Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC–DC Converter

Abstract

A decoupled model predictive control (MPC) power balancing strategy for a coupled inductor-based flying capacitor DC–DC converter (FCDC) is a proposed to solve the power imbalance caused by the parameter differences in the coupled inductor. The decoupled mathematical model of coupled inductor FCDC is firstly derived by analyzing the converter operation state under various modes. On this basis, the control relationship between inductor current and flying capacitor (FC) voltage is redefined and an MPC power balance strategy based on the inductor current with single-step optimization is proposed. The proposed MPC strategy not only achieves decoupled power balancing control but also solves multi-objective dynamic optimization control of the inductor current and FC voltage, greatly reducing the computation load. A detailed theoretical analysis of the proposed strategy is presented and the balancing performance is effectively verified through the experiments.

1. Introduction

Renewable energy generation represented by photovoltaic and wind power has become a global trend in the development of electricity [1,2,3]. However, the intermittency of renewable energy has brought challenges to the stable operation of power grids. As an effective way to stabilize the power fluctuations, the electrochemical energy storage technique has been widely used in renewable energy generation systems [4,5,6], of which a bi-directional DC–DC converter is the core equipment. Due to the low voltage stress, small passive components, and common ground between input and output terminals, the flying capacitor three-level topology has certain advantages in medium- and high-voltage energy storage systems (ESSs) [7,8].

Interleaved parallel technology is often used in high-power energy storage applications [9,10]. To further improve power density, a coupled inductor is proposed to replace independent inductors to reduce system volume and weight [11]. However, in actual systems, manufacturing errors and differences in hardware circuits may cause impedance deviations between coupled circuits, resulting in phase-to-phase power imbalance. In [12,13,14], the power self-balancing characteristic of coupled inductor topology is researched. However, in cases of large perturbation, external strategies are required to achieve power balancing control. Ref. [15] proposed an integrated current balancing transformer for a two-phase DC–DC converter and the switches were achieved ZVS operation, yet this method is suitable only for low-power applications and the transformer design is complicated. To find an optimal DC–DC converter topology suitable for high-power applications, a three-phase interleaved DC–DC converter with coupled technique was proposed in [16], and a power balancing strategy was proposed by solving the matrix model of the converter [17]. To reduce the control order, a symmetrical coupled inductor design scheme for three-phase circuits was proposed in [18]. However, compared to that in [17], the overall inductor volume was not effectively optimized. Through the above analysis, although the coupled inductor helps to improve the system’s power density, the phase-to-phase power imbalance caused by the difference in the coupled parameters is still a key issue that needs to be addressed.

For FC topology, in addition to current control, it is also necessary to control the voltage of the FC to ensure it is half of the port-side voltage. Ref. [18] studied FC voltage self-balancing characteristics with small perturbations. Ref. [19] proposed an FC voltage control strategy based on an RLC circuit, which showed acceptable reliability. However, the auxiliary circuit weakens system reliability and increases system cost. Based on the average state equation, refs. [20,21] proposed the decoupled control strategy of output voltage and FC voltage sharing control using a PI algorithm for three-level and five-level FC topology. However, with the dynamic response speed of the linear regulator, it is hard to satisfy abrupt load changes. To address this issue, ref. [22] proposed a FC topology control method based on MPC. However, the multi-objective control brings a heavy computational load to the MPC strategy. Therefore, refs. [23,24,25] use mean square error, dynamic optimization, and artificial neural network methods, respectively, to estimate the optimal cost function, but these algorithms are complex to apply. Ref. [26] proposed an improved MPC algorithm for a bidirectional four-quadrant flying capacitor bi-directional DC–DC converter and derived a decoupled control model for the dual-side capacitor and inductor current, and achieved desired control performance. However, the three control objectives still impose a large computational load on system control.

Coupled inductor FCDC effectively improves the power density of an ESS, but the power imbalance issue caused by the parameter differences in the coupled inductor has not been solved. In this paper, a decoupled MPC power balancing strategy with single-step optimization for a two-phase coupled inductor FCDC is proposed. The proposed strategy not only achieves decoupled power balancing control but also solves the multi-objective dynamic optimization control issue of inductance current and FC voltage and greatly reduces the computational load. The proposed strategy has favorable power, balancing performance and dynamic performance, and has been effectively verified through experimental results.

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_a and L_b represent self-inductance, M is mutual inductance. The self and mutual inductance satisfy L_a = L_b = L, −1 < k = M/L < 0, k is the coupled coefficient, and r_a and r_b are the equivalent resistance of the inductor. C_fa and C_fb represent the flying capacitor. i_a and i_b are the current flowing through each phase.

