Modeling and Centralized-ZVS Control for Wireless Charging Electric Vehicles Supplied by Parallel Modular Multi-Inverters

Abstract

In this paper, a parallel modular multi-inverter (PMMI) topology is proposed to supply high power for wireless charging electric vehicles (EVs). A major challenge in the implementation of PMMI topology is zero-voltage switching (ZVS) for all inverters to avoid high-frequency switching losses. Therefore, a centralized-ZVS control and master–slave frequency following (MSFF) strategy are presented to realize ZVS for all PMMIs by a single controller structure without extra controllers needed on the slave inverters. Meanwhile, a modeling method directly related to the ZVS angle and operating frequency for an arbitrary number of PMMIs is proposed and linearized to analyze the system dynamic characteristics at the operating point. Additionally, to obtain the desired dynamic performance, an optimal controller coefficient (OCC) configuration method is proposed for the design of controller parameters. Finally, a laboratory wireless power transfer (WPT) prototype supplied by three PMMIs is designed, built, and tested to verify the correctness of the theoretical analysis. Experimental results show that the ZVS angle of all PMMIs can maintain at the preset value with the desired settling time under designed the PI controller parameters. The measured whole system power transmission efficiency is $94.1\%$ at a 10 $\mathsf{\Omega }$ load.

1. Introduction

Wireless power transfer (WPT) has been widely used in EVs [1,2,3,4,5] due to the advantages of safety and convenience. High-power WPT is an attractive option for fast charging EVs because no high-power equipment needs to be operated by the user [6,7].

Limited by the rated voltage and current of the semiconductor components, high-power WPT is generally realized by multiple inverters [8,9,10] or inverter-legs [11,12,13,14], which both have the problem of current suppression. Current circulation in multi inverter-legs for WPT is suppressed by the use of inter-cell transformers (ICTs) [11,12]. However, the flexibility of a multiple inverter-legs WPT system is not good, as it is limited by the fixed number of inverter-legs. High-power WPT also can be achieved by upgrading the current level through multiple inverters connecting in parallel. Current balance is realized by the voltage-type droop control method [8] and the dual-loop based harmonic control strategy and admittance modeling method [10], which are mainly applied in fixed and low frequency (i.e., 50 Hz) power grid occasions and are not suitable for high-frequency WPT scenarios. A current circulation suppression method based on phase synchronization control [9] is applied for the multiple inverters by simulating different inverter output voltage phases. However, when analyzing the dynamic performance, the dynamic equation of the WPT system is not established.

Alternatively, this paper proposes a high-power WPT system supplied by PMMIs with good power requirement satisfaction due to the modularity of multi-inverters in which ICTs are applied to suppress circulation current. Furthermore, it is worth noting that ZVS is vital for system efficiency improvement as it can significantly reduce the switching loss, especially for high-power applications [15,16,17,18,19,20,21,22].

A dynamic dead-time control [15], a transistor-controlled variable capacitor [16], and a pulse density modulated full-bridge converter [17] are proposed for ZVS control. However, they are only applied in tens of watts situations, which are not suitable for high-power wireless charging EVs. Variable ZVS angles are realized by joint control for dynamic efficiency optimization in [18]. Nevertheless, controllers are needed on both sides to implement respective ZVS angles. A triple-phase-shift control strategy [19] and a hybrid frequency pacing technique [20] are presented to implement ZVS operations and are only applicable to the LCC network, not other compensation topologies. Based on the variable frequency phase shift control strategy, [21] proposes an optimal operating frequency range for ZVS control, but only a single inverter system is analyzed and the multi-inverter WPT system is not taken into consideration. Furthermore, the ZVS control methods proposed in [15,16,17,18,19,20,21,22] are all designed for a single inverter system, which are not suitable for a multi-inverter WPT system. Additionally, note that the ZVS control, which means the inverter output current lags behind the voltage [23] of the PMMI system, is much different from that of a single inverter system due to the parallel connection of the PMMIs.

In this paper, to acquire the ZVS operation for all PMMIs of high-power wireless charging EVs, a modeling and centralized-ZVS control strategy is proposed by regulating the operating frequency of the main inverter followed by other slave modular inverters. In contrast to the existing works, the contributions of this article are as follows.

(1)

The PMMI topology is proposed and applied to high-power wireless charging EVs. Meanwhile, on the basis of impedance angle analysis, the modular inverter output ZVS angle varies with the number of PMMIs, and load resistance is simulated and analyzed. Moreover, by introducing the MSFF strategy, a novel centralized-ZVS control is put forward to make all inverter modules operate at the required ZVS state.

(2)

A common dynamic model method considering the proposed PMMIs, ICTs, and the entire WPT system is proposed, which can directly obtain the slowly changing operating frequency and ZVS angle variables instead of time-varying voltage and current variables. Moreover, the OCC configuration method is proposed for the PI controller parameter design, which can dynamically regulate operating frequency to make all PMMIs always operate at the desired ZVS angle within the required setting time.

The remainder of this paper is organized as follows. Section 2 presents the ZVS angle analysis of the WPT system supplied by PMMIs and introduces the centralized-ZVS control scheme for PMMIs. Then, an equivalent modeling method and circuit of a WPT system supplied by PMMIs is analyzed and modeled in Section 3. Additionally, the simulation of the designed PI controller based on the OCC configuration is verified. Section 4 builds an experimental prototype and the dynamic performances of the ZVS control are validated. Finally, conclusions are drawn in Section 5.

