Parameter Estimation-Based Output Voltage or Current Regulation for Double-LCC Hybrid Topology in Wireless Power Transfer Systems

Abstract

In Wireless Power Transfer Systems (WPTS), variations in a load connected to a receiver can cause instability in the waveforms of output voltage and current due to their sensitivity to changes in load impedance. To overcome such drawbacks, this paper presents a control scheme for regulating voltage and current at the output of a WPTS system with the Double-LCC topology. The proposed method is based on estimating secondary-side parameters while assuming a constant coupling coefficient that remains close to its intended value during operation. The methodology begins with the mathematical modeling of the primary and secondary resonant circuits. By measuring the input voltage and current, the system estimates the load impedance, which is then used to derive the expected output voltage and a reference for the input voltage. To maintain a stable output, the system dynamically adjusts the input voltage, ensuring that it aligns with the theoretical reference value. Analytical calculations and simulations were performed using the MATLAB/Simulink platform to validate the proposed approach. Simulations confirmed the theoretical predictions for a wireless system operating at 120 kHz with a power transfer of 100 W. The results demonstrated that the load voltage remains stable at 32 V, even under varying load conditions, while the output current remains at 3 A despite fluctuations in battery voltage.

1. Introduction

Electric vehicles associated with micro-mobility encompass light automobiles designed for short trips and offer a sustainable, economical, and practical solution to urban transportation challenges. In such systems, wireless energy transfer emerges as an innovative, convenient, and safe technology for charging these vehicles. Beyond micro-mobility, this technology can also be applied to charging electric vehicles [1,2], electronic devices, and a wide range of applications across multiple fields, including implantable medical equipment [3,4], as well as in biology and medicine. In [5], the authors proposed an extendable power transmission platform for moving objects, with feasible applications in neurobehavior studies and brain–machine interface research. The success of WPT lies in its wire-free energy transfer process, which enhances safety and allows for reliable operation in humid and harsh environments [6].

The near-field resonant inductive coupling scheme is based on the principle of electromagnetic induction, where wireless energy transmission occurs between two magnetically coupled coils, one acting as a transmitter that generates magnetic flux and the other as a receiver, where voltage or current is induced according to Ampère’s and Faraday’s laws [7]. Due to the lack of a ferromagnetic core to confine and guide the flux lines, flux dispersion is an inherent characteristic of this design. As a result, viable solutions must be implemented to achieve adequate efficiency for practical applications. To address this, compensation topologies are employed to minimize the reactive power demand on the source while enhancing power transmission capacity [8,9].

The arrangement of compensation elements in the primary and secondary circuits of the system can vary, with classic configurations including SS (Series–Series), SP (Series–Parallel), PS (Parallel–Series), and PP (Parallel–Parallel) [10]. Each acronym denotes the positioning of the compensation capacitor on the primary and secondary sides. While these classical topologies have several advantages, such as high efficiency, low power loss, low cost, and reduced complexity and volume, they have inherent limitations that hinder their widespread adoption, as they are prone to instability due to parameter variations and suffer from unsafe operation when the secondary side deviates [11]. For instance, even a slight misalignment between the coupling coils can lead to impractically high current levels, compromising system performance. To overcome these challenges, hybrid topologies that mix classical configurations have gained relevance in enhancing system robustness and efficiency.

Hybrid topologies have various advantages by integrating the benefits of multiple compensation networks [11]. A series–parallel–series SPS topology was proposed in [12], and it combines characteristics of both SS and PS topologies. However, the resonant condition may vary due to changes related to the load condition. The inductance–capacitance–inductance LCL topology has advantages when the system works at the resonant frequency. The inverter only supplies the active power required by the load, and the current in the transmitter coil is independent of the load condition [13]. Nevertheless, this topology requires the same value for both inductors. In order to reduce the additional inductor size and cost, a capacitor is put in series with the transmitter coil, which forms an LCC compensation network [13]. According to the literature, Double-LCC compensation is preferred since it reduces current stress in the inverter while offering high misalignment tolerance [14].

In this article, the system under analysis is the Double-LCC topology, originally proposed in [13]. This topology features an LCC resonant network on both the primary and secondary sides, ensuring minimal variations in resonance frequency despite changes in the coupling coefficient or load conditions [13]. This topology can work at a constant frequency, which eases the control. Furthermore, nearly unit power factors can be achieved on both converter sides across the entire range of coupling and load conditions, so that high efficiency is easily achieved [13]. Additionally, it offers the advantage of bidirectional power transfer, making it a compelling solution for high-performance wireless power transmission systems.

When implementing inductive power transfer for charging batteries in electric micro-mobility vehicles, it is essential to ensure a charging process that follows a constant current followed by a constant voltage phase. As described in [15,16], the battery charging cycle occurs in distinct stages. Initially, the battery is charged with a constant current while its voltage gradually increases. Once the voltage reaches its maximum threshold, the charging mode shifts to constant voltage, allowing the current to decrease significantly. The cycle is complete when the current reaches a predefined cutoff value. Analyzing these two stages through the lens of Ohm’s Law, we see that the load resistance increases progressively throughout the charging process, meaning it is not constant. Consequently, variations in load resistance lead to fluctuations in output voltage, which must be carefully managed to ensure efficient and stable power transfer.