Basic operating principle: As shown in Figure 1, u_i represents the voltage on the high-voltage side and u_o represents the voltage on the low-voltage side. The carrier of phase a and b shifts 90°. When the energy flows from u_i to u_o, FCDC operates under Buck mode, S_a1, S_a2, S_b1, and S_b2 are alternately turned on, and S_a3, S_a4, S_b3, and S_b4 are turned off. When the energy flows from u_o to u_i, FCDC operates under Boost mode. Then, S_a3, S_a4, S_b3, and S_b4 are alternately turned on and S_a1, S_a2, S_b1, and S_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_a1 conducts, energy flows through C_fa and supplies power to u_o, while the diodes of S_b3 and S_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_fb. Phase a operates in a freewheeling state, and the current flows through S_b3 and S_b4. In Figure 2d, C_fa is discharged through S_a2 and the body diodes of S_a4, while phase b operates in freewheeling mode. In Figure 2f, C_fb is discharged through S_b2 and the body diodes of S_b4, while phase a operates in freewheeling mode. C_fb discharges in Figure 2f, and the energy on the u_i side is directly supplied to u_o through S_a1 and S_a2. In Figure 2g, C_fb is discharged and u_i charges C_fa. In Figure 2h, u_i continues to charge C_fa, while the energy flows through the upper bridge arm switch of phase b to supply power to u_o. In Figure 2i, u_i simultaneously charges C_fa and C_fb. In Figure 2j, while charging C_fb, u_i directly supplies power to u_o through the upper bridge arm of phase a. In Figure 2k, u_i charges C_fb and the energy in C_fa is released to the u_o side. In Figure 2l, C_fa discharges while u_i supplies power to u_o through the upper bridge arm of phase b. C_fa and C_fb discharge simultaneously in Figure 2m. In Figure 2n, u_i supplies power to u_o simultaneously through the switches on the upper bridge arms of phase a and b.

Actually, the converter operating state is determined by the region of the duty ratio. Taking 0 < D < 0.5 as an example, it can be seen from Figure 3 that the coupled inductor FCDC contains a total of eight operating states.

According to the analysis method shown in Figure 3, the operating states of coupled inductor FCDC with other duty ratios are shown in

Table 1. Operating states of coupled inductor FCDC with different duty ratios.

Duty Ratio Range	Operating State
−1 < D < −0.5	a, b, c, d, e
D = −0.5	a, c, d, e
−0.5 < D < 0	a, c, d, e, g, i, k, m
D = 0	g, i, k, m
0 < D < 0.5	f, g, h, i, j, k, l, m
D = 0.5	f, h, j, l
0.5 < D < 1	f, h, j, l, n

.

Power imbalance issue: According to Figure 1, when the energy flows from the left side to the right side and the switches S_a1 and S_b1 are conducting simultaneously, the voltage equation of the coupled inductor FCDC can be expressed as:

$$\left\{\begin{array}{l}{u}_i=L_{\mathrm{a}}\frac{d{i}_{\mathrm{a}}}{dt}+M\frac{d{i}_{\mathrm{b}}}{dt}+r_{\mathrm{a}}{i}_{\mathrm{a}}+u_{\mathrm{o}}\\ u_i=M\frac{d{i}_{\mathrm{a}}}{dt}+L_{\mathrm{b}}\frac{d{i}_{\mathrm{b}}}{dt}+r_{\mathrm{b}}{i}_{\mathrm{b}}+u_{\mathrm{o}}\end{array}\right.$$

(1)

It can be seen that only when L_a = L_b and r_a = r_b is the two-phase current equal and the power between the two phases balanced. However, manufacturer errors and other reasons lead the parameter differences in the coupled inductor in practical applications, resulting in imbalanced power between phases. It not only leads to a current stress deviation of the switching devices but also easily leads to saturation of the magnetic component. In addition, the mutual inductance leads to the power coupling between the two phases, which increases the difficulty of system power control. Thirdly, the control strategy design based on the converter model should not only consider the operating states shown in Table 1 but should also consider the impact of the coupled inductor on the performance of the control strategy. Therefore, it is necessary to further research the decoupled model and study the decoupled power control strategy.

3. Decoupled MPC Power Balancing Strategy for Coupled Inductor FCDC

3.1. Decoupled Mathematical Model of Coupled Inductor FCDC

The mathematical model of the coupled inductor FCDC should be firstly derived to propose the decoupled MPC power balancing strategy. Taking 0 < D < 0.5 as an example, as shown in

Table 2. Experimental parameters.

Item	Value	Unit
Maximum input voltage	100	V
Maximum operating power	300	W
Switching frequency	10	kHz
Self-inductance of phase a and b	1	mH
Mutual inductance	−0.33	mH
Flying capacitor	220	μF
Output voltage under Buck mode	50	V
Load	12	Ω

, the converter consists of eight operating states. Taking i_a, i_b, u_fa, and u_fb as the state variables, T_s is the switching period. Thus, the operation time of mode i, h, g, f, m, l, k, and j in each switching period can be derived as (0.75-d_b1)T_s, (d_b1-0.5)T_s, (0.75-d_a1)T_s, (d_a1-0.5)T_s, (0.75-d_b2)T_s, (d_b2-0.5)T_s, (0.75-d_a2)T_s, (d_a2-0.5)T_s. According to the operating states shown in Figure 2, writing the circuit state equations and combine them with the corresponding operating time, the average state equation of the coupled inductor FCDC with 0 < D < 0.5 can be derived as:

$$\frac{d}{dt}\left[\begin{array}{c}{i}_{\mathrm{a}}\\ i_{\mathrm{b}}\\ u_{\mathrm{fa}}\\ u_{\mathrm{fb}}\end{array}\right]=\left[\begin{array}{cccc}-\frac{r_{\mathrm{a}}}{L_{\mathrm{a}}}& 0& \frac{d_{\mathrm{a}2}-d_{\mathrm{a}1}}{L_{\mathrm{a}}}& 0\\ 0& -\frac{r_{\mathrm{b}}}{L_{\mathrm{b}}}& 0& \frac{d_{\mathrm{b}2}-d_{\mathrm{b}1}}{L_{\mathrm{b}}}\\ \frac{d_{\mathrm{a}1}-d_{\mathrm{a}2}}{C_{\mathrm{fa}}}& 0& 0& 0\\ 0& \frac{d_{\mathrm{b}1}-d_{\mathrm{b}2}}{C_{\mathrm{fb}}}& 0& 0\end{array}\right]\times \left[\begin{array}{c}{i}_{\mathrm{a}}\\ i_{\mathrm{b}}\\ u_{\mathrm{fa}}\\ u_{\mathrm{fb}}\end{array}\right]+\left[\begin{array}{cc}\frac{d_{\mathrm{a}1}}{L_{\mathrm{a}}}& -\frac{1}{L_{\mathrm{a}}}\\ \frac{d_{\mathrm{b}1}}{L_{\mathrm{b}}}& -\frac{1}{L_{\mathrm{b}}}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{u}_{\mathrm{i}}\\ u_{\mathrm{o}}\end{array}\right]-\left[\begin{array}{c}\frac{M}{L_{\mathrm{a}}}\frac{d{i}_{\mathrm{b}}}{dt}\\ \frac{M}{L_{\mathrm{b}}}\frac{d{i}_{\mathrm{a}}}{dt}\\ 0\\ 0\end{array}\right]$$

(2)

where u_fa and u_fb are the voltages of flying capacitors C_fa and C_fb, respectively.

Similarly, the average state equation of the coupled inductor FCDC operating in other duty ratio intervals can be obtained by the above method, and the result is the same as (2). From (2), the mutual inductance M leads to a coupling relationship between the two phases, affecting the power control performance and complicating the control strategy design. Therefore, it is necessary to decouple the converter model.

Writing (1) with Laplace transform, it yields:

U(s) = Z(s)I(s)

(3)

where Z(s) is the impedance matrix and it is expressed as:

$$\mathit{Z}(s)=\left[\begin{array}{cc}{r}_{\mathrm{a}}+L_{\mathrm{a}}s& Ms\\ Ms& r_{\mathrm{b}}+L_{\mathrm{b}}s\end{array}\right]$$

(4)

For the interleaved converter with independent inductor, the inverse matrix of the impedance matrix is a diagonal matrix. Thus, in order to decouple the mathematical model of the coupled inductor FCDC, the inverse matrix of Z(s) shown in (4) should be transformed into a second-order diagonal matrix.

Performing a similar diagonalization transformation on (4) yields:

Y(s) = Z⁻¹(s) = P⁻¹H(s)P

(5)

where:

$$\begin{array}{c}\mathit{P}=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right],{\mathit{P}}^{-1}=\left[\begin{array}{cc}1/2& 1/2\\ -1/2& 1/2\end{array}\right]\\ \mathit{H}(s)=\left[\begin{array}{cc}{r}_{\mathrm{a}}/[r_{\mathrm{a}}+s(L_{\mathrm{a}}+M)]& 0\\ 0& r_{\mathrm{b}}/[r_{\mathrm{b}}+s(L_{\mathrm{b}}-M)]\end{array}\right]\end{array}$$

Thus, the expressions for the voltage and current in (3) under the s domain can be rewritten as:

$$\mathit{U}(s)=\left[\begin{array}{cc}{r}_{\mathrm{a}}+s(L_{\mathrm{a}}+M)& 0\\ 0& r_{\mathrm{b}}+s(L_{\mathrm{b}}-M)\end{array}\right]\mathit{I}(s)$$

(6)

Substituting (6) into (2) and rewriting the result with inverse Laplace transform, the expressions of inductor voltage and current in the time domain are obtained as follows:

$$\left[\begin{array}{c}{u}_{\mathrm{a}}\\ u_{\mathrm{b}}\end{array}\right]=\left[\begin{array}{cc}{L}_{\mathrm{a}}+M& 0\\ 0& L_{\mathrm{b}}-M\end{array}\right]{\left[\begin{array}{cc}\frac{d{i}_{\mathrm{a}}}{dt}& \frac{d{i}_{\mathrm{b}}}{dt}\end{array}\right]}^T+\left[\begin{array}{c}{r}_{\mathrm{a}}{i}_{\mathrm{a}}\\ r_{\mathrm{b}}{i}_{\mathrm{b}}\end{array}\right]$$

(7)

