Data-Driven Characteristic Prediction and Output Optimization for Wireless Power Transfer Systems

Abstract

Constant current/voltage (CC/CV) output of wireless power transfer (WPT) systems deviates due to increased load resistance during charging and mutual inductance variations caused by misalignment. Dynamically regulating the DC input voltage can maintain a stable output at the preset value, and predicting the mutual inductance and load resistance can help monitor charging status. However, joint prediction of characteristics and regulation degree can be nonlinear and complicated. This work proposes a data-driven method for characteristic prediction and output optimization for WPT systems based on the current waveform from only the transmitter side. A Multi-Scale Parallel Convolutional (MSPC) neural network is applied to simultaneously predict the load resistance, mutual inductance, output deviation factor and regulation coefficient. By leveraging its multi-scale feature extraction capabilities, it can accurately estimate the aforementioned parameters based on only the AC current waveform at the transmitter side. To improve the model’s generalizability under practical conditions, transfer learning (TL) is utilized to minimize the discrepancy between simulated and physical data. Finally, a 140 W prototype of the series-series (SS)-compensated WPT system is built to validate the effectiveness of the proposed method.

1. Introduction

Wireless power transfer (WPT) technology has emerged as a transformative solution to eliminate the constraints of physical connectors, revolutionizing power delivery across industries by offering unparalleled convenience, safety, and flexibility. Its applications span electric transportation [1,2], portable electronics [3,4], medical implants [5,6,7], etc., where reliable and contactless power transfer is critical.

Constant-current (CC) and constant-voltage (CV) charging modes [8,9] of WPT systems are widely used in plenty of scenarios. Previous studies have achieved load-independent CC or CV output with input zero-phase angle (ZPA) by incorporating compensation networks operating at natural resonance frequencies, such as series–series (SS), parallel–parallel (PP), series–parallel (SP), parallel–series (PS) [10], and higher-order compensation networks, such as LCC-S, LCC–LCC, LCL-S, and S-LCC [11,12].

However, the properties of such topologies are derived from the fundamental harmonic approximation (FHA) method, which neglects the higher-order harmonic components of the square-wave voltage generated by DC/AC inverters. Thus, the output DC current in CC mode exhibits deviation under varying load conditions. The effective resistance of the battery increases as the charging process progresses. When the load resistance rises, the output current in CC mode will attenuate due to heavy load, as shown in Figure 1. This current drop entails consequences beyond prolonged charging time. Since the accuracy of the Coulomb Counting Method ($\Delta Q=\int I\left(t\right)dt$) heavily relies on a precise current profile [13,14], this attenuation will introduce significant State of Charge (SOC) estimation errors in the Battery Management System (BMS) if not taken into account. And it triggers premature CC-to-CV mode transitions, preventing the batteries from being fully charged [15]. Prolonged exposure to suboptimal charging conditions accelerates battery degradation [16].

Moreover, the change in the mutual inductance caused by misalignment [17,18] also leads to the deviation of the output. Numerous effective methods have been proposed to tackle this problem by optimizing the coupler structure [19,20,21] and designing topologies with high tolerance to misalignment [22,23]. However, the methods mentioned above can only mitigate the effects of load variation and misalignment to a certain extent, owing to the lack of control.

Therefore, implementing basic closed-loop control is a widely adopted strategy to realize stable output for WPT systems. As shown in Figure 2, a typical WPT system is composed of a DC/AC inverter, a coupler, an AC/DC rectifier and a compensation network. To maintain the DC output at a preset value, a DC/DC converter is cascaded with the inverter, such as Boost [24] and Buck–Boost [25]. An analog-to-digital converter (ADC) sampling module is placed at the output stage to acquire the feedback signal for the closed-loop control, which adjusts the conversion ratio of the DC/DC module. Some classical and effective control methods, including Proportional-Integral (PI) control, Model Predictive Control (MPC) and Sliding Mode Control (SMC), are widely adopted.

However, these methods require additional detection devices and a communication channel to transfer data between the transmitter side (TX) and the receiver side (RX), which increases cost and complexity of both hardware and software. Moreover, the variation of load resistance and mutual inductance can hardly be measured by detection devices at the RX. Consequently, considerable research efforts have been directed towards WPT control strategies based on information only from the TX [26,27,28].

Deep learning (DL) is an emerging technology designed for nonlinear complex issues, which has achieved outstanding success in computer vision, natural language processing and advanced control theories, etc. Reference [29] proposed a machine learning method using random forest and AdaBoost to estimate load resistance and coupling coefficient from transmitter-side measurements. However, this approach relies on manually extracted harmonic features via fast Fourier transform (FFT), which introduces additional computational overhead. While DL can bypass manual FFT, existing network architectures remain insufficient for analyzing harmonic-rich WPT waveforms. Image-based CNN [30] relies on external hardware and inherently neglects electrical dynamics. Furthermore, although LSTM [31] and TCN [32] process temporal data, their structures are misaligned with WPT waveform analysis. LSTM focuses on sequential temporal dependencies. TCN prioritizes long-term dependencies. Both lack parallel multi-scale receptive fields. Thus, they fail to concurrently extract harmonic-rich features. Moreover, few prior works simultaneously predict the load resistance and mutual inductance using only TX information while achieving output regulation.

Moreover, the data used in the aforementioned studies to train and validate DL models is mostly generated by offline simulation tools such as MATLAB/Simulink, LTspice or PLECS, etc. Such data has an inherent discrepancy from physical scenarios due to component parameter tolerances, parasitic effects, and switching dynamics [33,34]. This limitation reduces the prediction accuracy under practical operating conditions. Transfer learning (TL) [35] is a machine learning paradigm where knowledge acquired from solving the source domain is leveraged to improve learning efficiency and performance on a different but related target domain. A growing number of studies have utilized this algorithm in power converters [36,37], while its application in WPT parameter prediction and control remains relatively scarce.

To tackle these problems, this work proposes a Multi-Scale Parallel Convolutional (MSPC) deep learning model to achieve joint prediction of load resistance and mutual inductance, as well as deviation factor of the DC output and regulation coefficient of the DC input voltage in WPT systems utilizing the current waveform from only the TX. The MSPC model adopts a parallel structure of convolutional kernels with various dilation rates, which can capture multi-scale features from AC waveforms. This model is tailored to the harmonic-rich waveforms produced by power semiconductor switching. And TL is applied to enhance the prediction accuracy by minimizing the discrepancy between simulation data and physical data. Specifically, the MSPC model is first pre-trained using simulation datasets. The subsequent connection layers of the model are then updated via fine-tuning. With characteristic prediction finished, the regulation coefficient is used to optimize the output of the WPT system by regulating the input DC voltage. An SS-compensated experimental prototype is built to demonstrate that our method shows advantages and potential in joint characteristic prediction and output optimization using information from only the TX of WPT systems.