2. Analysis of the System Structure

2.1. Structure of the WPT System Supplied by PMMIs

The typical scheme of a high-power wireless charging EV system supplied by PMMIs is depicted in Figure 1. It consists of n PMMIs, parallel connecting coupling ICTs, magnetic resonance coupling parts, a full-bridge rectifier, and battery equivalent series resistance (ESR). Each PMMI is composed of four half-bridge switching legs, which are connected in parallel to acquire high output current. All the PMMIs are connected in parallel by coupling ICTs to suppress circulating current [11]. All PMMIs are fed by a common DC voltage source $V_{dc}$ to get the identical amplitude of output voltages. For the k-th PMMI, $v_{pk}$ and $i_{pk}$ are the output voltage and current, respectively, where $1\le k\le n$. The symbols $i_p$ and $v_p$ are the joint output current and voltage of the PMMIs, respectively. The symbol $i_s$ is the induced current at the secondary side.

Each ICT (${\mathrm{ICT}}_k$, for example) connected in parallel is made up of a primary inductor $L_{kp}$, a secondary inductor $L_{ks}$, a leakage inductor $L_{leak}$ for both sides, and a parasitic resistance $r_{ICT}$ for both sides shown in the equivalent circuit of Figure 1. $L_p$ and $L_s$ are the self-inductances of the primary resonant coil and secondary coil, respectively. M is the mutual inductance. $C_p$ and $C_s$ are the resonant capacitors at both sides, respectively. The equivalent series resistances of the two sides are represented by $r_p$ and $r_s$. On the secondary side, $C_f$ is the filter capacitor and R is the load resistance.

2.2. Analysis of ZVS for PMMIs Topology

Based on Kirchhoff’s law in the frequency domain, the output voltage and current for n PMMIs shown in Figure 1 can be expressed by

$$\left[\begin{array}{c}{V}_{p1}\\ V_{p2}\\ ⋮\\ V_{pn-1}\\ V_{pn}\end{array}\right]=\left[\begin{array}{ccccc}{Z}_C& -j{\omega }_s{M}_C& \cdots & 0& -j{\omega }_s{M}_C\\ -j{\omega }_s{M}_C& Z_C& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & Z_C& -j{\omega }_s{M}_C\\ -j{\omega }_s{M}_C& 0& \cdots & -j{\omega }_s{M}_C& Z_C\end{array}\right]\left[\begin{array}{c}{I}_{p1}\\ I_{p2}\\ ⋮\\ I_{pn-1}\\ I_{pn}\end{array}\right]+{\left[\begin{array}{c}{V}_p\\ V_p\\ ⋮\\ V_p\\ V_p\end{array}\right]}_{n\times 1},$$

(1)

where

$$\begin{array}{cc}\hfill V_p& =I_p{Z}_{in},\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill I_p& =\sum _{k=1}^n{i}_{pk},\hfill \end{array}$$

(2b)

where $V_{pk}$ and $I_{pk}$ are the amplitude of the k-th inverter module output voltage and current, k = 1∼n, $V_p$ and $I_p$ are the amplitude of the joint primary voltage and current, ${\omega }_s$ is the operating angular frequency, $M_C$ is the mutual inductance between two ICTs, and all windings for the ICTs have the identical number of turns. Hence, $M_C$ is equal to the magnetizing inductance $L_m$, namely, $M_C=L_m=L_{kp}=L_{ks}$, $Z_C$ represents the branch impedance of the ICTs, and $Z_{in}$ is the total equivalent impedance and can be expressed by

$$\begin{array}{cc}\hfill & Z_C=2j{\omega }_s{L}_m+2j{\omega }_s{L}_{leak}+2r_{ICT},\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & Z_{in}=r_p+j{\omega }_s{L}_p+\frac{1}{j{\omega }_s{C}_p}+Z_{ref},\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill & Z_{ref}=\frac{{\left({\omega }_{s}M\right)}^2}{r_s+j\left({\omega }_s{L}_s-\frac{1}{{\omega }_s{C}_s}\right)+R_e},\hfill \end{array}$$

(3c)

$$\begin{array}{cc}\hfill & R_e=\frac{8}{{\pi }^2}R.\hfill \end{array}$$

(3d)

After phase synchronization compensation and circuit current suppression [9], the output voltages and currents of different inverter modules are identical, because all PMMIs have the same hardware configuration, i.e., $I_{pi}=I_{pj},V_{pi}=V_{pj},\left\{i,j\in \left(1,n\right),i\ne j\right\}$. Substituting (2a) and (2b) into (1) yields the $k-\mathrm{th}$ modular inverter output current:

$$\begin{array}{c}\begin{array}{c}\hfill I_{pk}=\frac{V_{pk}}{2j{\omega }_s{L}_{leak}+2r_{ICT}+n{Z}_{in}}.\end{array}\end{array}$$

(4)

Hence, the output equivalent impedance of all inverter modules is identical and can be expressed as

$$\begin{array}{c}\begin{array}{c}\hfill Z_{pk}=\frac{V_{pk}}{I_{pk}}=2j{\omega }_s{L}_{leak}+2r_{ICT}+n{Z}_{in}.\end{array}\end{array}$$

