A Receiver Position Estimation Method Based on LSTM for Multi-Transmitter Single-Receiver Wireless Power Transfer Systems

Abstract

The multi-transmitter single-receiver wireless power transfer (MTSR-WPT) system has good tolerance for coil misalignment because the magnetic fields generated by multiple transmitters can be shaped to adapt to position changes in the receiver coil. In order to achieve magnetic field shaping of the MTSR-WPT system and increase power transfer efficiency, accurately estimating the position of the receiver coil is a key issue that needs to be addressed. In this article, a receiver position estimation method based on long short-term memory (LSTM) is proposed, which utilizes a data-driven approach to establish a neural network model. By learning the relationship between the measured time-series voltage data of the transmitter coils and the position of the receiver coil, the proposed model can achieve accurate position estimation of the receiver. Compared with previous works, the proposed method does not require communication between the transmitter and receiver, which is conducive to simplifying the system structure and reducing costs. In addition, the proposed LSTM-based method requires less derivation of complex formulas and the internal mechanism analysis of the system. Finally, a MTSR-WPT prototype is built to verify the proposed method. The experimental results show that the proposed LSTM-based method can achieve high-accuracy position estimation of the receiver. When the receiver moves within a range of 160 mm × 160 mm, the average error between the estimated receiver coil position using the proposed method and the actual receiver coil position is less than 2.40 mm.

1. Introduction

Wireless power transfer (WPT) using coupled magnetic resonance can achieve energy transmission between the charging equipment and power supply in a completely isolated state [1,2], which has been widely applied in various fields, such as wireless charging for wearable devices [3,4,5], medical devices [6,7,8], electric vehicles [9,10,11], unmanned aerial vehicles [12,13,14,15], etc.

In single-transmitter single-receiver (STSR) WPT systems, coil misalignment often occurs, which can significantly reduce the transmission efficiency of the system [16]. To improve the tolerance of the STSR-WPT system for coil misalignments, some researchers studied the optimization design methods of coils and developed DD and DQD coil structures [17,18,19]. The DD and DQD coils have good adaptability to lateral coil misalignment but have poor adaptability to longitudinal and angular coil misalignment.

In order to adapt to different types of coil misalignment, some researchers investigated the topology of the WPT system and proposed multi-transmitter single-receiver (MTSR) WPT systems. In the MTSR-WPT system, the magnetic fields generated by multiple transmitters can be shaped to adapt to position changes in the receiver coil in different directions [20,21]. Therefore, the MTSR-WPT system has good tolerances for lateral, longitudinal, and angular coil misalignments.

In order to achieve optimal magnetic field shaping and improve the overall efficiency of the MTSR-WPT system, accurately estimating the position of the receiver coil is a key issue that needs to be addressed. In engineering applications, the position estimation of the receiver can be realized by installing cameras or ranging devices such as LiDAR. But these methods will introduce additional hardware components, which will significantly increase the structural complexity and cost of the WPT system. To address these issues, receiver coil position estimation methods based on the online parameter measurement of the MTSR-WPT system were studied [22,23,24,25,26,27].

In [22], a STSR-WPT system for electric vehicle charging was studied. By adding eight sensor coils to the receiver coil and cooperating with the detection circuit, the receiver position estimation in the horizontal direction was realized. In [23], a WPT system for a multi-DOF motor was studied. Based the mechanism model of the WPT system, a mathematical formula for mutual inductance between the transmitter and receiver coils was derived. Then, the position of the receiver coil can be estimated based on the mutual inductance value. In [24], a 3 × 3 array multi-transmitter wireless power transfer system was studied. Similarly, the mutual inductance parameters were calculated first, and then the position was estimated based on the mutual inductance. However, the above methods still have some limitations. Using additional sensor coils and detection circuits to estimate the receiver position will significantly increase system size and cost, and may also affect the transmission efficiency of the WPT system. Estimating the receiver position through the mathematical formula of mutual inductance is highly dependent on the accuracy of the established mechanism model of the WPT system. Moreover, in the MTSR-WPT system, as the number of coupling coils increases, the mechanism model of the system becomes very complex. It is difficult to estimate the position of the receiver using the mathematical formula of mutual inductance. Therefore, it is necessary to find a receiver position estimation method, without increasing the complexity of the WPT system, that does not rely on the accuracy of the mechanism model.

