Mutual Inductance Estimation Using an ANN for Inductive Power Transfer in EV Charging Applications

Abstract

In the context of inductive power transfer (IPT) for electric vehicle (EV) charging, the precise determination of the mutual inductance between the magnetic pads is of critical importance. The value of this inductance varies depending on the EV positioning, affecting the power transfer capability. Therefore, the precise determination of its value yields various advantages, particularly by contributing to the optimization of the charging process of the EV batteries, since it offers the possibility of adjusting the position of the vehicle depending on the level of misalignment. Within this framework, algorithms grounded in artificial intelligence (AI) techniques emerge as promising solutions. This research work revolves around the estimation of the mutual inductance in a wireless inductive power transfer system using a resonant converter topology, implemented in MATLAB/Simulink^® R2021b. The system output was developed to emulate the behavior of a battery charger. To estimate this parameter, an artificial neural network (ANN) was developed. Given the characteristics of the system, the features were chosen in a way that they could provide a clear indication to the ANN if the vehicle position changed, independently of the charging power. In the pursuit of creating a robust AI model, the training dataset contained approximately 1% of the available data. Upon the analysis of the results, it was verified that the largest estimation error observed was around 3%, occurring at the lowest charging power considered. Hence, it can be inferred that the proposed ANN exhibits the capability to accurately estimate the value of mutual inductance in this type of system.

1. Introduction

Every inductive power transfer (IPT) system has limitations in terms of dependence on the coupling factor of the magnetic couplers. According to the Faraday law of electromagnetic induction, the induced voltage on the receiver pad depends on the rate of change in the magnetic flux that traverses it [1]. Generally, the coupling factor decreases from the nominal operating conditions when a misalignment between the transmitter and receiver pads occurs. This results in a decrease in both the mutual inductance and the output power. Consequently, the overall system efficiency will decrease, too. Particularly in the case of dynamic charging systems, these misalignments are unavoidable due to possible lateral ($m_x$), longitudinal ($m_y$) or vertical misalignments ($m_z$) [2], as can be seen in Figure 1. Hence, having access to this information in real time brings high value to these systems since it enables the possibility of adjusting the position of the vehicle and, thus, maximizing the power transfer. Having this in mind, this paper proposes the use an ANN capable of estimating, with high accuracy, the mutual inductance value between the transmitter and receiver pads of an IPT system used in electric vehicle (EV) charging systems for a wide range of charging powers.

This paper is structured into seven sections. Section 1 offers a broad introduction to the topic. Section 2 presents a literature review of the existing solutions regarding the estimation of different parameters in IPT systems using AI techniques. Section 3 presents the prototype used for the development of this work, as well as the resonant topology used in the simulation model. In Section 4, based on the conditions of operation of the simulation model, the process of assembling the training dataset and extracting features is explained, as well as data normalization. Additionally, this section also presents the AI model used for the purpose of this work, which, in this case, is an ANN. Section 5 presents the obtained results, along with the improvements made. Section 6 provides a discussion of the obtained results, including a direct comparison between the developed work and the literature. Finally, Section 7 presents the main conclusions drawn from this work.

2. Background

Mutual inductance can be obtained through an estimation process, which includes possibilities such as mathematical and artificial intelligence (AI) models. Traditionally, this estimation process has been carried out using mathematical models that combine electrical variable information applied primarily in dynamic IPT systems, i.e., in systems where the charging process of EV batteries occurs while the vehicle is moving [3,4,5]. To accomplish this, the developed models are based on the acknowledgment of various variables on the transmitter and receiver sides, including their structural parameters [6,7,8,9,10]. Hence, it is evident that these models require a substantial amount of information about the IPT system, and since the relationships between the parameters are significantly nonlinear, this will lead to high computational costs and long calculation times.

As an alternative to mathematical models, AI models have been used in various applications over the past few decades in various fields [11]. At present, there is a growing recognition of their utility and value, especially in power electronics [12]. For this reason, and since IPT systems have garnered increasing attention due to the continuous growth of interest in wireless charging for EV batteries [2], it becomes clear that AI models have emerged as a superior solution regarding parameter estimation in IPT systems. This is owed to the capability of adjusting to different configurations and also to the faster delivery of results with less computational power.

Table 1. Comparison between mathematical and AI models for parameter estimation in IPT systems.

Characteristics	Mathematical Models	AI Models	References
Parameter estimation	$●$	$●$
Accuracy	$●$	$●$
Calculation time	$●$	$●$	[8,13]
Dependency on structural parameters	$●$	$●$	[6,7,8,9,10,14,15,16,17,18,19]

presents a direct comparison between mathematical and AI approaches for estimation purposes, where the green and red circles are attributed to the model type that exhibits better and worse performance, respectively, and it can be concluded that AI models are superior to mathematical models. Both the calculation time and dependency on structural parameters are characteristics that differentiate AI models from mathematical models. Regarding the calculation time, mathematical models typically take longer to complete the estimation process and often rely on the structural parameters of the IPT system to complete these estimations, which, in a real-world application, would not be reliable.

