# Multi-Objective Optimization of LCC-S-Compensated IPT System for Improving Misalignment Tolerance
## 3 Multi-Objective Optimization of Compensation Parameters
### Optimization Objectives  
To achieve optimized parameters for the IPT system, the objectives of the optimization must be provided, which are defined as follows:  
1. Minimize output fluctuations: In any state where \(k\in(k_{\text{min}},\,k_{\text{max}})\) and \(R_{\text{O}}\in(R_{\text{Omin}},\,R_{\text{Omax}})\), the current \(I_{\text{RO}}\)\([k(i),\,R_{\text{O}}(j)]\) across the load is most approximate to the design value \(I_{\text{RO,D}}\) for the CC mode, and the voltage \(U_{\text{RO}}\)\([k(i),\,R_{\text{O}}(j)]\) over the load should be close to the design value \(U_{\text{RO,D}}\) for the CV mode. \(k(i)\) and \(R_{\text{O}}(j)\) represent the \(i\)th \(k\) and \(j\)th \(R_{\text{O}}\), respectively. Here, the square of the current difference is used to represent output fluctuations, and its mathematical description is defined as \[f_{\text{RO}}=\left\{\begin{array}{ll}\left\{\begin{array}{ll}\left\{I_{ \text{RO}}[k(i),R_{\text{O}}(j)]-I_{\text{RO,D}}\right\}^{2}&\text{for CC mode }\\ \left\{U_{\text{RO}}[k(i),R_{\text{O}}(j)]-U_{\text{RO,D}}\right\}^{2}&\text{for CV mode }\end{array}\right.\end{array}\right.\] (16) 2. Reduce current stresses across inductor and coils: The voltage stresses over capacitors and current stresses through inductors result in high power loss due to ESRs in the passive compensation components. In particular, the larger ESRs in the inductor and coils have a great impact on system efficiency. Therefore, the secondary objective is to minimize the current stress across the inductor and coils. The sum of current stresses through the inductor and coils are involved in the optimization function as follows: \[f_{\text{IL}}=I_{\text{LP1}}[k(i),R_{\text{O}}(j)]+I_{\text{LP}}[k(i),R_{ \text{O}}(j)]+I_{\text{LS}}[k(i),R_{\text{O}}(j)]\] (17) 3. Lessen magnetizing power in LCT: For the loosely coupled coils, neglecting the magnetic losses and coil resistance, the active power transfer from the transmitter to the receiver can be given as \[P_{\text{T}}=\omega MI_{\text{LP}}I_{\text{LS}}\sin\theta_{\text{PS}}\] (18) where \(I_{\text{LP}}\) and \(I_{\text{LS}}\) are the RMS value of the primary and secondary coil currents, respectively, and \(\theta_{\text{PS}}\) is the phase difference of the fundamental current phasors between the transmitting and receiving winding. The reactive power flowing between couple coils can be given as \[Q_{\text{T}}=\omega\left(L_{\text{P}}I_{\text{LP}}^{2}+L_{\text{S}}I_{\text{ LS}}^{2}+2 MI_{\text{LP}}I_{\text{LS}}\cos\theta_{\text{PS}}\right)\] (19) For a traditional compensated LCT, when the induced current \(I_{\text{S}}\) lags \(I_{\text{P}}\) by a quarter cycle, the maximum power can be transferred from \(L_{\text{P}}\) to \(L_{\text{S}}\). However, after anti-misalignment optimization, the detuned compensation parameters make \(\theta_{\text{PS}}\) no longer equal to \(\pi/2\). The reactive power causes coil magnetization, bringing more copper and core losses. In order to improve the efficiency of LCT, the ratio between the reactive power and active power should be kept to a minimum. The ratio, taken as the third objective function, is given as  
\[f_{\text{OLS}}=\frac{|Q_{\text{T}}|}{|P_{\text{T}}|}=\left|\frac{L_{\text{P}}I_{ \text{LP}}^{2}+L_{\text{S}}I_{\text{LS}}^{2}+2 ML_{\text{LP}}I_{\text{LS}}\cos \theta_{\text{PS}}}{ML_{\text{LP}}I_{\text{LS}}\sin\theta_{\text{PS}}}\right| \tag{20}\]  
The targets of the optimization consist of three parts. The most important part is to minimize the output current and voltage fluctuation for different coupling coefficients and loads. The other two parts are designed to improve transmission efficiency. Taken together, the objective function is  
\[F=\text{min}\sum_{i=1}^{N_{\text{K}}}\sum_{j=1}^{N_{\text{BO}}}(f_{\text{RO}}, f_{\text{LL}},f_{\text{OLS}}) \tag{21}\]  
where \(N_{\text{k}}\) and \(N_{\text{RO}}\) are set as the sampling number of coupling coefficients and loads.