Significant efforts have been dedicated to output voltage control in wireless power transfer (WPT) systems with time-varying loads [17,18,19]. In most cases, the literature suggests communication between the secondary and primary circuits to achieve regulation. However, this approach increases system complexity and cost, as it typically requires a dedicated wireless communication system between both stages.

To address this challenge, implementing a control strategy that relies solely on a primary-side controller, without feedback from the secondary circuit, can be a valuable alternative. This approach reduces overall cost, complexity, and system size while enhancing performance and reliability [20]. In [17], a control method was proposed for the SP topology, which was applied to implantable devices. By analyzing the input impedance, mutual inductance, and load, the system could estimate the output voltage, enabling control without direct measurements on the secondary side. A similar technique was applied in [18] for the S-LCL topology. Meanwhile, in [19], the author proposed an estimation method for voltage and current in the load of a system utilizing an LCC resonant network on both primary and secondary sides. It was possible to assess secondary circuit behavior and execute control adjustments accordingly by monitoring the currents in the primary inductors $L_1$ and filter $L_{f1}$, along with the inverter output voltage. However, the output voltage and current estimations presented in [19] depend on guaranteeing the inverter’s operation in Zero Voltage Switching to calculate the estimations. Moreover, although the work assures an output regulation despite variations in mutual inductance and load impedance, the results were obtained for different fixed values and did not analyze step variations through operation.

In order to reduce system complexity while ensuring high performance, this work proposes a method for estimating the load and output voltage in a wireless power transfer system for charging electric micro-mobility vehicles. The main advantage of this approach lies in simplifying the monitoring and control process without any communication between the primary and secondary circuits by measuring only the system’s primary voltage and current. The approach utilizes a Double-LCC hybrid compensation topology, relying on a stable and well-defined coupling factor. This topology is suited for charging electric vehicles due to its robustness. In addition, the resonant frequency of the coils is independent of the coupling coefficient and the load condition. Power factor close to unity can be easily achieved for primary-side and secondary-side converters throughout a wide range of coupling and load conditions, which ensures high efficiency for the overall WPT system [13]. Additionally, in this proposal, the usage of concentric solenoid coils is crucial in maintaining a consistent coupling coefficient, even in the presence of misalignments.

Once the coupling is established, voltage or current control at the load is achieved based on primary-side estimations. These estimations are derived from the total equivalent circuit impedance, the inverter’s output current, and its output voltage. Finally, variations in load demand or battery voltage are accounted for by dynamically adjusting the duty cycle of the single-phase full-bridge square-wave inverter. By modulating the output voltage of the DC-AC converter, the system ensures a stable and precisely regulated load voltage or current, guaranteeing the required operational conditions. Simulations involving load step-up and step-down were conducted in order to verify the controller’s performance, utilizing the proposed parameter estimation method.

2. Project Analysis

2.1. Principles

The diagram of the proposed wireless power transfer system is depicted in Figure 1. This system consists of three fundamental sections: the single-phase square-wave DC-AC converter, the resonant network, and the load, constituting a Double-LCC topology. The power switches, S1–S4, are MOSFETs that compose the inverter. $L_1$ and $L_2$ represent the self-inductances, while $R_{L1}$ and $R_{L2}$ denote the intrinsic resistances of the transmitting and receiving coil windings, respectively. The mutual inductance between the coils is represented by M. The resonant circuit is then formed by adding $C_1$ and $C_2$, which act as the compensation capacitors. $C_{f1}$ and $C_{f2}$ represent the filter capacitances, while $L_{f1}$ and $L_{f2}$ correspond to the filter inductance. $R_{Lf1}$ and $R_{Lf2}$ represent the resistances of the filter inductances. The indices ‘1’ and ‘2’ denote components in the primary and secondary sections, respectively. The DC input voltage source is denoted as $u_{cc}$, while $u_1$ represents the square-wave AC output voltage of the inverter. The system load, which may be a resistive load or a battery, is connected to the secondary terminals with a voltage denoted as $u_c$.

To help in the analytical development of the system, the T-equivalent model can be used to represent the schematic of Figure 1 [13], as shown in Figure 2. In this representation, the components of the receiving side are reflected onto the transmitting side, and are denoted by an apostrophe “ ´ ”. The parameter n is defined as the turn ratio between $L_1$ and $L_2$, as given by Equation (1). The T-model simplification represents the magnetizing inductance $L_m$ referred to the primary side, with k as the coupling factor. Additionally, $L_{d1}$ and $L_{d2}$ denote the leakage inductances associated with the coils $L_1$ and $L_2$, respectively. In this model, a resistive load R with voltage $u_r$ is considered. The formulation for the corresponding variables is expressed in Equations (2)–(8) [13].

$$\begin{array}{cc}\hfill n& =\sqrt{{\displaystyle \frac{L_2}{L_1}}}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill L_m& =k{L}_1\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill L_{d1}& =(1-k)L_1\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill L_{d2}^{\prime }& =(1-k){\displaystyle \frac{L_2}{n^2}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill L_{f2}^{\prime }& ={\displaystyle \frac{L_{f2}}{n^2}}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill C_2^{\prime }& =n^2{C}_2\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill C_{f2}^{\prime }& =n^2{C}_{f2}\hfill \end{array}$$