As shown in (7), there is no coupling relationship for the basic voltage equation of the coupled inductor. Thus, substituting (7) into (2), we can have:

$$\frac{d}{dt}\left[\begin{array}{c}{i}_{\mathrm{a}}\\ i_{\mathrm{b}}\\ u_{\mathrm{fa}}\\ u_{\mathrm{fb}}\end{array}\right]=\left[\begin{array}{cccc}\frac{-r_{\mathrm{a}}}{(L_{\mathrm{a}}+M)}& 0& \frac{d_{\mathrm{a}2}-d_{\mathrm{a}1}}{(L_{\mathrm{a}}+M)}& 0\\ 0& \frac{-r_{\mathrm{b}}}{(L_{\mathrm{b}}-M)}& 0& \frac{d_{\mathrm{b}2}-d_{\mathrm{b}1}}{(L_{\mathrm{b}}-M)}\\ \frac{d_{\mathrm{a}1}-d_{\mathrm{a}2}}{C_{\mathrm{fa}}}& 0& 0& 0\\ 0& \frac{d_{\mathrm{b}1}-d_{\mathrm{b}2}}{C_{\mathrm{fb}}}& 0& 0\end{array}\right]\times \left[\begin{array}{c}{i}_{\mathrm{a}}\\ i_{\mathrm{b}}\\ u_{\mathrm{fa}}\\ u_{\mathrm{fb}}\end{array}\right]+\left[\begin{array}{cc}\frac{d_{\mathrm{a}1}}{(L_{\mathrm{a}}+M)}& \frac{-1}{(L_{\mathrm{a}}+M)}\\ \frac{d_{\mathrm{b}1}}{(L_{\mathrm{b}}-M)}& \frac{-1}{(L_{\mathrm{b}}-M)}\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{u}_{\mathrm{i}}\\ u_{\mathrm{o}}\end{array}\right]$$

(8)

It can be seen from (8) that there is no coupling between the two phase circuits under the coupled inductor, that is, the mathematical model of coupled inductor FCDC has been decoupled.

3.2. 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_a, i_b, u_fa and u_fb as the control variables, the discrete models can be obtained as shown in Equation (10):

$$\left\{\begin{array}{}\text{(9a)}{i}_{\mathrm{a}}^{k+1}=i_{\mathrm{a}}^k-\frac{i_{\mathrm{a}}^k{r}_{\mathrm{a}}{T}_s}{L_{\mathrm{a}}+M}+\frac{(d_{\mathrm{a}2}-d_{\mathrm{a}1})T_s}{L_{\mathrm{a}}+M}{u}_{\mathrm{fa}}^k+\frac{T_s}{L_{\mathrm{a}}+M}\times \left(d_{\mathrm{a}1}{u}_{\mathrm{i}}^k-u_{\mathrm{o}}^k\right)\hfill \text{(9b)}{i}_{\mathrm{b}}^{k+1}=i_{\mathrm{b}}^k-\frac{i_{\mathrm{b}}^k{r}_{\mathrm{b}}{T}_s}{L_{\mathrm{b}}-M}+\frac{(d_{\mathrm{b}2}-d_{\mathrm{b}1})T_s}{L_{\mathrm{b}}-M}{u}_{\mathrm{fb}}^k+\frac{T_s}{L_{\mathrm{b}}-M}\times \left(d_{\mathrm{b}1}{u}_{\mathrm{i}}^k-u_{\mathrm{o}}^k\right)\hfill \text{(9c)}{u}_{\mathrm{fa}}^{k+1}=u_{\mathrm{fa}}^k+\frac{(d_{\mathrm{a}1}-d_{\mathrm{a}2})T_s}{C_{\mathrm{fa}}}{i}_{\mathrm{a}}^k\hfill \text{(9d)}{u}_{\mathrm{fb}}^{k+1}=u_{\mathrm{fb}}^k+\frac{(d_{\mathrm{b}1}-d_{\mathrm{b}2})T_s}{C_{\mathrm{fb}}}{i}_{\mathrm{b}}^k\hfill \end{array}\right.$$

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:

$$\begin{array}{cc}\hfill J^k& ={\lambda }_1{(i_{\mathrm{a}}^{\ast }-i_{\mathrm{a}}^{k+1})}^2+{\lambda }_2{(i_{\mathrm{b}}^{\ast }-i_{\mathrm{b}}^{k+1})}^2\hfill \\ & +{\lambda }_3{(u_{\mathrm{fa}}^{\ast }-u_{\mathrm{fa}}^{k+1})}^2+{\lambda }_4{(u_{\mathrm{fb}}^{\ast }-u_{\mathrm{fb}}^{k+1})}^2\hfill \end{array}$$

(10)

where the variables marked with * are the reference values and λ₁~λ₄ are the weight coefficients.

As shown in (11), in order to obtain the minimum J^k, it is not only necessary to calculate all combinations of the four duty ratios d_a1, d_a2, d_b1 and d_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:

$$\left\{\begin{array}{l}△i_{\mathrm{a}}=-\frac{r_{\mathrm{a}}}{(L_{\mathrm{a}}+M)}{i}_{\mathrm{a}}+\frac{d_{\mathrm{a}2}-d_{\mathrm{a}1}}{(L_{\mathrm{a}}+M)}{u}_{\mathrm{fa}}+\frac{d_{\mathrm{a}1}}{(L_{\mathrm{a}}+M)}{u}_{\mathrm{i}}-\frac{1}{(L_{\mathrm{a}}+M)}{u}_{\mathrm{o}}\\ △u_{\mathrm{fa}}=\frac{d_{\mathrm{a}1}-d_{\mathrm{a}2}}{C_{\mathrm{fa}}}{u}_{\mathrm{fa}}=\frac{△d_{\mathrm{c}}}{C_{\mathrm{fa}}}{u}_{\mathrm{fa}}\end{array}\right.$$

