# A nonlinear state-space model and control algorithm for a dynamic wireless power transfer system electric vehicle charger application

Zariff M Gomes, Hassan Moussa, Yann Le Gall, Edemar O Prado, Gilney

Damm, José Renes Pinheiro, Christophe Ripoll

► To cite this version:

Zariff M Gomes, Hassan Moussa, Yann Le Gall, Edemar O Prado, Gilney Damm, et al.. A nonlinear state-space model and control algorithm for a dynamic wireless power transfer system electric vehicle charger application. Control Engineering Practice, 2025, 158, pp.106270. 10.1016/j.conengprac.2025.106270. hal-04991883

HAL Id: hal-04991883

https://univ-eiffel.hal.science/hal-04991883v1

Submitted on 14 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# A nonlinear state-space model and control algorithm for a dynamic wireless power transfer system electric vehicle charger application ☆

Zariff M. Gomes<sup>a,c</sup>, Hassan Moussa<sup>a</sup>, Yann Le Gall<sup>a</sup>, Edemar O. Prado<sup>a,c,d ID</sup>, Gilney Damm<sup>a,b ID,*</sup>, José Renes Pinheiro<sup>c,d,e ID</sup>, Christophe Ripoll<sup>f</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

- Control design

- Coupling factor

- Dynamic wireless charging

- Electric vehicle battery charger

- Inductive power transfer

- System modeling

## ABSTRACT

This paper introduces a nonlinear state-space model for a Dynamic Wireless Power Transfer System (DWPT). DWPT enables the continuous delivery of power to an electric vehicle while it moves along the road, allowing long-haul vehicles to operate with a relatively small battery. The system comprises primary side coils, which are supplied by DC/AC converters. These coils transmit power through induction to an electric vehicle. Within the vehicle (secondary side), a power receiver coil is connected to a full bridge diode rectifier, followed by the standard electric vehicle system. In the second step, the paper presents the design of a control system that ensures the seamless transfer of power to the vehicle as it moves along the road. The chosen controller is an Extremum Seeking algorithm, capable of implementation in high-speed systems compatible with real-time control for fast-moving vehicles. The following step involves conducting test-bed experiments using the designed controller to verify the alignment of the proposed mathematical model with the real system. Furthermore, the controller achieves maximum power transfer and maintains power and voltage across the load during transient states. The proposed model effectively captures the relevant dynamics and is well-suited for control design. Additionally, it offers faster simulation times compared to the full electrical model. Due to its good fit with experimental results, the model can be utilized for sizing, design, and tuning of both the system and control algorithms. This is especially valuable before conducting more detailed simulations on the electrical model and, eventually, test-bed experiments and real-life deployment.

## 1. Introduction

Electric Vehicles (EVs) have become a reliable alternative for transportation (Caillierez, Sadarnac, Jaafari, & Loudot, 2014) for their several advantages as mitigating environmental issues such as carbon emissions, their remarkable efficiency, and being less noisy vehicles (Khaligh & Dusmez, 2012; Yilmaz & Krein, 2012). As presented by Khaligh and D'Antonio (2019) and Reimers, Dorn-Gomba, Mak, and Emadi (2019), countries like the USA, France, Canada, China, India, and others in the European Union have already established government incentive policies for the development of EVs. These EVs have battery packs that can be charged by onboard chargers (Bin, Chien-Yu, Baifeng,

Dominic, & Jih-Sheng, 2013; de Melo, Trovao, Pereirinha, Jorge, & Antunes, 2018; Deng, Li, Hu, Mi, & Ma, 2014) directly connected to the grid by wired technologies.

Recently, the development of wireless power transfer systems, and even for charging EVs while moving along the road (Dynamic Wireless Power Transfer - DWPT) has witnessed a gradual yet significant increase, particularly in the last decade, as addressed in Lee and Han (2015), Pang, Deng, Liu, and Wang (2017), Tran, Vu, and Choi (2018), Wang, Pratik, Jovicic, Hasan, and Pantic (2021), and Kadem, Meira, Damm, and Moussa (2022). The integration of wireless charging systems into vehicles potentially reduces the number of batteries required

<footnote><sup>a</sup> Institut VEDECOM, 23 bis Allée des Marronniers, Versailles, 78000, France</footnote>

<footnote><sup>b</sup> COSYS-IMSE, IFSTTAR, University Gustave Eiffel, Marne-la-Vallée, 77447, France</footnote>

<footnote><sup>c</sup> Energy Efficiency Lab, LABEFEA, Federal University of Bahia, Salvador, 40170-110, BA, Brazil</footnote>

<footnote><sup>d</sup> Power Electronics and Control Research Group, GEPOC, Federal University of Santa Maria, Santa Maria, 97105-900, RS, Brazil</footnote>

<footnote><sup>e</sup> University of Vale do Itajá, Univali, Itajá, 88302-901, SC, Brazil</footnote>

<footnote><sup>f</sup> Renault SAS, 1 avenue du golf, Guyancourt, 78288, France</footnote>

<footnote>☆ This work has received funding from the European Union's Horizon 2020 research and innovation program under Grant Agreement No 875683 (Incit EV project), and by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001.</footnote>

<footnote>* Corresponding author at: COSYS-IMSE, IFSTTAR, University Gustave Eiffel, Marne-la-Vallée, 77447, France.</footnote>

<footnote>E-mail addresses: zariff_meira@hotmail.com (Z.M. Gomes), yann.le-gall@vedecom.fr (Y.L. Gall), edemar.prado@ufba.br (E.O. Prado), gilney.damm@univ-eiffel.fr (G. Damm), jrenes@gepoc.ufsm.br (J.R. Pinheiro), christophe.ripoll@renault.com (C. Ripoll).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Typical WPT charging system for EVs.</figcaption></figure>

and allows for long-distance travel. These factors significantly reduce the need for stationary charging on the road, reducing dependence on the battery (Budhia, Covic, & Boys, 2011; Covic, Budhia, Huang, et al., 2013).

Overall, as presented in Meira, Damm, Kadem, and Moussa (2022), the DWPT system is based on resonant magnetic coupling between the primary coils on the ground and the secondary coil on the vehicle (inductive coupling). Fig. 1 illustrates the typical structure of a DWPT system (Serge & Antoine, 2017).

