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Solving Transformer Design Optimization Problem Using Multiobjective Lognormal-Beta Differential Evolution Approach Marina A. Tsili1, Eleftherios I. Amoiralis1, Jean V. Leite2, and Leandro dos S. Coelho3,4 1
Faculty of Electrical and Computer Engineering, National Technical University of Athens, Greece 2 GRUCAD-EEL-CTC, Federal University of Santa Catarina (UFSC), Florianopolis, SC, Brazil 3 Department of Electrical Engineering, Federal University of Parana, Curitiba, PR, Brazil 4 Industrial and Systems Eng. Graduate Program, Pontifical Catholic University of Parana, Curitiba, PR, Brazil Emails:
[email protected],
[email protected],
[email protected],
[email protected]
TABLE I PERFORMANCE METRICS (MEAN OF 30 RUNS WITH NORMALIZED OBJECTIVE FUNCTIONS VALUES) Performance metrics UPS-EMOA UPS-DELFBC Spacing (f1,f2) 0.13⋅10-4 5.03⋅10-4 Euclidean distance (f1,f2) 0.959 0.946 Pareto solutions (feasible 82 317 solutions filtered in 30 runs) 4
x 10 2.12
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I. INTRODUCTION Transformer design optimization (TDO) seeks a constrained minimum cost solution by setting the transformer geometry parameters and the relevant electrical and magnetic quantities considering that manufacturing materials are highly variable stock exchange commodities. Moreover, the TDO problems can be viewed as constrained multiobjective optimization problems (MOPs). In the multiobjective optimization, the number of possible solutions (nondominated solutions) is presented as a Paretooptimal front, instead of a single optimal solution. Evolutionary algorithms seem particularly suitable to solve MOPs, because they deal simultaneously with a set of possible solutions (Pareto front) and are less susceptible to the shape or continuity of Pareto front. In [1], an unrestricted population-size evolutionary multiobjective optimization algorithm (UPS-EMOA) based on differential evolution (DE) was presented. For a classical DE, the values of control parameters are usually predefined and do not change during the evolutionary process. The aim of this paper is to introduce an improved UPS-EMOA approach based on DE using lognormal distribution tuning of the scale factor and the beta distribution to adjust the crossover rate (UPS-DELFBC) in range [0,1] during the evolutionary cycle. The proposed UPS-DELFBC algorithm is applied for the design optimization of a distribution transformer. The optimization seeks the minimum of two objective functions, i.e. the manufacturing cost (f1) and the total owing cost (f2) that includes purchasing price and cost of losses while ensuring the
operational requirements. The design variables are the windings turns, the magnetic induction magnitude (B), the width of core leg (D) and the core window height (G) [2]. Optimization results of TDO showed in Table I and Figure 1 demonstrate the effectiveness of the proposed UPSDELFBC when compared with the UPS-EMOA. According to this figure, as the manufacturing cost (f1) of the optimal solutions increases, the total owning cost (f2) decreases and vice versa. This is expected since larger manufacturing costs correspond to better materials and lower losses, thus reducing the cost of losses included in the total owing cost. It is upon the designer’s choice to select the solution that provides the best compromise between both costs.
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Abstract—In this paper, the transformer design optimization is treated as a multiobjective problem, with the aim to minimize both the manufacturing cost and the total owing cost taking into consideration design constraints. For the solution of this problem, we propose the unrestricted population-size evolutionary multiobjective optimization algorithm (UPS-EMOA) approach combined with differential evolution using lognormal distribution tuning of the scale factor and the beta distribution to adjust the crossover rate (UPS-DELFBC). The proposed UPS-DELFBC is useful to maintain the adequate diversity in the population and avoid the premature convergence during the generational cycle. Preliminary optimization results using UPS-DELFBC are promising in terms of spacing and convergence. Index Terms—Transformer design optimization, multiobjective optimization, differential evolution.
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Fig. 1. Pareto front of UPS-EMOA and UPS-DELFBC.
In this case, UPS-DELFBC outperformed the classical UPSEMOA with respect to the spacing, Euclidean distance metrics and solutions number in Pareto front in 30 runs. REFERENCES [1] T. Aittokoski and K. Miettinen, “Efficient evolutionary approach to approximate the Pareto optimal set in multiobjective optimization, UPSEMOA,” Optimization Methods and Software, vol. 25, no. 6, pp. 841858, 2010. [2] E. I. Amoiralis, P. S. Georgilakis, M. A. Tsili, A. G. Kladas, “Global transformer optimization method using evolutionary design and numerical field computation,” IEEE Transactions on Magnetics, vol. 45, no. 3, pp. 1720-1723, 2009.