2. Analysis of SS-Compensated WPT System

The proposed data-driven method is designed and tested for an example of an SS-compensated WPT system. The schematic of the whole system is given in Figure 3, where $L_{\mathrm{P}}$ and $L_{\mathrm{S}}$ are the self-inductances of the coupler and M is the mutual inductance between two coils. $C_{\mathrm{P}}$ and $C_{\mathrm{S}}$ represent the compensation capacitors of the TX and RX, respectively. The full-bridge inverter consists of $Q_1$–$Q_4$, converting a DC input voltage $U_{\mathrm{DC}}$ to an AC input voltage $u_{\mathrm{AB}}$ with an operation frequency of angular frequency $\omega$. And $i_{\mathrm{AB}}$ refers to the current flowing through the capacitor $C_{\mathrm{P}}$. A full-bridge rectifier consists of $D_1$–$D_4$ and the filtering capacitor $C_{\mathrm{O}}$. It generates the DC output current $I_{\mathrm{O}}$ flowing through the load resistance $R_{\mathrm{L}}$.

The coupler can be modeled as a loosely coupled transformer. The equivalent circuit of the SS compensated WPT system is shown in Figure 4. To ensure input ZPA and load-independent ZPA, the values of $C_{\mathrm{P}}$ and $C_{\mathrm{S}}$ are subject to

$$\omega ={\displaystyle \frac{1}{\sqrt{L_{\mathrm{P}}{C}_{\mathrm{P}}}}}={\displaystyle \frac{1}{\sqrt{L_{\mathrm{S}}{C}_{\mathrm{S}}}}}=2\pi f_{\mathrm{sw}}$$

(1)

where $f_{\mathrm{sw}}$ is the resonant frequency of the system. The SS compensation topology is able to achieve CC output operating at $f_{\mathrm{sw}}$, which is the frequency of the fundamental wave of $u_{\mathrm{AB}}$. Based on the FHA method, $u_{\mathrm{AB}}$ is given as

$$u_{\mathrm{AB}}\left(t\right)={\displaystyle \frac{4U_{\mathrm{DC}}}{\pi }}sin{\displaystyle \frac{\pi D}{2}}sin\omega t$$

(2)

where D is the duty cycle of the PWM wave. And the output current $I_{\mathrm{O}}$ is a load-independent expression given as

$$I_{\mathrm{O}}={\displaystyle \frac{8U_{\mathrm{DC}}}{{\pi }^2}}sin{\displaystyle \frac{\pi D}{2}}·{\displaystyle \frac{1}{\omega M}}$$

(3)

However, Equation (3) is derived under the condition that the higher-order harmonics of $u_{AB}$ are neglected, whose standard notation should be given as

$$u_{AB}\left(t\right)={\displaystyle \frac{4U_{\mathrm{DC}}}{\pi }}sin{\displaystyle \frac{\pi D}{2}}(sin\omega t+{\displaystyle \frac{1}{3}}sin3\omega t+{\displaystyle \frac{1}{5}}sin5\omega t+\cdots )$$

(4)

Therefore, the AC input of the subsequent stage can be viewed as a parallel combination of a fundamental voltage source and higher-order harmonic voltage sources. As shown in Figure 4, the input current $i_{\mathrm{r}}$ of the rectifier can be expressed as

$${\mathit{i}}_{\mathbf{r}}={\displaystyle \frac{j\omega M{\mathbf{u}}_{\mathbf{AB}}}{(j\omega L_{\mathrm{P}}+\frac{1}{j\omega C_{\mathrm{P}}})(j\omega L_{\mathrm{S}}+\frac{1}{j\omega C_{\mathrm{S}}}+\frac{8}{{\pi }^2}{R}_{\mathrm{L}})+{\omega }^2{M}^2}}$$

(5)

The higher-order harmonic components are not at resonance. Therefore, the output current consists of superimposed fundamental and harmonic components following rectification, which exhibits attenuation under heavy load conditions. The content of harmonic components can be calculated by

$$H_n(100\%)={\displaystyle \frac{U_n}{U_1}}\times 100\%$$

(6)

where $U_n$ and $U_1$ denote the amplitude of the n-th harmonic and fundamental components of $i_{\mathrm{AB}}$, respectively. This work simulates the third and fifth harmonic contents of $i_{\mathrm{AB}}$ in the SS-compensated WPT system using MATLAB/Simulink 2024a. The simulation parameters are based on the design values listed in

Table 1. Experimental parameter details.

Description	Design Value	Experiment Value
Input DC voltage ($U_{\mathrm{DC}}$/V)	24.0	24.0
TX self-inductance ($L_{\mathrm{P}}$/μH)	13.715	13.654
RX self-inductance ($L_{\mathrm{S}}$/μH)	12.300	12.181
Mutual inductance (M/μH)	3.0 to 6.0	2.9 to 6.7
TX capacitor ($C_{\mathrm{P}}$/nF)	46.173	45.051
RX capacitor ($C_{\mathrm{S}}$/nF)	51.484	44.107
Resonance frequency ($f_0$/kHz)	200	198–200
Load resistance ($R_{\mathrm{L}}$/$\Omega$)	1 to 30	1 to 30

, with $C_{\mathrm{S}}$ adjusted to 45 nF to ensure zero-voltage switching (ZVS). The variations in $R_{\mathrm{L}}$ and M are set from 1 to 30 Ω and 3 to 7 μH, with steps of 1 Ω and 0.1 μH, respectively. The contents of the third and fifth harmonics are illustrated in Figure 5. The result indicates that the harmonic content of $i_{\mathrm{AB}}$ decreases as the load resistance and mutual inductance increase.

The majority of research treats the rectifier as an equivalent pure resistance, whose value equals ${\displaystyle \frac{8}{{\pi }^2}}{R}_{\mathrm{L}}$. In practice, the input impedance of the rectifier is inductive due to the nonlinearity of the diodes [38,39]. This occurs because the conduction of diodes introduces a phase shift, known as the rectifier angle $\gamma$ between the fundamental voltage and current. Therefore, the input impedance $Z_{\mathrm{e}}$ can be modeled as an effective resistance $R_{\mathrm{e}}$ and an effective inductance $L_{\mathrm{e}}$ in series, whose value can be expressed as

$$Z_{\mathrm{e}}=R_{\mathrm{e}}+j{L}_{\mathrm{e}}$$

(7)

As shown in Figure 3, $u_{\mathrm{r}}$ and $i_{\mathrm{r}}$ represent the input voltage and current of the rectifier. Assuming the WPT system operates in steady state, the output voltage $U_{\mathrm{O}}$ remains constant. Therefore, assuming D equals 1, $u_{\mathrm{r}}$ is a square wave with an amplitude of $U_{\mathrm{r}}$, which can be expressed as

$$u_{\mathrm{r}}\left(t\right)=\left\{\begin{array}{cc}{U}_{\mathrm{r}},\hfill & 0\le t\le 0.5T_{\mathrm{S}}\hfill \\ -U_{\mathrm{r}},\hfill & 0.5T_{\mathrm{S}}\le t\le T_{\mathrm{S}}\hfill \end{array}\right., U_{\mathrm{r}}=U_{\mathrm{O}}+2U_{\mathrm{D}}$$

