Computationally efficient topology optimization for the SynRMs based on the torque curve interpolation

Abstract

This study proposes the SIMP-based topology optimization which can design a high-torque synchronous reluctance motor (SynRM) in an efficient way. Although it is crucial to simultaneously optimize a SynRM design and the corresponding current-phase angle for the maximum performance, most studies to date have conducted design optimization at a fixed current-phase angle for simplicity. It is also difficult to consider the magnetic nonlinearity and complicated design features in topology optimization. To solve the above issue, this study expresses the optimal current-phase angle (β^(MTPA)) for the SynRM as a function of design variables (ρ). To efficiently determine β^(MTPA)(ρ), the average torque curve is interpolated by using the modified Akima method. For interpolation, three-time electromagnetic finite element (FE) analyses are conducted at every iteration. The structural FE analysis is also performed at every iteration to obtain a structurally meaningful design. Through the above procedure, the SynRM design and the corresponding current excitation can be simultaneously optimized. Optimization results are discussed in terms of computational accuracy and efficiency, compared with other methods.

Appendix A: Analytical sensitivity analysis

1.1 Modified Akima method-based sensitivity

In Fig. 4, the analytical sensitivities of the average torque at β₄, β₅, and β₆ can be calculated using the adjoint variable method (Choi and Yoo 2008). Note that the sensitivities of the average torque at β₃ = 0° and β₇ = 90° become zero. In addition, the sensitivities of the average torque at β₁, β₂, β₈, and β₉ can be obtained by using the modified Akima method (Akima 1970), as follows:

$$\frac{{\partial \Delta T_{{{\text{avg}}}} ({{\varvec{\uprho}}},\beta_{{\text{k + 4}}} ,\beta_{k + 3}^{{}} )}}{{(\beta_{k + 4}^{{}} - \beta_{k + 3}^{{}} )\partial \rho_{e}^{{}} }} - \frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 3}^{{}} ,\beta_{k + 2}^{{}} )}}{{(\beta_{k + 3}^{{}} - \beta_{k + 2}^{{}} )\partial \rho_{e}^{{}} }} = \frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 3}^{{}} ,\beta_{k + 2}^{{}} )}}{{(\beta_{k + 3}^{{}} - \beta_{k + 2}^{{}} )\partial \rho_{e}^{{}} }} - \frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 2}^{{}} ,\beta_{k + 1}^{{}} )}}{{(\beta_{k + 2}^{{}} - \beta_{k + 1}^{{}} )\partial \rho_{e}^{{}} }} = \frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 2}^{{}} ,\beta_{k + 1}^{{}} )}}{{(\beta_{k + 2}^{{}} - \beta_{k + 1}^{{}} )\partial \rho_{e}^{{}} }} - \frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 1}^{{}} ,\beta_{k}^{{}} )}}{{(\beta_{k + 1}^{{}} - \beta_{k}^{{}} )\partial \rho_{e}^{{}} }}$$

(A.1)

1.2 Sensitivity of the interpolated average torque (corresponding to Step 1 in Sect. 3.3 )

Within each interval of [β_k, β_i+1), the analytical sensitivity of the interpolated curve (T_avg(ρ, β)) with respect to a design variable (ρ_e) can be expressed, as follows:

$$\frac{{\partial T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta )}}{{\partial \rho_{e}^{{}} }} = (\beta - \beta_{k}^{{}} )_{{}}^{3} \frac{{\partial p_{3}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + (\beta - \beta_{k}^{{}} )_{{}}^{2} \frac{{\partial p_{2}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + (\beta - \beta_{k}^{{}} )\frac{{\partial p_{1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{\partial p_{0}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \left. {\frac{{\partial T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta )}}{\partial \beta }} \right|_{{\beta_{{}}^{{\text{(MTPA)}}} ({{\varvec{\uprho}}})}}^{{}} \frac{{\partial \beta_{{}}^{{\text{(MTPA)}}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }},$$

(A.2)

where

$$\partial p_{0}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} = \partial T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k}^{{}} )\partial \rho_{e}^{{}} ,$$

(A.3)

$$\partial p_{1}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} = \partial t_{k}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} ,$$

(A.4)

$$\frac{{\partial p_{2}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \frac{3}{{(\beta_{k + 1}^{{}} - \beta_{k}^{{}} )_{{}}^{2} }}\frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 1}^{{}} ,\beta_{k}^{{}} )}}{{\partial \rho_{e}^{{}} }} - \left( {\frac{{\partial t_{k + 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + 2\frac{{\partial t_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}} \right)\frac{1}{{\beta_{k + 1}^{{}} - \beta_{k}^{{}} }},$$

(A.5)

$$\frac{{\partial p_{3}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \frac{ - 2}{{(\beta_{k + 1}^{{}} - \beta_{k}^{{}} )_{{}}^{3} }}\frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 1}^{{}} ,\beta_{k}^{{}} )}}{{\partial \rho_{e}^{{}} }} + \left( {\frac{{\partial t_{k + 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{\partial t_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}} \right)\frac{1}{{(\beta_{k + 1}^{{}} - \beta_{k}^{{}} )_{{}}^{2} }}.$$

(A.6)