In order to build a foundation for the solutions available concerning parameter estimation in IPT systems using AI models, a literature review is presented, delving into the significant advancements made in leveraging AI in IPT systems and showcasing its potential across various domains. In this regard, to the best of the knowledge acquired, different AI techniques, including long short-term memory (LSTM), random forest (RF), decision trees (DTs), Adabooster with DT, eXtreme Gradient Boosting (XGBoost), random forest regression (RFR) and support vector machines (SVMs), have been implemented to estimate the parameters of IPT systems, as depicted in

Table 2. AI models used for parameter estimation in IPT systems.

AI Model	Parameter Estimation	References
ANN	Design parameters	[20]
	Load impedance	[21]
	Mutual inductance	[13,21,22]
	Position of receiver pad	[14,15,23]
	Quality factor	[24]
	Self-inductance	[21,22]
LSTM	Ferrite core structures	[25,26]
ML models 1	Coupling coefficient
	Load impedance	[16]
	Power transfer
RFR	Load impedance	[17,18]
	Mutual inductance	[17,19]
	Resonant frequency	[18]
SVM	Coupling coefficient
	Load impedance	[16]
	Power transfer
	Position of the receiver pad	[27]

¹ RF, DT, Adabooster with DT, XGboost.

.

Depending on the application, parameter estimation using AI techniques can have different purposes, like the optimization of performance by reducing the period of time required for the estimation, the optimization of the features used, the characterization of the quality factor of an IPT system and the actual estimation of various parameters of IPT systems, as will be further presented.

The artificial neural network (ANN) is a model with a wide range of applications regarding the estimation of parameters in an IPT system. In [13], the authors focused their attention on building a deep neural network (DNN) to design a wireless power transfer (WPT) system for an EV to posteriorly compare its performance with the calculation results of a classic Monte Carlo algorithm. Using a DNN divided into three parts that uses spatial variables as features, as well as the mutual inductance measured between the transmitter and receiver sides, the equivalent resistance of the primary and secondary loops and the load resistance, it was possible to reduce the estimation time from 1369 s, in the case of the Monte Carlo algorithm, to only 152 s. Alternatively, in [20], with the information from the structural parameters of an IPT system, an ANN was employed with the goal of not just reducing the training time and training data but also comparing the estimation performance with that of the finite element method (FEM) for different variables, like power transfer, self-inductance and mutual inductance, among others.

Furthermore, in [24], an ANN was designed to characterize the quality factor of spiral coils used in IPT systems. The features used were collected through the Ansys-Q3D simulator. Regarding general estimation, ANNs deliver great performance, encompassing a variety of parameters. In [22], using, once again, information from physical measurements of the coils, an ANN capable of estimating the values of self- and mutual inductances was developed.

Additionally, in [21], based on information from the input current and the transmission distance, the authors proposed an ANN that estimates not just the self- and mutual inductances but also the load. Moreover, in [23], using the information from the magnetic field, as well as the outputs of sensing coils, an ANN was developed with the goal of estimating the level of misalignment between the transmitter and receiver coils.

Alternatively, in [14], the authors took a different approach to the estimation of the lateral misalignment between the transmitter and receiver coils in a dynamic IPT system, since the features relied on the profile of the DC-link current and also on the speed of the vehicle. To enable the recognition of patterns in the waveform of the DC-link current, it was necessary to develop a time-series AI model, which, in this case, was a time-series ANN. Still regarding the estimation of the position of the receiver pad, in this case, in a system with multiple transmitters [27], an ANN together with a support vector machine (SVM) algorithm was proposed, enabling the ability to control the switching cycles of the transmitter pads. An SVM algorithm together with random forest, decision tree, Adabooster with decision tree and XGboost was tested, and their performance was compared regarding the estimation of the power delivered to the receiver pad, the load resistance and the coupling coefficient [16]. As features, these ML models used the first five harmonic current components and the RMS value of the current that flows in the transmitter coil. By using the data from the harmonic components, in [17,19], a random forest regression (RFR) algorithm was employed with the goal of estimating the mutual inductance between the transmitter and receiver coils. Nonetheless, in [17], the value of the load resistance was also estimated.