(8)

where $T_{\mathrm{S}}$ is the switching period and $U_D$ is the forward voltage drop of each diode. Based on the FHA method, the fundamental component $u_{\mathrm{r},1}$ of $u_{\mathrm{r}}$ can be expressed as

$$u_{\mathrm{r},1}\left(t\right)={\displaystyle \frac{4(U_{\mathrm{O}}+2U_{\mathrm{D}})}{\pi }}sin\omega t$$

(9)

As shown in Figure 6, $i_{\mathrm{o},1}$ is the rectified current of $i_{\mathrm{r},1}$. And $I_{\mathrm{O},1}$ is the fundamental component of $I_{\mathrm{O}}$, which can be calculated by

$$I_{\mathrm{O},1}={\displaystyle \frac{1}{T_{\mathrm{S}}/2}}{\int }_0^{{\displaystyle \frac{T_{\mathrm{S}}}{2}}}{i}_{\mathrm{o},1}\left(t\right)dt={\displaystyle \frac{2}{T_{\mathrm{S}}}}{\int }_0^{{\displaystyle \frac{T_{\mathrm{S}}}{2}}}{I}_{\mathrm{r},1}sin(\omega t-\gamma )dt={\displaystyle \frac{2}{\pi }}{I}_{\mathrm{r},1}cos\gamma$$

(10)

where $I_{\mathrm{r},1}$ is the amplitude of $i_{\mathrm{o},1}$, which can then be given as

$$I_{\mathrm{r},1}={\displaystyle \frac{\pi I_{\mathrm{O},1}}{2cos\gamma }}$$

(11)

Hence, by substituting Equations (9) and (11), the phasor ${\mathbf{Z}}_{\mathbf{e}}$ can be expressed as

$${\mathbf{Z}}_{\mathbf{e}}={\displaystyle \frac{{\mathit{U}}_{\mathbf{r},\mathbf{1}}}{{\mathit{I}}_{\mathbf{r},\mathbf{1}}}}={\displaystyle \frac{U_{\mathrm{r},1}}{I_{\mathrm{r},1}}}\angle \gamma ={\displaystyle \frac{8(U_{\mathrm{O}}+2U_{\mathrm{D}})}{{\pi }^2{I}_{\mathrm{O},1}}}cos\gamma \angle \gamma$$

(12)

By substituting Equation (12), $R_{\mathrm{e}}$ can be expressed as

$$R_{\mathrm{e}}=\left|{\mathbf{Z}}_{\mathbf{e}}\right|cos\gamma ={\displaystyle \frac{8(U_{\mathrm{O}}+2U_{\mathrm{D}})}{{\pi }^2{I}_{\mathrm{O},1}}}{cos}^2\gamma$$

(13)

By substituting Equation (12), $L_{\mathrm{e}}$ can be expressed as

$$L_{\mathrm{e}}=\left|{\mathbf{Z}}_{\mathbf{e}}\right|sin\gamma ={\displaystyle \frac{4(U_{\mathrm{O}}+2U_{\mathrm{D}})}{{\pi }^2{I}_{\mathrm{O},1}}}sin2\gamma$$

(14)

Hence, the effective circuit of the SS-compensated WPT system can be modeled as shown in Figure 7.

Under actual operating conditions, it is almost impossible for an SS-compensated WPT system to achieve strictly load-independent CC output. Both simulation and experimental studies confirm that the actual output current $I_{\mathrm{O}}$ decreases with increasing load resistance $R_{\mathrm{L}}$.

3. MSPC Model and Transfer Learning Framework

To achieve characteristic prediction via information from the TX, as well as optimization of the decayed output current $I_{\mathrm{O}}$, this work proposes an MSPC model assisted by TL. To precisely quantify the extent of output deviation and input regulation, this paper defines two scaling factors that serve as the prediction target of the deep learning model. Firstly, we define the output deviation factor of $I_{\mathrm{O}}$ as

$$\alpha ={\displaystyle \frac{I_{\mathrm{act}}}{I_{\mathrm{th}}}}$$

(15)

where $I_{\mathrm{act}}$ and $I_{\mathrm{th}}$ represent the output current under actual and theoretical conditions derived from the FHA method, respectively. Then, we define the regulation coefficient of $U_{\mathrm{DC}}$ as

$$\beta ={\displaystyle \frac{U_{\mathrm{reg}}}{U_{\mathrm{th}}}}$$

(16)

where $U_{\mathrm{reg}}$ denotes the regulated DC input voltage required to maintain the CC output, and $U_{\mathrm{th}}$ denotes the initial DC input voltage. By scaling the input voltage by a factor of $\beta$, the output current can be calibrated to approach its theoretical value. To achieve the joint prediction of $R_{\mathrm{L}}$, M, $\alpha$ and $\beta$, the data-driven method is discussed in the following sections.

3.1. Multi-Scale Parallel Convolutional Model

In this paper, the MSPC model is proposed to jointly predict load resistance $R_{\mathrm{L}}$, mutual inductance M, deviation factor $\alpha$ and regulation coefficient $\beta$ from the transmitter-side current waveform $i_{\mathrm{AB}}$. As illustrated in Figure 8, the proposed network employs a parallel multi-branch architecture where each branch adopts distinct dilation rates to capture dynamic information across different temporal scales. The model consists of two core modules: a Multi-Scale Feature Extractor and a Global Regressor.

3.1.1. Multi-Scale Feature Extractor

The Multi-Scale Feature Extractor serves as the core module of the MSPC model, utilizing one-dimensional dilated convolutions to capture multi-scale temporal dynamics in power electronics waveforms. For a given input sequence $\mathbf{X}\in {\mathbb{R}}^{B\times 1\times L}$, where B denotes the batch size and L denotes the sequence length, the dilated convolution operation at position t with dilation rate d is defined as

$$\left(\mathbf{X}{\ast }_d\mathbf{K}\right)\left(t\right)={\displaystyle \sum _{i=0}^{k-1}}\mathbf{K}\left(i\right)·\mathbf{X}(t-d·i)$$

(17)

where $\mathbf{K}\in {\mathbb{R}}^k$ is the convolution kernel of size k and d controls the spacing between elements in the input sequence. This formulation enables the network to expand its receptive field exponentially without increasing the number of parameters or sacrificing resolution.

As shown in Figure 8, the extractor adopts a parallel three-branch structure. Each branch comprises two stacked dilated convolutional layers with distinct dilation rates. As discussed in Section 2, $i_{\mathrm{AB}}$ contains superimposed fundamental and higher-order harmonic components, whose contents vary significantly with $R_{\mathrm{L}}$ and M. The dilation rates are designed to cover the temporal scales corresponding to these components. The high-frequency branch employs dilation rates $d=1$ and $d=1$, capturing rapid transients associated with switching events. The mid-frequency branch utilizes dilation rates $d=2$ and $d=4$, extracting intermediate-scale features related to envelope variations. The low-frequency branch applies dilation rates $d=8$ and $d=16$, capturing long-term trends and gradual changes in the waveform.