The energy from the electrical grid, initially in the form of alternating current (AC), is first converted to direct current (DC) through an AC/DC rectifier. Subsequently, the obtained DC energy is converted into high-frequency AC energy to power the transmitting coil (transmitter circuit). The compensation network reduces reactive power by operating at the resonance frequency (Kabbara et al., 2022). The high-frequency current flowing through the primary coil generates an alternating magnetic field, which in turn, induces AC current in the receiving coil. The transferred AC energy is then rectified and, through a DC/DC converter, charges the EV's battery.

To control and optimize the operation of such converters, the modeling and dynamic analysis of the overall system becomes imperative. Modeling a complete system that includes both the inverter and rectifier is inherently complex and time-consuming when using conventional simulation tools. To address this, Hu et al. (2019) proposes a discrete-time modeling method that effectively simulates a three-level inverter. Furthermore, Zhang, Wong, Tse, and Chen (2014) introduces a simplified frequency model for a single-stage (SS) circuit, which, by concentrating on the fundamental frequency, accurately represents the behavior of resonant systems. For the widely used Pulse Width Modulated (PWM) converters, a prevalent modeling technique is the state-space averaging approach. This method involves a change of coordinates converting sinusoidal AC state variables to DC ones with slow dynamics, by implementing a synchronous dq reference frame transformation (Aboushady, Ahmed, Finney, & Williams, 2013; Luk, Aldhaher, Fei, & Whidborne, 2015; Malan, Vilathgamuwa, & Walker, 2016). This approach is well-known for its efficacy in modeling resonant converters under large signal variations, and their controller design (Zahid et al., 2015). However, it is used in literature to model wireless power transfer systems under static conditions, lacking analysis under dynamic conditions.

The control strategies for dynamic wireless charging systems, as outlined in Bagchi, Kamineni, Zane, and Carlson (2021) and Behnamfar, Olowu, Tariq, and Sarwat (2024), can be broadly categorized into two primary objectives: segmentation control, and power flow control. A third objective can be introduced by further delineating power flow control into two components: power flow control, and resonance control.

The segmentation control strategy is designed to identify the vehicle's position and activate the appropriate coil or segment to optimize energy transfer. The power flow control strategy focuses on delivering the required power to the secondary side, which can be implemented in

either an open-loop or closed-loop configuration, on the primary or secondary side. Resonance control aims to maintain resonance conditions, such as Zero Voltage Switching (ZVS) and/or Zero Current Switching (ZCS), in the converter switches to enhance system efficiency.

In Wang et al. (2017), a power control method is presented that adjusts the duty cycle (phase-shift) of the output voltage, with secondary-side parameters mapped to predict the optimal phase-shift values, thereby maintaining consistent output power. Babaki, Vaez-Zadeh, Zakerian, and Covic (2021) introduces a control strategy based on a communication system between the primary and secondary sides, aimed at tracking the optimal operating frequency by comparing input and output power, thereby maximizing efficiency. In contrast, Zhang et al. (2014) describes a system design with a fixed frequency on the primary side, focused on stabilizing the output voltage under varying load conditions. Meanwhile, Wang, Stielau, and Covic (2005) proposes a design and control method that adjusts the inverter frequency according to analytical equations to accommodate varying loads, though this study does not report measurements related to phase, power, or efficiency.

This paper aims to apply the state-space averaging method to the equivalent circuit of a DWPT system, developing a mathematical model that represents the system under dynamic conditions. The model is further used to design a phase shift controller for the resonant converters. This controller autonomously tracks the resonance frequency to achieve maximum power delivery while ensuring soft switching of the power electronic components. A straightforward segmentation control strategy is implemented that does not rely on communication or require the installation of additional sensors in the system.

The remaining paper is first focused on simulating in Matlab/Simulink the dynamic charging, by emulating the car movement by changing the coupling coefficient between secondary and primary coils. In the following, the real charging system from the simulated model and its controller are experimented on a test bed. The experimental and simulated results are then compared, to show a good match between the mathematical model and the real circuits. Either simulations or experiments use the same designed control algorithm, illustrating its good performance and robustness.

## 2. Dynamic wireless power transfer system

### 2.1. General description

The DWPT explored in this study is shown in Fig. 1, where several DC/AC converters and their coils on the floor represent the primary system, while the embedded coil with its AC/DC converter represents the secondary. The system is free of communication between the primary and secondary circuits, intending to minimize latency for control purposes while the vehicle is in motion. Due to the large air gap between these primary and secondary coils, the transformer's leakage inductance is large. To compensate for this large inductance, capacitors are added to both sides of the transformer. A series-series compensation is mostly used since its design is independent of the mutual inductance and load. The system endures power variation when the coupling factors, system parameters, and load change (Caillierez, 2016; Gori, 2019). To address this issue, the model of the DWPT system operating at different parameters is here considered.

### 2.2. Studied DWPT system

The circuit diagram representing the series-series DWPT system is shown in Fig. 2.

The full-bridge converter on the primary side converts the DC voltage ($V_{DC}$) into a full square wave voltage at a switching frequency ($\omega_s$) equal to the resonant frequency ($\omega_r$) of the LC network. The converter at the secondary side is a rectifier that converts the secondary AC voltage ($v_s$) to an output DC voltage ($V_0$) after adding an output filter capacitor ($C_f$) followed by a resistive load ($R_{ch}$). For simplicity
<figure><img src="image_4.png" /><figcaption>Fig. 2. Equivalent circuit of DWPT system.</figcaption></figure>

and due to the repeating sequence, the DWPT system is considered to have two primary coils as the transition from one to another is similar during the vehicle movement on the charging track. The studied system is designed for a rated power of 30 kW. Without loss of generality, the primary and secondary coils are assumed to be identical so that their self-inductance is the same ($L_p = L_s = L$). To simplify the model, the analysis is carried out at the fundamental frequency of the resonant network ($\omega_r$), and the output voltage of the inverters 1 and 2 ($v_1$ and $v_2$ in Fig. 2, respectively) is considered as a perfect sine wave, which are represented as:

$$v_1 = \frac{4V_{DC}}{\pi} \sin(\omega_s t)$$

$$v_2 = \frac{4V_{DC}}{\pi} \sin(\omega_s t)$$

### 3. Average state-space model

In this section, a nonlinear state-space model is derived at the fundamental resonant frequency, then all the periodic variables are represented by Fourier series terms to derive the dq model. These dq terms represent the magnitudes of the sine and cosine of each electric variable.