This literature review has revealed that the self- and mutual inductances, coupling coefficient, load resistance, position of the receiver pad, power transferred and quality factor of the coils are examples of parameters that have been estimated through AI models in IPT systems. Typically, AI implementation relies on ML models or ANNs. Although some authors refer to DNNs as models that are difficult to understand and hard to train due to the necessity of a large dataset, their capacity to adapt to highly nonlinear situations, such as the estimation of mutual inductance, continues to be the reason why DL is considered the best approach. Moreover, to the best of the authors’ knowledge, there have not been any approaches described that utilize harmonic components measured from the secondary side as training data. Usually, the training data rely on measurements from the transmitter side or even on the structural parameters of the coils, which would be challenging to acquire in a real-world application. Hence, this factor constitutes a significant barrier for the scientific community to overcome in order to enhance the performance of these estimation models. To actively contribute to the ongoing advancements, this paper introduces an ANN model that proficiently estimates the mutual inductance under misalignment and varying charging power conditions. As will be further explained, although the dataset consisted of information corresponding to a defined set of misalignment positions and charging powers, this model delivered an outstanding performance using a training dataset comprising only 1.5% of the available data.

3. IPT System

One of the most employed resonant topologies in IPT systems is the series–series (SS) topology. Nevertheless, in situations without coupling or when the EV battery is fully charged, the current on the transmitter side tends toward infinity [28], which, in a real-world scenario, is not recommended. For this reason, a second coupling system can be added to the previous topology, thus avoiding this problem. Therefore, in this implemented solution, a double-coupling series–series–series (SSS) configuration is considered in the simulation model developed in MATLAB/Simulink^®. In Figure 2, a block diagram characterizing this system is shown. The system is fed from the grid, followed by a high-frequency (HF) diode rectifier, which is responsible for generating the DC bus voltage. Then, a high-frequency inverter generates a high-frequency voltage that feeds a resonant topology, where the first magnetic coupling (MC) is referred to as an isolation transformer, and the second MC is referred to as the transmitter and receiver pads. To charge the batteries of the EV, a high-frequency diode rectifier is used to convert the AC current into DC current as well a DC-DC converter to control the battery charging process.

To evaluate the behavior of the mutual inductance between the transmitter and receiver pads of an IPT system, a simulation environment able to perform a realistic characterization of the system is needed. In order to address this question, the collection of mutual inductance values across a range of misalignment positions from a characterized system is also required so that the results obtained in the simulation environment can be based on real-world measurements. Therefore, since a prototype of the mentioned IPT system was previously assembled, it was used in this research work. Figure 3 shows this prototype.

This prototype uses a small electric motor controlled to emulate the EV displacement, i.e., the longitudinal position along the $m_y$-axis, with an interval spanning between −250 mm and 250 mm, where 0 mm corresponds to the position of perfect alignment between the transmitter and receiver pads. The lateral position of the transmitter pad was adjusted manually between 0 mm and 175 mm. The airgap between the transmitter and receiver pads was kept constant at 50 mm. In relation to the structure of the transmitter and receiver pads, both were installed with a bipolar pad (BPP) topology with a 280 mm length and a 195 mm width. In regard to the enhancement of magnetic permeability, only the secondary pad was equipped with ferrite cores. Using this prototype and an LCR meter, the self-inductance of the transmitter pad ($L_t$), the self-inductance of the receiver pad ($L_r$) and the total inductance obtained through the series ($L_{T+}$) and anti-series ($L_{T-}$) connections of both coils for each considered position of misalignment were measured. With this information, it was possible to perform a characterization of the mutual inductance for different misalignment levels by using the relation presented in Equation (1):

$$M=\frac{L_t+L_r\pm L_T}{2}.$$

(1)

To perform a global characterization of the mutual inductance for all regions of interest, the longitudinal position was varied in increments of 10 mm, whereas the lateral position was varied in increments of 25 mm. Also, it is worth emphasizing the fact that measurements were taken only for half of the lateral misalignments (0–175 mm), given the symmetrical behavior of the mutual inductance regarding the axis direction [2]. With this information, a comprehensive three-dimensional table was constructed, encompassing all values of mutual inductance that were employed in the simulation. Figure 4 shows the measured characteristic of mutual inductance. Figure 4a represents the characteristic of mutual inductance from a 3D perspective, and Figure 4b shows the characteristic of mutual inductance for longitudinal and lateral misalignments when the vehicle is perfectly aligned with the road, i.e., when $m_x=0$ and $m_y=0$, respectively.

Figure 4, as expected, shows the Gaussian characteristics inherent in these systems [29]. Additionally, it can be verified that, for certain values of misalignment, the mutual inductance presents negative values. This happens because both the transmitter and receiver coils have a bipolar topology, and there will be situations where the magnetic fluxes generated by the two coils have different directions, which will cause a negative value of mutual inductance.

