Transformers G1 2x25
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The next sections of this article have been organized in the following way: Section 2 describes the detailed models for the numerical solution of the problem. Section 3 describes the proposed method for the simplified analytical solution, where the autotransformers are considered ideal (i.e., their impedances are null, in contrast with the real autotransformers, whose impedances are not null). Section 4 describes the numerical results, and it is split into two parts: (a) subsection 4.1 shows the results for linear cases with ideal autotransformers, which are identical using the analytical deductions of Section 3 or the numerical solution of Section 2 and (b) subsection 4.2 shows the results for linear and nonlinear cases with real autotransformers, which can only be accurately computed using the numerical solution of Section 2, and are useful to highlight the suitability of the developed analytical method as an approximate solution. Section 5 is devoted to the conclusion.
The numerical method described in Section 2, considering multiple trains and real autotransformers, was successfully verified with the help of the solution shown in [15]. The numerical results for this article are obtained using the system illustrated in Figure 1, whose main data were taken from [15] and are included in Appendix B. These numerical results are split into two subsections: (4.1) linear cases with ideal autotransformers and (4.2) linear and nonlinear cases with real autotransformers. The results of subsection 4.1 are identical when using the analytical deductions of Section 3 or the numerical solution of Section 2. The results of subsection 4.2 can only be accurately computed using the numerical solution of Section 2 and are useful to show the suitability of the developed analytical method as an approximate solution. Finally, subsection 4.3 is devoted to offering a summarized analysis of results related to the purpose of this article.
In this section, the autotransformer impedances (ZAT) are included in the simulation; therefore, the results only can be numerically obtained. That is, the results of this section are not predictable with accuracy using analytical equations, unlike the results of the previous section. For example, in cases with only one train, the currents are not mathematically equal to zero for those autotransformers which are not at each end of the cell with the train. The results are herein shown as a function of ZAT in order to show the influence of ZAT on the distribution of currents. In practice, ZAT tends to be very low (nearly 1%). The highest considered value of ZAT (5%), however, is much higher than the typical values in real life. The resistive part of ZAT was assumed to be 0.5% in all these examples.
The results with ideal autotransformers, as shown in Table 1, were obtained through two independent methods (the analytical solution shown in Section 3 and the linear numerical solution explained in Section 2). Both methods are alternative ways of obtaining the accurate solution in the case of ideal autotransformers. The analytical method can be easily implemented by the simple writing of formulas of Section 3 in a worksheet, whereas the linear numerical solution explained in Section 2 can be easily obtained by coding the correspondent program.
The distribution of currents among autotransformers is simple because currents circulate mainly through autotransformers located on both sides of one train. The superposition method is useful to explain the distribution of currents in cases with multiple trains. 59ce067264
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