For the j-th branch, the feature extraction process can be expressed as

$${\mathbf{F}}_j=\sigma \left({\mathbf{K}}_{j,2}{\ast }_{d_{j,2}}\sigma \left({\mathbf{K}}_{j,1}{\ast }_{d_{j,1}}\mathbf{X}\right)\right)$$

(18)

where ${\mathbf{K}}_{j,1}$ and ${\mathbf{K}}_{j,2}$ denote the convolution kernels of the first and second layers in branch j, respectively, while $d_{j,1}$ and $d_{j,2}$ are their corresponding dilation rates. The function $\sigma (·)$ represents the ReLU activation function. Each branch outputs a feature map ${\mathbf{F}}_j\in {\mathbb{R}}^{B\times 8\times L}$, with 8 representing the number of output channels. WPT waveforms exhibit structured and sparse frequency-domain properties, making eight channels sufficient to encode essential harmonic information. Furthermore, this compact dimension acts as implicit regularization to prevent overfitting given the limited training data.

The multi-scale features from all three branches are then concatenated along the channel dimension:

$${\mathbf{F}}_{\mathrm{concat}}=\left[{\mathbf{F}}_{\mathrm{high}},{\mathbf{F}}_{\mathrm{mid}},{\mathbf{F}}_{\mathrm{low}}\right]\in {\mathbb{R}}^{B\times 24\times L}$$

(19)

where ${\mathbf{F}}_{\mathrm{concat}}$ represents the concatenated feature map, while ${\mathbf{F}}_{\mathrm{high}}$, ${\mathbf{F}}_{\mathrm{mid}}$, and ${\mathbf{F}}_{\mathrm{low}}$ denote the outputs of the high-frequency, mid-frequency, and low-frequency branches, respectively. A pointwise convolution (kernel size $1\times 1$) is subsequently applied to fuse the multi-scale features:

$${\mathbf{F}}_{\mathrm{fused}}=\sigma \left({\mathbf{K}}_{\mathrm{fuse}}{\ast }_1{\mathbf{F}}_{\mathrm{concat}}\right)\in {\mathbb{R}}^{B\times 16\times L}$$

(20)

where ${\mathbf{K}}_{\mathrm{fuse}}\in {\mathbb{R}}^{1\times 24\times 16}$ is the fusion kernel, and $\ast 1$ denotes standard convolution with stride 1. This design enables the model to simultaneously capture high-frequency transients from switching events, mid-frequency variations from control dynamics, and low-frequency power trends.

3.1.2. Global Regressor

The Global Regressor transforms the multi-scale features into task-specific predictions. First, an adaptive global average pooling layer aggregates the temporal dimension:

$${\mathbf{f}}_{\mathrm{pool}}={\displaystyle \frac{1}{L}}{\displaystyle \sum _{t=1}^L}{\mathbf{F}}_{\mathrm{fused}}[:,:,t]\in {\mathbb{R}}^{B\times 16}$$

(21)

This operation compresses the temporal information into a fixed-size vector while preserving the channel-wise characteristics. The pooled features are then passed through a fully connected layer:

$$\mathbf{Y}=({\mathbf{W}}_{\mathrm{fc}}{\mathbf{f}}_{\mathrm{pool}}+{\mathbf{b}}_{\mathrm{fc}})\in {\mathbb{R}}^{B\times D}$$

(22)

where $\mathbf{W}\mathrm{fc}\in {\mathbb{R}}^{D\times 16}$ and $\mathbf{b}\mathrm{fc}\in {\mathbb{R}}^D$ are learnable parameters and $D=4$ denotes the number of prediction targets. The final output vector is given as

$$\mathrm{output}={[R_{\mathrm{L}},M,\alpha ,\beta ]}^{\mathrm{T}}$$

(23)

To ensure fair and balanced learning across all tasks without bias, the loss weights for the four output variables ($R_{\mathrm{L}}$, M, $\alpha$ and $\beta$) are kept equal. The overall MSPC model thus achieves joint prediction by leveraging multi-scale temporal features extracted from the TX current waveform, enabling accurate characterization of system states under varying load and coupling conditions. The detailed architecture and parameters of the MSPC model are shown in Appendix A,

Table A1. Detailed architecture and parameters of the MSPC model.

Layer	Kernel Size	Dilation Rate	Input Dim	Output Dim
High-frequency branch (Short-time scale)
Conv1d	3	1	$1\times 2000$	$8\times 2000$
ReLU	–	–	$8\times 2000$	$8\times 2000$
Conv1d	3	1	$8\times 2000$	$8\times 2000$
ReLU	–	–	$8\times 2000$	$8\times 2000$
Mid-frequency branch (Mid-time scale)
Conv1d	3	2	$1\times 2000$	$8\times 2000$
ReLU	–	–	$8\times 2000$	$8\times 2000$
Conv1d	3	4	$8\times 2000$	$8\times 2000$
ReLU	–	–	$8\times 2000$	$8\times 2000$
Low-frequency branch (Long-time scale)
Conv1d	3	8	$1\times 2000$	$8\times 2000$
ReLU	–	–	$8\times 2000$	$8\times 2000$
Conv1d	3	16	$8\times 2000$	$8\times 2000$
ReLU	–	–	$8\times 2000$	$8\times 2000$
Feature Concatenation and Fusion
Concatenate	–	–	$8\times 2000$ ($\times 3$)	$24\times 2000$
Conv1d ($1\times 1$)	1	1	$24\times 2000$	$16\times 2000$
ReLU	–	–	$16\times 2000$	$16\times 2000$
Global Regressor
AdaptiveAvgPool1d	–	–	$16\times 2000$	$16\times 1$
Flatten	–	–	$16\times 1$	16
Linear	–	–	16	4

The dimensions are expressed as (Channel × Sequence Length). The sequence length L is 2000. The output dimension is 4, corresponding to ${[R_L,M,\alpha ,\beta ]}^T$.

.

