Proceedings of the 7th World Congress on Mechanical, Chemical, and Material Engineering (MCM'21)
Prague, Czech Republic Virtual Conference – August, 2021
Paper No. ICMIE 105
DOI: 10.11159/icmie21.105

Magnetic Flux Density Distribution of 1.5 kW Permanent Magnet
Synchronous Generator by using Response Surface Methodology
Deniz Perin1, Mehmet Baki Dogru1, Demet Gonen Ocaktan2, Ali Oral3, Aslan Deniz Karaoglan2
ISBIR Electric Co.
Department of R&D, 10150, Balikesir, Turkey
tsr16@isbirelektrik.com.tr; tsr03@isbirelektrik.com.tr
2Balikesir University
Department of Industrial Engineering, 10145, Balikesir, Turkey
dgonen@balikesir.edu.tr; deniz@balikesir.edu.tr
3Balikesir University
Department of Mechanical Engineering, 10145, Balikesir, Turkey
alioral@balikesir.edu.tr
1Abstract - In this paper, design optimization of 1.5 kW permanent magnet synchronous generator (PMSG) is performed. The aim is toobtain the target magnetic flux density distributions for the responses namely stator-teeth flux density (Tesla), stator-yoke flux density(Tesla), and air-gap flux density (Tesla). For this purpose response surface methodology (RSM) – which is a well-known design ofexperiments technique – is used for mathematically modelling the relations between the mentioned responses and the designparameters (embrace, outer diameter, and thickness). Maxwell simulation results are used for experimental calculations. Optimization
results are also confirmed by using Maxwell program. The results indicate that the RSM is an effective method for design optimizationof this type of magnetic devices.
Keywords: magnetic flux density distribution (MFDD), permanent magnet synchronous generator (PMSG), response
surface methodology (RSM), design optimization.

1. Introduction

It is important for the generators to keep the magnetic flux density distribution in a particular range to provide the highefficiency for the electric machine. Magnetic device design optimization by the aid of RSM was investigated by many
researchers in the last decades. Gillon and Brochet [1] researched electric motor design optimization using RSM. Jolly etal. [2] studied permanent magnet motor (PMM) design optimization using RSM and genetic algorithm (GA). For
numerical experiments, they used finite element analysis (FEA). Fang et al. [3] optimized the design of an interior PM
synchronous motor (PMSM) by using RSM. Hasanien et al [4] used RSM and GA together to optimize the design of PMtransverse flux linear motor (TFLM). FEA computations are used for numerical experiments. Hasanien, and Muyeen [5]studied on the design optimization of frequency converter of a variable speed wind turbine (VSWT) driven PM
synchronous generator (PMSG) by using GA and RSM. Wen et al. [6] used RSM and particle swarm optimization (PSO)
together for optimizing the coil design of wind turbine generator. They considered air gap magnetic flux density (AGMFD)
as response. Zhang et al. [7] used PSO and RSM for optimization of transverse flux PMM (TFPMM) design. Asef et al. [8]used RSM and Booth’s algorithm using simulated annealing (BA-SA) together to optimize the design of PMSG. Soleimaniet al. [9] proposed RSM for optimizing high power TFPMG design. Fekri et al. [10] presented optimal design of surface
mounted 3-phase axial flux switching PMG (SMAFSPM) by using RSM and FEA together. Semon et al. [11] studied onrotor design optimization of a V-type interior PMSM. They used RSM to optimize the air-gap magnetic flux density.
Karaoglan et al. [12] studied on stator slot design optimization of PMG. They used RSM for multi-objective optimizationof the magnetic flux for air-gap, stator teeth, and stator yoke, and also maximizing the efficiency.
In this study design optimization of 1.5 kW 16-poled PMSG is performed. The responses are selected as the magnetic
flux density distributions at the stator-teeth, stator-yoke, and air-gap. The factors those will be optimized are the embrace,ICMIE 105-1

outer diameter, and thickness. The selected performance criterions (responses) and the factors did not investigate
together previously and this is the novelty aspect of this research. Next section describes the materials and methods ofthis study.