In order to overcome the shortcomings of mechanism models, data-driven models have been proposed by researchers [25,26,27]. In [25], a WPT system with two orthogonally placed transmitter coils was studied. By using a backpropagation (BP) neural network to explore the relationship between receiver angle and transmitter coil current, the prediction of the receiver angle was achieved. In [26], a neural network (NN) model was proposed for coil misalignment prediction and metal object detection. The established NN model can output the horizontal position of the receiver coil based on the measured voltage of the transmitter coil. In [27], aiming at the multi-transmitter multi-receiver (MTMR) WPT system for Automatic Guided Vehicle (AGV) charging, the BP algorithm and support vector regression (SVR) algorithm were studied for AGV position estimation. By comparing the position estimation accuracy of the two algorithms, the BP algorithm estimation model with smaller errors was selected. Although the methods proposed in [25,26,27] can be used to estimate the receiver position, the final estimation accuracy can still be improved.

To overcome the limitations of previous receiver position estimation methods, this article proposes a long short-term memory (LSTM)-based position estimation model to address the receiver position estimation issue in the MTSR-WPT system. By learning the relationship between the measured time-series voltage data of the transmitter coils and the position of the receiver coil, the proposed model can achieve accurate position estimation of the receiver.

The rest of this article is structured as follows. Section 2 describes the structure of the MTSR-WPT system and reveals the relationship between the receiver position and the voltage of transmitter coils. Section 3 provides a detailed introduction to the proposed LSTM-based receiver position estimation method. The experimental verification of the proposed receiver position estimation method is introduced in Section 4, and the conclusions are drawn in Section 5.

2. Analysis of the MTSR-WPT System

Figure 1 shows the equivalent circuit model of the MTSR-WPT system with series compensation capacitors, which contains n transmitters and one single receiver. In the k-th transmitter loop, $V_k$ represents the open circuit voltage of the power supply. $R_{T,k}$ and $C_{T,k}$ represent the equivalent resistance and compensation capacitance of the transmitter circuit. $L_{T,k}$ is the inductance of the transmitter coil. $I_{T,k}$ represents the current of the transmitter circuit. $U_k$ represents the voltage of the transmitter coil. $R_R$ and $C_R$ represent the equivalent resistance and compensation capacitance of the receiver circuit. $L_R$ is the inductance of the receiver coil. $R_L$ is the load resistance. $M_{i,j}$ represents the mutual inductance between the i-th and j-th transmitter coils, and $M_{R,k}$ represents the mutual inductance between the k-th transmitter coil and the receiver coil.

To further investigate the relationship between the position of the receiver coil and mutual induction $M_{R,k}$, assuming the relative position of the k-th transmitter coil and the receiver coil is shown in Figure 2. $R_1$ and $R_2$ are the radii of the transmitter and receiver coils, respectively. $C_R(x_R,y_R,z_R)$ is the center of the receiver coil, and the receiver coil is parallel to the plane $ax+by+cz=0$. The current in the transmitter coil is $I$.

Based on [28], the mutual inductance between the coils can be calculated as follows: Firstly, the magnetic potential $\overrightarrow{A}$ is calculated from $I$. Secondly, the magnetic flux $\mathsf{\Phi }$ can be calculated according to $\overrightarrow{A}$ by the Stokes theorem. Finally, according to the definition of mutual inductance $M=\mathsf{\Phi }/\mathrm{I}$, the mutual inductance between the transmitter and receiver coils $M_{R,k}$ can be calculated as