(5)

The $k-\mathrm{th}$ modular inverter output phase ${\varphi }_{pk}$ and primary phase ${\varphi }_{in}$ can be defined as

$$\begin{array}{c}\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\varphi }_{pk}=\frac{180}{\pi }\mathrm{arc}tan\left[{\displaystyle \frac{\mathrm{Im}\left[Z_{pk}\right]}{\mathrm{Re}\left[Z_{pk}\right]}}\right]}\hfill \\ {\displaystyle {\varphi }_{in}=\frac{180}{\pi }\mathrm{arc}tan\left[{\displaystyle \frac{\mathrm{Im}\left[Z_{in}\right]}{\mathrm{Re}\left[Z_{in}\right]}}\right]}\hfill \end{array}\right..\end{array}\end{array}$$

(6)

Note that

$$\begin{array}{c}\begin{array}{c}\hfill {\varphi }_{ZVS}={\varphi }_{pk}={\varphi }_{vpk}-{\varphi }_{ipk},\end{array}\end{array}$$

(7)

where ${\varphi }_{ZVS}$ is the ZVS angle, and ${\varphi }_{vpk}$ and ${\varphi }_{ipk}$ are the k-th inverter module output voltage and current phase, respectively. Figure 2 depicts the primary phase ${\varphi }_{in}$ and ${\varphi }_{pk}$ varying with different numbers of PMMIs at operating frequencies ranging from 80 kHz to 90 kHz. The system parameters are listed in

Table 1. Parameters for Simulation.

Symbol	Parameter	Value
$V_{dc}$	Output voltage of the DC source	50 V
$f_s$	Operating frequency	80∼90 kHz
M	Mutual inductance between two coils	7.25 $\mathsf{\mu }$H
$L_{p,s}$	Inductances of both resonant coils	32.5 $\mathsf{\mu }$H
$C_{p,s}$	Capacitances of both resonant loops	113.0 nF
R	Load resistance	10 $\mathsf{\Omega }$
$L_m$	Mutual inductance between two ICTs	14 $\mathsf{\mu }$H
$L_{leak}$	Leakage inductance between two ICTs	2 $\mathsf{\mu }$H
$r_{ICT}$	Parasitic resistance of single ICT	20 $\mathrm{m}\mathsf{\Omega }$
$r_{p,s}$	Parasitic resistance at both resonant sides	50 $\mathrm{m}\mathsf{\Omega }$
n	Number of the PMMIs	3∼7

. Conclusions can be drawn from Figure 2.

(1)

${\varphi }_{pk}$ is larger than ${\varphi }_{in}$ at the same operating frequency, meaning that the operating frequency of the PMMI system is lower than that of the single inverter WPT system at the same ZVS angle. Additionally, when the ZVS state is implemented, ${\varphi }_{in}$ can be negative.

(2)

Moreover, ${\varphi }_{pkn}$ decreases with the number of PMMIs and gradually approaches ${\varphi }_{in}$. When n is large enough, ${\varphi }_{pkn}$ approaches ${\varphi }_{in}$ indefinitely, which means ${\varphi }_{ZVS}\approx {\varphi }_{in}$.

(3)

Additionally, the same operating frequency change of the PMMI system can cause a larger ZVS angle change compared with a single inverter system, meaning that ${\varphi }_{pkn}$ is more sensitive to operating frequency than ${\varphi }_{in}$.

With the number of PMMIs fixed, ${\varphi }_{pk}$ as a function of operating frequency $f_s={\omega }_s/2\pi$ and load resistance R is calculated and depicted in Figure 3a. When n and R are both fixed, ${\varphi }_{pk}$ as a function of $f_s$ is plotted in Figure 3b. It can be seen that ${\varphi }_{pk}$ increases with $f_s$ under a given load resistance R and number of PMMIs. In other words, the ZVS control for the WPT system supplied by PMMIs can be achieved by adjusting the operating frequency.

2.3. Control Scheme

Based on the analysis in the previous section, a centralized-ZVS control applying an MSFF strategy for a PMMI WPT system with n inverter units is proposed as shown in Figure 4. A master–slave scheme is employed for the frequency following communication in which one inverter module k acts as the master unit (MU) while other modules follow the switching frequency of the MU as slave units (SUs). A square wave signal is generated by the MU, driving the inverter module k directly, which is also transmitted to other SUs through physical signal wires and corresponding transceivers. Therefore, all the inverter modules operate at the identical frequency by the application of MSFF.

For the MU (i.e., inverter module k), ${\varphi }_{pk}$ is obtained by the phase detection circuit of output voltage $v_{pk}$ and current $i_{pk}$ [24]. Based on the preset angle ${\varphi }_{ref}$ and detection angle ${\varphi }_{pk}$, the operating frequency of the MU can be calculated via the centralized PI controller. Hence, the ZVS state of the MU can be implemented by adjusting the operating frequency of the $k-\mathrm{th}$ PMMI.

Additionally, according to (5) and (6), all PMMIs have the same inverter output phases because of the identical operating frequency and hardware configuration, which is regulated by adjusting the operating frequency of the MU. In consequence, a centralized-ZVS PI controller can fully realize ZVS control for all PMMIs. It should be mentioned that the master unit can be any inverter module from 1 to n.

