# Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC-DC Converter
## 2 Basic Principle Analysis of Coupled Inductor FCDC  
The topology of the coupled inductor FCDC applied in an ESS is shown in Figure 1. The coupled inductor is introduced to the traditional flying capacitor topology constructing the two-phase interleaved structure [27]. \(L_{\text{a}}\) and \(L_{\text{b}}\) represent self-inductance, \(M\) is mutual inductance. The self and mutual inductance satisfy \(L_{\text{a}}=L_{\text{b}}=L\), \(-1<k=M/L<0\), k is the coupled coefficient, and \(r_{\text{a}}\) and \(r_{\text{b}}\) are the equivalent resistance of the inductor. \(C_{\text{fa}}\) and \(C_{\text{fb}}\) represent the flying capacitor. \(i_{\text{a}}\) and \(i_{\text{b}}\) are the current flowing through each phase. **Basic operating principle:** As shown in Figure 1, \(u_{\text{i}}\) represents the voltage on the high-voltage side and \(u_{\text{o}}\) represents the voltage on the low-voltage side. The carrier of phase a and b shifts 90\({}^{\circ}\). When the energy flows from \(u_{\text{i}}\) to \(u_{\text{o}}\), FCDC operates under Buck mode, \(S_{\text{a1}}\), \(S_{\text{a2}}\), \(S_{\text{b1}}\), and \(S_{\text{b2}}\) are alternately turned on, and \(S_{\text{a3}}\), \(S_{\text{a4}}\), \(S_{\text{b3}}\), and \(S_{\text{b4}}\) are turned  
Figure 1: Topology of coupled inductor FCDC. off. When the energy flows from \(u_{\rm o}\) to \(u_{\rm\mu}\), FCDC operates under Boost mode. Then, \(S_{\rm a3}\), \(S_{\rm a4}\), \(S_{\rm b3}\), and \(S_{\rm b4}\) are alternately turned on and \(S_{\rm a1}\), \(S_{\rm a2}\), \(S_{\rm b1}\), and \(S_{\rm b2}\) are turned off. The coupling inductor helps to reduce the inductor volume, but it does not change the voltage gain of the FCDC, and the gain characteristics are the same as a traditional flying capacitor DC-DC converter [28]. Taking Buck mode as the example, based on the above modulation principle, the coupled inductor FCDC has a total of 14 operating states, as shown in Figure 2.  
Under phase- shifting control, the switches take turns conducting. As shown in Figure 2a, when \(S_{\rm a1}\) conducts, energy flows through \(C_{\rm fa}\) and supplies power to \(u_{\rm o}\), while  
Figure 2: Operating states of coupled inductor FCDC. (**a**) Mode 1. (**b**) Mode 2. (**c**) Mode 3. (**d**) Mode 4. (**e**) Mode 5. (**f**) Mode 6. (**g**) Mode 7. (**h**) Mode 8. (**i**) Mode 9. (**j**) Mode 10. (**k**) Mode 11. (**l**) Mode 12. (**m**) Mode 13. (**n**) Mode 14.  
the diodes of S\({}_{\text{b3}}\) and S\({}_{\text{b4}}\) in phase b provide the current freewheel path. In Figure 2b, all the switches on the upper bridge arm are turned off, and the energy in the coupling inductor flows through the lower bridge arm diode for freewheeling. In Figure 2c, the high-voltage-side energy flows through the upper bridge arm of phase b to charge \(C_{\text{fb}}\). Phase a operates in a freewheeling state, and the current flows through S\({}_{\text{b3}}\) and S\({}_{\text{b4}}\). In Figure 2d, \(C_{\text{fa}}\) is discharged through S\({}_{\text{a2}}\) and the body diodes of S\({}_{\text{a4}}\), while phase b operates in freewheeling mode. In Figure 2f, \(C_{\text{fb}}\) is discharged through S\({}_{\text{b2}}\) and the body diodes of S\({}_{\text{b4}}\), while phase a operates in freewheeling mode. \(C_{\text{fb}}\) discharges in Figure 2f, and the energy on the \(u_{\text{i}}\) side is directly supplied to \(u_{\text{o}}\) through S\({}_{\text{a1}}\) and S\({}_{\text{a2}}\). In Figure 2g, \(C_{\text{fb}}\) is discharged and \(u_{\text{i}}\) charges \(C_{\text{fa}}\). In Figure 2h, \(u_{\text{i}}\) continues to charge \(C_{\text{fa}}\), while the energy flows through the upper bridge arm switch of phase b to supply power to \(u_{\text{o}}\). In Figure 2i, \(u_{\text{i}}\) simultaneously charges \(C_{\text{fa}}\) and \(C_{\text{fb}}\). In Figure 2j, while charging \(C_{\text{fb}}\), \(u_{\text{i}}\) directly supplies power to \(u_{\text{o}}\) through the upper bridge arm of phase a.