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Optimization of multiphase single-layer winding end-connections by differential evolution

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Abstract

In this paper, the authors propose a flexible automatic procedure for optimizing multiphase single-layer winding end-connections by the differential evolution algorithm. By using the winding distribution in slots as a predefined input, connections between phase conductors located in different slots are determined to minimize the winding average coil pitch and to simultaneously maximize the number of coils with identical coil pitch. For an adopted geometrical shape and number of turns of individual coils, minimization of winding average coil pitch results in minimized end-connection length, winding resistances, end-leakage inductances, winding mass and copper losses, while machine efficiency and power factor are maximized. However, maximizing the number of identical coils simplifies and speeds up the process of coil fabrication during machine assembly, but potentially leads to higher average coil pitch. The proposed method is applied to single-layer windings with different slot, pole and phase combinations, where it is proven that the method automatically provides end-connection arrangements which are better or equal in terms of copper usage and simplicity of fabrication than the ones found in the literature. Verification is conducted by comparing overall performance of several Rotor Permanent Magnet Flux Switching motors with different stack lengths, outer diameters and optimized end-windings by means of 2D finite element analysis and 3D analytical approach. The proposed method is conceived as an automatic tool for optimization of single-layer winding end-connections during comprehensive electrical machine design optimization.

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Abbreviations

A :

Winding distribution table/matrix

\(a_{n,k}^{( + )}\), \(a_{n,k}^{( - )}\) :

Elements of the winding distribution table

a p :

Number of winding parallel paths

A s :

Total slot area

c :

Weighing factor

c 0 :

Critical value of the weighing factor

C n :

Connection matrix for n-th phase

CR :

Differential evolution crossover rate

DE:

Differential evolution

f(Y n):

Objective function

F :

Differential evolution scaling factor

FEA:

Finite element analysis

G :

Number of generations/iterations of the differential evolution algorithm

i :

Population member index

J :

Current density in conductors

j :

Optimization variable index

k :

Coil index

k Cu :

Slot filling factor

l ew :

Average length of a single coil end-connection

L ew :

End-leakage inductance

L sew :

End-winding self-inductance

m :

Number of phases (phase windings)

m Cu :

Copper mass

M ew :

End-winding mutual inductance

MMF:

Magnetomotive force

n :

Phase index

N c :

Number of coils per turn

n lay :

Number of winding layers

N p :

Number of population members in the differential evolution algorithm

n wc :

Number of wound coils per phase

p :

Number of pole pairs

P Cu :

Copper losses

Q s :

Number of slots

R ew :

End-winding resistance

RPMFS:

Rotor Permanent Magnet Flux Switching

T n :

Vector containing all positive elements from n-th row of the winding distribution table

u :

Number of coils with different coil pitch

U n , i , G :

i-th trial vector for n-th phase on G-th generation of the differential evolution algorithm

V n , i , G :

i-th donor vector for n-th phase on G-th generation of the differential evolution algorithm

WDT:

Winding distribution table

X n , i , G :

i-th variable vector for n-th phase on G-th generation of the differential evolution algorithm

\(x_{k}^{(n)}\) :

Element of n-th phase variable vector

Y n :

Coil pitch vector for n-th phase

\(y_{{{\text{avg}}}}^{(n)}\) :

Average coil pitch of n-th phase

\(y_{k}^{(n)}\) :

Coil pitch of k-th coil in n-th phase

γ Cu :

Mass density of copper

ρ Cu75 :

Resistivity of copper at 75 °C

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Authors and Affiliations

  1. School of Electrical Engineering, University of Belgrade, Belgrade, 11000, Serbia

    Đorđe M. Lekić & Slobodan N. Vukosavić

  2. Faculty of Electrical Engineering, University of Banja Luka, 78000, Banja Luka, Republic of Srpska, Bosnia and Herzegovina

    Đorđe M. Lekić

Authors
  1. Đorđe M. Lekić
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  2. Slobodan N. Vukosavić
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Contributions

Đorđe M. Lekić developed the theoretical formalism, performed the analytic calculations and performed the numerical simulations. Both authors discussed the results and contributed to the final version of the manuscript. Slobodan N. Vukosavić supervised the findings of this work.

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Correspondence to Đorđe M. Lekić.

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Lekić, Đ.M., Vukosavić, S.N. Optimization of multiphase single-layer winding end-connections by differential evolution. Electr Eng 104, 2589–2602 (2022). https://doi.org/10.1007/s00202-022-01498-4

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  • DOI: https://doi.org/10.1007/s00202-022-01498-4

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