Here, the sensitivity of t_k(ρ) can be evaluated, as follows:

$$\begin{gathered} \frac{{\partial t_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \left( {\frac{1}{{w_{k,\,1}^{{}} ({{\varvec{\uprho}}}) + w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}} - \frac{{w_{k,\,1}^{{}} ({{\varvec{\uprho}}})}}{{(w_{k,\,1}^{{}} ({{\varvec{\uprho}}}) + w_{k,\,2}^{{}} ({{\varvec{\uprho}}}))_{{}}^{2} }}} \right)\frac{{\partial w_{k,\,1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}\delta_{k - 1}^{{}} ({{\varvec{\uprho}}}) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {\frac{1}{{w_{k,\,1}^{{}} ({{\varvec{\uprho}}}) + w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}} - \frac{{w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}}{{(w_{k,\,1}^{{}} ({{\varvec{\uprho}}}) + w_{k,\,2}^{{}} ({{\varvec{\uprho}}}))_{{}}^{2} }}} \right)\frac{{\partial w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}\delta_{k}^{{}} ({{\varvec{\uprho}}}) + \frac{{w_{k,\,1}^{{}} ({{\varvec{\uprho}}})}}{{w_{k,\,1}^{{}} ({{\varvec{\uprho}}}) + w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}}\frac{{\partial \delta_{k - 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}}{{w_{k,\,1}^{{}} ({{\varvec{\uprho}}}) + w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}}\frac{{\partial \delta_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}, \hfill \\ \end{gathered}$$

(A.7)

$$\frac{{\partial \delta_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \frac{{\partial \Delta T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta_{k + 1}^{{}} ,\beta_{k}^{{}} )}}{{(\beta_{k + 1}^{{}} - \beta_{k}^{{}} )\partial \rho_{e}^{{}} }},$$

(A.8)

$$\frac{{\partial w_{k,\,1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \left( {\frac{{\partial \delta_{k + 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} - \frac{{\partial \delta_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}} \right) \times sign\left( {\delta_{k + 1}^{{}} ({{\varvec{\uprho}}}) - \delta_{k}^{{}} ({{\varvec{\uprho}}})} \right) + \frac{1}{2}\left( {\frac{{\partial \delta_{k + 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{\partial \delta_{k}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}} \right) \times sign\left( {\delta_{k + 1}^{{}} ({{\varvec{\uprho}}}) + \delta_{k}^{{}} ({{\varvec{\uprho}}})} \right),$$

(A.9)

$$\frac{{\partial w_{k,\,2}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \left( {\frac{{\partial \delta_{k - 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} - \frac{{\partial \delta_{k - 2}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}} \right) \times sign\left( {\delta_{k - 1}^{{}} ({{\varvec{\uprho}}}) - \delta_{k - 2}^{{}} ({{\varvec{\uprho}}})} \right) + \frac{1}{2}\left( {\frac{{\partial \delta_{k - 1}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{\partial \delta_{k - 2}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }}} \right) \times sign\left( {\delta_{k - 1}^{{}} ({{\varvec{\uprho}}}) + \delta_{k - 2}^{{}} ({{\varvec{\uprho}}})} \right).$$

(A.10)

1.3 Sensitivity of the average torque at the optimal current phase angle (corresponding to Step 2 in Sect. 3.3 )

If β_c is selected as β^(MTPA)(ρ), the final term in Eq. (A.2) can be expressed, as follows:

$$\left. {\left. {\frac{{\partial T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta )}}{\partial \beta }} \right|_{{\beta_{{}}^{{\text{(MTPA)}}} ({{\varvec{\uprho}}})}}^{{}} \frac{{\partial \beta_{{}}^{{\text{(MTPA)}}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \frac{{\partial T_{{{\text{avg}}}}^{{}} ({{\varvec{\uprho}}},\beta )}}{\partial \beta }} \right|_{{\beta_{c}^{{}} ({{\varvec{\uprho}}})}}^{{}} \frac{{\partial \beta_{c}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }},$$

(A.11)

where

$$\frac{{\partial \beta_{c}^{{}} ({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} = \frac{{\partial \beta_{c}^{{}} ({{\varvec{\uprho}}})}}{{\partial a({{\varvec{\uprho}}})}}\frac{{\partial a({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{\partial \beta_{c}^{{}} ({{\varvec{\uprho}}})}}{{\partial b({{\varvec{\uprho}}})}}\frac{{\partial b({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }} + \frac{{\partial \beta_{c}^{{}} ({{\varvec{\uprho}}})}}{{\partial c({{\varvec{\uprho}}})}}\frac{{\partial c({{\varvec{\uprho}}})}}{{\partial \rho_{e}^{{}} }},$$

(A.12)

$$\partial a({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} = 3\partial p_{3}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} ,$$

(A.13)

$$\partial b({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} = - 6\beta_{k}^{{}} \partial p_{3}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} \beta_{k}^{{}} + 2\partial p_{2}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} ,$$

(A.14)

$$\partial c({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} = 3\beta_{k}^{2} \partial p_{3}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} - 2\beta_{k}^{{}} \partial p_{2}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} + \partial p_{1}^{{}} ({{\varvec{\uprho}}})/\partial \rho_{e}^{{}} .$$

(A.15)

In the above equations, ∂β_c/∂a, ∂β_c/∂b, and ∂β_c/∂c can be easily obtained by using Eqs. (17)–(20).