(11)

where △d_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 △d_c is zero. At this time, the phase currents i_a and i_b are determined by d_a1 and d_b1, respectively, and are not affected the by flying capacitor. A small perturbation in flying capacitor voltage can cause variation in △d_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:

$$\left\{\begin{array}{l}{J}_{\mathrm{a}}^k={(i_{\mathrm{a}}^{*}-i_{\mathrm{a}}^{k+1})}^2\\ J_{\mathrm{b}}^k={(i_{\mathrm{b}}^{*}-i_{\mathrm{b}}^{k+1})}^2\end{array}\right.$$

(12)

However, in order to achieve the minimum value of (13), it is necessary to calculate the predicted current values with different duty ratios according to (10). Even if the two cost functions are independent of each other, there is still a significant computational load.

Actually, (13) is a non-negative function. Assuming u_i = 100 V, u_o = 60 V, $i_a^{*}$ = i_ak = 5 A and u_fa = 50 V, the curves of different duty ratios based on (13) can be obtained and are shown in Figure 4.

It can be seen from Figure 4 that when the derivative value of J_a^k is zero, the minimum value is obtained. So, taking derivatives of J_a^k for d_a1^k, d_a2^k and taking derivatives of J_b^k for d_b1^k and d_b2^k, respectively, we can have:

$$\left\{\begin{array}{c}{\left.\frac{\partial J_{\mathrm{a}}^k}{\partial d_{\mathrm{a}1}^k}\right|}_{d_{\mathrm{a}2}^k=1}=2(i_{\mathrm{a}}^{*}-i_{\mathrm{a}}^{k+1})(\frac{\partial i_{\mathrm{a}}^{k+1}}{\partial d_{\mathrm{a}1}^k}-\frac{\partial i_{\mathrm{a}}^{*}}{\partial d_{\mathrm{a}1}^k})\\ {\left.\frac{\partial J_{\mathrm{a}}^k}{\partial d_{\mathrm{a}2}^k}\right|}_{d_{\mathrm{a}1}^k=1}=2(i_{\mathrm{a}}^{*}-i_{\mathrm{a}}^{k+1})(\frac{\partial i_{\mathrm{a}}^{k+1}}{\partial d_{\mathrm{a}2}^k}-\frac{\partial i_{\mathrm{a}}^{*}}{\partial d_{\mathrm{a}2}^k})\end{array}\right.$$

(13)

$$\left\{\begin{array}{c}{\left.\frac{\partial J_{\mathrm{b}}^k}{\partial d_{\mathrm{b}1}^k}\right|}_{d_{\mathrm{b}2}^k=1}=2(i_{\mathrm{b}}^{*}-i_{\mathrm{b}}^{k+1})(\frac{\partial i_{\mathrm{b}}^{k+1}}{\partial d_{\mathrm{b}1}^k}-\frac{\partial i_{\mathrm{b}}^{*}}{\partial d_{\mathrm{b}1}^k})\\ {\left.\frac{\partial J_{\mathrm{b}}^k}{\partial d_{\mathrm{b}2}^k}\right|}_{d_{\mathrm{b}1}^k=1}=2(i_{\mathrm{b}}^{*}-i_{\mathrm{b}}^{k+1})(\frac{\partial i_{\mathrm{b}}^{k+1}}{\partial d_{\mathrm{b}2}^k}-\frac{\partial i_{\mathrm{b}}^{*}}{\partial d_{\mathrm{b}2}^k})\end{array}\right.$$

(14)

When the derivative value of J_a^k and J_b^k to d_a1^k, d_a2^k and d_b1^k, d_b2^k is zero, the minimum value of the cost function shown in (13) can be obtained, and thus the duty ratio to control i_a^k and i_b^k tracking the current reference can be derived. Assuming the equations shown in (14) are 0, we can have:

$$\left\{\begin{array}{c}(\frac{\partial i_{\mathrm{a}}^{k+1}}{\partial d_{\mathrm{a}1}^k}-\frac{\partial i_{\mathrm{a}}^{*}}{\partial d_{\mathrm{a}1}^k})\ne 0\\ (\frac{\partial i_{\mathrm{a}}^{k+1}}{\partial d_{\mathrm{a}2}^k}-\frac{\partial i_{\mathrm{a}}^{*}}{\partial d_{\mathrm{a}2}^k})\ne 0\end{array}\right.$$

(15)

Due to the predicted value being approximately equal to the steady-state reference, this yields:

$$i_x^{*}\approx i_x^k-\frac{i_x^k{r}_x{T}_s}{L_x\pm M}+\frac{(d_{x2}-d_{x1})T_s}{L_x\pm M}{u}_{\mathrm{f}x}^k+\frac{T_s}{L_x\pm M}\times \left(d_{x1}{u}_{\mathrm{i}}^k-u_{\mathrm{o}}^k\right)$$

(16)