(7)

$$\begin{array}{c}\hfill R^{\prime }={\displaystyle \frac{R}{n^2}}\end{array}$$

(8)

To parameterize the equivalent input impedance of the circuit in Figure 2, an analytical simplification of each branch is performed based on Thévenin’s theorem. Consequently, the equivalent impedances are defined by Equations (9)–(15). For clarity, the analyzed branches are indicated in Figure 2. By considering the contribution of each branch in the equivalent model, the total theoretical impedance of the system can be determined. Through the algebraic manipulations described in Equations (16) and (17), the resulting expression is given by Equation (18).

$$\begin{array}{cc}\hfill a& ={\displaystyle \frac{R}{n^2}}+{\displaystyle \frac{j\omega L_{f2}}{n^2}}+{\displaystyle \frac{R_{Lf2}}{n^2}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill b& ={\displaystyle \frac{1}{n^{2}j\omega C_{f2}}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill c& =j\omega (1-k){\displaystyle \frac{L_2}{n^2}}+{\displaystyle \frac{1}{n^{2}j\omega C_2}}+{\displaystyle \frac{R_{Ld2}}{n^2}}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill d& =kj\omega L_1\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill e& =j\omega (1-k)L_1+{\displaystyle \frac{1}{j\omega C_1}}+R_{Ld1}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill f& ={\displaystyle \frac{1}{j\omega C_{f1}}}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill g& =j\omega L_{f1}+R_{Lf1}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill A& ={\displaystyle \frac{ab}{a+b}}+c\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill B& ={\displaystyle \frac{Ad}{A+d}}+e\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill Z_{input}& ={\displaystyle \frac{Bf}{B+f}}+g\hfill \end{array}$$

(18)

2.2. Coupling Coil Design

Initially, the fundamental design parameters for sizing the coupling coils must be defined. These include the design frequency ($f_s$), in Hertz, the peak current in the coil ($I_{peak}$), in Amperes, the desired inductance value ($L_{desired}$), in Henries, and the internal radius ($R_{internal}$), in meters. The copper resistivity is $2\times {10}^{-8} \Omega$/A and the magnetic permeability (${\mu }_0$), corresponds to $4\pi \times {10}^{-7}$ H/m.

Next, the cross-sectional area of the AWG conductor wire is calculated, along with the number of conductors in parallel. In alternating current applications at high frequencies, the skin effect significantly increases the resistance of the conductors. As a result, Joule heating losses rise, and the effective cross-sectional area of the cable decreases. To mitigate these undesirable effects, multiple twisted cables, known as Litz wires, are used to form the coupling coil conductor. Therefore, the cable cross-sectional area, in mm², must be determined according to Equation (19) [21]. Based on Equation (19), the Litz conductor should be composed of several wires, each with a cross-sectional area $S_{litz}$ equal to or less than $2\delta$.

$$\delta =\sqrt{{\displaystyle \frac{\rho }{\pi \times f_0\times {\mu }_0}}}$$

(19)

After selecting the AWG wire for the project, its values for the diameter with insulation of the chosen conductor $d_{litz}$, in meters, and the wire’s current-carrying capacity $I_{litz}$, in Amperes, must be used. The minimum number of conductors in parallel that satisfies the design requirements can then be calculated using Equation (20), yielding the smallest integer value greater than or equal to $N{c}_{min}$ (minimum number of conductors) [22].

$$N{c}_{min}=\lceil {\displaystyle \frac{I_{peak}}{I_{litz}}}\rceil$$

(20)

The calculation of the inductance of the coupling coils primarily depends on the chosen geometry. For both the primary and secondary coils, a solenoid configuration is selected, where a conductive wire is wound in a spiral, forming a cylindrical shape. This choice is made because solenoid coils exhibit minimal variation in the coupling factor, even when there are misalignments. Given the defined geometry, the inductance value $L_{conductor}$ is calculated, in Henries, which depends on its physical dimension and construction, using Equation (21) [23]. Here, ${\mu }_r$ represents the relative permeability of the solenoid core material, which, in this case, is the air, and it is approximately equal to 1. N denotes the number of solenoid turns. A indicates the cross-sectional area of the solenoid based on the previously defined internal radius $R_{internal}$, in square meters, and $l_{solenoid}$ is the length of the solenoid, in meters.

$$L_{condutor}={\displaystyle \frac{{\mu }_0\times {\mu }_r\times N^2\times A}{l_{solenoid}}}$$

(21)

To determine the total length of the solenoid $l_{solenoid}$, the number of turns N must be multiplied by the total diameter of the litz wire $D_{total}$. Additionally, 30% should be added to the calculated length to account for non-idealities in practical implementation. Equation (22) is then derived. Additionally, the approximate total length of the conductor $l_{cable}$ can be calculated by multiplying the number of turns of the coil by the length of a single circumference, which is equal to $2\pi R_{internal}$, as shown in Equation (23).

$$l_{solenoid}=N\times D_{total}\times 1.3$$

(22)

$$l_{cable}=N\times 2\times \pi \times R_{internal}$$

(23)

The total diameter of the set of conductors in parallel $D_{litz}$ can be determined using Equation (24), where $S_{total}$ represents the total area of the conductor set in parallel, calculated according to Equation (25).

$$D_{total}=\sqrt{{\displaystyle \frac{4\times S_{total}}{\pi }}}$$

(24)

$$S_{total}=N{c}_{min}\times {\displaystyle \frac{\pi \times d_{\mathit{litz}}^2}{4}}$$

(25)

Given that the process of modeling and constructing the coils involves several interdependent steps, a pseudo-code is provided in Algorithm 1 to help clarify the presented methodology.