2. Response Surface Methodology

Optimization by using design of experiment techniques has three main steps. These are: (i) constructing theexperimental design and performing the experiments, (ii) mathematical modelling by using these experimental data,(iii) performing the optimization by using the mathematical models. RSM is one of the widely used design ofexperiment technique. It is used for modelling the mathematical relations between the factors (input variables) and theresponse (output variable) by using the experimental results (which are obtained from an orthogonal design). Centralcomposite design, face centered design, and Box-Behnken design are the well-known RSM experimental design. In
this study we used face centered design [13, 14].
Mathematical models in RSM can be linear or second order polynomials. These polynomials are calledregression equation. In this study, full quadratic regression model is used. This model includes linear, square, andinteraction terms. General representation of full quadratic model is given in Eq. (1) [13, 14]:
2𝑌𝑌𝑢𝑢 = 𝛽𝛽0 + ∑𝑛𝑛𝑖𝑖=1 𝛽𝛽𝑖𝑖 𝑋𝑋𝑖𝑖𝑖𝑖 + ∑𝑛𝑛𝑖𝑖=1 𝛽𝛽𝑖𝑖𝑖𝑖 𝑋𝑋𝑖𝑖𝑖𝑖+ ∑𝑛𝑛𝑖𝑖<𝑗𝑗 𝛽𝛽𝑖𝑖𝑖𝑖 𝑋𝑋𝑖𝑖𝑖𝑖 𝑋𝑋𝑗𝑗𝑗𝑗 + 𝑒𝑒𝑢𝑢

(1)

Following the calculation of mathematical models, coefficient of determination (R2) is calculated. R2 is desired tobe closer to 1 (in other words: 100%). Then this means the factors (X terms) used in the mathematical model issufficient and the model has the ability of explaining the response (Y). Then significance for the regression equation
is tested by using analysis of variance (ANOVA). In this study we used Minitab statistical package for designing,modelling and optimizing the problem. To perform ANOVA, using P-value approach is a common way. According tothis approach if the P-value calculated by the aid of Minitab is less than 0.05 (5%) (Type-I error is 5% for 95%
confidence level), then this means the mathematical model represented in Eq. (1) is significant and can be used for
optimization [13, 14].
In the third phase, optimization is performed to determine the optimum factor levels those provide us to obtain
the desired response values. RSM uses gradient search algorithm for this purpose. In this study we used gradientsearch algorithm by the aid of ‘Minitab Response Optimizer Module’.

3. Experimental Results

The aim of this research is to calculate the optimum design values for embrace, outer diameter, and thickness toobtain the desired magnetic flux distributions for the 1.5 kW 16-poled PMSG. The performance criteria are
determined as 1.8 Tesla, 1.1 Tesla, and 0.9 Tesla values for stator-teeth, stator-yoke, and air-gap flux density values,respectively. The structure of the PMSG that is used for the experiments are presented in Figure 1. Embrace is a ratiothat shows how much of the rotor surface the magnet is enveloping. So it has no unit. The measurements for the othertwo factors are expressed in millimetres (mm).
RSM face centered design is used for designing the experiment. The factor levels are determined (as [min, max]values) as [0.45, 0.95], [210, 250], and [3, 10] for the embrace, outer diameter, and thickness respectively. The center
points are 0.7, 230, and 6.5 respectively. The experimental design is given in Table 1. In Table 1, the factor level ofthickness in the RSM experiment design (run numbers: 9, 10, 13, 14, 15) was 6.5 mm in the original experimental
design. However, this value was taken as 7 mm as it does not comply with mass production conditions.

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Fig.1: Design of the PMSG that will be optimized.
Table 1: Simulation results of Maxwell for the RSM face centered design.