$$\left\{\begin{array}{l}{M}_{R,k}={\displaystyle \frac{{\mu }_0{R}_2}{\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{[p_1\cos \phi +p_2\sin \phi +p_3]\Psi (k)}{k\sqrt{V_0{}^3}}d\phi },\\ \mathsf{\alpha }={\displaystyle \frac{R_2}{R_1}},\beta ={\displaystyle \frac{x_R}{R_1}},\gamma ={\displaystyle \frac{y_R}{R_1}},\delta ={\displaystyle \frac{z_R}{R_1}},L=\sqrt{a^2+b^2+c^2},\mathcal{l}=\sqrt{a^2+c^2},\\ p_1=\pm {\displaystyle \frac{\gamma c}{\mathcal{l}}},p_2=\mp {\displaystyle \frac{\beta {\mathcal{l}}^2+\gamma ab}{\mathcal{l}L}},p_3={\displaystyle \frac{\alpha c}{L}}, p_4=\pm {\displaystyle \frac{\beta ab-\gamma {\mathcal{l}}^2+\delta bc}{\mathcal{l}L}},p_5=\mp {\displaystyle \frac{\beta c-\delta a}{\mathcal{l}}},\\ V_0{}^2={\alpha }^2\left[\left(1-{\displaystyle \frac{b^2{c}^2}{{\mathcal{l}}^2{L}^2}}\right){\cos }^2\phi +{\displaystyle \frac{c^2}{{\mathcal{l}}^2}}{\sin }^2\phi +{\displaystyle \frac{abc}{{\mathcal{l}}^{2}L}}\sin 2\phi \right]+{\beta }^2+{\gamma }^2\mp 2\alpha {\displaystyle \frac{\beta ab-\gamma {\mathcal{l}}^2}{\mathcal{l}L}}\cos \phi \mp {\displaystyle \frac{2\alpha \beta c}{\mathcal{l}}}\sin \phi ,\\ F(\rho ,k)={\displaystyle \underset{0}{\overset{\rho }{\int }}\frac{1}{\sqrt{1-k^2{\sin }^2\theta }}d\theta },E(\rho ,k)={\displaystyle \underset{0}{\overset{\rho }{\int }}\sqrt{1-k^2{\sin }^2\theta }d\theta },\\ A_0=1+{\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2+2\alpha \left(p_4\cos \phi +p_5\sin \phi \right),k=\sqrt{{\displaystyle \frac{4V_0}{A_0+2V_0}}},\mathsf{\Psi }\left(k\right)=\left(1-{\displaystyle \frac{k^2}{2}}\right)K\left(k\right)-E\left(k\right)\end{array}\right.$$

(1)

where ${\mu }_0$ is vacuum permeability, $K\left(k\right)=F(\rho =\pi /2,k)$ is the complete elliptic function of the first kind, and $E(k)=E(\rho =\pi /2,k)$ is the complete elliptic function of the second kind. The detailed derivation process can be seen in [28].

The change in mutual inductance $M_{R,k}$ caused by the receiver coil position change can affect the state of the circuit system. Assuming that the operating frequency of the MTSR-WPT system is $f$, the KVL equations of the transmitter and receiver circuits can be derived as

$$\left[\begin{array}{ccccc}{Z}_1& -j\omega M_{1,2}& \cdots & -j\omega M_{1,n}& -j\omega M_{R,1}\\ -j\omega M_{1,2}& Z_2& \cdots & -j\omega M_{2,n}& -j\omega M_{R,2}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -j\omega M_{1,n}& -j\omega M_{2,n}& \cdots & Z_n& -j\omega M_{R,n}\\ -j\omega M_{R,1}& -j\omega M_{R,2}& \cdots & -j\omega M_{R,n}& Z_R\end{array}\right]\left[\begin{array}{c}{I}_{T,1}\\ I_{T,2}\\ ⋮\\ I_{T,n}\\ I_R\end{array}\right]=\left[\begin{array}{c}{V}_1\\ V_2\\ ⋮\\ V_n\\ 0\end{array}\right]$$