The values obtained for the dimensioning of the primary and secondary coils are summarized in

Table 1. Designed parameters for primary and secondary coils.

Variable	Primary Coil	Secondary Coil
$I_{peak}$	1.72 A	1.72 A
$D_{internal}$	0.07 m	0.10 m
Conductor	6 × [AWG 26]	6 × [AWG 26]
N	110	54
$l_{solenoid}$	16 cm	8 cm
$l_{wire}$	24 m	17 m
$L_{desired}$	360 μH	360 μH
$L_{conductor}$	363.17 μH	363.84 μH

.

Algorithm 1 Pseudo-algorithm for physical coil design.

1: Declare variables ($f_0$, $I_{peak}$, $L_{desired}$, $R_{internal}$, $\rho$, ${\mu }_0$) 2: Select conductor based on skin effect 3: $\delta \leftarrow \sqrt{{\displaystyle \frac{\rho }{{\mu }_0\times \pi \times f_0}}}$ (calculate penetration depth) 4: $S_{litz}\leftarrow max(AWG\le 2\delta )$ (maximum allowed section) 5: Obtain $d_{litz}$, $I_{litz}$ 6: Design litz wire 7: $N{c}_{min}\leftarrow \lceil {\displaystyle \frac{I_{peak}}{I_{litz}}}\rceil$ (minimum number of parallel conductors) 8: $S_{total}\leftarrow N{c}_{min}\times {\displaystyle \frac{\pi \times d_{\mathit{litz}}^2}{4}}$ (total area of parallel conductor set) 9: $D_{total}\leftarrow \sqrt{{\displaystyle \frac{S_{total}\times 4}{\pi }}}$ (total diameter of parallel conductor set) 10: Inductance calculation 11: $N\leftarrow 1$ (initially, number of turns equals 1) 12: while true do 13:    $l_{solenoid}\leftarrow N\times D_{total}\times 1,3$ (solenoid length) 14:    $l_{wire}\leftarrow N\times 2\times \pi \times R_{internal}$ (total conductor length) 15:    $L_{conductor}\leftarrow {\displaystyle \frac{{\mu }_0\times {\mu }_r\times N^2\times A}{l_{solenoid}}}$ (calculated coil inductance) 16:    if $L_{conductor}\ge L_{desired}$ (verify if coil inductance reached design value) then 17:       break 18:    else 19:       $N\leftarrow N+1$ (add one turn to the solenoid) 20:    end if 21: end while 22: return  $L_{conductor}$

2.3. Load Estimation

The input impedance of the system can be determined by measuring the voltage $u_1$ and the current $i_1$, with their respective amplitudes represented by $U_1$ and $I_1$. The impedance is related to the phase angle between $u_1$ and $i_1$. Therefore, the magnitude and phase angle of the input impedance can be defined as shown in Equation (26).

$$\begin{array}{cc}\hfill |Z_{input}|& = {\displaystyle \frac{U_1}{I_1}}\hfill \\ \hfill \angle Z_{input}& = \theta \hfill \end{array}$$

(26)

The real and imaginary components can be expressed as shown in Equation (27).

$$\begin{array}{cc}\hfill |Z_{input}|\angle \theta & = |Z_{input}|cos\angle \theta +j|Z_{input}|sin\angle \theta \hfill \end{array}$$

(27)

Using Equations (18) and (27), the equivalent input impedance can ultimately be rewritten as given in Equation (28).

$$\begin{array}{cc}\hfill |Z_{input}|\angle \theta & = {\displaystyle \frac{Bf}{B+f}}+g\hfill \end{array}$$

(28)

Equation (28) is algebraically manipulated to isolate the variable R. This allows for the calculation of the estimated load resistance of the system, denoted as $R_{est}$, as given in Equation (29).