According to (17), it can be seen that the predicted value is approximately equal to the reference value, so:

$$\left\{\begin{array}{c}{i}_{\mathrm{a}}^{*}-i_{\mathrm{a}}^{k+1}\approx 0\\ i_{\mathrm{b}}^{*}-i_{\mathrm{b}}^{k+1}\approx 0\end{array}\right.$$

(17)

Thus, d_a1^k and d_a2^k can be derived from (15) and (18). And the expression of d_a1^k and d_a2^k under Boost mode is shown in Appendix A.

$$\left\{\begin{array}{l}{d}_{\mathrm{a}1}^k=\frac{u_{\mathrm{o}}^k-u_{\mathrm{fa}}^k+r_{\mathrm{a}}{i}_{\mathrm{a}}^k}{u_{\mathrm{i}}^k-u_{\mathrm{fa}}^k}+\frac{\left(L_{\mathrm{a}}+M\right)\left(i_{\mathrm{a}}^{*}-i_{\mathrm{a}}^k\right)}{T_s\left(u_{\mathrm{i}}^k-u_{\mathrm{fa}}^k\right)}\\ d_{\mathrm{a}2}^k=\frac{u_{\mathrm{o}}^k+u_{\mathrm{fa}}^k-u_{\mathrm{i}}^k+r_{\mathrm{a}}{i}_{\mathrm{a}}^k}{u_{\mathrm{fa}}^k}+\frac{\left(L_{\mathrm{a}}+M\right)\left(i_{\mathrm{a}}^{*}-i_{\mathrm{a}}^k\right)}{T_s{u}_{\mathrm{fa}}^k}\end{array}\right.$$

(18)

Similarly, the duty ratio for phase b can be expressed as:

$$\left\{\begin{array}{l}{d}_{\mathrm{b}1}^k=\frac{u_{\mathrm{o}}^k-u_{\mathrm{fb}}^k+r_{\mathrm{b}}{i}_{\mathrm{b}}^k}{u_{\mathrm{i}}^k-u_{\mathrm{fb}}^k}+\frac{\left(L_{\mathrm{b}}-M\right)\left(i_{\mathrm{b}}^{*}-i_{\mathrm{b}}^k\right)}{T_s\left(u_{\mathrm{i}}^k-u_{\mathrm{fb}}^k\right)}\\ d_{\mathrm{b}2}^k=\frac{u_{\mathrm{o}}^k+u_{\mathrm{fb}}^k-u_{\mathrm{i}}^k+r_{\mathrm{b}}{i}_{\mathrm{b}}^k}{u_{\mathrm{fb}}^k}+\frac{\left(L_{\mathrm{b}}-M\right)\left(i_{\mathrm{b}}^{*}-i_{\mathrm{b}}^k\right)}{T_s{u}_{\mathrm{fb}}^k}\end{array}\right.$$

(19)

According to (18) and (19), the following conclusions can be drawn.

(1)

Taking phase a as an example, when the converter operates under steady-state i_a^k ≈ i_a^*, u_fa^k ≈ u_i^k/2 and u_o^k ≈ u_o^*, we can have d_a1^k ≈ d_a2^k. Under this situation, d_a1^k and d_a2^k are derived by optimizing control i_a, and u_fa cannot be controlled by (19). The operation principle of phase b is the same as that of phase a.

(2)

With 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_fa* is the FC voltage reference, its expression can be written as:

$$u_{\mathrm{fa}}^{*}=u_{\mathrm{fa}}^k-\frac{1}{2}\Delta u_1^k+\frac{1}{2}\Delta u_2^k$$

(20)

According to the FC charging and discharging circuit states, Δu₁^k and Δu₂^k can be expressed as:

$$\left\{\begin{array}{l}\Delta u_1^k=\frac{(1-d_{\mathrm{a}1}^k-d_{\mathrm{ca}}^k)T_s}{2C_{\mathrm{fa}}}{i}_{\mathrm{a}}^k\\ \Delta u_2^k=\frac{(1-d_{\mathrm{a}1}^k+d_{\mathrm{ca}}^k)T_s}{2C_{\mathrm{fa}}}{i}_{\mathrm{a}}^k\end{array}\right.$$

(21)

where d_ca^k is the duty ratio to control the flying capacitor voltage.

Substituting (22) into (21), the duty ratio d_ca^k to control u_fa can be derived as:

$$d_{\mathrm{ca}}^k=\frac{2C_{\mathrm{fa}}}{T_s{i}_{\mathrm{a}}^k}(u_{\mathrm{fa}}^{*}-u_{\mathrm{fa}}^k)$$

(22)

Similarly, the duty ratio d_cb^k to control u_fb can be derived as:

$$d_{\mathrm{cb}}^k=\frac{2C_{\mathrm{fb}}}{T_s{i}_{\mathrm{b}}^k}(u_{\mathrm{fb}}^{*}-u_{\mathrm{fb}}^k)$$

(23)

Even if d_ca^k and d_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 ∆i_Llim, ∆d_f1^k and ∆d_f2^k must be limited to ∆D_lim, ensuring that the deviation of i_L is within ∆i_Llim. The relationship between ∆i_Llim and ∆D_lim can be expressed as:

$$\Delta D_{\lim }\le (\Delta i_{L\lim }-\frac{1}{2}\Delta i_L)\frac{4L}{T_s(u_2^{*}-u_1)}$$

(24)

Thus, we can get the control block diagram of the proposed MPC power balancing strategy, as shown in Figure 5.