Factors
Run Embrace
(X1)
12
34
5678
910111213
1415

0.45
0.95
0.45
0.95
0.45
0.95
0.45
0.95
0.45
0.95
0.7
0.7
0.7
0.7
0.7

Outer
Diameter(X2) (mm)
210
210
210
210
250
250
250
250
230
230
230
230
210
250
230

Thickness
(X3)
(mm)
3
3
1010
3
3
10107
7
3
10777

Responses
Stator -Teeth
Flux Density(Y1) (Tesla)
1.54
1.56
1.87
1.96
1.54
1.56
1.87
1.96
1.8
1.86
1.55
1.92
1.83
1.83
1.83

Stator -Yoke
Flux Density(Y2) (Tesla)
0.73
1.47
0.921.73
0.22
0.44
0.27
0.52
0.4
0.79
0.52
0.66
1.35
0.41
0.63

Air-Gap FluxDensity(Y3) (Tesla)
0.78
0.79
0.94
0.99
0.78
0.79
0.94
0.99
0.91
0.94
0.78
0.97
0.92
0.92
0.92

The mathematical models those represents the relations between the responses and the factors are calculated byusing Minitab statistical package and given in Eqs. (2) - (4) below:
Y1 (Stator − Teeth Flux Density ) = 1.078344444444 + 0.004888888889X1 + 0.001277777778X2 +0.114084656085X3 − 0.017777777778𝑋𝑋12 − 0.00000277777777777768𝑋𝑋22 − 0.005830687831𝑋𝑋32 −0.00000000000000000755X1 X2 + 0.02X1 X3 + 0.00000000000000000040X2 X3(2)Y2 (Stator − Yoke Flux Density) = 31.19413306 + 8.05066125X1 − 0.27747873X2 + 0.18763905X 3 −(3)0.68444444𝑋𝑋12 + 0.00060556𝑋𝑋22 − 0.00312434𝑋𝑋32 − 0.027X1 X2 + 0.01235772X1 X3 − 0.0005874X2 X3ICMIE 105-3

Y3 (Air − Gap Flux Density ) = 0.358474255 − 0.077344173X1 + 0.002555556X2 + 0.055106594X3 +0.044444444𝑋𝑋12 − 0.00000556𝑋𝑋22 − 0.002851852𝑋𝑋32 − 0.00000000000000000382X1 X2 +0.011382114X1 X3 + 0.00000000000000000041X2 X 3(4)The statistical analysis results obtained from Minitab are presented in Table 2. According to these results the R2
values are very good (close to 100%) which means these three factors are sufficient to explain the responses. Also theP-values are lover then the 5% which means these models are significant and can be used for optimization.
Response

Table 2: Statistical analysis results for the mathematical models.

Stator Teeth Flux Density
Stator Yoke Flux Density
Air Gap Flux Density

R2 Results
R2 (%) R2 (Prediction) (%)
100
99.98
99.71
97.31
99.95
99.58

R2 (Adjusted) (%)
99.99
99.18
99.86

ANOVA Results
P-Value
Result0.000<0.05 Significant
0.000<0.05 Significant
0.000<0.05 Significant

Table 3 presents the prediction performances of the models. In Table 3, 𝑌𝑌𝑖𝑖 values represents the observed values(Maxwell results), and the 𝑌𝑌�𝑖𝑖 values are the expected values (Minitab predictions for the mathematical models). PE is theprediction error (𝑃𝑃𝑃𝑃𝑖𝑖 (%) = ��𝑌𝑌𝑖𝑖 − 𝑌𝑌�𝑖𝑖 �/𝑌𝑌�𝑖𝑖 �100):Run(i)123456789101112131415

Table 3: Prediction performances.