#### 3.1. Nonlinear state space model

Applying Kirchhoff Voltage Law (KVL) to the circuit of Fig. 2 and deriving the induced voltage due to the mutual impedance, one obtains:

$$v_1 = v_{C1} + r_1 i_1 + L \frac{di_1}{dt} - M_1 \frac{di_s}{dt} \quad (1)$$

$$v_2 = v_{C2} + r_2 i_2 + L \frac{di_2}{dt} - M_2 \frac{di_s}{dt} \\ v_s = -v_{Cs} - r_s i_s - L \frac{di_s}{dt} + M_1 \frac{di_1}{dt} + M_2 \frac{di_2}{dt}$$

where, $i_1$ and $i_2$ represent the output current of converter 1 and 2 respectively, while $i_s$ is the secondary rectifier input current, and $r_1$, $r_2$ and $r_s$ are the primary and secondary resistances. $v_{C1}$, $v_{C2}$, and $v_{Cs}$ are the voltage in series capacitors for converters 1, 2, and secondary side input, respectively. $M_1$ and $M_2$ represent the mutual inductance between each primary and secondary coil, and are represented as:

$$M_1 = k_1 L \quad M_2 = k_2 L \quad (2)$$

where, $k_1$ and $k_2$ are the coupling coefficients.

Eq. (1) are rearranged and solved to get the expressions:

$$\frac{di_1}{dt} = \frac{1}{\xi} \left[ \frac{-r_1}{\sigma_1 L} i_1 - \frac{r_1}{\sigma_1 \sigma_2 L} k_1 i_s - \frac{r_1}{\sigma_1 \sigma_2 L} k_1 k_2 i_2 - \frac{k_1}{\sigma_1 \sigma_2 L} (v_s + v_{Cs}) + \frac{1}{\sigma_1 L} (v_1 - v_{C1}) \right]$$

$$\frac{di_2}{dt} = \frac{k_1 k_2}{\sigma_1 \sigma_2 L} (v_2 - v_{C2}) \quad (3)$$

$$\frac{di_2}{dt} = \frac{1}{\xi} \left[ \frac{-r_2}{\sigma_2 L} i_2 - \frac{r_2}{\sigma_1 \sigma_2 L} k_2 i_s - \frac{r_2}{\sigma_1 \sigma_2 L} k_1 k_2 i_1 - \frac{k_2}{\sigma_1 \sigma_2 L} (v_s + v_{Cs}) + \frac{1}{\sigma_1 L} (v_2 - v_{C2}) + \frac{k_1 k_2}{\sigma_1 \sigma_2 L} (v_1 - v_{C1}) \right]$$

$$\frac{di_s}{dt} = \frac{1}{\xi} \left[ \frac{-r_1}{\sigma_1 \sigma_2 L} k_1 i_1 - \frac{r_2}{\sigma_1 \sigma_2 L} k_2 i_2 - \frac{r_s}{\sigma_1 \sigma_2 L} i_s - \frac{1}{\sigma_1 \sigma_2 L} (v_s + v_{Cs}) + \frac{k_1}{\sigma_1 \sigma_2 L} (v_1 - v_{C1}) + \frac{k_2}{\sigma_1 \sigma_2 L} (v_2 - v_{C2}) \right]$$

given, $\sigma_1 = 1 - k_1^2$, $\sigma_2 = 1 - k_2^2$ and $\xi = 1 - \frac{(k_1 k_2)^2}{\sigma_1 \sigma_2}$.

The remaining state variables, the series capacitor voltages $v_{C1}$, $v_{C2}$, $v_{Cs}$ and the output voltage $V_0$ of the complete system are described by:

$$\begin{aligned} \frac{dv_{C1}}{dt} &= \frac{1}{C_1} i_1 \\ \frac{dv_{C2}}{dt} &= \frac{1}{C_2} i_2 \\ \frac{dv_{Cs}}{dt} &= \frac{1}{C_s} i_s \\ \frac{dV_0}{dt} &= \frac{1}{C_f} [i_{rec} - i_0] \end{aligned} \quad (4)$$

where, $C_f$ is the output capacitor, $i_{rec} = |i_s| = \frac{2}{\pi} i_{s,peak}$ is the rectifier output current, and $i_0 = \frac{V_0}{R_{ch}}$, the load current. Eqs. (3)–(4) represent the exact nonlinear state-space model of the DWPT system represented by Fig. 2.

#### 3.2. Model in dq synchronous frame

All the state variables are periodic and can be represented by a Fourier series. The power transfer mainly occurs at the fundamental frequency of the resonant network and therefore, modeling is done at the fundamental frequency $\omega_s$. Each AC state variable is represented by its own sine and cosine terms described by:

$$x_i = x_{id} \sin(\omega_s t) + x_{iq} \cos(\omega_s t) \quad (5)$$

The amplitudes $x_{id}$ and $x_{iq}$ represent the new system state variables replacing the AC terms of the nonlinear model. Substituting (5) on each AC state variable of the model represented in (3) and (4), a new dq model is obtained as:

$$\frac{di_{1d}}{dt} = \frac{1}{\xi} \left[ \frac{(-r_1)}{(\sigma_1 L)} i_{1d} - \frac{r_1}{(\sigma_1 \sigma_2 L)} k_1 i_{sd} - \frac{r_1}{(\sigma_1 \sigma_2 L)} k_1 k_{2d} - \frac{k_1}{(\sigma_1 \sigma_2 L)} (v_{sd} + v_{Csd}) + \frac{1}{(\sigma_1 L)} (v_{1d} - v_{C1d}) + \frac{(k_1 k_2)}{(\sigma_1 \sigma_2 L)} (v_{2d} - v_{C2d}) \right] + \omega_s i_{1q}$$

