanent magnet radial flux surface mounted generator, with a speed of 22.5rpm and frequency of 11.25Hz, and compares the performance of the GA and the DE in terms of their accuracy and robustness to optimize the design.

2MW RFPM Benchmark Machine - Rating EPC Generate geometry -Slots/pole/phase -No. armature turns, tooth width, slot dimensions, stator back iron dimensions, coil pitch & winding end turn geometry Generate geometry -Slots/pole/phase -No. armature turns, tooth width, slot dimensions, stator back iron dimensions, coil pitch & winding end turn geometry Electrical Freq & rotor surface speed Winding skew & magnetic gap factors are estimated Airgap magnetic flux (accounting for slots, varying reluctances & flux leakage) Magnetic flux, back voltage and internal voltage. Terminal voltage and current, taking into account conductor and windage losses. Efficiency is found SPECIFICATIONsymbolunitvalue MAIN DIMENSIONS Output coefficient K’K’ Internal diameter of stator Dm5.541 Gross length of stator Lm0.870 Pole pitch pp m0.290 Peripheral speed vm/s6.528 STATOR WINDING Flux per pole  wb0.089 Turns per phase T ph -90 Number of slots--540 Slot pitch ss cm0.032 Air gap lengthlgcm0.71 ROTOR DIMENSIONS Number of poles P 60 Magnet heighthmhm mm18.4 Depth of the rotor core d rc m0.601 PERFORMANCE Efficiency  Percent89

IMPLEMENTATION OF GA AND DE i Objective function: –Single performance index for both optimization routines, efficiency. –factors which affect efficiency: Current density J a, Airgap flux density B g, Length of Airgap l g, Magnet- fraction α m, and Slot- fraction α s.

IMPLEMENTATION OF GA AND DE ii Genetic Algorithm The Genetic Algorithm (GA) mimics some aspects in the natural process of evolution. It is a search procedure which emulates the mechanics of evolution. Population Creation –Random process of choosing permutations of design variables. Population Evaluation Selection Crossover Mutation Termination Differential Evolution DE is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Mutation –Plays a more prominent role –DE produces a mutation vector (v1,g+1) in the mutation operation, by adding the weighted (F) difference between two randomly chosen vectors (xr2,g-xr3,g) to a third vector(xr1,g).

NUMERICAL RESULTS AND COMPARISONS (Optimal Solution Searchability)  The performance of the GA and DE both vary case by case, however useful observations can be seen by running each algorithm multiple times.  Due to both algorithms making use of randomized and probability driven processes, statistic-based results are necessary.  Optimal Solution Searchability : –DE randomly generated population size of 20 while being limited to 10 generations. It was run repeatedly, until a value of (highest efficiency recorded). –GA was then repeated for the same number of times, and the results were compared

 it can be seen that after 36 generations, the DE reaches a value of for the objective function.  the GA manages  The standard deviation of the fitness values for the algorithm is 0.19 and 0.23 respective  Because the DE had a higher overall fitness value and had a lower standard deviation, from a stochastic point of view, it indicates the DE to have better performance NUMERICAL RESULTS AND COMPARISONS (Optimal Solution Searchability) Pop. Size Genetic Algorithm Best fitness Worst fitness Times Run Standard deviation of fitness Differential Evolution Best fitness Worst fitness Times Run Standard deviation of fitness

NUMERICAL RESULTS AND COMPARISONS (Computational Efficiency)  Machine design optimization, the majority of the process involves running a machine design model to evaluate each iterated design.  Intelligent algorithm find an optimal solution by running the algorithm a sufficient number of times with a large enough diverse population  Computational efficiency = F (number of iterations, population size and no. Algorithm runs), before an optimal solution is found.

NUMERICAL RESULTS AND COMPARISONS (Computational Efficiency) Generations and population size Both the GA and the DE use randomly generated populations, of sizes 2 and 3, while being limited to 20 generations. Pop. Size Genetic Algorithm Best fitness Worst fitness Average fitness Standard deviation of fitness Differential Evolution Best fitness Worst fitness Average fitness Standard deviation of fitness i.the GA suffers more degradation as compared with the DE, which is still able to give reasonable solutions, with a population size of 2. ii.The standard deviation of the DE is however in most cases double that of the GA

NUMERICAL RESULTS AND COMPARISONS (Computational Efficiency) Number of Algorithm Executions  the GA and the DE were given random tuning parameters ( Mutation Rate, F,CR) and were run for 10 generations, with a population limited to 20 individuals. Pop. Size Genetic Algorithm Best fitness Worst fitness Average fitness Standard deviation of fitness Differential Evolution Best fitness Worst fitness Average fitness Standard deviation of fitness i.DE outputs a best fitness value of 90.25, where as the GA’s best fitness value is ii.DE has the lowest fitness value of 89.62, as compared with the GA’s iii.The average fitness value for the DE is marginally higher, with a value of compared to the GA’s iv.Finally, the DE has a higher standard deviation of 0.23, compared with the GA’s standard deviation of Although the DE seems to have better results than the GA, it should be noted that these results are only marginally different, with the exception of the standard deviation. It should also be noted that the GA and the DE share many fundamental similarities and for this reason, it may be said that they have similar tuning parameter sensitivities.

CONCLUSION  Both the GA and DE are able to arrive at optimal solutions  In many cases the DE outperforms the GA in terms of the best fitness scoring individual and the average fitness from multiple runs of the algorithm  DE however excels in the area of computational efficiency, which is important in the computationally intensive practice of machine design and optimization  The GA and DE show many similarities, however the results show that when computational efficiency and time is a limiting factor, the DE should be preferred over the GA.