# 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  
The power balancing strategy needs to control both the current of each phase and the voltage of two flying capacitors. Under this requirement, by discretizing (8) with \(i_{\text{a}}\), \(i_{\text{b}}\), \(u_{\text{fa}}\) and \(u_{\text{fb}}\) as the control variables, the discrete models can be obtained as shown in Equation (10):  
\[\left\{ \begin{array}{l}i_{\text{a}}^{k+1}=i_{\text{a}}^{k}-\frac{i_{\text{ a}}^{k}T_{\text{a}}T_{\text{a}}}{L_{\text{a}}+M}+\frac{(d_{\text{a}}-d_{ \text{a}})T_{\text{a}}}{L_{\text{a}}+M}u_{\text{fa}}^{k}+\frac{T_{\text{a}}}{ L_{\text{a}}+M}\times\Big{(}d_{\text{a}1}u_{\text{i}}^{k}-u_{\text{o}}^{k} \Big{)}\end{array} \right. \tag{9a}\] \[\left\{ \begin{array}{l}i_{\text{b}}^{k+1}=i_{\text{b}}^{k}-\frac{i_ {\text{b}}^{k}T_{\text{b}}}{L_{\text{b}}-M}+\frac{(d_{\text{a}}-d_{\text{b}})T _{\text{a}}}{L_{\text{b}}-M}u_{\text{fb}}^{k}+\frac{T_{\text{b}}}{L_{\text{b}} -M}\times\Big{(}d_{\text{b}1}u_{\text{i}}^{k}-u_{\text{o}}^{k}\Big{)}\end{array} \right.\] (9b) \[\left.u_{\text{fa}}^{k+1}=u_{\text{fa}}^{k}+\frac{(d_{\text{a}}-d_ {\text{a}})T_{\text{a}}}{C_{\text{fa}}}{L_{\text{a}}}\right.\] (9c) \[\left.u_{\text{fb}}^{k+1}=u_{\text{fb}}^{k}+\frac{(d_{\text{b}}-d_ {\text{b}})T_{\text{a}}}{C_{\text{fb}}}{L_{\text{b}}}\right. \tag{9d}\]  
where the variables marked with \(k\) are the sampling value at the current time, and the symbols marked with \(k+1\) represent the predicted value at the next time step. According to the traditional MPC algorithm, the cost function can be written as:  
\[ \begin{array}{l}J^{k}&=\lambda_{1}\big{(}i_{\text{a}}^{*}-i_{\text{a}}^{k+1} \big{)}^{2}+\lambda_{2}\big{(}i_{\text{b}}^{*}-i_{\text{b}}^{k+1}\big{)}^{2} \\ &+\lambda_{3}\big{(}u_{\text{fa}}^{*}-u_{\text{fa}}^{k+1}\big{)}^{2}+\lambda_{4} \big{(}u_{\text{fb}}^{*}-u_{\text{fb}}^{k+1}\big{)}^{2}\end{array}  \tag{10}\]where the variables marked with * are the reference values and \(\lambda_{1}\)-\(\lambda_{4}\) are the weight coefficients. As shown in (11), in order to obtain the minimum \(\hat{l}^{k}\), it is not only necessary to calculate all combinations of the four duty ratios \(d_{\text{a1}}\), \(d_{\text{a2}}\), \(d_{\text{b1}}\) and \(d_{\text{b2}}\) but also to find the optimal solution for the four weight coefficients, which greatly increases the computational complexity. So, the traditional MPC control algorithm is not suitable for the control of the FCDC with coupled inductor. **Current control optimization:** According to (8), the variation values of inductor current and flying capacitor voltage can be expressed as:  
\[\left\{ \begin{array}{l}\triangle i_{\text{a}}=-\frac{\tau_{\text{a}}}{(L_{ \text{a}}+M)}i_{\text{a}}+\frac{d_{\text{a2}}-d_{\text{a1}}}{(L_{\text{a}}+M)}u _{\text{a}}+\frac{d_{\text{a1}}}{(L_{\text{a}}+M)}u_{\text{i}}-\frac{1}{(L_{ \text{a}}+M)}u_{\text{o}}\\ \triangle u_{\text{fa}}=\frac{d_{\text{a1}}-d_{\text{a2}}}{C_{\text{fa}}}u_{ \text{fa}}=\frac{\triangle d_{\text{c}}}{C_{\text{fa}}}u_{\text{fa}}\end{array} \right. \tag{11}\]  
where \(\triangle d_{\text{c}}\) is the duty ratio to control FC voltage. As shown in (12), when the converter is under the steady state, the flying capacitor voltage is unchanged and \(\triangle d_{\text{c}}\) is zero. At this time, the phase currents \(i_{\text{a}}\) and \(i_{\text{b}}\) are determined by \(d_{\text{a1}}\) and \(d_{\text{b1}}\), respectively, and are not affected the by flying capacitor. A small perturbation in flying capacitor voltage can cause variation in \(\triangle d_{\text{c}}\), but its impact on the average current can be treated as affecting the current ripple, which can be negligible. Thus, when the current is taken as the control objective, the flying capacitor voltage in (11) can be simplified, and based on the decoupled mathematical model of the FCDC, the cost function in (11) can be rewritten as:  
\[\left\{ \begin{array}{l}J_{\text{a}}^{k}=(i_{\text{a}}^{*}-i_{\text{a}}^{k+1 })^{2}\\ J_{\text{b}}^{k}=(i_{\text{b}}^{*}-i_{\text{b}}^{k+1})^{2}\end{array} \right.