$$\frac{di_{1q}}{dt} = \frac{1}{\xi} \left[ \frac{-r_1}{\sigma_1 L} i_{1q} - \frac{r_1}{\sigma_1 \sigma_2 L} k_1 i_{sq} - \frac{r_1}{\sigma_1 \sigma_2 L} k_1 k_{2q} - \frac{k_1}{\sigma_1 \sigma_2 L} (v_{sq} + v_{Csq}) + \frac{1}{\sigma_1 L} (v_{1q} - v_{C1q}) + \frac{k_1 k_2}{\sigma_1 \sigma_2 L} (v_{2q} - v_{C2q}) \right] - \omega_s i_{1d}$$

$$\frac{di_{2d}}{dt} = \frac{1}{\xi} \left[ \frac{-r_2}{\sigma_2 L} i_{2d} - \frac{r_2}{\sigma_1 \sigma_2 L} k_2 i_{sd} - \frac{r_2}{\sigma_1 \sigma_2 L} k_1 k_{2i} - \frac{k_2}{\sigma_1 \sigma_2 L} (v_{sd} + v_{Csd}) + \frac{1}{\sigma_1 L} (v_{2d} - v_{C2d}) + \frac{k_1 k_2}{\sigma_1 \sigma_2 L} (v_{1d} - v_{C1d}) \right] + \omega_s i_{2q}$$
$$ \frac{di_{2q}}{dt} = \frac{1}{\xi} \left[ \frac{-r_2}{\sigma_2 L} i_{2q} - \frac{r_2}{\sigma_1 \sigma_2 L} k_2 i_{sq} - \frac{r_2}{\sigma_1 \sigma_2 L} k_1 k_2 i_{1q} \right. \\ \left. - \frac{k_2}{\sigma_1 \sigma_2 L} (v_{sq} + v_{Csq}) + \frac{1}{\sigma_1 L} (v_{2q} - v_{C2q}) \right. \\ \left. + \frac{k_1 k_2}{\sigma_1 \sigma_2 L} (v_{1q} - v_{C1q}) \right] - \omega_s i_{2d} $$

$$ \frac{di_{sd}}{dt} = \frac{1}{\xi} \left[ \frac{-r_1}{\sigma_1 \sigma_2 L} k_1 i_{1d} - \frac{r_2}{\sigma_1 \sigma_2 L} k_2 i_{2d} - \frac{r_s}{\sigma_1 \sigma_2 L} i_{sd} \right. \\ \left. - \frac{1}{\sigma_1 \sigma_2 L} (v_{sd} + v_{Csd}) + \frac{k_1}{\sigma_1 \sigma_2 L} (v_{1d} - v_{C1d}) \right. \\ \left. + \frac{k_2}{\sigma_1 \sigma_2 L} (v_{2d} - v_{C2d}) \right] + \omega_s i_{sq} $$

$$ \frac{di_{sq}}{dt} = \frac{1}{\xi} \left[ \frac{-r_1}{\sigma_1 \sigma_2 L} k_1 i_{1q} - \frac{r_2}{\sigma_1 \sigma_2 L} k_2 i_{2q} - \frac{r_s}{\sigma_1 \sigma_2 L} i_{sq} \right. \\ \left. - \frac{1}{\sigma_1 \sigma_2 L} (v_{sq} + v_{Csq}) + \frac{k_1}{\sigma_1 \sigma_2 L} (v_{1q} - v_{C1q}) \right. \\ \left. + \frac{k_2}{\sigma_1 \sigma_2 L} (v_{2q} - v_{C2q}) \right] - \omega_s i_{sd} $$

$$ \frac{dv_{C1d}}{dt} = \frac{1}{C_1} i_{1d} + \omega_s v_{C1q} $$

$$ \frac{dv_{C1q}}{dt} = \frac{1}{C_1} i_{1q} - \omega_s v_{C1d} $$

$$ \frac{dv_{C2d}}{dt} = \frac{1}{C_2} i_{2d} + \omega_s v_{C2q} $$

$$ \frac{dv_{C2q}}{dt} = \frac{1}{C_2} i_{2q} - \omega_s v_{C2d} $$

$$ \frac{dv_{Csd}}{dt} = \frac{1}{C_s} i_{sd} + \omega_s v_{Csq} $$

$$ \frac{dv_{Csq}}{dt} = \frac{1}{C_s} i_{sq} - \omega_s v_{Csd} $$

$$ \frac{dV_0}{dt} = \frac{1}{C_f} \left[ \frac{2}{\pi} i_{s,peak} - \frac{V_0}{R_{ch}} \right] \qquad (6) $$

where, $i_{s,peak} = \sqrt{i_{sd}^2 + i_{sq}^2}$.

Eqs. (6) represent the dq model of the nonlinear system of Fig. 2. This model represents the envelope of AC currents and voltages of the resonant network in both steady-state and transient-state conditions. The dq model can be written in the form:

$$ \frac{d}{dt}x = f(x, u) $$

$$ y = g(x, u) $$

where,

$$ x = [i_{1d} \ i_{1q} \ i_{2d} \ i_{2q} \ i_{sd} \ i_{sq} \ v_{C1d} \ v_{C1q} \ v_{C2d} \ v_{C2q} \ v_{Csd} \ v_{Csq} \ V_0]^T $$

$$ u = [\omega_s \ v_1 \ v_2]^T $$

$$ y = V_0 $$

## 4. Model results in simulation, and phase shift control design

The model is built on MATLAB with the parameters shown in Table 1. First, a simulation is performed for one converter supplying a primary coil. This simulation is performed considering the system in static condition and coupling factor $k_1 = 0.14$, $k_1 = 0.2$, and $k_1 = 0.26$, while $k_2 = 0$. The switching frequency ($F_{SW}$) is varied from 75 kHz to 95 kHz. Although the inferior limit for $F_{SW}$ presented in Table 1 is 79 kHz, 75 kHz is used in Fig. 3 for a better understanding and visualization of $F_{SW}$ influence in the energy transferred. Fig. 3 shows the output voltage ($V_O$) versus $F_{SW}$, as well as the phase shift ($\delta$) between voltage and current on the primary side. These steady-state curves give a prediction at which resonant frequency the phase shift control algorithm will settle to obtain a maximum power transfer. It