3.3. Stability Analysis

The direct Lyapunov method is employed in this paper to verify the stability of the proposed MPC strategy. This method indicates that when the function L(x) defined according to the system state variables x satisfies that L(x) is positive, $\dot{L}(\mathrm{x})$ is negative and ${\lim }_{‖x‖\to \infty }L(\mathrm{x})=\infty$, and thus the system is stable. So, according to the cost function, the discrete Lyapunov function can be defined as follows:

$$L{(x)}^k=\frac{1}{2}{[x_{L-\mathrm{err}}^{k+1}]}^{\mathrm{T}}[x_{L-\mathrm{err}}^{k+1}]$$

(25)

where x is [i_a i_b]^T.

Define the error variables between predicted and reference values as:

$$\left\{\begin{array}{l}{i}_{\mathrm{a}-\mathrm{err}}^{k+1}=i_{\mathrm{a}}^{k+1}-i_{\mathrm{a}}^{\ast }\\ i_{\mathrm{b}-\mathrm{err}}^{k+1}=i_{\mathrm{b}}^{k+1}-i_{\mathrm{b}}^{\ast }\end{array}\right.$$

(26)

According to (10), (26), and (27), the variation rate of L(x) as a function of the inductor current can be expressed as:

$$\begin{array}{cc}\hfill \Delta L{(i_x)}^k& =L{(i_x)}^{k+1}-L{(i_x)}^k\hfill \\ & =\frac{1}{2}\left[\begin{array}{l}{i}_x^k-\frac{i_x^k{r}_x{T}_s}{L_x\pm M}+\frac{(d_{x1}-d_{x2})T_s}{L_x\pm M}{u}_{\mathrm{f}x}^k\\ +\frac{T_s}{L_x\pm M}\times \left[u_{\mathrm{i}}^k+\left(d_{x2}-1\right)u_{\mathrm{o}}^k\right]-i_x^{\ast }\end{array}\right]\hfill \\ & \times \left[\begin{array}{l}{i}_x^k-\frac{i_x^k{r}_x{T}_s}{L_x\pm M}+\frac{(d_{x1}-d_{x2})T_s}{L_x\pm M}{u}_{\mathrm{f}x}^k\\ +\frac{T_s}{L_x\pm M}\times \left[u_{\mathrm{i}}^k+\left(d_{x2}-1\right)u_{\mathrm{o}}^k\right]-i_x^{\ast }\end{array}\right]\hfill \\ & -\frac{1}{2}\left[i_{x\_\mathrm{err}}^k\right]\left[i_{x\_\mathrm{err}}^k\right]\hfill \end{array}$$

(27)

where x represents phase a and b, respectively. When x represents phase a, the symbol ± should be written as +, and when x represents phase b, the symbol ± should be written as −.

When ∆d_f1^k and ∆d_f2 do not reach the limit, the predicted value is approximately equal to the steady-state reference due to the tracking characteristics of MPC, as shown in (17). Substituting (17) into (28), we can have:

$$\Delta L{(i_x)}^k=-\frac{1}{2}\left[i_{x\_\mathrm{err}}^k\right]\left[i_{x\_\mathrm{err}}^k\right]$$

(28)

However, when ∆d_f1^k or ∆d_f2^k reaches the limit, the predicted current value will fail to track the reference. To address this issue, the adjustment of ∆i_Llim should align with the changes in ΔL(x) to guarantee the stability and convergence of the variable’s control. The maximum current deviation can be optimized as follows:

$$\Delta i_{L\lim \_\max }=\left\{\begin{array}{l}\Delta i_{L\lim }+s,if\Delta L{(i_x)}^k>0;\\ \Delta i_{L\lim },if\Delta L{(i_x)}^k\le 0.\end{array}\right.$$

(29)

where s represents the increment step of current deviation.

The relaxation of ∆i_Llim leads to a corresponding relaxation of ∆D_lim, thereby ensuring the sustained validity of Equation (29) within the system under these conditions.

As shown in (26) and (29), L(x) > 0 and $\dot{L}(x)$ < 0. Thus, based on the direct Lyapunov method, it can be known that the MPC strategy based on the cost function shown in (13) is stable.

4. Experimental Verification

In order to verify the effectiveness of the proposed MPC power balancing strategy, an experimental platform is built in the laboratory, which is shown in Figure 6. A dual-core control architecture based on TMS320F28335 + XC3S500E is adopted, TMS320F28335 is responsible for sampling and calculation, while XC3S500E is responsible for generating PWM signals and achieving system protection. The experimental parameters are shown in Table 2. To simulate the imbalanced operation conditions, an inductor with 1 mH is connected in series in phase b.