3.2. Transfer Learning

Transfer learning is a machine learning paradigm that aims to transfer knowledge learned from one task (source domain) to a different but related task (target domain) to improve learning efficiency and performance. The core concept of TL can be formalized as

$$f_{\mathrm{T}}^{*}=\mathcal{A}({\mathcal{D}}_{\mathrm{T}};{\mathcal{K}}_{\mathrm{S}})$$

(24)

where $f_{\mathrm{T}}^{*}$ is the optimal model for the target task; ${\mathcal{D}}_{\mathrm{T}}$ represents the target domain training data; ${\mathcal{K}}_{\mathrm{S}}$ denotes the knowledge extracted from the source domain ${\mathcal{D}}_{\mathrm{S}}$; $\mathcal{A}$ represents the transfer learning algorithm that integrates target data with source knowledge. In this work, the source domain ${\mathcal{D}}_{\mathrm{S}}$ consists of abundant simulation data. The target domain ${\mathcal{D}}_{\mathrm{T}}$ contains limited experimentally detected waveforms of $i_{\mathrm{AB}}$. A fine-tuning strategy is employed to bridge the gap between simulated and physical scenarios. As shown in Figure 9, the MSPC model is first pre-trained on ${\mathcal{D}}_{\mathrm{S}}$ to extract general waveform features. The model parameters are divided into two parts based on their functions. The Multi-Scale Feature Extractor parameters ${\theta }_f$ include the multi-scale parallel convolutional layers. The Global Regressor parameters ${\theta }_r$ include the global averaging and fully connected layers. The pre-training stage on ${\mathcal{D}}_{\mathrm{S}}$ solves

$$({\widehat{\theta }}_f,{\widehat{\theta }}_r)=arg\underset{{\theta }_f,{\theta }_r}{min}{\mathcal{L}}_{\mathrm{S}}({\theta }_f,{\theta }_r;{\mathcal{D}}_{\mathrm{S}})$$

(25)

where ${\mathcal{L}}_{\mathrm{S}}$ is the loss function on the source domain. $({\widehat{\theta }}_f,{\widehat{\theta }}_r)$ are the pre-trained parameters. In this work, the Mean Squared Error (MSE) is adopted as the loss function; thus, the above equation can be formulated in detail as

$${\mathcal{L}}_{\mathrm{S}}\left({\theta }_f,{\theta }_r;{\mathcal{D}}_{\mathrm{S}}\right)={\displaystyle \frac{1}{N_S}}{\displaystyle \sum _{i=1}^{N_S}}{∥{\mathbf{y}}_i-{\widehat{\mathbf{y}}}_i∥}_2^2$$

(26)

where $N_S$ is the number of samples in the source domain dataset ${\mathcal{D}}_{\mathrm{S}}$, ${\mathbf{y}}_i={[R_{L,i},M_i,{\alpha }_i,{\beta }_i]}^{\mathrm{T}}$ denotes the ground-truth vector for the i-th sample as defined in Equation (26), ${\widehat{\mathbf{y}}}_i$ represents the corresponding predicted vector, and ${∥·∥}_2^2$ denotes the squared L2 norm.

The model then adapts to the target domain using the knowledge ${\mathcal{K}}_{\mathrm{S}}=\{{\widehat{\theta }}_f,{\widehat{\theta }}_r\}$. The Multi-Scale Feature Extractor ${\widehat{\theta }}_f$ is frozen to preserve its general feature extraction capability. Only the Global Regressor parameters are fine-tuned with the limited target data. This fine-tuning process is formulated as

$${\theta }_r^{*}=arg\underset{{\theta }_r}{min}{\mathcal{L}}_{\mathrm{T}}({\widehat{\theta }}_f,{\theta }_r;{\mathcal{D}}_{\mathrm{T}})$$

(27)

where ${\mathcal{L}}_{\mathrm{T}}$ is the loss function on the target domain. ${\theta }_r^{*}$ represents the optimized regressor parameters adapted to the real-world data distribution. Similarly, ${\mathcal{L}}_{\mathrm{T}}$ shares the same MSE formulation as ${\mathcal{L}}_{\mathrm{S}}$, calculated over ${\mathcal{D}}_{\mathrm{T}}$.

The final model for the target task is $f_{\mathrm{T}}^{*}\left(x\right)=f(x;{\widehat{\theta }}_f,{\theta }_r^{*})$. It combines a frozen feature extractor with a fine-tuned regressor. Pre-trained on abundant simulation data, the Multi-Scale Feature Extractor captures the parameter-dependent harmonic characteristics with respect to $R_{\mathrm{L}}$ and M. The discrepancy of $i_{\mathrm{AB}}$ between ${\mathcal{D}}_{\mathrm{S}}$ and ${\mathcal{D}}_{\mathrm{T}}$ mainly lies in amplitude and initial phases, while the variation laws of $i_{\mathrm{AB}}$ in distinct scenarios remain constant. Therefore, only the Global Regressor requires adjustment to compensate for these physical offsets. This approach mitigates the discrepancy between ${\mathcal{D}}_{\mathrm{S}}$ and ${\mathcal{D}}_{\mathrm{T}}$ without requiring extensive experimental data. The overall framework of the MSPC model is illustrated in Figure 9.

3.3. Data Augmentation

All fine-tuning data from ${\mathcal{D}}_{\mathrm{T}}$ is derived from actual measurements taken on each WPT prototype. When the ranges of $R_{\mathrm{L}}$ and M variations are relatively wide, obtaining sufficiently valid ${\mathcal{D}}_{\mathrm{T}}$ data for fine-tuning still requires a substantial amount of work. To reduce the workload of collecting experimental data, data augmentation is applied to increase the diversity of data from ${\mathcal{D}}_{\mathrm{T}}$. Although these methods slightly alter the original information in the target domain data, they simulate the inevitable errors encountered in real-world conditions. This enhances the practical applicability of this data-driven approach. The specific data augmentation methods deployed are phase shifting, frequency distortion and amplitude scaling. Figure 9 shows the framework of TL assisted by data augmentation. Details and mechanisms are as follows.

To begin with, since the input data is obtained using sliding windows, the initial phase of the current waveform is typically random. Phase shifting increases the amount of valid data by copying the waveform starting from different phase points in the range of $[0,2\pi ]$. Moreover, the PWM signal driving inverter transistors is generated by a digital controller like DSP or FPGA. There is a slight, unavoidable discrepancy between the actual and preset resonance frequency. Therefore, data points are randomly added to or removed from the ${\mathcal{D}}_{\mathrm{T}}$ arrays to simulate a deviation of approximately $1\%$ in frequency. Finally, since the detection of the waveform has inevitable minor errors in amplitude, the amplitude of waveforms in ${\mathcal{D}}_{\mathrm{T}}$ is randomly scaled up or down by 0.01–0.1 times.

4. Online Validation of the Data-Driven Method

4.1. Benchmark Models

In this work, we compare four benchmark DL models with the proposed MSPC model to evaluate its predictive performance. The benchmark models are: Convolutional Neural Network (CNN), Temporal Convolutional Network (TCN), Long Short-Term Memory (LSTM) and Bidirectional Long Short-Term Memory (BiLSTM). They serve as basic and commonly used DL models for sequential prediction.

CNN is a widely used model initially designed for computer vision, which is also highly effective for time series processing in industrial scenarios. In this work, the 1D CNN consists of two convolutional layers (kernel sizes 7 and 5, filters = 16) with batch normalization, ReLU, and dropout (0.1); a max pooling layer (kernel size = 2) for downsampling; and global average pooling followed by a fully connected layer for final prediction.