Figure 5 illustrates a simplified representation of the implemented simulation model. The elements labeled as $L_p$ and $L_s$ represent the self-inductances of the isolation transformer, while $L_t$ and $L_r$ denote the self-inductances of MC. Furthermore, $C_p$, $C_t$ and $C_r$ refer to the resonant capacitors. All elements are connected in series to guarantee the SSS configuration. Moreover, the blue and yellow circles represent the voltage and current sensors, respectively, that were used to acquire the necessary data to subsequently train the AI model. Finally, the block identified with the letter M portrays the 3D table with the values in Figure 4a, whereas the block identified with the letter P represents the charging power defined by the user.

The various elements in the circuit were parameterized in order to guarantee the operation at resonance, which includes the reactive elements of the circuit. Thus, considering that the values of the self-inductance of each coil were measured, Equation (2) allows the values of the capacitors to be calculated and guarantees the operation of the circuit in the resonant condition [30,31,32], where f represents the switching frequency of the inverter:

$$f=\frac{1}{2·\pi \sqrt{L_p·C_p}}=\frac{1}{2·\pi \sqrt{(L_s+L_t)·C_t}}=\frac{1}{2·\pi \sqrt{L_r·C_r}}.$$

(2)

Table 3. Simulation parameters.

Parameter	Value
$V_{DC}$	100 V
f	20 kHz
$V_{ak}$	0.70 V
$L_p$	41.78 μH
$L_s$	42.29 μH
$L_t$	45.75 μH
$L_r$	60.20 μH
$M_{ps}$	30.27 μH
$M_{tr}$	15.83 μH
$C_p$	1.52 μF
$C_t$	0.72 μF
$C_r$	1.05 μF
$C_{out}$	6400 μF
$r_{p,s,t,r}$	0.10 Ω

represents the system parameter values of the most relevant elements of this circuit. The $M_{tr}$ parameter corresponds to the mutual inductance measured between the transmitter and receiver pads; thus, it will vary between its maximum and minimum values (−10.76 μH ≤ $M_{tr}$ ≤ 15.83 μH) depending on the misalignment of the pads.

The acquired data based on real-world measurements were given as input to a MATLAB/Simulink^® lookup table block, enabling the possibility of extracting the mutual inductance value for any given point of interest, given the range of values available. Subsequently, it was possible to emulate the variation in the mutual inductance of the second MC in the simulation environment by adding a transformer equivalent model with two controlled voltage sources to replace the second MC. In MATLAB/Simulink^® R2021b, these blocks can generate a voltage that is driven by an input signal. Defining $v_{rt}$ and $v_{tr}$ as the imposed voltage on the primary and secondary sides, respectively, the control inside the subsystem “Voltage Source Control” has the function of guaranteeing the following conditions:

$$v_{rt}=-L_{tr}·\frac{d{i}_r}{dt},$$

(3)

$$v_{tr}=L_{tr}·\frac{d{i}_t}{dt},$$

(4)

which entails knowing the mutual inductance value. Additionally, for simplification purposes, although the self-inductances of the MC have slight variations in real-world measurements, in the simulation environment, these values were considered constant.

Finally, in order to emulate the output of a conventional battery charger, a controlled current source was integrated into the system. The function used to control this current source was implemented with the goal of reproducing, in a simpler form, a behavior similar to that of a back-end DC-DC converter. A block diagram of this control system is displayed in Figure 6.

The current source value is based on the power output of the charger, defined by the user, and the output voltage, measured at the terminals of the rectifier. This allows the output current to adapt according to the power demand, guaranteeing that the output of the system aligns with the characteristics of a standard battery charger in a current control stage, which is the first stage when charging a discharged battery. Also, it must be acknowledged that the reliability of this function is only guaranteed in a simulation environment. In a real-world situation, it would not be possible to impose a charging current and, consequently, the corresponding charging power independently of the misalignment level. In situations of high misalignments, since the output voltage is directly proportional to the mutual inductance between the magnetic couplers [2], this will result in a significantly high charging current, which is not recommended. Nevertheless, although the function has its limitations regarding power management, the focus of this work remained on the ability to properly estimate the mutual inductance measured between the transmitter and receiver pads.

4. Mutual Inductance Estimation Using AI

To develop an AI model that can accurately estimate the value of mutual inductance under misalignment and charging power variations, it is necessary to choose the ideal features and assemble the dataset.

4.1. Dataset Assembly and Feature Selection

Starting with the assembly of the dataset, there are various considerations to take into account, including the dimension of the dataset and also the specific positions to consider. To sum up these factors, two criteria were established:

(i)

A higher number of misalignment positions must be ensured in regions where the mutual inductance has a nonlinear characteristic in order to accurately test the performance of the AI model.

(ii)

The considered misalignment position boundaries are based on the operating limits of the IPT system.

