Optimization Techniques and Mathematical Modeling Applied to Reluctance Motors

Miranda, B.B.; Malagoli, J. A.; Camacho, J. R.

doi:10.1590/2179-10742022v21i3261941

B.B. Miranda

Postgraduate Program in Electrical Engineering, FEELT, Universidade Federal de Uberlândia, Uberlândia MG, Brazil, brenobm@ufu.br, jrcamacho@ufu.br

https://orcid.org/0000-0002-2334-6659

J. A. Malagoli

Postgraduate Program in Electrical Engineering, PPGEE, Universidade Federal do Paraná, Curitiba PR, Brazil, juliana.malagoli@ufpr.br

https://orcid.org/0000-0002-8723-033X

J. R. Camacho

Postgraduate Program in Electrical Engineering, FEELT, Universidade Federal de Uberlândia, Uberlândia MG, Brazil, brenobm@ufu.br, jrcamacho@ufu.br

https://orcid.org/0000-0001-5019-5342

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Fig. 1
Dimensions of the Single-Phase Variable Reluctance Motor.

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Fig. 2
Design of the 4-Phase 8/6 Variable Reluctance Motor.

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Fig. 3
Flowchart of DE algorithm.

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Fig. 4
Flowchart of SMPSO algorithm.

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Fig. 5
Surface Σ between the two continuous medium, Ω₁ and Ω₂.

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Fig. 6
Flowchart of case study steps to minimize VRM copper losses.

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Fig. 7
Magnetic flux density at airgap before optimization.

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Fig. 8
Magnetic flux density at airgap after DE optimization.

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Fig. 9
Magnetic flux density at airgap after PSO optimization.

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Fig. 10
Magnetic flux density in motor at 31.34 A before optimization.

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Fig. 11
Magnetic flux density in motor at 31.34 A after DE optimization.

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Fig. 12
Magnetic flux density in motor at 31.34 A after PSO optimization.

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Fig. 13
Pareto front obtained from the multi-objective optimization through SMPSO algorithm.

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Fig. 14
Magnetic flux density in the reference 8/6 VRM.

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Fig. 15
Magnetic flux density in the 8/6 VRM optimized through SMPSO.

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Fig. 16
Convergence of the DE algorithm.

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Fig. 17
Magnetic flux density in the 8/6 VRM optimized through DE (CR=0.9 with dither).

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Fig. 18
Static torque obtained from de reference and optimized motors.

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Table I
ELECTRICAL CHARACTERISTICS OF THE SINGLE-PHASE VARIABLE RELUCTANCE MOTOR

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Table II
SINGLE-PHASE VARIABLE RELUCTANCE MOTOR DIMENSIONS BEFORE OPTIMIZATION

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Table III
ELECTRICAL CHARACTERISTICS OF REFERENCE 8/6 VRM

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Table IV
DIMENSIONS OF REFERENCE 8/6 VRM

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Table V
DIMENSIONS OF THE SINGLE-PHASE VRM AFTER DE OPTIMIZATION

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Table VI
DIMENSIONS OF THE SINGLE-PHASE VRM AFTER PSO OPTIMIZATION

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Table VII
COPPER LOSSES B EFORE AND AFTER OPTIMIZATION.

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Table VIII
DIMENSIONS OF 8/6 VRM AFTER SMPSO OPTIMIZATION

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Table IX
DIMENSIONS OF 8/6 VRM AFTER DE OPTIMIZATION

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Table X
FLUX LINKAGE, MAGNETIC VOLUME AND FLUX DENSITY FOR THE THREE 8/6 VRM MODELS

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

Parameters	Values	Parameters	Values
k1	1.44	n	0.80
TRV	16000	Nr	6
P	5,516.25 W	Bsat	1.85 T
RPM	1760 rpm	μ ₀	4π.10⁻⁷ H/m
Ns	6	Ip	31.34 A
V	220 V	Jc	6.0 A/mm²

Parameters	Values	Parameters	Values
βs	30 º	hs	31.20 mm
βr	30.72 º	hr	17.50 mm
Dr	134.60 mm	ts	35.00 mm
Ds	236.04 mm	tr	35.65 mm
L	193.20 mm	ac	3.31 mm²
Ys	19.205 mm	g	0.32 mm
Yr	21.40 mm	NE	20 turns

Parameters	Values	Parameters	Values
k1	–	n	–
TRV	–	Nr	6
P	2200 W	Bsat	1.69 T
RPM	3500 rpm	μ ₀	4π.10⁻⁷ H/m
Ns	8	Ip	10A
V	220 V	Jc	3.60 A/mm²

Parameters	Values	Parameters	Values
βs	22.5 º	hs	22.0 mm
βr	24.5 º	hr	15.0 mm
Dr	90.5 mm	ts	17.8 mm
Ds	160.0 mm	tr	18.3 mm
L	63.0 mm	ac	2.08 mm²
Ys	12.45 mm	g	0.30 mm
Yr	13.0 mm	NE	55 turns

Parameters	Values	Parameters	Values
βs	30 º	hs	27.30 mm
βr	30.74 º	hr	15.60 mm
Dr	120 mm	ts	31.20 mm
Ds	210.50 mm	tr	31.80 mm
L	174.24 mm	ac	3.31 mm²
Ys	17.65 mm	g	0.30 mm
Yr	23.25 mm	NE	18 turns

Parameters	Values	Parameters	Values
βs	30 º	hs	25.50 mm
βr	30.74 º	hr	15.70 mm
Dr	120 mm	ts	31.20 mm
Ds	210.46 mm	tr	31.80 mm
L	172.40 mm	ac	3.31 mm²
Ys	19.43 mm	g	0.30 mm
Yr	22.06 mm	NE	18 turns

Current (A)	FEMM (W)	Analytical (W)	Difference (%)
Original
31.34	118.68	119.92	1.034
47.00	266.98	269.80	1.045
DE
31.34	96.33	94.75	1.668
47.00	216.70	213.15	1.665
PSO
31.34	95.32	94.75	0.602
47.00	214.42	213.15	0.596

Parameters	Values (mm)	Parameters	Values (mm)
g	0.3	hs	18.05
Dr	100.5	hr	22.51
Ys	11.39	ts	19.72
Yr	11.66	tr	21.32
Deix	32.151	–	–

Parameters	Values (mm)	Parameters	Values (mm)
g	0.3	hs	18.34
Dr	100.5	hr	27.76
Ys	11.12	ts	19.72
Yr	10.04	tr	21.32
Deix	24.86	–	–

Parameters	Values at 10 A	Values at 15 A
Reference
Fc (Vs)	0.2065	0.2257
Vmag (L)	0.8251	0.8251
Fc/Vmag (Vs/L)	0.2503	0.2735
BYs (T)	1.198	1.3110
SMPSO
Fc (Vs)	0.2267	0.2474
Vmag (L)	0.8228	0.8228
Fc/Vmag (Vs/L)	0.2755	0.2958
BYs (T)	1.434	1.519
DE
Fc (Vs)	0.2253	0.2428
Vmag (L)	0.8101	0.8101
Fc/Vmag (Vs/L)	0.2786	0.2997
BYs (T)	1.461	1.571

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Optimization Techniques and Mathematical Modeling Applied to Reluctance Motors

Abstract