3. Dynamic Modeling and Controller Design

3.1. The Dynamic Differential Equations

For a half-bridge inverter with four phases under a duty cycle of 0.5, as shown in Figure 1, the fundamental components of the k-th inverter output voltage by Fourier series expansion are

$$\begin{array}{c}\hfill v_{pk}=\frac{2V_{dc}}{\pi }cos\left({\omega }_{s}t\right).\end{array}$$

(8)

According to KCL and KVL laws, the dynamic equations of the equivalent circuit for the WPT system supplied by n PMMIs depicted in Figure 1 can be expressed by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle v_{pk}-v_p=2i_{pk}{r}_{ICT}+\left(2L_m+2L_{leak}\right)\frac{d{i}_{pk}}{dt}}\hfill \\ {\displaystyle -M_C\frac{d\left(i_{pk-1}+i_{pk+1}\right)}{dt},k=1\sim n}\hfill \\ i_p={\displaystyle \sum _{k=1}^n}{i}_{pk},\left\{i_{pj}=i_{pi},j\ne i\right\}\hfill \\ {\displaystyle v_p=i_p{r}_p+v_{Cp}+L_p\frac{d{i}_p}{dt}-M\frac{d{i}_s}{dt}}\hfill \\ {\displaystyle i_p=C_p\frac{d{v}_{Cp}}{dt}}\hfill \\ {\displaystyle M\frac{d{i}_p}{dt}=L_s\frac{d{i}_s}{dt}+v_{Cs}+i_s\left(r_s+R_e\right)}\hfill \\ {\displaystyle i_s=C_s\frac{d{v}_{Cs}}{dt}}\hfill \end{array}\right..\end{array}$$

(9)

By adding the first n formulas of (9) and eliminating $v_p$ and $i_p$, the simplified dynamic equations are obtained as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle v_{pk}=\left(n{r}_p+2r_{ICT}\right)i_{pk}+v_{Cpe}-M\frac{d{i}_s}{dt}}\hfill \\ {\displaystyle +\left(n{L}_p+2L_{leak}\right)\frac{d{i}_{pk}}{dt}}\hfill \\ {\displaystyle i_{pk}=C_{pe}\frac{d{v}_{Cpe}}{dt}}\hfill \\ {\displaystyle M\frac{d{i}_{pk}}{dt}=\frac{L_s}{n}\frac{d{i}_s}{dt}+v_{Cse}+i_s\frac{\left(r_s+R_e\right)}{n}}\hfill \\ {\displaystyle i_s=C_{se}\frac{d{v}_{Cse}}{dt}}\hfill \end{array}\right.,\end{array}$$

(10)

where $C_{pe}=C_p/n$, $C_{se}=n{C}_s$ are the equivalent capacitances, respectively. Hence, the equivalent circuit for the k-th inverter module with input voltage $v_{pk}$ and current $i_{pk}$ can be depicted in Figure 5.

To solve the problems of time-variance and the rapid changing of the state model expressed by (10), the resonant currents and voltages are represented by slowly varying amplitudes and phases of the coupled-mode theory (CMT) [25]. From the energy point of view, the resonant voltages and currents can be expressed as

$$\begin{array}{cc}\hfill & i_{pk}=\sqrt{2/\left(n{L}_p+2L_{leak}\right)}·a_p\left(t\right)cos\left({\omega }_{s}t+{\theta }_p\right),\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill & v_{Cpe}=\sqrt{2n/C_p}·a_p\left(t\right)sin\left({\omega }_{s}t+{\theta }_p\right),\hfill \end{array}$$

(11b)

$$\begin{array}{cc}\hfill & i_s=\sqrt{2n/L_s}·a_s\left(t\right)cos\left({\omega }_{s}t+{\theta }_s\right),\hfill \end{array}$$

(11c)

$$\begin{array}{cc}\hfill & v_{Cse}=\sqrt{2/n{C}_s}·a_s\left(t\right)sin\left({\omega }_{s}t+{\theta }_s\right),\hfill \end{array}$$

(11d)

where $v_{Cpe,s}$ are the resonant voltages of the equivalent capacitors at primary and secondary sides, respectively, ${\theta }_{p,s}$ are the phases of the coupled modes, and $a_{p,s}$ denotes the energy contained in both resonators, which can be replaced as follows [26]:

$$\begin{array}{cc}\hfill & a_p\left(t\right)=\sqrt{\left(n{L}_p+2L_{leak}\right)/2}·I_{mpk}\left(t\right),\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill & a_s\left(t\right)=\sqrt{L_s/2n}·I_{ms}\left(t\right),\hfill \end{array}$$

(12b)

where $I_{mpk}\left(t\right)$ and $I_{ms}\left(t\right)$ are the amplitude of $i_{pk}$ and $i_s$. It should be noted that the coefficients of resonant voltages and currents are different from that of a single inverter WPT system.