Based on the limitations of the control system operation, the referenced boundaries were established based on the behavior of the output current. Thus, in conditions where a high value of charging power is required and the vehicle is considerably misaligned, the system may not operate properly. Subsequently, it was determined that, since the output voltage was the only parameter that could be constrained in order to guarantee a minimum of 500 W of charging power at the position of greater misalignment, the limit positions would need to be selected accordingly. With $m_x$ and $m_y$ measured in millimeters, Figure 7a presents the group of misalignment positions considered, referred to in this work as a grid of misalignments, which entails a total of 108 possible misalignment positions. In the positions of perfect alignment and the greatest misalignment, mutual inductances of 15.83 μH and 5.5 μH, respectively, were measured. Additionally, Figure 7b presents the characteristic of the mutual inductance for these positions, adapted to encompass the four quadrants, assuming symmetry. It is also worth mentioning that, since, in the prototype, the distance between measurement points regarding the lateral misalignment was significant, the resultant characteristic had poor resolution. Thus, in the simulation environment, this situation was compensated for by adding more misalignment positions in these intervals.

Feature extraction is certainly one of the most important steps for obtaining AI models with high performance. Essentially, it consists in extracting relevant information from raw data [33]. Therefore, there are some criteria that have to be taken into account. Firstly, the chosen features must have a pronounced variation when a misalignment occurs, and secondly, they must have a constant trend across different charging power levels. This means that this model must be able to predict the mutual inductance value as accurately as possible for all misalignments within the grid and also under all allowed charging powers.

Therefore, a careful analysis was carried out regarding the waveforms and amplitudes of the harmonics measured at specific points on both the transmitter and receiver sides, which are highlighted in Figure 5. From the waveforms of the signals, it was not possible to extract relevant features, which led to the analysis of the amplitude of the harmonic components of voltage and current signals. The process of selecting the most suitable harmonics as features for the AI model was divided into three main steps:

(i)

Comparing the amplitudes of the different harmonics, including the fundamental component, for the chosen charging powers;

(ii)

Selecting the harmonic components that presented similar amplitudes for all charging powers and had the greater influence, in terms of amplitude, on the signal under analysis;

(iii)

Verifying whether the selected harmonic components exhibit noticeable variations in the event of a misalignment occurrence and opting for those that demonstrate such variations. However, although the amplitude of some harmonic components is not the same across different charging power levels, the percentage variations at all charging powers under misalignment variations exhibit similar patterns.

Synthesizing the information extracted from this detailed analysis, the features that are more suitable for the purpose of this work are as follows:

• The amplitudes of the third and fifth harmonics of voltage $v_4$;
• The amplitudes of the fundamental component (FC) and the third and fifth harmonics of voltage $v_5$;
• The amplitude of the third harmonic of current $i_{sec}$;
• The amplitude of the third harmonic of the reactive power, measured at the HF rectifier input.

4.2. Data Normalization

A crucial step before training any AI model is without a doubt data pre-processing, which includes removing outliers and performing normalization procedures, which helps improve the nature of the information [34]. Normalization by itself is the process where the attributes of the dataset are categorized to increase the bond between them, making the dataset more flexible [35]. For this application in particular, the data were extracted for different misalignment positions and charging powers, ranging between 500 W and 3000 W. Since the AI model to be implemented must be able to predict the value of mutual inductance for various misalignments independently of the charging power value, the data must be normalized for a specific value of charging power. The objective of the developed AI model is to operate in a wide range of charging powers without the need for training for each charging power in particular. Hence, considering the preference for AI models to accurately estimate the value of mutual inductance in real-world scenarios, particularly in systems operating at a higher charging power, and also taking into account the performance results of the conducted tests, it was decided to normalize the data for 2000 W.

The adopted procedure of normalization consisted of collecting the maximum value of each feature when the charging power of the system was 2000 W. Consequently, by employing Equation (5), all the values of each feature were normalized. In this equation, $n_{feat,cp,mp}$ refers to the normalized value for a given feature $feat$, charging power $cp$ and misalignment position $mp$. The variable $max\left(v_{feat,2000}\right)$ corresponds to the maximum value among all misalignment positions of a given feature when the system works at 2000 W of charging power, and $v_{feat,cp,mp}$ is the real value of a given feature for a specific charging power and misalignment position.

$$n_{feat,cp,mp}=\frac{max\left(v_{feat,2000}\right)-v_{feat,cp,mp}}{max\left(v_{feat,2000}\right)}.$$

(5)

4.3. Developed ANN Model

The estimation of mutual inductance is a highly nonlinear and complex task to perform with mathematical models. As cited in [24], ML models are reliable as long as the data samples available appropriately represent the end-to-end relationship, whereas DL networks yield exceptional performance in classification and regression tasks [36] without the constraints of ML models [37] since they have the ability to adapt. With that said, it was decided to implement Multi-Layer Perceptron (MLP), i.e., a fully connected multi-layer neural network [38] in MATLAB^® software R2021b, using the fitrnet function. This AI model has been widely employed in IPT systems [14,15,21,22,23,24,25,27,39,40,41,42] due to its high reliability and performance in fulfilling regression tasks.