<figure><img src="image_5.png" /><figcaption>Fig. 3. System output voltage (top) and phase shift (bottom) versus switching frequency.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Flow chart of the phase-shift control algorithm.</figcaption></figure>

<table><caption>Table 1
DWPT system simulation parameters.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Transferred power</td><td>30 kW</td></tr><tr><td>DC input voltage</td><td>430 V</td></tr><tr><td>Coupling coefficient</td><td>from 0.1 to 0.26</td></tr><tr><td>Primary and secondary inductances</td><td>L<sub>p</sub> = L<sub>s</sub> = L = 135 µH</td></tr><tr><td>Primary and secondary resistance</td><td>r<sub>1</sub> = r<sub>2</sub> = r<sub>s</sub> = 0.1 Ω</td></tr><tr><td>Load Resistance</td><td>R<sub>L</sub> = 5.2 Ω</td></tr><tr><td>Output Filter Capacitor</td><td>C<sub>f</sub> = 600 µF</td></tr><tr><td>Primary and secondary resonant capacitor</td><td>33 nF</td></tr><tr><td>Switching frequency</td><td>79 to 90 kHz</td></tr></tbody></table>

can be seen that the maximum output voltage obtained is around 400 V for a resistive load of $R_L=5.2 Ω$.

According to Fig. 3, the phase that maximizes the power transfer is 16.2 degrees for $k_1 = 0.14$, 9.3 degrees for $k_1 = 0.2$, and 7.7 degrees for $k_1 = 0.26$. Based on this, a phase shift control algorithm is implemented on the primary side of the DWPT system, to tune the resonant frequency by tracking the phase shift between the output voltage and current of the converter between 10 and 15 degrees. Fig. 4 shows the program flow chart of the phase shift control algorithm. It can be understood as a modified *Extremum Seeking Control* (see Meira et al. (2022), Moussa, Gomes, Kadem, and Damm (2023) for previous versions).
<figure><img src="image_7.png" /><figcaption>Fig. 5. Phasor diagram of one converter.</figcaption></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Currents and series capacitor voltage. (a) dq output currents of converter 1. (b) dq voltage of series capacitor C1.</figcaption></figure>

First, the frequency of the converter starts at its maximum value $f_0$, higher than the resonant frequency. The measured phase angle is compared to a given interval and the controller will determine the optimal correcting value of the switching frequency of the converter. The program executes until the phase shift is between the limits defined, maximizing the power transferred, and maintaining a Zero Current Switching (ZCS), and Zero Voltage Switching (ZVS) condition of the PWM converter. ZCS and ZVS ensure switching loss reduction in the inverter.

To calculate the phase shift from the dq model, Fig. 5 describes the phasor diagram of one converter with the assumption that the phase angle of the voltage $v_1$ is the reference. The phase shift is given as:

$$ \delta_1 = tan^{-1} \left( \frac{i_{1q}}{i_{1d}} \right) \qquad (7) $$

To assess the performance of the phase-shift controller, a setup simulation is developed for $k_1 = 0.26$ and $k_2 = 0$. In this case, one converter is supplying one primary coil with the secondary coil facing the primary coil with a 15 cm air gap. Fig. 6(a) shows the output dq currents passing through the primary coil. The d-component is different from zero, which indicates a phase shift between the current vector $i_1$ and the voltage vector $v_1$ which is aligned to the d-axis (i.e., $v_{1d} = (4V_{DC})/\pi$ and $v_{1q} = 0$). The peak current can be calculated as $i_{(1,peak)} = \sqrt{(i_{1d}^2 + i_{1q}^2)}=111.4$ A.

Fig. 6(b) shows the dq voltages of the series capacitor used to compensate for the leakage inductance. The values show a peak voltage equal to 5.74 kV across the capacitor $C_1$. A high-voltage film capacitor is used for this application that can support an RMS voltage of 8500 V.

Fig. 7(a) shows the switching frequency, the output power, and the output DC voltage. At steady state, the switching frequency is equal

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 7. Static simulation. (a) Switching frequency. (b) Output power. (c) Output voltage.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 8. Primary and secondary coils for the DWPT system.</figcaption></figure>

to 86.65 kHz which is slightly greater than 86.5 kHz (as presented in Fig. 3), since the closed-loop phase-shift controller tends to hold a positive phase between current and voltage from 10 to 15 degrees. The output power is 28.36 kW, and the output voltage is 384 V, as shown in Fig. 7(b) and (c), respectively.

The dynamic wireless power transfer system is evaluated by simulating the system with a time-varying coupling factor between the primary and secondary coils. This emulates a scenario where a vehicle moves along the charger track. Fig. 8 shows the DWPT system under investigation. The secondary coil (S) is initially aligned with coil N of the primary side. Converter N operates in the Master state to transfer power to the load ($v_1 \neq 0$), while the second converter (N+1) remains with the coil in a short circuit ($v_2 = 0$). The induced current on the N+1 coil is used to track the secondary movement. This is explained in detail in Kamineni, Neath, Zaheer, Covic, and Boys (2016) and Kadem et al. (2019).
<figure><img src="image_14.png" /><figcaption>Fig. 9. Coupling coefficients variation as a function of the secondary coil movement.</figcaption></figure>

As the secondary coil (installed in the car) transitions from one primary coil to another, the magnitude of the currents circulating in coil N+1 increases. In an ideal system, when the current in coil N equals the induced current in coil N+1, a switching command is issued to transition the coils to a new state. However, to accommodate non-ideal scenarios, the switching time is determined by a ratio between the currents. For example, when the current in coil N+1 surpasses 75% of the current in coil N, the switching command is activated, resulting in a change in the switching state. When the coils transition to a new state, converter N+1 assumes the master state ($v_2 \neq 0$), transferring power to the load. Meanwhile, the preceding converter N and the subsequent converter N+2 are set to the OFF state ($v_1 = 0$) by short-circuiting their primary coils.

The coupling coefficient values for two adjacent primary coils are shown in Fig. 9, with the secondary coil moving alongside them at a 15 cm air gap. These values are obtained using the Ansys Maxwell simulator for both the primary and secondary coils. For the system analysis, the coupling coefficient between the primary coils is not included, as it is significantly lower compared to the coupling factor between the primary and secondary coils, and only one primary coil is active at a time. This design choice simplifies the system analysis and keeps the focus on maximizing energy transmission between the primary and secondary coils.