For simplification purposes, it is considered $|Z_{input}|\angle \theta =Z.$

The parameters in Equation (29) are calculated using Equations (9)–(15). It can be observed that the load estimate can be derived from the system’s input impedance, which is solely dependent on parameters obtained from the primary circuit. For this purpose, the characteristic parameters of the wireless power transfer system—namely, $L_1$, $L_2$, $L_{f1}$, $L_{f2}$, $R_{L1}$, $R_{L2}$, $R_{Lf1}$, $R_{Lf2}$, $C_1$, $C_2$, $C_{f1}$, $C_{f2}$, $\omega$, and k—are constants and are known. It is worth pointing out that despite the parameters being considered constant, there are studies that evaluate the impact of fabrication tolerances and uncertainties on the maximal rectification efficiency of the WPT application [24]. In [25], the authors noted that variations in series capacitors in a double-sized LCC topology WPT system can affect the impedance characteristics and power factor, which can impact the system’s efficiency and the achievement of Zero Voltage Switching.

$$R_{est}=-n^2\left({\displaystyle \frac{R_{Lf2}+j\omega L_{f2}}{n^2}}+{\displaystyle \frac{bcdf-Zbce-Zbcf-Zbde-Zbdf-Zbcd+bcdg+bcef+bceg+bdef+bcfg+bdeg+bdfg}{bdf-Zbe-Zcd-Zbf-Zce-Zcf-Zde-Zdf-Zbd+bdg+bef+cdf+beg+cdg+cef+bfg+ceg+def+cfg+deg+dfg}}\right)$$

(29)

The system’s output voltage, characteristic of the load, can be projected based on a defined coupling factor. It is important to note that k must remain constant or very close to the design value, thus excluding conditions where $k\ne k_{project}$. To achieve this, the estimated load resistance is used to determine the equivalent output voltage.

By applying consecutive current dividers in the schematic of Figure 2, as shown in Equations (30)–(33), the currents $i_1$, $i_2$, and $i_3$ can be calculated. Consequently, the load current $i_{out}$ is determined, and the estimated load voltage $u_{c_{est}}$ is obtained, as expressed in Equation (34).

$$\begin{array}{cc}\hfill i_1& ={\displaystyle \frac{u_1}{Z_{input}}}\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill i_2& ={\displaystyle \frac{f}{f+B}}{i}_1\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill i_3& ={\displaystyle \frac{d}{d+A}}{i}_2\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill i_{out}& ={\displaystyle \frac{b}{b+a}}{i}_3\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill u_{c_{est}}& =|R_{est}{i}_{out}|\hfill \end{array}$$

(34)

2.4. Output Voltage Control

A single-phase full-bridge inverter is used to power the circuit. The relationship between the DC input voltage, $u_{cc}$, and the magnitude of the high-frequency AC output voltage of the inverter, $U_1$, can be expressed as shown in Equation (35) [26].

$$\begin{array}{c}\hfill U_1={\displaystyle \frac{4}{\pi }}{u}_{cc}\end{array}$$

(35)

In wireless power transfer systems, the goal is to charge the load in a manner that ensures a constant voltage or current, providing optimal performance and protecting the load from undesirable voltage fluctuations [16]. However, due to frequent load variations during charging, the output voltage and current may become unstable. To simplify system design and to eliminate the need for sensors or feedback communication, a control scheme is proposed that maintains a stable and appropriate load voltage or current, based on the estimation of the secondary circuit parameters. To achieve this, a control scheme that ensures no communication between the primary and secondary sides of the system is implemented. For this, the control scheme relies only on the primary-side data, $u_1$ and $i_1$, to ensure a stable and regulated output.

To keep $u_r$ or the current $i_{out}$ stable, the evaluation is based on the inverse analysis of the equations presented in (30)–(33). In this sense, it is of utmost importance to evaluate the required input voltage $u_1$ to maintain a stable output voltage $u_r$. By setting a predetermined and required value for $u_r$, and knowing the estimated load $R_{est}$, according to (29), the output current $i_{out}$ can be determined.

From the development of Equations (30)–(33), $i_1$ is calculated by current dividers in the T-equivalent model of the Double-LCC topology, and consequently, the required output voltage of the reference inverter, $u_{1_{ref}}$, is determined. After comparing $u_{1_{ref}}$ with $u_1$, the error e is sent to a PI controller. The controller then adjusts the duty cycle of the inverter PWM, modifying $u_1$ and ensuring that $u_r$ remains constant at the proposed design value. Consequently, despite load variations, the output voltage or current does not fluctuate. It remains stable by adjusting the input voltage.

The schematic of the proposed control strategy is illustrated in Figure 3a, while Figure 3b presents the details of the applied PI controller. In order to enable system modeling for control design, modifications were made to the load. Consequently, at the output, the load was represented as a parallel combination of a filter capacitance $C_d$ equal to 110 nF and a resistance R, which represents the equivalent load. Moreover, $u_{ab}$ represents the first-order effect of the output voltage before the rectifier bridge, where $u_{ab}\left(t\right)=g_s\left(t\right)u_o\left(t\right)$. $i_r$ can be expressed as $i_r=g_s\left(t\right)i_{out}\left(t\right)$. The output voltage of the system is denoted by $u_o$, and the current flowing through the inductor $L_{f2}$ is denoted by $i_{\mathrm{out}}$. It is important to note that $g_s\left(t\right)$ denotes the nonlinear energy transformation function associated with the secondary edge, whose first harmonic component in the Fourier domain is given by ${〈g_s\left(t\right)〉}_1=-{\displaystyle \frac{2j}{\pi }}$.