The basic TCN consists of causal convolution and dilated convolution modules, serving as an enhanced CNN variant specifically designed for time series processing. It uses causal convolutions with dilation rates of 1 and 2 (kernel size = 3, filters = 16) to expand receptive fields without pooling. The remaining structure (global average pooling and fully connected layer) is identical to CNN.

LSTM employs unique gating units to control memory retention and updates, allowing it to effectively learn long-term dependencies in sequences, which has established it as a classic solution for various time-series tasks. In this work, the LSTM model is configured with a hidden size of 64, 2 layers, and a dropout rate of 0.3 to serve as a recurrent baseline for comparison.

As an extension of the standard LSTM, BiLSTM processes input sequences in both forward and backward directions, enabling it to capture contextual information from past and future states simultaneously. In this work, the BiLSTM is configured with a hidden size of 64, 2 layers, and a dropout rate of 0.3, serving as a bidirectional recurrent baseline for comparison.

4.2. Generation of Training Data in ${\mathcal{D}}_{\mathrm{S}}$

Data from ${\mathcal{D}}_{\mathrm{S}}$ is generated by MATLAB/Simulink 2024a, with $R_{\mathrm{L}}$ varied from 1 to 30 $\Omega$ in steps of 1 $\Omega$ and M varied from 3 to 7 μH in steps of 0.1 μH. $i_{AB}$ in each scenario has a sequence length of 5000. A sliding window method is taken to segment the initial data, whose window size is 2000 and stride is 100. This method turns the shape of ${\mathcal{D}}_{\mathrm{S}}$ into ${\mathbb{R}}^{38130\times 2000}$, and the shape of labels is ${\mathbb{R}}^{38130\times 4}$ according to Equation (23).

To generate highly accurate ground-truth labels for the neural network, an automated Simulink-based simulation framework combined with Brent’s method is developed. Specifically, for a given combination of $R_{\mathrm{L}}$ and M, Brent’s method is employed as an iterative optimizer to dynamically adjust $U_{\mathrm{DC}}$ in the simulation model. The algorithm intelligently switches between inverse quadratic interpolation, the secant method, and bisection based on convergence conditions. For instance, when the interpolation fails or only two points are valid, the next candidate is approximated by the secant-based update rule as follows:

$${\beta }_{k+1}={\beta }_k-f\left({\beta }_k\right){\displaystyle \frac{{\beta }_k-{\beta }_{k-1}}{f\left({\beta }_k\right)-f\left({\beta }_{k-1}\right)}}$$

(28)

where $f\left({\beta }_k\right)=I_{\mathrm{act}}\left({\beta }_k\right)-I_{\mathrm{th}}$ represents the output current error. This process of running a full simulation cycle and dynamically updating the input voltage repeats automatically until $\left|f\right({\beta }_{k+1}\left)\right|$ converges to a predefined tolerance threshold. In this work, the iteration terminates when the voltage $U_{\mathrm{DC}}$ variation between consecutive steps falls below 0.01 V, corresponding to a current error of approximately 1 mA. Through this mechanism, the exact $\beta$ required to counteract the non-ideal current droop is reliably obtained for every operating scenario.

4.3. Validation Results of Prediction Accuracy

To evaluate the performance of time series predictive models, the following metrics are utilized to quantify the prediction accuracy. Mean Absolute Error (MAE) quantifies the average absolute value of prediction error, which is expressed as

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}|y_i-\widehat{y_i}|$$

(29)

where n denotes the sample size and $\overline{y}$ denotes the average value of the actual samples. Root Mean Squared Error (RMSE) is defined as the square root of the mean squared errors, which is expressed as

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-\widehat{y_i})}^2}$$

(30)

Coefficient of Determination ($R^2$) quantifies how well the data fit the regression model. A value closer to 1 indicates superior predictive performance. A low $R^2$ is generally an unacceptable sign for predictive models. It is calculated by

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-\widehat{y_i})}^2}{{\sum }_{i=1}^n{(y_i-\overline{y_i})}^2}}$$

(31)

Data from ${\mathcal{D}}_{\mathrm{S}}$ is split into a training set and a validation set at a 7:3 ratio. To prevent data leakage caused by the overlapping sliding windows, this split is performed at the scenario level (combinations of $R_{\mathrm{L}}$ and M) rather than the window level, ensuring no overlap between training and validation scenarios. Both the training data and the labels were normalized. The hardware and software environment for training can be seen in

Table 2. Training hardware and software environment.

Hardware	Intel Core i9-14900HX CPU
	NVIDIA GeForce RTX 4060 GPU
	32 GB RAM
Software	Python 3.13.5
	Pytorch 2.7.1
	Numpy 2.1.3
	Adam Optimizer

. To ensure the fairness of the comparison, all baseline models have been tuned to their optimal configurations. The adaptive average pooling layer, flatten operation and final fully connected layer are kept identical across the MSPC model with CNN and TCN. All models are trained for 500 epochs with a learning rate of 0.001. The validation results are shown in

Table 3. Validation results among tested models.

Model	MAE	RMSE	${\mathit{R}}^2$
CNN	0.0799	0.1188	0.8296
TCN	0.0597	0.1006	0.8793
LSTM	0.0472	0.0789	0.8803
BiLSTM	0.0321	0.0446	0.9246
MSPC	0.0105	0.0156	0.9969

. All three loss metrics perform best in the MSPC model. Moreover, $R^2$ is extremely close to 1. The findings indicate that the MSPC model outperforms other models in prediction loss, due to its ability to extract dynamic features at multiple time scales of the WPT waveform, which contains high-frequency current components induced by power electronic switching devices.

4.4. Sensitivity Analysis on Dilation Rates

To justify the selection of dilation rates of $\left\{\right(1,1),(2,4),(8,16\left)\right\}$ mentioned in Section 3.1.1, a sensitivity analysis is conducted by comparing the proposed scheme with two alternative configurations. The architecture and other hyperparameters of the MSPC model are kept identical. The validation results are presented in

Table 4. Prediction results of the MSPC model with distinct dilation rates.

Dilation Rates	Description	MAE	RMSE	${\mathit{R}}^2$
$\left\{\right(1,1),(2,2),(4,4\left)\right\}$	Scheme 1	0.0214	0.0332	0.9812
$\left\{\right(1,1),(4,8),(16,32\left)\right\}$	Scheme 2	0.0178	0.0285	0.9885
$\left\{\right(1,1),(2,4),(8,16\left)\right\}$	Proposed Scheme	0.0105	0.0156	0.9969

The dilation rates in each scheme are assigned to ${\mathbf{F}}_{\mathrm{high}}$, ${\mathbf{F}}_{\mathrm{mid}}$, ${\mathbf{F}}_{\mathrm{low}}$, respectively.

.