The coil dimensions were defined based on the restrictions presented by the automakers and road construction partners participating in the project, as well as on the application requirements, such as electromagnetic emissions and maximizing the coupling coefficient. A width-to-height ratio of 1:0.6 was chosen to define the coil's dimensions, and a rectangular shape was selected to increase the area and improve the coupling coefficient (Dehui, Fan, Chao, & Fang, 2019).

Fig. 10 illustrates the amplitude of the peak current ($I_{peak}$) flowing through both primary coils with a secondary velocity of 80 km/h. Initially, the $I_{peak}$ in the first converter is maximum while the $I_{peak}$ in the second converter is zero. As previously mentioned, the transition to the master state on the primary side occurs when the current in coil 2 reaches approximately 75% of the current in coil 1. This transition aligns with the secondary coil nearing a centered position between the two primary coils.

When the switching state occurs, the second converter supplies the load and reaches its maximum output peak current when the secondary coil is aligned to it. Fig. 11(a) displays the resonant frequency of the secondary coil, while Fig. 11(b) and (c) show the output power and voltage, respectively, during the transition from one primary coil to another.

Due to the transition between converters, the output power decreases to a value determined by the resonant frequency. It returns to its maximum value when completely shifting to another coil. The

<figure><img src="image_15.png" /><figcaption>Fig. 10. Peak current circulating through the first and second primary coils.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 11. Simulation results. (a) Switching Frequency. (b) Output power. (c) Output voltage.</figcaption></figure>

regulation of the output voltage at the secondary coil can be designed using a buck-boost DC-DC converter to maintain a constant output voltage during battery charging. However, this aspect is not in the scope of this paper.

## 5. Experimental validation

The experiments are performed using a wireless inductive charging systems testing bench composed of a robotic arm, power analyzers, electromagnetic field probes, and oscilloscopes, all synchronized by
<figure><img src="image_17.png" /><figcaption>Fig. 12. Wireless inductive charging systems testing bench.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 13. DSP measurements for current circulating through the first and second primary coils, sampling time 50 ms.</figcaption></figure>

a supervision system, as presented in Fig. 12. The primary coils are placed on the table forming the charging lane. The secondary coil is attached to the robotic arm, which can emulate the movement of a vehicle at a maximum speed of 13 km/h. Each primary coil is equipped with an inverter. Inverters are supplied with a 430 VDC power source, provided by ITECH IT6000D with 7 modules connected in parallel. Fig. 12 also presents a zoom of the inverter control board. A resistive load of 5.32 Ω is used in the experiments.

Frequency, current, voltage, and power are measured using sensors and the DSP. The DSP control sample is 100 µs, while measurements are stored each 50 ms. Fig. 13 presents the currents measured in primary coils 1 and 2, for a velocity of 0.41 km/h. Initially, the secondary is well-aligned with coil 1. As the secondary moves, the induced current in coil 2 increases. When the ratio between the current in coil 1 and coil 2 reaches ≈ 75%, coil 2 is activated, and coil 1 is short-circuited.

Figs. 14(a), (b), and (c) show the measurements for frequency, power, and output voltage, respectively. As indicated in Fig. 14(a), when the secondary is well-aligned with coil 1, the frequency is in the maximum value. When the secondary starts to move, the frequency is adjusted to compensate for the reduction in the coupling factor. It can be observed in Figs. 14(b) and (c), that there are sags in power and output voltage during the transition. At this moment, frequency is at its minimum value (inferior limit of 79 kHz). The efficiency results are obtained for the power in the input and output presented in Fig. 14(b), where the power in the input (DC bus) is 28.5 kW and in the secondary side (rectifier output) is 26.4 kW, resulting in an efficiency of 92.6%.

## 6. Comparison among model and experimental results

To compare the results obtained by the proposed model with those obtained experimentally, the model is simulated for the same operating

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 14. DSP measurements, sampling time 50 ms. (a) Switching Frequency. (b) Power. (c) Output voltage.</figcaption></figure>

point as the experiment, considering a resistive load of 5.32 Ω and a velocity of 0.41 km/h. Losses are also included in the model, calculated as the difference between the measured power in the DC bus and the power in the rectifier output. Consequently, the equivalent resistance, initially considered as 0.1 Ω in Table 1, is updated to 0.2 Ω. It is worth mentioning that the most important values for comparison are presented at the start position. Since the robotic arm position is manually controlled, the final position is not necessarily well-aligned, as it depends on the operator.

The comparison for frequency, output power, and output voltage is presented in Figs. 15(a), (b), and (c), respectively. The measurements are depicted in black, while the model results are shown in gray.
<table><caption>**Table 2**
Comparison between results obtained by the model and measurements.</caption><thead><tr><th></th><th>Secondary position</th><th>Measured</th><th>Model</th><th>Error</th></tr></thead><tbody><tr><td rowspan="2">F<sub>SW</sub></td><td>start</td><td>88.9 kHz</td><td>86.7 kHz</td><td>2.47%</td></tr><tr><td>stop</td><td>87.8 kHz</td><td>86.7 kHz</td><td>1.25%</td></tr><tr><td rowspan="2">Power</td><td>start</td><td>26.4 kW</td><td>25.7 kW</td><td>2.6%</td></tr><tr><td>stop</td><td>22.8 kW</td><td>25.8 kW</td><td>13.15%</td></tr><tr><td rowspan="2">Voltage</td><td>start</td><td>375.1 V</td><td>370.2 V</td><td>1.31%</td></tr><tr><td>stop</td><td>370.6 V</td><td>349.1 V</td><td>5.8%</td></tr></tbody></table>

Table 2 summarizes the numerical results for the start and stop points, including the error/difference between the model and measurement results.

For the frequency (Fig. 15(a)), there is an error of 2.47% for the start position and 1.25% for the stop position. The parameter that can influence this difference between the model and measurements is mainly the coupling factor, which may deviate from the FEA simulation and the coupling factor in the experimental setup. Another aspect to be analyzed in this figure is that, in the stop position, the measured frequency is lower compared to the start position. In practice, this suggests that the coupling factor is lower in this position, indicating that the secondary side is not well-aligned with coil 2.