$$\begin{array}{c}\hfill A=\left[\begin{array}{cccc}{\displaystyle \frac{-K_4}{2\pi D_1}}& {\displaystyle \frac{{\Delta }_1{x}_{30}-2K_1{\Phi }_{40}}{2\pi D_1}}& {\displaystyle \frac{{\Delta }_2}{2\pi D_1}}& {\displaystyle \frac{-{\Delta }_1}{2\pi D_1}}\\ {\displaystyle \frac{{\Delta }_1{x}_{30}-2K_2{\Phi }_{40}}{-2\pi D_1{x}_{10}^2}}& {\displaystyle \frac{{\Delta }_2{x}_{30}-2K_2{\Phi }_{30}}{-2\pi D_1{x}_{10}}}& {\displaystyle \frac{{\Delta }_1}{2\pi D_1{x}_{10}}}& {\displaystyle \frac{{\Delta }_2{x}_{30}}{2\pi D_1{x}_{10}}}\\ {\displaystyle \frac{{\Delta }_3}{-2\pi D_2}}& {\displaystyle \frac{{\Delta }_4{x}_{10}}{-2\pi D_2}}& {\displaystyle \frac{A_3{K}_8}{-2D_2}}& {\displaystyle \frac{-{\Delta }_4{x}_{10}+K_5{\Phi }_{60}}{-2\pi D_2}}\\ {\displaystyle \frac{{\Delta }_4}{2\pi D_2{x}_{30}}}& {\displaystyle \frac{{\Delta }_3{x}_{10}}{-2\pi D_2{x}_{30}}}& {\displaystyle \frac{{\Delta }_4{x}_{10}+K_5{\Phi }_{60}}{-2\pi D_2{x}_{30}^2}}& {\displaystyle \frac{{\Delta }_3{x}_{10}-K_5{\Phi }_{50}}{2\pi D_2{x}_{30}}}\end{array}\right].\end{array}$$

(13)

Substituting (11a)–(11d) into (10) and taking the average values of both sides during a switching period yields the non-linear time-invariant averaged model, which can be expressed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=f\left(x\right)+g\left(x\right)u\hfill \\ y=h\left(x\right)\hfill \end{array}\right.,\end{array}$$

(14)

where the state variable x is expressed by $x=\left[a_p,{\theta }_p,a_s,{\theta }_s\right]$. The non-linear functions $f\left(x\right)$, $g\left(x\right)$ and the output ZVS angle $h\left(x\right)$ can be expressed by

$$\begin{array}{cc}\hfill & f\left(x\right)=\hfill \\ \hfill & \left(\begin{array}{c}{\displaystyle \frac{K_{1}cos{x}_2}{\pi D_1}+\frac{\left(-K_2{x}_3{\Phi }_1+K_3{x}_3{\Phi }_2-K_4{x}_1\right)}{2\pi D_1}}\\ {\displaystyle \frac{-n{A}_1}{2A_2{C}_p}+\frac{A_2{L}_s}{2D_1}+\frac{\left(K_3{x}_3{\Phi }_1+K_2{x}_3{\Phi }_2-2K_{1}sin{x}_4\right)}{2\pi D_1{x}_1}}\\ {\displaystyle \frac{1}{2\pi D_2}\left(-\pi K_8{A}_3{x}_3+K_{5}cos{x}_4-K_6{x}_1{\Phi }_1-K_7{x}_1{\Phi }_2\right)}\\ {\displaystyle K_{10}+\frac{n{K}_9}{2D_2}-\frac{\left(K_{5}sin{x}_4-K_7{x}_1{\Phi }_1+K_6{x}_1{\Phi }_2\right)}{2\pi D_2{x}_3}}\end{array}\right),\hfill \end{array}$$

(15)

$$\begin{array}{c}\hfill g\left(x\right)={\left(\begin{array}{cccc}0& -2\pi & 0& -2\pi \end{array}\right)}^T,\end{array}$$

(16)

$$h\left(x\right)=\left(-180/\pi \right)x_2,$$

(17)

where the variables $A_1$–$A_4$, $D_1$–$D_2$, ${\Phi }_1$–${\Phi }_2$, and $K_1$–$K_{10}$ are listed in the Appendix A.

3.2. Small-Signal Modeling and Linearization

The non-linear time-invariant averaged model contains both a steady-state operating point and a small-signal model, which is linear. Hence, by linearizing around the operating point, the state variable x in (14) and (15) can be expressed by the sum of the steady-state value $x_0$ and small-signal $\widehat{x}$ as follows:

$$x=x_0+\widehat{x},$$

(18)

where $x_0=\left[a_{p0},{\theta }_{p0,}{a}_{s0},{\theta }_{s0}\right]$, which can be acquired by setting the derivative terms depicted in (14) and (15) to zero, and $\widehat{x}={\left[{\widehat{a}}_p,{\widehat{\theta }}_p,{\widehat{a}}_s,{\widehat{\theta }}_s\right]}^T$ is the linearized small-signal model. Hence, the linearized state-space model can be expressed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d\widehat{x}}{dt}=A\widehat{x}+B\widehat{u}}\hfill \\ \widehat{y}=C\widehat{x}\hfill \end{array}\right.,\end{array}$$

(19)

where $\widehat{u}=f_s$ and $\widehat{y}$ are the input and output vector of the model; $f_s$ is the operating frequency; A, B, and C are the partial differential matrices of (15)–(17); and A is given by (13) as shown at the bottom of this page, where ${\Phi }_{10}=cos\left({\theta }_{p0}-{\theta }_{s0}\right),{\Phi }_{20}=sin\left({\theta }_{p0}-{\theta }_{s0}\right),{\Delta }_1=K_2{\Phi }_{20}+K_3{\Phi }_{10},{\Delta }_2=K_3{\Phi }_{20}-K_2{\Phi }_{10},{\Delta }_3=K_7{\Phi }_{20}+K_6{\Phi }_{10},{\Delta }_4=K_7{\Phi }_{10}-K_6{\Phi }_{20}$.