Figure 8 illustrates the structure of the ANN implemented in this work. To arrive at this configuration, the performance of the network was tested for different combinations. In the first instance, it was evaluated with a lower number of neurons, and subsequently, it was analyzed with a higher number of hidden layers and neurons.

The inputs of the ANN represent the values of each feature that were selected according to the requirements identified in Section 4.1. The output corresponds to the predicted mutual inductance value. For the ANN be able to compute an output value with the information available from the features, it needs to be composed of multiple layers. These layers are usually known as fully connected layers since each neuron of a given layer is connected to all neurons of the next layer [25]. Essentially, this connection between neurons can be formally expressed as in Equation (6) [24], where $z_{l,i}$ denotes the output of the i-th neuron of the l-th layer, $\sigma (·)$ is a nonlinear activation function, $\langle ·,·\rangle$ represents the inner product, ${\mathsf{\Theta }}_i^{\left(l\right)}$ is the weight vector for the i-th neuron in the l-th hidden layer, $u_l$ refers to the input vector of the l-th layer, and finally, $b_i^{\left(l\right)}$ denotes the bias term.

$$z_{l,i}=\sigma \left(\langle {\mathsf{\Theta }}_i^{\left(l\right)},u_l\rangle +b_i^{\left(l\right)}\right).$$

(6)

In this particular case, the final version of the implemented ANN consists of four layers: an input layer, two fully connected layers, one with 1024 neurons and another with a single neuron, and finally, the output layer, which corresponds to the predicted mutual inductance values. The activation function used was the logistic sigmoid. The training phase stopped when the validation loss was greater than or equal to the minimum validation loss computed for a hundred consecutive occurrences. These parameters were selected by trial and error, depending on the performance of the network.

4.4. Methodology

The objectives of the implemented model rely on accurately predicting the mutual inductance value, under misalignment and charging power variations, with minimal training requirements by using only information from the receiver side in order to simplify the deployment in a real-world scenario where a communication link between the transmitter and receiver sides is not available. In simpler terms, it aims to achieve this with as little information from misalignment positions as possible, simulating a real-world scenario where network training can be more straightforward, thus bolstering the robustness of the model. Also, in order to test the robustness of the network, an additional dataset was created with data collected from noisy signals, i.e., signals with the inclusion of white noise with 5% of the amplitude of the respective signal.

Hence, to achieve the best possible performance, the tuning process was based on the diagram in Figure 9. The first step involves providing a training dataset to the network, which then attempts to identify patterns and relationships within the data in order to predict mutual inductance values. The second step is related to the testing phase, where the ability of the network to generalize the learned information to unseen data can be verified. In this step, the feature contribution is also analyzed. In the final step, based on the results of the second step, adjustments are made to the parameterization of the ANN, including the number of neurons per hidden layer, as well as the size of the training dataset.

Typically, for regression tasks [17,22,43,44,45], the metrics that are commonly used are MAE, MAPE and $R^2$. Therefore, these metrics were selected to evaluate the overall performance of the network. Nevertheless, the $R^2$ metric was the main reference since it has been recurrently and independently used to accurately evaluate the performance of regression models [37,43,44,45]. Furthermore, to better understand the improvements applied to the network,

Table 4. Stages for training the ANN.

Description	1st Stage	2nd Stage	3rd Stage	4th Stage
Features	$v_4^3$, $v_4^5$	$v_4^3$, $v_4^5$	$v_4^3$, $v_4^5$	$v_4^3$, $v_4^5$
	$v_5^{FC}$, $v_5^3$, $v_5^5$	$v_5^{FC}$, $v_5^3$, $v_5^5$	$v_5^{FC}$, $v_5^3$, $v_5^5$	$v_5^{FC}$, $v_5^3$, $v_5^5$
	$i_{sec}^3$	$i_{sec}^3$	$i_{sec}^3$	$i_{sec}^3$
	$Q^3$
Number of positions	2	2	8	9
Noise Signals	No	No	No	Yes

provides a compilation of the various stages involved in the training and tuning of the network. Firstly, the row described as “features” represents the features that were used in the training phase. Secondly, the ”number of positions” refers to the number of misalignment positions that were included in the training dataset. Lastly, “noise signals” clarifies whether, in the specified stage, the current and voltage signals used to obtain the harmonic components had white noise introduced or not.