According to [27], the state-space equation of the small signal model for the Double-LCC WPT system can be expressed according to Equation (36). $\widehat{x}\left(t\right)$ represents the state variable vector, $\widehat{\tilde{u}}\left(t\right)$ refers to the input variable vector, and $\widehat{y}\left(t\right)$ represents the output vector. A, $\tilde{B}$, and C correspond to the state, control, and output state coefficient matrices of the system, respectively.

$$\left\{\begin{array}{c}\dot{\widehat{x}}\left(t\right)=A\widehat{x}\left(t\right)+\tilde{B}\widehat{\tilde{u}}\left(t\right)\hfill \\ \widehat{y}\left(t\right)=C\widehat{x}\left(t\right)\hfill \end{array}\right.$$

(36)

According to Equation (36), the small-signal open-loop transfer function $G\left(s\right)$ of the WPT system can be derived in Equation (37). The system’s high order and complexity make controller design difficult. As a result, the system model must be reduced.

$$G\left(s\right)={\displaystyle \frac{\begin{array}{c}2.382\times {10}^{34}{s}^{11}+1.159\times {10}^{39}{s}^{10}+1.336\times {10}^{47}{s}^9-6.619\times {10}^{52}{s}^8\hfill \\ +2.228\times {10}^{59}{s}^7-7.339\times {10}^{64}{s}^6+9.959\times {10}^{70}{s}^5+6.681\times {10}^{76}{s}^4\hfill \\ -3.747\times {10}^{82}{s}^3+5.01\times {10}^{88}{s}^2-5.529\times {10}^{92}s+7.357\times {10}^{98}\hfill \end{array}}{\begin{array}{c}{s}^{17}+3.651\times {10}^4{s}^{16}+2.016\times {10}^{13}{s}^{15}-2.35\times {10}^{19}{s}^{14}+1.359\times {10}^{26}{s}^{13}\hfill \\ -2.396\times {10}^{32}{s}^{12}+5.624\times {10}^{38}{s}^{11}-8.637\times {10}^{44}{s}^{10}+1.348\times {10}^{51}{s}^9\hfill \\ -1.323\times {10}^{57}{s}^8+1.457\times {10}^{63}{s}^7-7.425\times {10}^{68}{s}^6+4.48\times {10}^{74}{s}^5\hfill \\ -1.961\times {10}^{79}{s}^4+1.205\times {10}^{85}{s}^3-1.236\times {10}^{89}{s}^2+8.283\times {10}^{94}s+6.394\times {10}^{97}\hfill \end{array}}}$$

(37)

To reduce the system order, the Balanced Model Reduction is applied. Equation (38) defines the resulting sixth-order reduced transfer function. Additionally, Figure 4 illustrates the Bode diagrams of both the 17th-order and 6th-order equivalent systems. By comparing the magnitude and phase frequency response curves, it is clear that the 6th-order model effectively represents the performance of the original system.

$$G_{\mathrm{reduced}}\left(s\right)={\displaystyle \frac{\begin{array}{c}-5.351s^6+1.586\times {10}^7{s}^5-2.178\times {10}^{13}{s}^4+1.479\times {10}^{19}{s}^3\hfill \\ -6.682\times {10}^{24}{s}^2-1.521\times {10}^{29}s+1.181\times {10}^{35}\hfill \end{array}}{\begin{array}{c}{s}^6+7.466\times {10}^6{s}^5-2.668\times {10}^{13}{s}^4+3.934\times {10}^{19}{s}^3\hfill \\ -2.739\times {10}^{25}{s}^2+1.329\times {10}^{31}s+1.026\times {10}^{34}\hfill \end{array}}}$$

(38)

The PI controller $G_c$ is designed based on the reduced-order small-signal model of the system. As a result of a series of simulations and tests carried out in MATLAB/Simulink software (R2018a), the most appropriate PI controller was the one operating with $K_p=0.025$ and $K_i=250$, as expressed in Equation (39). Additionally, Figure 5 shows the Bode diagram of the open-loop control system, represented by the dashed black line, where the reduced model achieves a gain margin of 17.5 dB and a phase margin of 36.9 degrees at $1.4\times {10}^3$ rad/s.

$$G_c=0.025+{\displaystyle \frac{250}{s}}$$

(39)

In order to analyze the system’s stability, simulations were carried out considering variations in the circuit parameters. The variation ranges in relation to nominal values were established as follows: $\pm 10\%$ for capacitances, $\pm 20\%$ for inductances, and up to $20\%$ for the load resistance.

A total of five cases were simulated with random variations within these ranges, and the stability of the system was analyzed using Bode diagrams, as illustrated in Figure 5. Furthermore,

Table 2. Parameter variations for each case.

Case	${\mathit{C}}_{\mathit{d}}$	${\mathit{C}}_{\mathit{f}1}$	${\mathit{C}}_{\mathit{f}2}$	${\mathit{C}}_{\mathit{p}}$	${\mathit{C}}_{\mathit{s}}$	${\mathit{L}}_{\mathit{f}1}$	${\mathit{L}}_{\mathit{f}2}$	${\mathit{L}}_{\mathit{p}}$	${\mathit{L}}_{\mathit{s}}$	${\mathit{R}}_{\mathit{L}}$
1	90%	107%	102%	110%	101%	99%	112%	89%	100%	118%
2	101%	107%	105%	102%	95%	107%	83%	105%	106%	115%
3	108%	110%	105%	102%	109%	103%	81%	85%	115%	110%
4	107%	94%	101%	103%	91%	105%	94%	82%	100%	104%
5	92%	94%	93%	94%	91%	105%	91%	102%	108%	110%

summarizes the percentage variations for each component in all five cases.