For the power (Fig. 15(b)), there is an error of 2.6% for the start position and 13.15% for the stop position. Regarding the voltage (Fig. 15(c)), there is an error of 1.31% for the start position and 5.8% for the stop position. These differences can be attributed to the same uncertainties discussed for Fig. 15(a).

In addition to the good fit of simulations using the *dq* model and experiments, it is important to remark that the simulation time using a computer whose configuration is an i7 12th Gen i7 – 12700H, 16 GB RAM 4800 MHz with the full electrical model of the vehicle running at 80 km/h takes 10.7 min. In contrast, the same simulation takes 3.08 s using the *dq* model. This illustrates the advantage of using the latter for developing and tuning phases, before tests in the full electrical model.

## 7. Conclusion

This paper has first proposed a nonlinear model of Dynamic Wireless Power Transfer (DWPT) for EV battery charging applications. An average state-space model is derived by mathematically analyzing the equivalent electric circuit of the DWPT. In the following, a *dq* model of the system is derived by converting all the AC state variables to DC states of slow dynamics, which is better suited for control design and simulations. An extremum-seeking control, implemented as a phase shift hysteresis controller, is then proposed for each converter to control the frequency and obtain maximum power transfer to the secondary side. The objective of this controller is to seek the resonant frequency by compensating the phase shift between the output voltage and current of the converter.

A dynamic change of the coupling coefficients is introduced to the model to emulate the movement of the car along the charging track. Simulation results show the control of power when passing from one primary coil to another by analyzing the magnitude of the output current of the present and following converters. The criteria for comparing the current magnitudes is important since it provides essential information about the secondary coil position related to the primary coils. It was demonstrated that the continuity of power transfer to the load is achieved in this manner.

The comparison of simulations and experiments shows the good accordance among them, validating the *dq* model. This one is well-fitted for control design and will be used in the future to propose other more sophisticated and possibly more performing control algorithms.

It is important to remark that the *dq* model is much faster to simulate than the full electrical one. It can then be used for development (not only of control algorithms but also structural ones) that can be tested and tuned using this *dq* model, before further validation using the electrical one.

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 15. Comparison among DSP measurements and model results. (a) Switching Frequency. (b) Power. (c) Output voltage.</figcaption></figure>

## CRediT authorship contribution statement

Zariff M. Gomes: Writing – review & editing, Writing – original draft, Validation, Investigation, Conceptualization. **Hassan Moussa:** Writing – original draft, Investigation, Conceptualization. **Yann Le Gall:** Validation, Software. **Edemar O. Prado:** Writing – review & editing, Writing – original draft, Validation, Investigation, Conceptualization. **Gilney Damm:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **José Renes Pinheiro:** Writing – original draft, Investigation, Conceptualization. **Christophe Ripoll:** Writing – review & editing, Validation.
Declaration of competing interest

The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Acknowledgments

This work has received funding from the European Union's Horizon 2020 research and innovation program under Grant Agreement No 875683 (Incit EV project) and from the CNPq (175271/2023-2) and CAPES - Finance Code 001.

References

Aboushady, A. A., Ahmed, K. H., Finney, S. J., & Williams, B. W. (2013). Linearized large signal modeling, analysis, and control design of phase-controlled series-parallel resonant converters using state feedback. *IEEE Transactions on Power Electronics, **28***, 3896-3911.

Babaki, A., Vaez-Zadeh, S., Zakerian, A., & Covic, G. A. (2021). Variable-frequency retuned WPT system for power transfer and efficiency improvement in dynamic EV charging with fixed voltage characteristic. *IEEE Transactions on Energy Conversion*, [ISSN: 15580059] 36, 2141–2151. http://dx.doi.org/10.1109/TEC.2020.3048196.

Bagchi, A. C., Kamineni, A., Zane, R. A., & Carlson, R. (2021). Review and comparative analysis of topologies and control methods in dynamic wireless charging of electric vehicles. *IEEE Journal of Emerging and Selected Topics in Power Electronics*, [ISSN: 21686785] 9, 4947–4962. http://dx.doi.org/10.1109/JESTPE.2021.3058968.

Behnamfar, M., Olowu, T. O., Tariq, M., & Sarwat, A. (2024). Comprehensive review on power pulsation in dynamic wireless charging of electric vehicles. *IEEE Access*, [ISSN: 21693536] 12, 66858–66882. http://dx.doi.org/10.1109/ACCESS.2024.3397583.

Bin, G., Chien-Yu, L., Baifeng, C., Dominic, J., & Jih-Sheng, L. (2013). Zero-voltage-switching PWM resonant full-bridge converter with maximized circulating losses and minimal voltage stresses of bridge rectifiers for electric vehicle battery chargers. *IEEE Transactions on Power Electronics, **28***, 1132–1144.

Budhia, M., Covic, G. A., & Boys, J. T. (2011). Design and optimization of circular magnetic structures for lumped inductive power transfer systems. *IEEE Transactions on Power Electronics, **26***, 10396–103108.

Caillierez, A. (2016). *Etude et mise en oeuvre du transfert de l'énergie électrique par induction: application à la route électrique pour véhicules en mouvement* (Ph.D. thesis), Université Paris Saclay (COmUE).

Caillierez, A., Sadarnac, D., Jaafari, A., & Loudot, S. (2014). Dynamic inductive charging for electric vehicle: Modelling and experimental results. In *7th IET international conference on power electronics, machines and drives (PEMD 2014)*. IET.

Covic, G. A., Budhia, M., Huang, C.-Y., et al. (2013). Development of a single-sided flux magnetic coupler for electric vehicle IPT charging systems. *IEEE Transactions on Industrial Electronics, **60***) (1), 318–328.

de Melo, H. N., Trovao, J. P. F., Pereirinha, P. G., Jorge, H. M., & Antunes, C. H. (2018). A controllable bidirectional battery charger for electric vehicles with vehicle-to-grid capability. *IEEE Transactions on Vehicular Technology, **67***, 114–123.

Dehui, W., Fan, Y., Chao, H., & Fang, C. (2019). Method for the calculation of coupling coefficient between two arbitrary-shaped coils. *IET Power Electronics*, URL https://apisemanticscholar.org/CorpusID:203992933.

