# FEM Based Preliminary Design Optimization in Case of Large Power Transformers
## 1 Introduction  
Large power transformers are generally specific, tailored to the unique customer requirements. In case of large machines their design process is a complex, labour intensive task, where many physical fields have to be considered simultaneously [1; 2; 3]. During the tendering procedure, a preliminary design is made to determine the final price and the key-design parameters of the cost optimal transformer design (Figure 1). It is important to consider not only the technical feasibility, but the economic aspects, as well. Generally, the total cost of ownership (TOC) is used as a goal function [4] to consider the lifetime costs of the transformer [2; 4; 5; 6; 7; 8]. The uniqueness is a very important factor during the design and optimization of very large machines. Generally, only one design is built with the given requirements, there is no other possibility to tune or refine the parameters after the measurements. Moreover, the manufacturing cost of these machines are very high, therefore a company can win (or loose) a lot of money if it can won the bidding procedure with a good preliminary design. The mathematical representation of this problem belongs to the most general branch of discrete, non-linear mathematical optimization problems [9]. During the preliminary design process, this good design have to be selected from several thousands of feasible transformer designs, in a very short time (Figure 2). Many different methodologies have been published in the literature, which use a lot of simplifications [10; 11; 12]. These can decrease the robustness of the solution, due to the modelling uncertainties [13]. In the case of very large power transformers, several winding layouts are used Figure 3, because of their benefits and drawbacks. However, the nowadays used algorithms replace the detailed winding layouts with copper filling factor based models, or do not consider these differences [10; 11; 12; 14; 15; 16; 17; 18; 19]. The knowledge of the conductor sizes and the winding layout are essential to make an accurate stray loss calculation and create a more robust solution [3; 9; 20]. Most of the existing algorithms are using copper filling factor based winding models to the optimization [10; 11; 12; 14; 15]. This modelling technique is widely used in the transformer industry, because it estimates well the losses, the outer dimensions of the winding and the related main electrical properties of the transformer [3; 9]. Some of them use FEM (Finite Element Method) techniques in their optimization loop to refine the results [11; 16; 18; 19; 21]. However, these algorithms uses the FEM method only to refine the best individual from the generation, they are not considers the short-circuit impedance and other important electrical parameters [18] during the calculation. Since the cost and constraints are generally non-linear functions of the design variables, the mathematical representation of the preliminary transformer design optimization problem is strongly non-linear. There can be several extremums, which has nearly the same TOC and the designer can think that these solutions are very similar. However their key-design parameters can be very different. Figure 1: Schematic view of a large power transformer design process. Figure 2: Schematic view the optimized key-design parameters of a three phase, core-form large power transformer. All of the optimized, independent key-design parameters have been noted on the picture. Therefore, it is important to check the feasibility of the windings and make precise short-circuit impedance calculation during this very first optimization stage to provide a more robust solution. This paper proposes an evolutionary algorithm based method, which can calculate the optimal conductor sizes and winding layout in the preliminary optimization stage, to provide more robust key-design parameters for the final design. The analytical part of the transformer model is used to calculate some electrical parameters and the shape of the core and winding system (Figure 3).