Scheme 1 yields the highest RMSE (0.0332) due to its insufficient receptive field to capture the long-term fundamental envelope of $i_{AB}$. Scheme 2 also degrades performance since the overly sparse dilation skips crucial sampling points of the higher-order harmonic components. The proposed exponential scheme ($\left\{\right(1,1),(2,4),(8,16\left)\right\}$) achieves the optimal balance, which best exploits the multi-scale feature extraction capability of the MSPC model.

4.5. Ablation Study on Multi-Scale Architecture

To verify the necessity of the multi-scale parallel design in the MSPC model, an ablation study is conducted. As given in Equation (19), ${\mathbf{F}}_{\mathrm{concat}}$ is formed by concatenating ${\mathbf{F}}_{\mathrm{high}}$, ${\mathbf{F}}_{\mathrm{mid}}$ and ${\mathbf{F}}_{\mathrm{low}}$ in parallel. The performance of the MSPC model employing one or two branches is assessed. All variants were trained and validated under identical conditions. The results are shown in

Table 5. Validation results of multi-scale architecture.

${\mathbf{F}}_{\mathbf{concat}}$	MAE	RMSE	${\mathit{R}}^2$
${\mathbf{F}}_{\mathrm{high}}$	0.0583	0.1042	0.8658
${\mathbf{F}}_{\mathrm{mid}}$	0.0625	0.1030	0.8683
${\mathbf{F}}_{\mathrm{low}}$	0.0265	0.0398	0.9492
$[{\mathbf{F}}_{\mathrm{high}},{\mathbf{F}}_{\mathrm{mid}}]$	0.0186	0.0292	0.9892
$[{\mathbf{F}}_{\mathrm{high}},{\mathbf{F}}_{\mathrm{low}}]$	0.0207	0.0327	0.9847
$[{\mathbf{F}}_{\mathrm{mid}},{\mathbf{F}}_{\mathrm{low}}]$	0.0155	0.0208	0.9901
$[{\mathbf{F}}_{\mathrm{high}},{\mathbf{F}}_{\mathrm{mid}},{\mathbf{F}}_{\mathrm{low}}]$	0.0105	0.0156	0.9969

. The validation results indicate that relying on a single temporal scale yields suboptimal prediction accuracy. Specifically, ${\mathbf{F}}_{\mathrm{low}}$ achieves the best single-branch performance ($R^2=0.9492$). However, its prediction error remains relatively high (MAE = 0.0265). Conversely, ${\mathbf{F}}_{\mathrm{high}}$ and ${\mathbf{F}}_{\mathrm{mid}}$ branches alone struggle to capture the comprehensive waveform dynamics, resulting in lower $R^2$ values. This confirms that a single receptive field is insufficient to characterize the harmonic-rich WPT waveforms. Combining any two branches leads to a substantial performance improvement. Notably, the combination of $[{\mathbf{F}}_{\mathrm{mid}},{\mathbf{F}}_{\mathrm{low}}]$ yields an $R^2$ of 0.9901.

Ultimately, the proposed full architecture $[{\mathbf{F}}_{\mathrm{high}},{\mathbf{F}}_{\mathrm{mid}},{\mathbf{F}}_{\mathrm{low}}]$ achieves the optimal performance across all schemes (MAE = 0.0105, RMSE = 0.0156, $R^2$ = 0.9969). This indicates that the multi-frequency features associated with switching events play a necessary role in completing the waveform representation and minimizing prediction errors. Therefore, the parallel multi-scale design is strictly necessary to comprehensively extract the multi-scale harmonic information in WPT systems.

4.6. Computational Complexity and Inference Latency

To evaluate the feasibility of deploying the MSPC model on resource-constrained embedded systems, a comparison of computational complexity is conducted. Floating Point Operations (FLOPs) represent the total number of addition and multiplication operations required to perform a single model inference. A lower FLOPs count indicates reduced computational demand, resulting in faster processing speed on embedded platforms. This work compares the FLOPs and parameter counts of the MSPC network with four benchmark models and three widely used algorithms for edge computing (TinyRNN, MiniRocket and 1D Transformer). The results are shown in

Table 6. Comparison results of computational complexity.

Model	FLOPs (M)	Parameters (K)
CNN	3.056	1.539
TCN	1.920	0.963
LSTM	102.912	50.627
BiLSTM	271.360	134.019
TinyRNN	1.760	0.899
MiniRocket	1.638	1.074
1D Transformer	133.248	67.011
MSPC	2.096	1.147

. The shapes of input and output data of validation models remain consistent.

As observed, LSTM, BiLSTM and 1D Transformer suffer from massive computational overhead, while the MSPC model maintains a lightweight profile with 2.096 $\mathrm{M}$ FLOPs and 1.147 $\mathrm{K}$ parameters. This demonstrates the potential of the MSPC model for future deployment on embedded devices. Furthermore, the inference latency of the proposed MSPC model was measured. The average single-inference time over 100 consecutive runs is merely 1.337 ms. This verifies the real-time feasibility of the MSPC model for embedded WPT controllers.

5. Experiments

5.1. Experimental Setup

To verify the above data-driven method for characteristic prediction and output optimization, an SS compensated WPT prototype is built, as shown in Figure 10. The parameters of the experimental setup are given in Table 1. To realize precise frequency hopping, Microcontroller TMS320F28335 is used to drive $Q_1$–$Q_4$. Operating at a kHz-level switching frequency, $Q_1$–$Q_4$ use MOSFET IPW65R080CFD and $D_1$–$D_4$ use diodes IDW15E65D2 from Infineon Inc. The initial duty cycle D of the PWM is set as 0.95. To ensure ZVS, $C_{\mathrm{S}}$ of 44.107 nF has a decrement compared to the calculated value.

A current sensor probe (CP1015) is used to detect the waveform of $i_{AB}$. During the experiment, the misalignment distance is varied within the range 0–55 mm, which results in a range of 2.9 μH to 6.7 μH for the mutual inductance variation. The current sensor probe is used to transmit the detected waveform to the host computer, which runs MATLAB/Simulink 2024a and Visual Studio Code for generating simulation data and operating DL models.

The initial 30 experimental scenarios for ${\mathcal{D}}_{\mathrm{T}}$ data are selected to cover the primary operational ranges, with M varying from approximately 4.46 to 6.53 μH and $R_{\mathrm{L}}$ ranging from 1.2 to 30 Ω. Subsequently, data augmentation is applied to expand the scenario numbers from 30 to 300. To apply transfer learning, the Multi-Scale Feature Extractor of MSPC is frozen to preserve the feature extraction capability learned from ${\mathcal{K}}_{\mathrm{S}}$. The parameters of the Global Regressor are fine-tuned using the data from ${\mathcal{D}}_{\mathrm{T}}$. Additionally, the output linear layer of the comparative models is unfrozen. The fine-tuning epoch of TL is 500. The hardware and software environment remains the same as what is shown in Table 2. Figure 11 shows the experimental and simulated $i_{\mathrm{AB}}$ waveforms when $R_{\mathrm{L}}=14$ Ω and $M=5.2$ μH. As depicted in the figure, the measured current waveform exhibits slight deviations in both amplitude and frequency compared to the simulated data, as mentioned in Section 3.3. The Total Harmonic Distortion (THD) also varies due to the non-ideal characteristics of practical circuit connections. Moreover, the inherent sensor noise from measurement equipment cannot be neglected.