Subsequently, to give a better visualization of the considered misalignment positions, Figure 10 illustrates the positions that were added to the training dataset in each stage of training. To move to the next training stage, the performance of the network was tested for every misalignment position on the grid of misalignments, and the metrics were analyzed. It is worth noting that, even in the last training stage, the number of positions considered in the training dataset was only 9 out of 108 possible positions of misalignment acquired for a charging power of 2000 W, which corresponds to approximately 1.5% of the data available (108 misalignment positions for six different charging power levels between 500 W and 3000 W with 500 W incremental steps), demonstrating the usefulness of both the features and the parameterization of the network.

In the first stage, the feature contribution was studied, and it was concluded that the amplitude of the 3rd harmonic of the reactive power contributes negatively to the performance of the network; thus, its removal was deemed necessary in the 2nd stage.

Subsequently, considering what was previously stated regarding the dimension of the training dataset, it was initially established to only use information from two positions of misalignment: the positions of perfect alignment and maximum lateral misalignment. In this way, in the second stage, the parameters of the ANN were modified. For performance analysis purposes,

Table 5. The results of MAE, MAPE and $R^2$ metrics for each training stage.

Training Stage	Charging Power [W]	MAE	MAPE [%]	${\mathit{R}}^2$	Improvement from Previous Stage [%]
1st Stage	500	1.75	15.0166	0.6086	-
	1000	1.0825	8.5072	0.8388	-
	1500	0.4960	3.9510	0.9651	-
	2000	0.3014	2.8317	0.9769	-
	2500	0.9784	7.7771	0.8585	-
	3000	1.5186	11.5341	0.6615	-
2nd Stage	500	1.6917	15.0094	0.6299	3.50
	1000	1.0166	8.0878	0.8569	2.16
	1500	0.4859	3.8393	0.9672	0.22
	2000	0.1968	1.9191	0.9867	0.10
	2500	0.7903	6.2707	0.9066	5.60
	3000	1.2618	9.5551	0.7650	15.65
3rd Stage	500	0.2575	2.5675	0.9886	56.95
	1000	0.1421	1.1794	0.9966	16.30
	1500	0.0660	0.5118	0.9992	3.31
	2000	0.0227	0.1820	0.9999	1.34
	2500	0.0370	0.2762	0.9997	10.27
	3000	0.0536	0.3835	0.9993	30.63
4th Stage 1	500	0.2061	2.3083	0.9892	0.06
	1000	0.1045	0.9622	0.9973	0.07
	1500	0.0475	0.4005	0.9995	0.03
	2000	0.0207	0.1786	0.9999	0
	2500	0.0292	0.2462	0.9998	0.01
	3000	0.0447	0.3582	0.9997	0.04
4th Stage 2	500	0.2921	2.6821	0.9873	−0.19
	1000	0.1652	1.3152	0.9958	−0.15
	1500	0.0760	0.5763	0.9990	−0.05
	2000	0.0186	0.1539	0.9999	0
	2500	0.0370	0.2745	0.9997	−0.01
	3000	0.0551	0.3923	0.9992	−0.10

¹ Model trained with clean data and 9 positions of misalignment. ² Model trained with white noise and 9 positions of misalignment.

presents the metrics obtained for each training phase. Moreover, it also presents the percentage improvement in estimation accuracy over the previous training stages. Therefore, the 1st stage does not apply to the improvement evaluation. Therefore, by inspecting the obtained results, it can be verified that, despite the improvement, the performance of the network in the second stage was still poor.

In the third stage, it was concluded that the only solution was to increase the dataset dimension, since the modification of the ANN parameters was not producing any better results. To give a clear insight into what misalignment positions should be included in the training dataset, Figure 11 displays the network performance from a different perspective. It provides a direct comparison between the predicted and measured values of mutual inductance for each situation of misalignment. Essentially, points on the reference line indicate correct predictions. In Figure 11a, by inspecting the behavior of the network, it can be concluded that from 8 μH onward, there was a clear divergence between the predicted and measured values of mutual inductance, which suggested that the network needed more information to achieve a better performance. Thus, it was decided to give the network the information obtained for the misalignment positions illustrated in Figure 10. The positions are identified with numbers between 1 and 5, defined as follows:

• Position n°1 correspond to the maximum mutual inductance value (aligned position).
• Positions n°2 corresponds to mutual inductance values between 15 μH and 15.5 μH.
• Positions n°3 corresponds to mutual inductance values between 12 μH and 12.5 μH.
• Position n°4 corresponds to mutual inductance values between 8 μH and 8.5 μH.
• Position n°5 correspond to the minimum mutual inductance value.

The metrics and the graphical comparison between the predicted and real mutual inductance values shown in Table 5 and Figure 11b, respectively, illustrate a clear improvement in the estimation performance.