Stator Teeth Flux Density
PEi1 (%)
𝑌𝑌𝑖𝑖1
𝑌𝑌�𝑖𝑖1
1.54 1.5396 0.03
1.56 1.5596 0.03
1.87 1.8706 0.03
1.96 1.9606 0.03
1.54 1.5396 0.03
1.56 1.5596 0.03
1.87 1.8706 0.03
1.96 1.9606 0.03
1.81.7998 0.01
1.86 1.8598 0.01
1.55 1.5518 0.11
1.92 1.9178 0.12
1.83 1.8298 0.01
1.83 1.8298 0.01
1.83 1.8309 0.05

Stator Yoke Flux Density
Air Gap Flux Density�PEi2 (%) 𝑌𝑌𝑖𝑖3
PEi3 (%)
𝑌𝑌𝑖𝑖2
𝑌𝑌𝑖𝑖2
𝑌𝑌�𝑖𝑖30.73
0.74271.71
0.78
0.77940.08
1.47
1.47250.17
0.79
0.78890.14
0.92
0.94732.88
0.94
0.94140.15
1.73
1.72030.56
0.99
0.99080.08
0.22
0.22934.06
0.78
0.77940.08
0.44
0.41915.00
0.79
0.78890.14
0.27
0.26940.21
0.94
0.94140.15
0.52
0.50243.49
0.99
0.99080.08
0.4
0.351213.89
0.91
0.90840.17
0.79
0.83575.47
0.94
0.94070.07
0.52
0.51640.69
0.78
0.78360.45
0.66
0.66040.07
0.97
0.96560.46
1.35
1.31712.49
0.92
0.91960.05
0.41
0.43976.76
0.92
0.91960.05
0.63
0.63620.98
0.92
0.9218
0.19

The results given in Table 3 show that the mathematical models good-fit the observations. The optimizationis performed by using ‘Minitab Response Optimizer Module’ by using the Eqs. (2) – (4). The Optimizationgraph obtained from Minitab is presented in Figure (2). According to this figure optimum factor levels arecalculated as 0.9, 220 mm, and 6 mm for the embrace, outer diameter, and thickness respectively. The responses
are predicted as 1.7976 Tesla (Target: 1.80), 1.1076 Tesla (Target: 1.1), and 0.9076 Tesla (Target: 0.9) for thestator-teeth flux density, stator-yoke flux density, and air-gap flux density.

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Fig.2: Optimization result calculated by ‘Minitab Response Optimizer’.

The optimum design is confirmed by Maxwell simulation. The confirmations and comparisons are given inTable 4.
Table 4: Confirmations and comparisons concerning the optimization results.
Maxwell ( Yi ) Minitab ( Yˆ )PE (%)

Responses

Stator-teeth flux density
Stator-yoke flux density
Air-gap flux density

1.81
1.04
0.91

1.7976
1.10760.9076

i

0.69
6.10
0.26

According to the results presented in Table 4, it can be clearly indicated that the predicted results are very
close to the Maxwell results and the overall prediction error is less than 6.1%. This means the design
optimization is completed and this design can be used in mass production. The magnetic flux density distribution
graph obtained from Maxwell simulation is presented in Figure 3.

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Fig.3: Maxwell simulation result for magnetic flux density distribution of the optimized PMSG.

4. Conclusion

In this study design optimization of 1.5 kW 16-poledPMSG is performed. For this purpose RSM is used to design
the experiments, mathematical modelling and optimization. The observations for the experimental design arecalculated from Maxwell simulations. Optimum factor levels are calculated by using ‘Minitab Response Optimizermodule’. Optimum factor levels are calculated as 0.9, 220 mm, and 6 mm for the embrace, outer diameter, andthickness respectively. According to the confirmations performed by using Maxwell simulations, the target magnetic
flux density distributions are reached for the responses. These results indicate that the RSM can be used effectively forthis type of problems. In the future research this study can be extended for additional performance criterions. Also theaccuracy of the results will be confirmed by producing prototype PMSG.

Acknowledgements

We would like to thank Isbir Electric Co. for providing us with the ability to use its facilities and softwares.

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ICMIE 105-7