4.1. Experimental Results

The output voltage under Buck mode is set to 50 V. The experimental results of the proposed power balancing MPC strategy under Buck mode are shown in Figure 7. Due to the unequal inductance between the two phases, i_a and i_b are imbalanced. As can be seen in Figure 7a, before the proposed strategy put into operation, i_a is 3 A and i_b is 2 A. After the control strategy is put into operation, the current reaches a balanced state within 10 ms. Compared to the initial state, the two-phase current is balanced by 20%. Figure 7b shows the voltage curves of C_fa and C_fb when the power balancing control is in action. It can be seen that the FC voltage is unchanged. Figure 7c shows the response curve of FC voltage from an imbalanced to a balanced state. Due to the direct regulation of flying capacitor voltage, the voltage difference between C_fa and C_fb quickly changes from 10 V to 0 V within 5 ms, achieving a balanced control state.

The input voltage under Boost mode is set to 50 V and the voltage on high voltage side is set to 60 V. The verification results are shown in Figure 8. Figure 8a shows that before the proposed strategy is put into operation, i_a is 4 A and i_b is 2.8 A. Due to the shoot-through state in Boost mode, the current change is more obvious than that in Buck mode, and it can be seen that i_a and i_b achieve balancing state within 20 ms at 3 A. The maximum current balancing regulation reaches 33%. Figure 8b shows the output voltage and FC voltage curves. Because the FC voltage is related to the output voltage under Boost mode, u_o, u_fa and u_fb finally remain at 60 V, 30 V, and 30 V, respectively, after a short fluctuation. Figure 8c shows the FC voltage control response curve. It can be seen that the voltage difference between C_fa and C_fb quickly changes from 10 V to 0 V less than 5 ms, achieving a balanced control state.

The abrupt load changing results under Buck mode are shown in Figure 9. Figure 9a shows the loading results from 12 Ω to 6 Ω. Due to the differences in inductance, the two-phase current has different response curves, increasing from 1.6 A to 3.2 A. After a short oscillation within 22 ms, the two-phase current reaches a new balancing state and the voltage of flying capacitor remains controllable. The abrupt unloading results are shown in Figure 9b. The load resistor is changed from 6 Ω to 12 Ω. It can be seen that the current between phases can always remain balanced while the voltage of u_fa and u_fb is kept at 50 V.

The experimental results with load changing from 12 Ω to 6 Ω and from 6 Ω to 12 Ω under Boost mode are shown in Figure 10a,b. The inductance current increased from 1.5 A to 3 A under the loading condition and recovered to 1.5 A after unloading. It can be seen that both the current and voltage have favorable response performance: the current remains balanced and the FC voltage is controllable.

According to the above experimental results, it can be concluded that the proposed MPC power balancing strategy has favorable power balance performance to control the coupled inductor FCDC with imbalanced coupled inductor and can achieve FC voltage control. It can quickly respond to load power demands, maintain current balancing control, and ensure system stability.

4.2. Performance Discussion

Efficiency analysis: The converter efficiency is mainly determined by the switching loss. According to Figure 2 and Figure 3, it can be seen that the coupled inductor FCDC studied in this paper achieves Buck operation by chopping the upper bridge arm and keeping the lower bridge arm switch off. In Boost mode, the lower bridge arm switch operates under the chopping state, and the upper bridge switches are turned off. Therefore, its basic modulation method is the same as the flying capacitor topology controlled by a traditional PI strategy [20]. The MPC control strategy proposed in the paper is fixed frequency control, and the switching frequency is not affected by the control algorithm, so there will be no additional losses due to frequency changes. In addition, the control variable generated by the MPC strategy is the duty ratio, and the switch does not operate in soft switching mode. Therefore, we can say that the loss of the proposed MPC strategy is basically the same as that of the traditional PI control strategy.

Comparison with other strategies: The MPC strategy has better dynamic performance compared to traditional PI control. The proposed MPC strategy in the paper mainly solves the power imbalance issue under coupled inductance and the control of flying capacitor voltage. Compared with the coupled inductance balancing strategy proposed in [16], according to the experimental results shown in Figure 7, Figure 8, Figure 9 and Figure 10, it can be seen that the proposed MPC strategy has favorable balancing performance, with a two-phase current difference close to 0 under a steady state and dynamic load changes. Meanwhile, according to the voltage response curve of the flying capacitor proposed in [18], it can be seen that under similar operating conditions, the response time of the strategy is longer than 500 ms, while the response time of the MPC strategy proposed in this paper is about 10 ms and the capacitor voltage response time is about 5 ms. In comparison, the proposed strategy has obvious advantages in both balancing performance and dynamic response performance.

5. Conclusions

This paper proposes a decoupled MPC power balancing strategy to address the power imbalance issue caused by parameter differences in a coupled inductor FCDC. The decoupled mathematical model of the converter is studied first, and the basic principle of the proposed MPC power balancing strategy is derived by analyzing the relationship between system current and FC voltage control. By solving the constraint function, independent single-step optimization control of system current is proposed, At the meantime, the effective control of FC voltage is realized based on the time deviation of capacitor charging and discharging and combined with current ripple constraints. The proposed MPC power balancing strategy optimizes the multi-objective control of a coupled inductor FCDC, realizes power balancing control and FC voltage control, and greatly reduces the computing load. Experimental results also fully demonstrate that the proposed strategy has favorable current and voltage control capabilities and can achieve fast power balancing control.