(2)

where

$$\left\{\begin{array}{l}{Z}_k=R_{T,k}+1/j\omega C_{T,k}+j\omega L_{T,k}, k=1,2,\cdots ,n\hfill \\ Z_R=R_L+R_R+1/j\omega C_R+j\omega L_R\hfill \\ I_{T,k}=U_k/j\omega L_{T,k}, k=1,2,\cdots ,n\hfill \\ \omega =2\pi f\hfill \end{array}\right.$$

(3)

Here, we have explored the connection among the receiver position, mutual inductance $M_{R,k}$, and the state of the circuit system, such as the transmitter coil voltage $U_k$. Assuming that the TX coil voltage $U_k$ is the state response of the MTSR-WPT circuit system when the RX coil position changes and is being measured, the relationship between TX coil voltage $U_k$ and mutual induction $M_{R,k}$ needs to be derived to reverse the change in the receiver coil position. By substituting (3) into (2), $M_{R,k}$ can be expressed as follows: when $n=1$,

$$M_{R,1}={\displaystyle \frac{Z_R{}^{(1/2)}{(-Z_1{U}_1+j\omega L_{T,1}{V}_1)}^{(1/2)}}{\omega U_1{}^{(1/2)}}}$$

(4)

when $n=2$,

$$\left\{\begin{array}{l}{M}_{R,1}={\displaystyle \frac{{(-Z_R/\alpha )}^{(1/2)}}{\omega {\gamma }_1/\beta {\gamma }_2}},M_{R,2}={\displaystyle \frac{{(-Z_R/\alpha )}^{(1/2)}}{\omega /\beta }}\hfill \\ \alpha =-Z_2{L}_{T,1}{}^2{U}_2{}^2-Z_1{L}_{T,2}{}^2{U}_1{}^2+j\omega L_{T,1}{}^2{L}_{T,2}{V}_2{U}_2+j\omega L_{T,1}{L}_{T,2}{}^2{V}_1{U}_1+2j\omega M_{1,2}{L}_{T,1}{L}_{T,2}{U}_1{U}_2\hfill \\ \beta =\omega L_{T,1}{L}_{T,2}{V}_2+\omega M_{1,2}{L}_{T,1}{U}_1+j{Z}_2{L}_{T,1}{U}_2\hfill \\ {\gamma }_1=-Z_2{L}_{T,1}{U}_2+j\omega L_{T,1}{L}_{T,2}{V}_2+j\omega L_{T,2}{M}_{1,2}{U}_1\hfill \\ {\gamma }_2=-Z_1{L}_{T,2}{U}_2+j\omega L_{T,1}{L}_{T,2}{V}_1+j\omega M_{1,2}{L}_{T,1}{U}_2\hfill \end{array}\right.$$

(5)

If $n\ge 3$, the expression of $M_{R,k}$ will become very complex due to the mutual coupling between multiple coils. Furthermore, if the receiver position is derived based on mutual inductance $M_{R,k}$, the analytical solution for the receiver position is difficult to obtain due to the use of elliptic integrals in (1). Additionally, since the measurements of the $U_k$ quantity involve extensive calculations, there is a risk of measurement errors being amplified due to multiple operations.

Therefore, it is necessary to explore a more effective and accurate position estimation method for the MTSR-WPT system.

3. Position Estimation Method Based on LSTM

Based on the analysis in Section 2, it can be concluded that there exists a certain internal relationship between $U_k$ and the receiver position. Utilizing a neural network model to establish the connection between the two is a highly feasible approach. In this article, a receiver position estimation method based on LSTM is proposed. By using the ability of the neural network to approximate any function, a receiver position estimation neural network model is established, where LSTM is used in the input layer to process the time-series voltage data of the transmitter coils $\{u_k(t)\}$.

3.1. Standard LSTM Model

As a variant of RNN, LSTM can overcome some limitations of RNN, such as long-term dependency issues. In addition, LSTM can effectively solve the problems of vanishing and exploding gradients in RNN by introducing memory gates and gating mechanisms. Therefore, LSTM can better capture and process long-term dependencies in time series data, especially when dealing with long sequences. The typical information flow within an LSTM unit is shown in Figure 3a, and the internal detailed structure of a single LSTM unit is shown in Figure 3b.

