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Optimal design of electrical machines: mathematical programming formulations. (English) Zbl 1282.78027

Summary: Purpose - The purpose of this paper is to investigate the impact of different mathematical formulations of the problem of optimal design of electrical machines on the results obtained using a local optimization solver. The aim is to investigate the efficiency and reliability of standard local solvers when handling different mathematical formulations. This could provide guidelines for designers in practical engineering applications.
Design/methodology/approach - The paper proposes six equivalent mathematical formulations of the optimal design problem of a slotless permanent-magnet electric rotating machine. The authors investigate the impact of these different mathematical formulations on the results obtained using a local optimization solver which is well-known in the engineering community: MatLab’s fmincon function. The paper first computationally compares the six proposed formulations with a fixed value for the number of pole pairs \(p\), that gives continuous optimization problems, then discusses some results when \(p\) is free on three mixed-integer formulations.
Findings - The paper shows that, even though the considered formulations are mathematically equivalent, their numerical performances are different when an optimization solver, such as the one proposed by MatLab in fmincon, is used. Thus, the designer must take care about the formulation of the design problem in order to make more efficient the use of these kind of algorithms.
Originality/value - In the context of engineering applications, one usually resorts to well known and easy to use optimization solvers. The same optimization problem can be often formulated in different ways. Furthermore, the formal description of optimization problems has an impact on the applicability and efficiency of the corresponding solution methods. This is usually not taken into account when optimization solvers are exploited. The originality of this paper is in building on the theory of reformulations in mathematical optimization to investigate and highlight the impact of formulation differences.

MSC:

78A55 Technical applications of optics and electromagnetic theory
78M50 Optimization problems in optics and electromagnetic theory

Software:

Matlab
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Full Text: DOI

References:

[1] Fitan, E., Messine, F. and Nogare‘de, B. (2003), ”A general analytical model of electrical permanent magnet machine dedicated to optimal design”, COMPEL – International Journal for Computation and Mathematics in Electrical and Electronic Engeneering, Vol. 22 No. 4. · Zbl 1054.78033
[2] DOI: 10.1109/TMAG.2004.827183 · doi:10.1109/TMAG.2004.827183
[3] DOI: 10.1145/1271.1276 · Zbl 0562.90079 · doi:10.1145/1271.1276
[4] DOI: 10.1051/ro/2009005 · Zbl 1158.90390 · doi:10.1051/ro/2009005
[5] DOI: 10.1051/ro:2004026 · Zbl 1114.90156 · doi:10.1051/ro:2004026
[6] DOI: 10.1109/20.650361 · doi:10.1109/20.650361
[7] Nogare‘de, B., Kone, A. and Lajoie-Mazenc, M. (1995), ”Optimal design of permanent-magnet machines using analytical field modeling”, Electromotion, Vol. 2 No. 1, pp. 25-34.
[8] DOI: 10.1109/20.497463 · doi:10.1109/20.497463
[9] DOI: 10.1080/07313569208909572 · doi:10.1080/07313569208909572
[10] DOI: 10.1109/20.497516 · doi:10.1109/20.497516
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