The results demonstrate that, within the defined parameter variations, the system remains stable, indicating robustness to parametric variations.

3. Simulation Results and Discussion

3.1. Voltage Source Operation

3.1.1. Load Step-Up

For the simulation analyses, MATLAB^®, in combination with Simulink^®, was utilized to validate the secondary data estimation and plant control. The design parameters defining the Double-LCC hybrid topology used in the system simulation are listed in

Table 3. System parameters.

Parameter	Value
Transferred power (P)	100 W
Switching frequency ($f_s$)	120 kHz
Input voltage ($u_{cc}$)	36 V
Coupling factor (k)	0.25
Transmitter coil inductance ($L_1$)	360 μH
Receiver coil inductance ($L_2$)	360 μH
Filter inductance ($L_{f1}$)	35.41 μH
Filter inductance ($L_{f2}$)	35.41 μH
Filter capacitance ($C_{f1}$)	49.67 nF
Filter capacitance ($C_{f2}$)	49.67 nF
Primary capacitance ($C_1$)	5.42 nF
Secondary capacitance ($C_2$)	5.42 nF
Transmitter coil resistance ($L_1$)	541.50 m$\Omega$
Receiver coil resistance ($L_2$)	541.50 m$\Omega$
Filter inductance resistance ($L_{f1}$)	3.10 m$\Omega$
Filter inductance resistance ($L_{f2}$)	3.10 m$\Omega$

. As stated in [13], the transferred power P can be determined using Equation (40).

$$P={\displaystyle \frac{\sqrt{L_1{L}_2}}{\omega L_{f1}{L}_{f2}}}k{u}_{1_{rms}}{u}_{R_{rms}}$$

(40)

With an input DC voltage of 36 V, the amplitude of the output voltage of the single-phase inverter is 45.8 V_peak, as given by Equation (35). By substituting the variables in Equation (40) with the corresponding values from Table 3, the designed secondary output voltage is determined to be 32.4 V_rms. Consequently, given the power and voltage, the coupled design load resistance R must be 10.505 $\Omega$.

As the load increases, the resistance rises accordingly. In this simulation scenario, an increment of 5 $\Omega$ is considered, resulting in a total resistance of 15.505 $\Omega$. The figures depicting the simulation results for the system under a load step-up condition, in open-loop, are shown in Figure 6. For $k=0.25$, it is observed that the load resistance can be accurately determined using the estimation given by Equation (29), as shown in Figure 6a. Figure 6b illustrates the inverter output voltage, which remains constant at 45.8 V_peak. According to Figure 6c,d, the estimated and simulated load voltages are presented, respectively. With the load step increase, the voltage $u_R$ rises from 32.2 V_rms to 47.3 V_rms. It is evident that the estimated voltage closely matches the simulated value. Consequently, implementing a control method to maintain a constant voltage becomes essential.

After applying the proposed control method, it is also possible to evaluate, as shown in Figure 7, the behavior of the secondary parameter estimates for a load step up from 10.505 $\Omega$ to 15.505 $\Omega$. In Figure 7a, the estimated load resistance is shown. Figure 7b illustrates the modulating signal with the insertion of the PI controller, which reduces the modulation index from 0.48 to 0.24. As a result, the inverter output voltage $u_1$ adjusts accordingly to maintain a constant load voltage $u_R$ at the load.

The calculation of $u_{1_{ref}}$ is essential for adjusting the inverter output voltage to ensure proper control of the output voltage. According to Figure 7c,d, the calculated reference inverter output voltage $u_{1_{ref}}$ and the simulated input voltage $u_1$ are presented. Before the variation in R, $u_1$ remains at 45.8 V_peak. After the change, $u_1$ adjusts to approximately 31.2 V_peak. The equivalence between $u_{1_{ref}}$ and $u_1$ is then demonstrated. The difference between these values represents the error fed into the PI controller.

Thus, in Figure 7e,f, the estimated and simulated output voltages are shown, respectively. The output voltage remains constant at approximately 32.2 V_rms, even after the load variation. It should be noted that the maximum voltage $u_1$ is constrained by the modulation index limit of 0.50. Consequently, load variations are accommodated within a restricted range. Whether increasing or decreasing, these variations must occur above the nominal load design value of 10.5050 $\Omega$.

A series of simulations was conducted to evaluate the variation of ripple with load resistance. The resistance ranged from 10.5050 $\Omega$ to 12.5050 $\Omega$, with increments of 1 $\Omega$. The results show that as the power decreases, the ripple also reduces, and the voltage fluctuations diminish accordingly. This analysis is illustrated in Figure 8a–f, where the relationship between load resistance and ripple behavior is clearly observed.

3.1.2. Load Step-Down

To evaluate the behavior of the system under a decreasing load, an initial load of 20.5050 $\Omega$ is assumed, with a variation of 5 $\Omega$, resulting in a final load of 15.5050 $\Omega$. The simulation results obtained are shown in Figure 9.