5.2. Characteristic Prediction Results

To evaluate the prediction accuracy of the MSPC model assisted with transfer learning, a full factorial experiment was conducted with 15 levels of $R_{\mathrm{L}}$ and four levels of M, resulting in 60 distinct test conditions.

Table 7. Validation error of tested models with and without TL.

Model	${\mathit{R}}_{\mathbf{L}}$	M	$\mathit{\alpha }$	$\mathit{\beta }$
CNN	32.88%	12.66%	60.17%	73.41%
TCN	33.47%	12.52%	53.40%	64.23%
LSTM	47.90%	19.11%	59.47%	65.97%
BiLSTM	39.65%	28.47%	53.21%	71.43%
MSPC	45.64%	12.14%	53.62%	63.84%
CNN+TL	30.76%	6.23%	18.64%	10.29%
TCN+TL	35.66%	9.98%	7.85%	15.94%
LSTM+TL	10.96%	8.63%	10.07%	8.36%
BiLSTM+TL	7.32%	3.18%	3.38%	6.29%
MSPC+TL	2.55%	0.91%	1.52%	2.64%

illustrates the prediction performance of tested models without and with TL, respectively. Models trained only on simulation data suffer from severe prediction errors in physical scenarios. For instance, the errors for $\alpha$ and $\beta$ exceed $50\%$ across all models. This clearly highlights the gap between simulation and real-world environments. However, assisted by TL, all models show significant improvement in accuracy. This indicates TL’s generalization capability in practical applications. The proposed MSPC model combined with TL achieves the lowest errors. Its average relative error remains below $2.55\%$ across all four predicted parameters. This better performance illustrates its effective multi-scale feature extraction and generalization ability.

Figure 12 shows the prediction results of MSPC assisted by TL. Most predicted values closely match the measured ones, with the average error remaining below 3% and 1% for $R_{\mathrm{L}}$ and M, respectively. The results comparison also highlights the advantage of TL in reducing the discrepancy between simulated and actual conditions, which is crucial for characteristic prediction with information only from TX in WPT systems.

5.3. Output Optimization Results

To validate the output optimization performance of the data-driven method, the predicted $\beta$ is used to compensate for $I_{\mathrm{O}}$ attenuation by multiplying $U_{\mathrm{DC}}$ by $\beta$. Regarding error propagation, the direct prediction of $\beta$ is decoupled from the estimation of $R_{\mathrm{L}}$ and M. This ensures that characteristic prediction errors do not propagate to the output optimization process, guaranteeing the robustness of the CC regulation. Figure 13 shows the result of output optimization under different load resistances when M is 4.98 μH. ${\alpha }_{\mathrm{real}}$ denotes the ratio of the actual output DC current to the theoretical one. By querying the MSPC model’s output results under each scenario, $U_{\mathrm{DC}}$ is manually multiplied by ${\beta }_{\mathrm{pred}}$. ${\beta }_{\mathrm{sim}}$ is derived from simulation using Brent’s method mentioned in Section 4.2. And ${\beta }_{\mathrm{real}}$ is obtained by continuously adjusting the voltage of the DC power supply until $I_{\mathrm{O}}$ is fully compensated. The results illustrate that simply using data from ${\mathcal{D}}_{\mathrm{S}}$ cannot reliably predict the deviation factor or optimize the CC output current attenuated by heavy load resistance. In contrast, the proposed MSPC model with TL effectively compensates for this nonlinear error.

Figure 14 and Figure 15 show the experimental waveforms of the prototype without and with output optimized under four operating conditions, respectively, where $u_{\mathrm{GS}}$ represents the voltage between the gate and source of MOSFET $Q_1$. As shown in Figure 14, $I_{\mathrm{O}}$ exhibits attenuation under increasing $R_{\mathrm{L}}$. For instance, when M is $6.53\mathsf{\mu }\mathrm{H}$, increasing the load resistance from $15\Omega$ to $30\Omega$ causes a noticeable drop in $I_{\mathrm{O}}$ from 2.11 A to 1.98 A. A similar degradation trend is observed when M is $5.63\mathsf{\mu }\mathrm{H}$.

The steady-state waveforms obtained after applying the proposed data-driven optimization are depicted in Figure 15. By scaling the DC input voltage according to the predicted $\beta$, $I_{\mathrm{O}}$ is successfully restored and maintained at a stable level. Specifically, under the same $6.53\mathsf{\mu }\mathrm{H}$ condition, the output at $15\Omega$ and $30\Omega$ are clamped at 2.38 A and 2.37 A, respectively, with a tiny deviation from the preset value (2.3707 A). These experimental results consistently validate that the proposed data-driven optimization method can effectively compensate for the nonlinear current attenuation. Figure 14 and Figure 15 show that $u_{\mathrm{AB}}$ and $i_{\mathrm{AB}}$ are free from ringing at the switching instants. This indicates that the system maintains clear ZVS across all scenarios. For instance, when $R_{\mathrm{L}}$ is $15\Omega$ and M is 6.53 $\mathsf{\mu }\mathrm{H}$, the zero-crossing point of $i_{\mathrm{AB}}$ lags behind that of $u_{\mathrm{AB}}$ by approximately 400 ns (phase angle = ${28.8}^{\circ }$). The lagging extent of the current $i_{\mathrm{AB}}$ decreases with increasing $R_{\mathrm{L}}$. When $R_{\mathrm{L}}$ reaches its maximum value of 30 $\Omega$, ZVS reaches its critical boundary. The soft switching performance is preserved after output optimization.

In future embedded deployments, the DSP will sample the TX current in real time. The lightweight MSPC model will then infer the value of $\beta$. The target DC input voltage required by the WPT system can be calculated as $U_{\mathrm{reg}}=\beta \times U_{\mathrm{th}}$. Subsequently, the PWM duty cycle of the front-end DC/DC converter (as shown in Figure 2) is regulated to track this target voltage. Through this continuous process, the output current of the WPT system is automatically stabilized at the preset value without manual intervention.

6. Conclusions

This work has proposed a data-driven method for joint characteristic prediction and output optimization for WPT systems relying solely on the TX AC current. A Multi-Scale Parallel Convolutional neural network is introduced to predict load resistance, mutual inductance, deviation factor, and regulation coefficient, while the output current is optimized by multiplying the input DC voltage by the regulation coefficient under heavy load conditions. Transfer learning and data augmentation are applied to minimize the discrepancy between simulation data and physical conditions. The MSPC model demonstrates excellent prediction performance, achieving an $R^2$ of 0.9969. In experimental validation, the average relative error for characteristic prediction is below $2.55\%$, and the output optimization effectively maintains the CC output against load variations.