In a single unit, $x_t$, $h_t$, $C_t$ respectively represent the input, output, and unit state at time t. Circular nodes represent pointwise operation. Rectangular nodes represent neural network layers. The three gates of LSTM are composed of circular and rectangular nodes, which are forget, input, and output gates. As shown in Figure 3b, the information flow within a single unit at time t is as follows: Firstly, based on the previous hidden state $h_{t-1}$ and the current input variable $x_t$, the forget gate determines how much long-term memory $C_{t-1}$ is inherited, with a ratio of $f_t$. Secondly, the input gate selects new information ${\tilde{C}}_t$ to be stored in the LSTM’s memory with a ratio of $i_t$. Then, the current unit state $C_t$ is calculated. Finally, the output gate selectively outputs a new hidden state $h_{ t}$ based on $C_t$ and the selection rate $o_t$. The derivation between these values is as follows:

$$\left\{\begin{array}{l}{f}_t=\sigma \left(W_f\cdot \left[h_{t-1},x_t\right]+b_f\right)\\ i_t=\sigma \left(W_i\cdot \left[h_{t-1},x_t\right]+b_i\right)\\ {\tilde{C}}_t=\tanh \left(W_C\cdot \left[h_{t-1},x_t\right]+b_C\right)\\ C_t=f_t\cdot C_{t-1}+i_t\cdot {\tilde{C}}_t\\ o_t=\sigma \left(W_o\cdot \left[h_{t-1},x_t\right]+b_o\right)\\ h_{ t}=o_t\cdot \tanh \left(C_t\right)\end{array}\right.$$

(6)

3.2. Position Estimation Method

The schematic diagram of the receiver position estimation method based on LSTM is shown in Figure 4. The raw voltage data of the receiver coils collected by sensors are selected as the system observations and the input data of the position estimation model. Thus, all acquisition processes are completed on the transmitter side, and the measurement on the receiver side can be avoided. The output data of the model is the estimated receiver position.

The main steps of the proposed position estimation method include:

(1) Circuit parameter initialization. The circuit parameters of the MTSR-WPT system need to be designed, including the operating frequency, power supply, coil inductance, resonant capacitance, load resistance, etc.

(2) Raw data collection. According to the left part of Figure 4, the collected raw data include the receiver position $P_R$ and the time-series voltage of the transmitter coils $\left\{{\left.u_k(t)\right|}_{t=m{T}_s}\right\}$. $\left\{{\left.u_k(t)\right|}_{t=m{T}_s}\right\}$ and $P_R$ are used as input data and output data for the model training. It is worth noting here that the dimension information of $\left\{{\left.u_k(t)\right|}_{t=m{T}_s}\right\}$ is

$$\left\{{\left.u_k(t)\right|}_{t=m{T}_s},k=1,2,\dots ,n,m=1,2,\dots N\right\}$$

(7)

where k represents the number of transmitters. Assuming the operating frequency of the MTSR-WPT system is $f$ and the sampling frequency is $f_s$, then $T_s=1/f_s$ represents the sampling interval, and $m{T}_s$ represents the sampling time. In order to cover the complete information in a voltage change cycle under the system operating frequency, at least $N=⌈f_s/f⌉$ sampling points are required. Therefore, the dimension of the input data is $n\times N$.

(3) Data normalization and classification. The raw data of the coils are normalized and divided into three groups: training data, validation data, and test data.

(4) Model initialization. Before training the position estimation model, it is necessary to determine the initial settings. In the input layer, LSTM is used, and two key parameters need to be determined.

① Input shape

The shape of input data is usually represented as a binary array (timesteps, input dim), where timesteps is the length of the input sequence and input dim is the number of input features. Therefore, the input shape of the MTSR-WPT system is (n, N).