In Figure 9a, the load resistance is accurately estimated. In Figure 9b, the modulation index increases from approximately 0.17 to 0.24. As shown in Figure 9c,d, the reference voltage $u_{1_{ref}}$ and the simulation voltage $u_1$ converge to initial values of 23.7 V_peak and final values of 31.2 V_peak.

Thus, by readjusting the inverter output voltage, the output voltage $u_r$ can be observed to remain constant at the predefined value. According to Figure 9e,f, the estimated and simulated output voltages are presented, clearly demonstrating that $u_R$ stays constant and close to the defined value despite the load step-down.

Additionally, simulations were performed to analyze the effect of load resistance on ripple. The resistance varied from 12.5050 $\Omega$ to 10.5050 $\Omega$ in decrements of 1 $\Omega$. As shown in Figure 10a–f, for small load steps, the ripple is observed to be reduced.

3.2. Current Source Operation

Moving on to practical applications, it is possible to propose battery charging using inductive wireless power transfer with the Double-LCC topology, as shown in Figure 11. An important characteristic of the hybrid topology under analysis is that the LCC resonant network on the secondary side functions as a current source, which is ideal for the charging process with constant current (an important feature of the actual battery charging systems). Given this, it is necessary to incorporate a rectifier circuit for coupling with the battery, as it provides a DC voltage that varies over the course of the charging time.

To simulate a battery charging process with increasing voltage, the load with varying voltage was modeled. Initially, $u_o$ is set to 36 V and gradually increases to 42 V. Unlike the previous analysis, which focused on maintaining a constant voltage, the objective now is to supply the load with a fixed current of 3 A_rms. Using Equations (30)–(33), the reference voltage $u_{1_{ref}}$ that ensures a constant load current $i_{out}$ at the required value can be determined. $u_{1_{ref}}$ is then compared to the actual voltage $u_1$, and the difference corresponds to the error e, which is fed to the PI controller. The controller adjusts the switching signal of switches $S_1$–$S_4$, thereby regulating the inverter output voltage and ensuring a constant output current.

The simulation results are presented in Figure 12. Figure 12a shows the curve of the estimated load resistance, which exhibits an increasing trend from 10.7 $\Omega$ to 12.5 $\Omega$. However, $R_{est}$ exhibits an error of approximately 1.5 $\Omega$ during the steady state, which does not significantly impact the control performance. In Figure 12b, the modulation index for controlling the switches shows a minimal increase, but it is sufficient to adjust the voltage $u_1$. As shown in Figure 12c,d, both $u_{1_{ref}}$ and $u_1$ exhibit equivalent and ascending behavior.

The battery voltage is satisfactorily estimated when compared to the simulation value, as shown in Figure 12e,f. Figure 12g,h illustrate the reference and simulation output $i_{ou{t}_{rms}}$ current, respectively. The simulated current converges to the required value of 3 A_rms. Finally, the constant value of the $i_{out}$ current is maintained, even after variations in the battery voltage during charging, thereby validating the proposed control strategy.

4. Conclusions

This paper proposes a control scheme for regulating the output voltage or current of a wireless power transfer system used for recharging the batteries of electric vehicles associated with micromobility. The analytical modeling of the parameters on the receiver side was carried out based on the measurement of the primary voltage and current, as well as the calculation of the total equivalent impedance observed by the source.

The Double-LCC hybrid topology, operating at a switching frequency of 120 kHz and designed to transmit 100 W, was used. The desired output voltage is 32.4 V, with a load resistance of 10.5050 $\Omega$. The load resistance of the system was successfully estimated, accounting for both increasing and decreasing variations while maintaining a coupling coefficient close to the design value.

Furthermore, it was possible to emulate a coupled battery with varying voltage during charging. In the proposed simulations, the charging steps were correctly evaluated. Given the load resistance and desired output voltage, the reference inverter output voltage was calculated to ensure that $u_r$ remained stable and constant even after load variations. Similarly, the charging current was controlled to the desired value by calculating and maintaining the reference voltage accordingly.

The difference between the reference inverter output voltage and the simulation voltage was fed to the PI controller, which adjusted the modulation index of the switches in the single-phase inverter. As a result, $u_1$ was adjusted, ensuring a constant and fixed output voltage or current. The simulation results ultimately validated the proposed analytical formulations, using MATLAB/Simulink^® software (R2018a).

As a continuation of this study, future work will be directed toward the development of a hardware prototype to validate the proposed wireless power transfer system under real operating conditions. Experimental implementation will allow us to assess the practical effectiveness of the control strategy, particularly in accurately estimating the load and maintaining performance in dynamic environments. In response to the inherent complexity of power electronic systems, future research will also incorporate state-space modeling to provide a more rigorous analysis of the system’s non-linear behavior and transient responses.

Special attention will be given to the influence of parameter variations and fluctuations in the coupling coefficient on overall system stability and control accuracy. Furthermore, integration of real-time monitoring techniques and adaptive control, potentially supported by machine learning algorithms, will be explored to improve the robustness of the system. These steps are essential not only to validate the theoretical findings presented here but also to pave the way for the practical deployment of intelligent and efficient wireless power transfer solutions.