In this paper, a WPT system supplied by three PMMIs, namely, $n=3$, is selected as an example. Considering the analysis of the efficiency and total power loss [27], the preset ZVS angle is selected to be ${20}^{\circ }$, which yields the operating frequency $f_s$ = 82.94 $\mathrm{kHz}$. With the system parameters listed in Table 1, one can obtain the operating point as

$$x_0={\left[\begin{array}{cccc}0.1175& -0.3491& 0.0178& 0.0054\end{array}\right]}^T.$$

(20)

By substituting $x_0$ and Table 1 into (19) and (13), the partial differential matrices $A_0$, $B_0$, and $C_0$ for the linear state-space model are acquired as

$$\begin{array}{cc}\hfill & A_0=\left[\begin{array}{cccc}-982.9& 1.22\times {10}^3& -4.78\times {10}^4& -821.2\\ -8.86\times {10}^4& -2.15\times {10}^3& 3.93\times {10}^5& -7.24\times {10}^3\\ 2.02\times {10}^4& -6.48\times {10}^3& -1.32\times {10}^5& 6.48\times {10}^3\\ 3.10\times {10}^6& 1.33\times {10}^5& -2.05\times {10}^7& -1.48\times {10}^5\end{array}\right],\hfill \end{array}$$

(21a)

$$\begin{array}{cc}\hfill & B_0={\left[\begin{array}{cccc}0& -6.28& 0& -6.28\end{array}\right]}^T,\hfill \end{array}$$

(21b)

$$\begin{array}{cc}\hfill & C_0=\left[\begin{array}{cccc}0& -57.3& 0& 0\end{array}\right].\hfill \end{array}$$

(21c)

Then, the open-loop transfer function of the system where the input is the operating frequency $f_s$ and the output is the ZVS angle ${\varphi }_{ZVS}$ can be expressed by

$$\begin{array}{cc}\hfill G\left(s\right)& =\frac{Y\left(s\right)}{U\left(s\right)}\hfill \\ \hfill & =\frac{360s^3+9.84\times {10}^7{s}^2+5.67\times {10}^{13}s+4.59\times {10}^{17}}{s^4+2.83\times {10}^5{s}^3+1.60\times {10}^{11}{s}^2+2.78\times {10}^{15}s+1.38\times {10}^{19}}.\hfill \end{array}$$

(22)

4. Controller Design and Simulation Verification

4.1. Controller Design

The unity feedback root locus and closed-loop zero-pole diagram of the ZVS angle transfer function (22) are plotted in Figure 6a,b. By analyzing the root locus and zero-pole distribution, the following conclusions can be drawn:

(1)

The influence of two dipoles can be neglected when analyzing the dynamic performance of the system, which mainly relies on the closed-loop dominant poles (CLDPs) that are close to the imaginary axis compared to other closed-loop poles [28].

(2)

A pole at the origin and a zero on the negative real axis will be added to the root locus due to the proportional and integral coefficients of the added PI controller, which means that a new CLDP will be added on the negative real axis. Hence, the required response dynamic performance of the ZVS angle control can be achieved by designing PI controller parameters to configure the new CLDP.

(3)

The dynamic performance with the setting time in tens of milliseconds is completely determined by the new CLDP, because in comparison, the original closed-loop poles of (22) are far away from the imaginary axis. Therefore, the dynamic performance of the transfer function with the PI controller is almost identical to that of the first-order system, meaning that they both have the same closed-loop characteristic root (CLCR).

The designed PI controller can be expressed as

$$\begin{array}{c}\hfill G_c\left(s\right)=K_p+\frac{K_i}{s},\end{array}$$

(23)

where $K_p$ and $K_i$ are the proportional and integral coefficients, respectively. The closed-loop characteristic equation after adding the PI controller can be expressed by

$$\begin{array}{cc}\hfill & 1+\overline{G}\left(s\right)=0,\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill & 1+K_p{G}_{equ}\left(s\right)=0,\hfill \end{array}$$

(24b)

where $\overline{G}\left(s\right)=G_c\left(s\right)G\left(s\right)$ is the equivalent open-loop transfer function (EOLTF) and $G_{equ}\left(s\right)$ is the EOLTF with $K_p$ as coefficient. The closed-loop transfer function of the first-order model can be described by

$$\begin{array}{c}\hfill \Phi \left(s\right)=\frac{1}{T_{s}s+1},\end{array}$$

(25)

where $T_s$ is the time constant, and the setting time can be obtained as $t_s=4.4T_s$ with an error band of $\mathsf{\Delta }=2\%$. Then, the CLCR can be obtained as

$$\begin{array}{c}\hfill s=a=-\frac{1}{T_s}=-\frac{4.4}{t_s}.\end{array}$$

(26)