② Units

The number of LSTM in the input layer determines the dimension of the internal states (including unit state and hidden state) of the LSTM layer. These LSTM units work together to capture long-term dependencies in the input data and generate a fixed size output vector. Then, this output vector is passed to the fully connected layer to generate the final estimated value. In the MTSR-WPT system, the collected raw voltage data can be regarded as a sine signal, including amplitude, phase, and other information. Thus, units should take a positive integer greater than 2n.

Both the hidden layer and the output layer use fully connected networks, and the number of output units at the end of the output layer remains consistent with the dimension of $P_R$. Before formal training, some hyper-parameters also need to be determined, including learning rate, batch size, epochs, activation function, loss, etc.

(5) Model training. After the model initialization, the training and validation data are used to train the model. The training process stops after reaching the preset epochs or other conditions. Before the entire process ends, the test data are used to perform loss testing on the trained model to verify output accuracy. If the output accuracy meets the standard, the process ends; otherwise, the model training is repeated.

According to the proposed position estimation method, the corresponding relationship between the observed transmitter voltage and the receiver position can be obtained. The output of the trained model can achieve the receiver position estimation and can also reduce output errors in practical applications through online learning.

4. Experimental Verification

4.1. Implementation and Measurement of the MTSR-WPT System

In order to verify the proposed position estimation method, an experimental MTSR-WPT system is built, as shown in Figure 5a, which mainly consists of four transmitter coils, a receiver coil, a high-frequency voltage source, multiple compensation capacitors, and non-inductive resistance. The details of the experimental system are shown blow.

(1)

High-frequency voltage source

The signal generator FY6900 (manufactured by FeelElec Corp., Zhengzhou, China) and power amplifier FPA301 (manufactured by FeelElec Corp., Zhengzhou, China) are used to form a high-frequency voltage source, as shown in Figure 5a, which can independently adjust the amplitude and phase of the input voltages required by the MTSR-WPT system.

(2)

Circuit boards of the transmitter and receiver

In Figure 5b, the circuit board of transmitter 1 (Tx-1) is shown as an example. The circuit connections are indicated using solid black lines. The voltage source is connected to the right port of the circuit board. The left port is connected to channel 1 of the oscilloscope. The circuit board of the receiver is shown in Figure 5c, which includes a receiver coil, a resonant capacitor, and a load resistor. The circuit connections are indicated using solid black lines.

(3)

Measurement equipment

The OWON VDS3104 oscilloscope (manufactured by Fujian Lilliput Optoelectronics Technology Co., Ltd., Zhangzhou, China) is used to measure the voltage of the transmitter coils. The oscilloscope with four independent channels can simultaneously record the voltage data of the transmitter coils, where channel-1, channel-2, channel-3, and channel-4 are used to record the voltage data of Tx-1 $u_1(t)$, Tx-2 $u_2(t)$, Tx-3 $u_3(t)$, Tx-4 $u_4(t)$, respectively. The impedance analyzer Agilent 4294A (manufactured by Agilent Technologies, Santa Clara, USA) is used to measure the values of the coil inductance, capacitance, and equivalent resistance. The measurement result of the Tx-1 coil is shown in Figure 5d. It can be seen that at 1.436 MHz, the coil inductance and equivalent resistance of the Tx-1 are 56.2 μH and 3.2 ohms, respectively. The main system parameters of the experimental MTSR-WPT system are listed in Table 1.

Figure 5. Layout of the experimental MTSR-WPT system and measurement result. (a) Overall hardware implementation of the MTSR-WPT System; (b) circuit board of the Tx-1; (c) circuit board of the receiver; (d) measurement result of the Tx-1 coil by Agilent 4294A at 1.436MHz.

Figure 5. Layout of the experimental MTSR-WPT system and measurement result. (a) Overall hardware implementation of the MTSR-WPT System; (b) circuit board of the Tx-1; (c) circuit board of the receiver; (d) measurement result of the Tx-1 coil by Agilent 4294A at 1.436MHz.

Table 1. Parameter values of the experimental MTSR-WPT system.