Deng, J., Li, S., Hu, S., Mi, C. C., & Ma, R. (2014). Design methodology of LLC resonant converters for electric vehicle battery chargers. *IEEE Transactions on Vehicular Technology, **63***, 1581–1592.

Gori, P.-A. (2019). transmission dynamique d'énergie par induction : application au véhicule électrique (Ph.D. thesis), (2019SACLC063), Université Paris Saclay (COmUE), URL https://theses.hal.science/tel-02413595.

Hu, X., Wang, Y., Lv, S., Dong, X., Chen, T., Jiang, Y., et al. (2019). Discrete time modeling of wireless power transfer system using lcc compensation topology. In *2019 10th international conference on power electronics and ECCE Asia (ICPE 2019 - ECCE Asia)* (pp. 968–973). http://dx.doi.org/10.23919/ICPE2019-ECCEAsia42246. 2019.8797095.

Kabbara, W., Bensetti, M., Phulpin, T., Caillierez, A., Loudot, S., & Sadarnac, D. (2022). A control strategy to avoid drop and inrush currents during transient phases in a multi-transmitters DIPT system. *Energies, **15***) (8), 2911.

Kadem, K., Cheriet, F., Laboure, E., Bensetti, M., Le Bihan, Y., & Debbou, M. (2019). Sensorless vehicle detection for dynamic wireless power transfer. In *2019 21st European conference on power electronics and applications (ePE'19 ECCE europe)* (pp. P-1). IEEE.

Kadem, K., Meira, Z., Damm, G., & Moussa, H. (2022). Efficient sequencing method of ground coils for dynamic wireless power transfer. In *IEEE wireless power week (WWP 2022)*. IEEE.

Kamineni, A., Neath, M. J., Zaheer, A., Covic, G. A., & Boys, J. T. (2016). Interoperable EV detection for dynamic wireless charging with existing hardware and free resonance. *IEEE Transactions on Transportation Electrification*, 3(2), 370–379.

Khaligh, A., & D'Antonio, M. (2019). Global trends in high-power on-board chargers for electric vehicles. *IEEE Transactions on Vehicular Technology*, 68(4), 3306–3324.

Khaligh, A., & Dusmez, S. (2012). Comprehensive topological analysis of conductive and inductive charging solutions for plug-in electric vehicles. *IEEE Transactions on Vehicular Technology*, 61(8), 3475–3489.

Lee, J.-Y., & Han, B.-M. (2015). A bidirectional wireless power transfer EV charger using self-resonant PWM. *IEEE Transactions on Power Electronics*, 30, 1784–1787.

Luk, P. C.-K., Aldhaher, S., Fei, W., & Whidborne, J. F. (2015). State-space modeling of a class E2 converter for inductive links. *IEEE Transactions on Power Electronics*, 30, 3242–3251.

Malan, W. L., Vilathamuwa, D. M., & Walker, G. R. (2016). Modeling and control of a resonant dual active bridge with a tuned cllc network. *IEEE Transactions on Power Electronics*, 31, 7297–7310.

Meira, Z., Damm, G., Kadem, K., & Moussa, H. (2022). A 30 kW dynamic wireless inductive charging system for EVs. In *24th European conference on power electronics and applications*. EPE.

Moussa, H., Gomes, Z. M., Kadem, K., & Damm, G. (2023). Modeling and control of dynamic wireless power transfer system for electric vehicle charger application. In *IFAC world congress 2023*. Yokohama, Japan: IFAC.

Pang, B., Deng, J., Liu, P., & Wang, Z. (2017). Secondary-side power control method for double-side LCC compensation topology in wireless EV charger application. In *43rd annual conference of the IEEE industrial electronics society (IECON 2017)*. IEEE.

Reimers, J., Dorn-Gomba, L., Mak, C., & Emadi, A. (2019). Automotive traction inverters: Current status and future trends. *IEEE Transactions on Vehicular Technology*, 68(4), 3337–3350.

Serge, L., & Antoine, C. (2017). Procedé de charge sans contact dynamique, et système correspondant. France Patent, FR3045231A1 URL https://worldwide.espacenet.com/patent/search/family/055646725/publication/FR3045231A1?q=pn%3DFR3045231A1 Renault SAS.

Tran, D. H., Vu, V. B., & Choi, W. (2018). Design of a high-efficiency wireless power transfer system with intermediate coils for the on-board chargers of electric vehicles. *IEEE Transactions on Power Electronics*, 33, 175–187.

Wang, H., Pratik, U., Jovicic, A., Hasan, N., & Pantic, Z. (2021). Dynamic wireless charging of medium power and speed electric vehicles. *IEEE Transactions on Vehicular Technology*, 70, 12552–12566.

Wang, C.-S., Stielau, O., & Covic, G. (2005). Design considerations for a contactless electric vehicle battery charger. *IEEE Transactions on Industrial Electronics*, 52(5), 1308–1314. http://dx.doi.org/10.1109/TIE.2005.855672.

Wang, C., Zhu, C., Song, K., Wei, G., Dong, S., & Lu, R. G. (2017). Primary-side control method in two-transmitter inductive wireless power transfer systems for dynamic wireless charging applications. In *2017 IEEE PELS workshop on emerging technologies: wireless power transfer (WoW)* (pp. 1–6). http://dx.doi.org/10.1109/WoW.2017.7959366.

Yilmaz, M., & Krein, P. T. (2012). Review of battery charger topologies, charging power levels, and infrastructure for plug-in electric and hybrid vehicles. *IEEE Transactions on Power Electronics*, 28(5), 2151–2169.

Zahid, Z. U., Dalala, Z. M., Zheng, C., Chen, R., Faraci, W. E., Lai, J.-S. J., et al. (2015). Modeling and control of series-series compensated inductive power transfer system. *IEEE Journal of Emerging and Selected Topics in Power Electronics*, 3, 111–123.

Zhang, W., Wong, S.-C., Tse, C. K., & Chen, Q. (2014). Design for efficiency optimization and voltage controllability of series-series compensated inductive power transfer systems. *IEEE Transactions on Power Electronics*, 29(1), 191–200. http://dx.doi.org/10.1109/TPEL.2013.2249112.