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.