Parameter Type	Parameter Value
Coil inductances (μH) $L_{T,1}$/$L_{T,2}$/$L_{T,3}$/$L_{T,4}$/$L_R$	56.2/56.0/55.9/56.1/56.3
Capacitances (pF) $C_{T,1}$/$C_{T,2}$/$C_{T,3}$/$C_{T,4}$/$C_R$	220.1/216.5/219.7/226.2/224.0
Resistances (Ω) $R_{T,1}$/$R_{T,2}$/$R_{T,3}$/$R_{T,4}$/$R_R$/	3.6/3.5/3.4/3.4/3.6/50
Height of the receiver plane (mm) $z_R$	50
Frequencies (MHz) $f$/$f_s$	1.436/250

Table 1. Parameter values of the experimental MTSR-WPT system.

Parameter Type	Parameter Value
Coil inductances (μH) $L_{T,1}$/$L_{T,2}$/$L_{T,3}$/$L_{T,4}$/$L_R$	56.2/56.0/55.9/56.1/56.3
Capacitances (pF) $C_{T,1}$/$C_{T,2}$/$C_{T,3}$/$C_{T,4}$/$C_R$	220.1/216.5/219.7/226.2/224.0
Resistances (Ω) $R_{T,1}$/$R_{T,2}$/$R_{T,3}$/$R_{T,4}$/$R_R$/	3.6/3.5/3.4/3.4/3.6/50
Height of the receiver plane (mm) $z_R$	50
Frequencies (MHz) $f$/$f_s$	1.436/250

(4)

Coils

The geometric dimensions of the transmitter coils and the receiver coil are the same. The coils are winded using 0.07 mm × 400 strands Litz wires. As shown in Figure 6a, the inner and outer diameters of the coil are 120 mm and 185 mm, respectively. The number of turns of the coil is 15. The arrangement of the transmitter and receiver coils is shown in Figure 6b. The geometric center of the transmitter coils is taken as the coordinate origin to establish a rectangular coordinate system. The receiver is parallel to the $Oxy$ plane with its center moving along the x and y directions.

Figure 6. Geometrical size and arrangement of the coils. (a) Geometrical size of the coil; (b) arrangement of the receiver and transmitter coils.

Figure 6. Geometrical size and arrangement of the coils. (a) Geometrical size of the coil; (b) arrangement of the receiver and transmitter coils.

4.2. Verification of the Proposed LSTM-Based Model

(1)

Data collection

In this paper, in order to simplify the experiment, only x and y directions are considered when the receiver position changes. Therefore, a data sample should contain one receiver position data and the corresponding transmitter voltage data.

The format of the data sample is

$$\mathrm{label}=\left(x_R,y_R\right), \mathrm{data}=\left\{{\left.u_k(t)\right|}_{t=m{T}_s}\right\}$$

The distribution of the labels on the $Oxy$ plane is shown in Figure 7a. The red dots represent the data for training and validation, while the yellow crosses represent the test data. Figure 7b shows the measured voltage waveforms using the oscilloscope at two different positions.

When located at $P_R$ = (−2.5 mm, −2.5 mm), the receiver is almost centered around the transmitters, so the voltage of each channel is almost the same.

When located at $P_R$ = (77.5 mm, 77.5 mm), the receiver is almost directly above the Tx-3, resulting in a decrease in the voltage amplitude of channel-3, while the amplitudes of other channels increase.

(2)

Construction and Training of the Estimation Model

This article utilizes the TensorFlow framework and its high-level API, Keras, to build and train the LSTM-based position estimation model. Based on the parameters of the experimental prototype of the MTSR-WPT system, some hyperparameters and initialization settings of the estimation model are shown in Table 2.

Figure 7. Sampling positions of the receiver coil and measured voltage waveforms of the transmitter coils. (a) Schematic diagram of sampling positions; (b) display of the oscilloscope at two different positions.