Substituting (26) into (24a) yields the relationship between $K_i$ and $K_p$

$$\begin{array}{c}\hfill K_i=\lambda -a{K}_p,\lambda =\frac{-a}{G\left(a\right)}.\end{array}$$

(27)

Substituting (27) into (24b), we can get

$$\begin{array}{c}\hfill G_{equ}\left(s\right)=\frac{\left(s-a\right)Y\left(s\right)}{\lambda U\left(s\right)+sU\left(s\right)}.\end{array}$$

(28)

The unit step response performance of the closed-loop transfer with the PI controller is depicted in Figure 7. One can see that the settling time of the curves gradually decreases with proportional and integration coefficient increasing, indicating that the dynamic performance of the controller can be further improved. An optimal controller coefficient (OCC) by configuring the coefficient of $K_p$ and $K_i$ is proposed to obtain the linear relationship with the settling time, which can be described by (29), based on (26)–(28).

$$OCC=\frac{4.4\left(1+K_{p}G\left(a\right)\right)}{K_{i}G\left(a\right)}\left|{}_{a=\frac{-4.4}{t_s},K_i=\lambda -a{K}_p}\right.$$

(29)

The relationship between settling time $t_s$ and OCC is depicted in Figure 8a based on (29). One can see that OCC increases strictly monotonically as $t_s$ increases and has nothing to do with the specific values of $K_p$ and $K_i$. Hence, considering the time needed for computing and transmitting data of the experimental prototype [12], with the required setting time $t_s$ = 10 ms (namely, a = −440) and $K_p$ = 0.50 and based on OCC, the parameters of the PI controller can be obtained as

$$\begin{array}{c}\hfill K_p=0.50,K_i=9.0\times {10}^3.\end{array}$$

(30)

The Bode diagrams of the EOLTF $\overline{G}\left(s\right)$ and the first-order model, which both have the same setting time of 10 ms, are depicted in Figure 8b. It can be seen that both models have almost the same frequency characteristics at all frequency bands, which verifies the effectiveness of the PI controller design method based on OCC.

4.2. Simulation Verification

Similarly, to verify the OCC configuration method for different settling times, the parameters of the designed PI controller for different $t_s$ are calculated and listed in

Table 2. Parameters For PI Controller.

Case	Setting Time	Parameter of ${\mathit{K}}_{\mathit{p}}$	Parameter of ${\mathit{K}}_{\mathit{i}}$
Case (I)	$t_s$ = 10 ms	0.50	9.00 × 10${}^3$
Case (II)	$t_s$ = 15 ms	0.50	6.04 × 10${}^3$
Case (III)	$t_s$ = 20 ms	0.50	4.55 × 10${}^3$

. Figure 9 depicts the step response curves of versions controlled within different settling times and the uncontrolled WPT system. It can be seen that the curves in all three cases can reach stability within the required settling times, which verifies the validity of PI parameter design based on the OCC configuration. It should be noted that for different required setting times, the designed parameters of $K_p$ and $K_i$ are not fixed, but they need to meet the proportional allocation of the OCC configuration expressed by (29).

5. Experimental Results

5.1. WPT Experimental Prototype Supplied by PMMIs

A series–series compensated WPT prototype supplied by three PMMIs is built to verify the proposed modeling method and controller performance shown in Figure 10. This prototype is composed of DC power modules, three PMMIs, the coupling ICTs, a resonant wireless bank consisting of two inductor coils and two capacitors, a class-D full-bridge rectifier, and three switchable power resistors.

Three adjustable DC voltage source modules INFY POWER REG75030 were connected in parallel to provide power of up to 45 kW for the inverters. The dimensions of the two coaxial coils were the same, whose rectangular outer ring was 90 cm × 70 cm. The gap between the primary and secondary coil was 20 cm. Each coil was built with four turns of Litz-wire. The diameter of the Litz-wire consisting of 2000 isolated strands was 6 mm, and the diameter of each strand was 0.1 mm. The resonant capacitor at both sides was composed of 100 film capacitors (1 nF each), connected in parallel. Four VS-UFB280FA40 diodes were used to build the full-bridge rectifier. For the load, different resistance values are obtained by connecting several 10 $\mathsf{\Omega }$ resistors connected in parallel or series with air switches. The main parameters are listed in

Table 3. Main Parameters of the WPT Prototype.

Symbol	Parameter	Value
$V_{dc}$	DC power source output voltage	200 V
M	Mutual inductance between two coils	7.89 $\mathsf{\mu }$H
$L_p$	Inductance at the primary side	35.92 $\mathsf{\mu }$H
$C_p$	Capacitance at the primary side	100.2 nF
$r_p$	Parasitic resistance at the primary side	31 $\mathrm{m}\mathsf{\Omega }$
$L_s$	Inductance at the secondary side	35.30 $\mathsf{\mu }$H
$C_s$	Capacitance at the secondary side	101.1 nF
$r_s$	Parasitic resistance at the secondary side	32 $\mathrm{m}\mathsf{\Omega }$
$L_m$	Inductance of the ICTs	13.8∼14.3 $\mathsf{\mu }$H
$L_{leak}$	Leakage inductance between two ICTs	1.95∼2.03 $\mathsf{\mu }$H
$r_{ICT}$	Parasitic resistance of ICTS	29.3∼30.5 $\mathrm{m}\mathsf{\Omega }$

, where resistances, inductances, and capacitances were measured with the Agilent E4980A LCR meter at 85 kHz. The detailed phase detection circuit has been published in [24] and is not repeated here.

