# Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC-DC Converter
## Appendix A  
The average state equation of the coupled inductor FCDC under Boost mode is shown in (A1).  
\[\begin{split}\frac{d}{dt}\left[\begin{array}{c}i_{\text{a}}\\ i_{\text{b}}\\ u_{\text{fa}}\\ u_{\text{fb}}\end{array}\right]&=\left[\begin{array}{cccc}\frac{-r_{\text{a}}} {(L_{\text{a}}+M)}&0&\frac{d_{\text{d}}-d_{\text{a}3}}{(L_{\text{a}}+M)}&0\\ 0&\frac{-r_{\text{b}}}{(L_{\text{b}}-M)}&0&\frac{d_{\text{b}}-d_{\text{b}3}}{( L_{\text{b}}-M)}\\ \frac{d_{\text{a}3}-d_{\text{a}3}}{C_{\text{a}}}&0&0&0\\ 0&\frac{d_{\text{h}3}-d_{\text{h}4}}{C_{\text{h}}}&0&0\end{array}\right] \times\left[\begin{array}{c}i_{\text{a}}\\ i_{\text{b}}\\ u_{\text{fa}}\\ u_{\text{fb}}\end{array}\right]\\ &+\left[\begin{array}{cccc}\frac{1}{(L_{\text{a}}+M)}&\frac{d_{\text{a}3}-1 }{(L_{\text{a}}+M)}\\ \frac{1}{(L_{\text{b}}-M)}&\frac{d_{\text{b}3}-1}{(L_{\text{b}}-M)}\\ 0&0\end{array}\right]\left[\begin{array}{c}u_{\text{i}}\\ u_{\text{o}}\end{array}\right]\end{split} \tag{10}\]  
According to (A1), the discrete model of the control variable under Boost mode can be obtained as shown in (A2):\[\left\{\begin{array}{ll}\dot{t}_{\rm a}^{k+1}&=\dot{t}_{\rm a}^{k}-\frac{\dot{t}_{ \rm a}^{k}T_{\rm a}}{L_{\rm a}^{k}+M}+\frac{(d_{\rm a1}-d_{\rm a2})T_{\rm a}}{L_{ \rm a}^{k}+M}u_{\rm fa}^{k}\\ &+\frac{T_{\rm a}}{L_{\rm a}+M}\times\left[u_{\rm i}^{k}+(d_{\rm a2}-1)u_{\rm o }^{k}\right]\\ \dot{t}_{\rm b}^{k+1}&=\dot{t}_{\rm b}^{k}-\frac{\dot{t}_{\rm b}^{k}T_{\rm b}}{ L_{\rm b}-M}+\frac{(d_{\rm b1}-d_{\rm b2})T_{\rm a}}{L_{\rm b}-M}u_{\rm fb}^{k}\\ &+\frac{T_{\rm a}}{L_{\rm b}-M}\times\left[u_{\rm i}^{k}+(d_{\rm b2}-1)u_{\rm o }^{k}\right]\\ u_{\rm fa}^{k+1}&=u_{\rm fa}^{k}+\frac{(d_{\rm a2}-d_{\rm a1})T_{\rm a}}{C_{ \rm fa}}\dot{t}_{\rm a}^{k}\\ u_{\rm fb}^{k+1}&=u_{\rm fb}^{k}+\frac{(d_{\rm b2}-d_{\rm b1})T_{\rm a}}{C_{ \rm fb}}\dot{t}_{\rm b}^{k}\end{array}\right. \tag{12}\]  
According to the cost function shown in (13), (19), and (20), we can obtain the partial derivatives of \(J_{\rm a}^{k}\) under Boost mode as shown in (13).  
\[\left\{\begin{array}{ll}\frac{\partial{t}_{\rm a1}^{k}}{\partial{t}_{\rm a2}^ {k}}\bigg{|}_{d_{\rm a2}^{k}=1}=2(\dot{t}_{\rm a}^{k}-\dot{t}_{\rm a}^{k+1})( \frac{\partial{t}_{\rm a1}^{k+1}}{\partial{t}_{\rm a1}^{k}}-\frac{\partial{t} _{\rm a1}^{k}}{\partial{t}_{\rm a1}^{k}})\\ \frac{\partial{t}_{\rm a1}^{k}}{\partial{t}_{\rm a2}^{k}}\bigg{|}_{d_{\rm a1}^{k }=1}=2(\dot{t}_{\rm a}^{k}-\dot{t}_{\rm a}^{k+1})(\frac{\partial{t}_{\rm a2}^ {k+1}}{\partial{t}_{\rm a2}^{k}}-\frac{\partial{t}_{\rm a2}^{k}}{\partial{t}_ {\rm a2}^{k}})\end{array}\right. \tag{13}\]  
Solving (13) with the method used under Buck mode, we can have:  
\[\left\{\begin{array}{ll}d_{\rm a1}^{k}=\frac{u_{\rm fa}^{k}-u_{\rm b}^{k}+ r_{\rm a1}^{k}}{u_{\rm fa}^{k}}+\frac{(L_{\rm a}+M)(\dot{t}_{\rm a}^{k}-\dot{t}_{ \rm a}^{k})}{u_{\rm fa}^{k}\,T_{\rm a}}\\ d_{\rm a2}^{k}=\frac{u_{\rm fo}^{k}-u_{\rm fb}^{k}-u_{\rm fa}^{k}+r_{\rm a2}^{k }}{u_{\rm o}^{k}-u_{\rm fa}^{k}}+\frac{(L_{\rm a}+M)(\dot{t}_{\rm a}^{k}-\dot{t }_{\rm a}^{k})}{(u_{\rm fo}^{k}-u_{\rm fa}^{k})T_{\rm a}}\end{array}\right. \tag{14}\]  
With the aid of the FC voltage control equation shown in (27) and (28), the FC voltage under Boost mode can be controlled, and thus the MPC power balancing control algorithm for the coupled inductor FCDC under Boost mode is derived.