# Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC-DC Converter
## 3 Decoupled MPC Power Balancing Strategy for Coupled Inductor FCDC
### Multi-Objective MPC Strategy Design
\begin{aligned} {d_{\text{a}1}^{k}}&=\frac{u_{ \text{b}}^{k}-u_{\text{b}}^{k}+r_{\text{a}}^{k}}{u_{\text{a}1}^{k}-u_{\text{b} }^{k}}+\frac{(L_{\text{a}}+M)\left(i_{\text{b}}^{*}-\dot{i}_{\text{b}}^{k} \right)}{T_{s}\left(u_{\text{a}}^{k}-u_{\text{b}}^{k}\right)}\\ {d_{\text{a}2}^{k}}&=\frac{u_{\text{b}}^{k}+u_{\text{b }}^{k}-u_{\text{b}}^{k}+r_{\text{a}}^{k}}{u_{\text{b}}^{k}}+\frac{(L_{\text{a}} +M)\left(i_{\text{a}}^{*}-\dot{i}_{\text{a}}^{k}\right)}{T_{s}u_{\text{b}}^{k}} \\ \end{aligned} \right. \tag{18}\]  
Similarly, the duty ratio for phase b can be expressed as:  
\[\left\{\begin{aligned} {d_{\text{b}1}^{k}}&=\frac{u_{ \text{b}}^{k}-u_{\text{b}}^{k}+r_{\text{b}}^{k}}{u_{\text{a}1}^{k}-u_{\text{b} }^{k}}+\frac{(L_{\text{b}}-M)\left(i_{\text{b}}^{*}-\dot{i}_{\text{b}}^{k} \right)}{T_{s}\left(u_{\text{a}}^{k}-u_{\text{b}}^{k}\right)}\\ {d_{\text{b}2}^{kWith the aid of (19) and (20), the optimal duty ratio to track the current reference can be obtained through a single calculation, greatly reducing the computational burden.  
**Flying capacitor voltage control:** Although the flying capacitor topology has voltage self-balancing ability [27], large perturbations will still cause flying capacitor voltage derivation. Hence, the FC voltage control has to be considered to ensure that the FC voltage is always controlled at half of the port voltage.  
According to the operation states shown in Figure 2 and Table 1, the FC has charging and discharging processes in each control period. Assuming \(u_{\text{fa}}\)* is the FC voltage reference, its expression can be written as:  
\[u_{\text{fa}}^{*}=u_{\text{fa}}^{k}-\frac{1}{2}\Delta u_{1}^{k}+\frac{1}{2} \Delta u_{2}^{k} \tag{20}\]  
According to the FC charging and discharging circuit states, \(\Delta u_{1}^{k}\) and \(\Delta u_{2}^{k}\) can be expressed as:  
\[\left\{ \begin{array}{l}\Delta u_{1}^{k}=\frac{(1-d_{\text{fa}}^{k}-d_{\text {ca}}^{k})T_{s}}{2C_{\text{fa}}}l_{\text{fa}}^{k}\\ \Delta u_{2}^{k}=\frac{(1-d_{\text{fa}}^{k}+d_{\text{ca}}^{k})T_{s}}{2C_{\text {fa}}}l_{\text{fa}}^{k}\end{array} \right. \tag{21}\]  
where \(d_{\text{ca}}^{k}\) is the duty ratio to control the flying capacitor voltage. Substituting (22) into (21), the duty ratio \(d_{\text{ca}}^{k}\) to control \(u_{\text{fa}}\) can be derived as:  
\[d_{\text{ca}}^{k}=\frac{2C_{\text{fa}}}{T_{s}i_{\text{a}}^{k}}(u_{\text{fa}}^ {*}-u_{\text{fa}}^{k}) \tag{22}\]  
Similarly, the duty ratio \(d_{\text{cb}}^{k}\) to control \(u_{\text{fb}}\) can be derived as:  
\[d_{\text{cb}}^{k}=\frac{2C_{\text{fb}}}{T_{s}i_{\text{b}}^{k}}(u_{\text{fb}}^ {*}-u_{\text{fb}}^{k}) \tag{23}\]  
Even if \(d_{\text{ca}}^{k}\) and \(d_{\text{ca}}^{k}\) only affect the current ripple, a large voltage deviation in the FC may affect the average value of the system current. Therefore, it is necessary to set constraints on (23) and (24) to obtain a satisfactory duty ratio for FC voltage control. Assuming the maximum current deviation is \(\Delta i_{\text{Llim}}\), \(\Delta d_{\text{f1}}^{k}\) and \(\Delta d_{\text{f2}}^{k}\) must be limited to \(\Delta D_{\text{lim}}\), ensuring that the deviation of \(i_{L}\) is within \(\Delta i_{\text{Llim}}\). The relationship between \(\Delta i_{\text{Llim}}\) and \(\Delta D_{\text{lim}}\) can be expressed as:  
\[\Delta D_{\text{lim}}\leq(\Delta i_{\text{Llim}}-\frac{1}{2}\Delta i_{L}) \frac{4L}{T_{s}(u_{2}^{*}-u_{1})} \tag{24}\]  
Thus, we can get the control block diagram of the proposed MPC power balancing strategy, as shown in Figure 5.