The PMMIs are composed of one master inverter module and two slave inverter modules, as depicted in Figure 10b. The MOSFETs used in each inverter are IPW65R041CFD. The controller is implemented by two MCUs, namely, an ARM of STM32F407VGT and an FPGA of XC6SLX9-3TQG1441, which generates the square wave driver signal in 199.5 MHz. The driver signal generated by FPGA in the master inverter is transmitted to the slave inverters via MAX485 transceivers to realize the MSFF strategy. All the inverter modules are connected in parallel with three ICTs. Each of the ICTs is with two toroidal cores T300-2, and the Litz-wire of 1000 × 0.1 mm${}^2$ consists of 24 winding turns.

5.2. Experiments of Steady State

In the experiment, the ZVS angle ${\varphi }_{ZVS}$ is preset as 20${}^{\circ }$, and the load resistor R is adjusted to verify the robust ability of the proposed controller. To validate the MSFF strategy, the master inverter output voltage $v_{p1}$ and current $i_{p1}$ together with one slave inverter output voltage $v_{p2}$ and current $i_{p2}$ were captured by a four-channel oscilloscope, Tektronix DPO 2004 B, and shown in Figure 11a under a common input DC voltage 200 V at R = 10 $\mathsf{\Omega }$. One can see that both PMMIs have the same ZVS angle ${\varphi }_{ZVS}=20.{14}^{\circ }$ and identical operating frequency $f_s$ = 84.04 kHz, which verifies the rationality of the proposed MSFF strategy. Hence, in the following experiments, the output voltage and current of the master inverter are captured to represent the output voltage and current of all three inverters.

The steady-state operation waveforms of the master inverter output voltage $v_{p1}$ and current $i_{p1}$ together with charging voltage $V_0$ and current $I_0$ on the load under a preset ZVS angle ${\varphi }_{ZVS}={20}^{\circ }$ and load resistance of 10, 15, and 20 $\mathsf{\Omega }$, respectively, are shown in Figure 11b,c. It can be seen from Figure 11b that the operating frequency $f_s$ is 83.95 kHz when the load resistance is 10 $\mathsf{\Omega }$, and the reference ZVS angle is $19.{51}^{\circ }$. When the load resistance increases to 15 $\mathsf{\Omega }$ and 20 $\mathsf{\Omega }$, as shown in Figure 11c,d, $f_s$ is decreased to 83.41 kHz and 83.12 kHz to get the preset ZVS angle, respectively.

5.3. Experiments Under Disturbances

To verify the effectiveness of the proposed controller design method, experiments with sharp load resistance changes under the designed parameters of the PI controller in cases (I)–(III) were conducted, respectively. The dynamic response waves for different experimental situations are shown in Figure 12 and Figure 13. Figure 12a–f shows the regulating process of the master inverter output voltage $v_{p1}$ and current $i_{p1}$ with a reference ZVS angle of ${20}^{\circ }$ and designed PI controller under load disturbances, i.e., from 10 to 15 and from 15 to 20 $\mathsf{\Omega }$, respectively. One can see that the settling times for case (I) are within 10 ms, those for case (II) are within 16 ms, and those for case (III) are within 22 ms, respectively, which are close to the required settling times shown in Table 2 and the simulated unit step response time depicted in Figure 9.

The regulating process of the ZVS angle ${\varphi }_{ZVS}$ between the main inverter output voltage and current under the designed PI controller parameters (case (I)$--$case (III)) are plotted in Figure 13, where (a) is for the load change of 10 to 15 $\mathsf{\Omega }$ and (b) is for 15 to 20 $\mathsf{\Omega }$. Although the ZVS angle rises rapidly when the load changes, it gradually stabilized at the reference angle of ${20}^{\circ }$, and the settling times are mainly consistent with the designed settling times in all three cases. The output phase data are uploaded from the ARM to the private computer via RS232 communication. Moreover, the system’s maximum transmission of 94.1% was observed under a 200 V input DC voltage of the inverters at a load resistance of R = 10 $\mathsf{\Omega }$. In this case, the measured voltage at the load resistance was 202.4 V, and the input DC current of the inverters was 21.79 A.

6. Conclusions

A centralized-ZVS control based on the MSFF strategy is presented for wireless charging EVs driven by the PMMI topology. Only one controller on the main inverter can realize ZVS for all inverter modules, which simplifies the complexity of the control system. A modeling method directly related to the ZVS angle and operating frequency for an arbitrary number of PMMIs is proposed to facilitate the analysis of dynamic performance. The non-linear time-invariant averaged model is linearized to a fourth-order linear model by adopting the small signal method. Based on the fourth-order model, an OCC configuration method is proposed for the parameter design of the PI controller, which is especially useful when the system is required to be stable within the desired settling time. A WPT prototype powered by three modular inverters is built and constructed with the designed PI controllers. Experiments show that by applying the MSFF strategy, all inverters have the same preset ZVS angle, and the experimental settling times are consistent with the desired ones under the designed controller parameters, which proves the effectiveness and accuracy of the proposed centralized-ZVS controller, modeling, and controller parameter design method.