Furthermore, in the fourth stage, the robustness of the network was tested when trained with data acquired from current and voltage signals with noise. White noise with 5% of the amplitude of the respective signal was added to each signal extracted from the simulations. In this document, the obtained data are referred to as noisy data. To obtain similar results to the ones presented with no noise, an additional misalignment position where the value of mutual inductance stood in a range between 15 μH and 15.5 μH needed to be added to the training data. Therefore, to compare the obtained results when the network was trained with information from the same number of misalignment positions, this stage was divided into two parts. The first part corresponds to the situation where the model was trained using a dataset with clean data, while the second part deals with noisy data, with both cases using information from nine positions of misalignment. The metrics obtained in this stage are presented in Table 5, and it demonstrates that, similarly to the previous situation, the estimation capability of the presented network is accurate, with an error percentage less than 1%. Additionally, a graphical comparison between the real and predicted values of mutual inductance is shown in Figure 11c,d.

5. Results

Regarding the obtained results, in order to verify that the model has not overfitted the training data, Figure 12 shows a comparison between the training mean square error (MSE) and the validation MSE at each iteration. It can be observed that both curves present a decreasing trend and that, after 200 iterations, the value of the loss function remained fairly constant. This clearly demonstrates that the network has not overfitted the training data.

Given the obtained results, the effects, although subtle, of the noisy data on the performance of the network can be observed. The addition of noise to the collected data (second part of fourth stage) only slightly degrades the performance of the ANN in comparison to the ideal situation (no noise—first part of fourth stage), thus demonstrating the high immunity of the developed system to this type of disturbance.

The training phases with clean and noisy data were completed after approximately 400 and 200 epochs, respectively, resulting in a process that lasted around 2 s.

6. Discussion

Given the topical issue of implementing reliable IPT systems for EVs, this work proposes an ANN capable of estimating, with noteworthy accuracy, the mutual inductance value under misalignment and charging power variations. The tuning process of the ANN consisted of adjusting the network parameterization and training dataset dimension and verifying the features’ contributions. It must be highlighted that, during this tuning process, the modification that led to the biggest improvement was the addition of more information to the training dataset. As a result, by using only 1.5% of the available data for training, it was possible to obtain a maximum error of less than 3%.

Thus, the developed work is considered to provide significant value to the scientific community in this field of investigation. To this end,

Table 6. A comparison between the contributions of the presented work and the solutions presented in the literature regarding mutual inductance estimation.

	*	[21]	[13]	[22]	[17]	[19]
IPT system	Static IPT	Static IPT	Static IPT	Static IPT	Static IPT	Static IPT
Charging power	Yes	No	No	No	No	No
Training dataset	1.5%	80%	70%	68%	66.6%	62.5%
Noise immunity	Yes	No	Yes	Not applied	No	No
AI approach	ANN	ANN	ANN	ANN	RFR	RFR
Estimation error	2.68%	2.66%	Not mentioned	9.47%	3.11%	2.55%

* Proposed Work.

presents a direct comparison between various characteristics of the developed work and what is stated in the literature in order to highlight its originality and relevance.

In future work, in light of the conducted research, implementing extensions such as adapting this model to dynamic IPT systems with various resonant topologies, integrating a buck-boost converter and a battery into the output of the simulation model, and deploying it in an experimental setup to validate its performance under real-world conditions would significantly enhance the optimization of IPT systems through AI.

7. Conclusions

The estimation of the mutual inductance between the transmitter and receiver pads of IPT systems under misalignment conditions is critical to maximizing the power transfer and efficiency while decreasing the charging time of EVs. Contrary to other works in the field, here, an ANN has been implemented using only information available on the receiver side, hence easing its deployment in a real-world scenario where a communication link between the transmitter and receiver sides is not available.

The obtained results have demonstrated that feeding the ANN with a high number of features does not necessarily lead to the improved performance of the network. This work demonstrates that the addition of the amplitude of the third harmonic of the reactive power exemplifies this observation, as, without its contribution, the accuracy of the model increased by approximately 4%, 2%, 1%, 1%, 6% and 16% for charging power levels of 500 W, 1000 W, 1500 W, 2000 W, 2500 W and 3000 W, respectively. To further verify the robustness of the proposed network, training data were derived from clean and noisy signals. In the first case, the highest observed error was 2.57%, occurring at a charging power of 500 W. Similarly, the highest error for the second situation was 2.68%, also at a charging power of 500 W.

Taking into account the range of charging powers supported by the IPT system under study, an optimal AI model performance at higher charging powers is preferred to ensure efficient charging in real-world scenarios. The proposed ANN configuration manages to deliver a good estimation performance across the entire charging power range. To accomplish this, the features chosen were variables with more pronounced variation during misalignment conditions while maintaining minimal variation when the charging power varies.

These results confirm that the proposed ANN effectively estimates the mutual inductance under misalignment and varying charging power conditions, thereby enhancing power transfer optimization in IPT systems.