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 20133

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

Electrical Freq & rotor surface speed

Winding skew & magnetic gap factors are estimatedWinding skew & magnetic gap factors are estimated

Airgap magnetic flux (accounting for slots,

varying reluctances & flux leakage)

Airgap magnetic flux (accounting for slots,

varying reluctances & flux leakage)

Magnetic flux, back voltage and internal

voltage.

Magnetic flux, back voltage and internal

voltage.

Terminal voltage and current, taking into

account conductor and windage losses.

Terminal voltage and current, taking into

account conductor and windage losses.

Efficiency is foundEfficiency is found

SPECIFICATION symbol unit valueMAIN DIMENSIONS

     

Output coefficient

K’ - 199.54

Internal diameter of stator

D m 5.541

Gross length of stator

L m 0.870

Pole pitch p m 0.290

Peripheral speed

v m/s 6.528

STATOR WINDING

     

Flux per pole wb 0.089Turns per phase

Tph - 90

Number of slots - - 540Slot pitch s cm 0.032

Air gap length lg cm 0.71ROTOR DIMENSIONS

     

Number of poles

P   60

Magnet height hm mm 18.4

Depth of the rotor core

drc m 0.601

PERFORMANCE      Efficiency Percent 89

IMPLEMENTATION OF GA AND DE i

Objective function:– Single performance

index for both optimization routines, efficiency.

– factors which affect efficiency: Current density Ja, Airgap flux density Bg, Length of Airgap lg, Magnet-fraction αm, and Slot-fraction αs .

IMPLEMENTATION OF GA AND DE iiGenetic 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).

Selection

Crossover

Mutation

Stop

Display Best Individuals

Fitness evaluation of each individual

Enough Generation?

Start

Initial population generation for GA

YES

NO

NO

YES

Random choice of two population members 𝑥𝑟2,𝑥𝑟3

𝐹.(𝑥𝑟2 − 𝑥𝑟3) Build weighted difference vector

Choose target vector 𝑆1

Initial population generation for DE

𝑣 = 𝑥𝑟1 + 𝐹.(𝑥𝑟2 − 𝑥𝑟3) Add a third randomly

chosen vector𝑥𝑟1:

Do crossover with target vector 𝑆1 to get trial vector u

u < 𝑆1 ?

Create new population generation

Replace 𝑆1 with u Keep 𝑆1

Enough Generations?

Stop

YES NO

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 90.30 (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 90.31 for the objective function.

the GA manages 90.28

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 RunStandard

deviation of fitness

20

90.25 89.44 36 0.23Differential Evolution

Best fitness

Worst fitness

Times RunStandard

deviation of fitness

90.31 89.43 36 0.19

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 AlgorithmBest

fitnessWorst fitness

Average fitnessStandard deviation

of fitness

2 89.90 89.45 89.76 0.143 89.90 89.44 89.64 0.15

Differential EvolutionBest

fitnessWorst fitness

Average fitnessStandard deviation

of fitness

2 90.31 89.31 89.99 0.313 90.27 89.22 89.92 0.32

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 fitnessStandard deviation

of fitness

20

90.23 89.78 89.93 0.19Differential Evolution

Best fitness

Worst fitness

Average fitnessStandard deviation

of fitness

90.25 89.62 89.97 0.23

i. DE outputs a best fitness value of 90.25, where as the GA’s best fitness value is 90.23ii. DE has the lowest fitness value of 89.62, as compared with the GA’s 89.78. iii. The average fitness value for the DE is marginally higher, with a value of 89.97 compared to

the GA’s 89.3. iv. Finally, the DE has a higher standard deviation of 0.23, compared with the GA’s standard

deviation of 0.19.

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.