Figure 7. Sampling positions of the receiver coil and measured voltage waveforms of the transmitter coils. (a) Schematic diagram of sampling positions; (b) display of the oscilloscope at two different positions.

Table 2. Hyperparameters and initialization settings of the estimation model.

Option	Setting
Input Shape	(174, 4)
LSTM Units	20
Initial Leaning Rate	0.001
Optimizer	Adaptive Moment Estimation (Adam)
Loss	Mean Absolute Error (MAE)

Table 2. Hyperparameters and initialization settings of the estimation model.

Option	Setting
Input Shape	(174, 4)
LSTM Units	20
Initial Leaning Rate	0.001
Optimizer	Adaptive Moment Estimation (Adam)
Loss	Mean Absolute Error (MAE)

(3)

Training Result

The loss curve during the training of the estimation model is shown in Figure 8. As the epoch increases, the loss on both training data and validation data decreases. The downward trend of the two curves remains consistent, which indicates that there is no overfitting during the model training.

Figure 9 shows the output results of the estimation model. The red dots represent the original points used as labels, and the blue dots represent the estimated positions of the receiver coil by the model.

Figure 9a–c respectively show the estimated positions of the receiver coil by the model in the early (Epochs = 20), middle (Epochs = 200), and late (Epochs = 400) stages of the training process. It can be seen that the estimation error of the model significantly decreases with the increase in epoch. It is evident that the LSTM-based estimation model exhibits higher output accuracy for positions closer to the center of the transmitter. Figure 9d shows the estimated positions of the receiver coil when replacing the model input layer with RNN. The error is the average Euclidean distance between the estimation point and the original point. The output errors of different models under various epochs are shown in

Table 3. Comparison of output error.

Input Layer	Epochs	Average Error (mm)
LSTM	20	13.15
	200	6.62
	400	2.40
RNN	400	19.33

. The estimation error of the model with RNN is much higher than the model with LSTM. Therefore, the proposed LSTM-based method is beneficial for improving the accuracy of the position estimation model.

The experimental results demonstrate that:

(1)

For the MTSR-WPT system, the position of the receiver coil can be estimated through a trained neural network model, simplifying the internal mechanism analysis and complex formula derivation of the system.

(2)

Through the proposed method, the position of the receiver coil can be estimated solely based on the voltage data of the transmitter coil. In this way, there is no need to add voltage measurement and communication units in the receiver circuit.

(3)

When the receiver moves within a range of 160 mm × 160 mm, the average error between the estimated receiver coil position using the proposed method and the actual receiver coil position is less than 2.40 mm.

The comparison between the previous works for the receiver position estimation and the LSTM-based method proposed in this article is shown in

Table 4. Comparison of estimation accuracy among different models.

Reference	Model	Average Error (mm)
[24]	mathematical model	16.24
[27]	BP	2.80
	SVR	4.64
This article	LSTM	2.40

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5. Conclusions

To solve the position estimation issue of the receiver in MTSR-WPT systems, a novel receiver position estimation method based on LSTM is proposed in this article. This method uses data-driven and machine learning to establish a position estimation neural network model with LSTM as the input layer. By learning the relationship between the measured time-series voltage data of the transmitter coils and the position of the receiver coil, the proposed model can achieve an accurate position estimation of the receiver. Compared with previous works, since the input data of the estimation model are only measured on the receiver side, no communication between the transmitter and receiver is required. In addition, because the neural network model is trained based on the measured voltage data, there is no need to analyze the internal mechanism of the MTSR-WPT system, nor to conduct complex formula derivation. Finally, an experimental MTSR-WPT system is built. The experimental results show that when the receiver coil moves within a range of 160 mm × 160 mm, the average position estimation error of the model is less than 2.40 mm. The proposed method can be used to obtain the accurate position of the receiver coil and to achieve optimal magnetic field shaping for the MTSR-WPT system.

Due to the data-driven nature of the LSTM-based method, it still faces the challenge of collecting a large amount of data. The issue of improving the accuracy of the receiver position estimation of the MTSR-WPT system under small sample data can be further studied.