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Article

Symbolic Regression Method for Estimating Distance Between Two Coils of an Inductive Wireless Power Transfer System

by
Davide Milillo
*,†,
Lorenzo Sabino
†,
Rafiq Asghar
† and
Francesco Riganti Fulginei
†
Department of Industrial, Electronic and Mechanical Engineering, University of Roma Tre, 00146 Rome, Italy
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Appl. Sci. 2025, 15(7), 3668; https://doi.org/10.3390/app15073668
Submission received: 20 February 2025 / Revised: 19 March 2025 / Accepted: 20 March 2025 / Published: 27 March 2025
(This article belongs to the Special Issue New Insights into Wireless Power Transmission Systems)
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Abstract

:
Symbolic regression (SR) has emerged as a powerful tool for the characterization of Wireless Power Transfer (WPT) systems, estimating the distance between coils and finding the relationship between frequency and phase so as to find the best frequency to increase the power factor. This study explores the application of SR on both simulated and experimental data, demonstrating its effectiveness with low prediction errors. SR employs a genetic algorithm to identify the analytical formula that best represents the input–output relationship, combining the strengths of traditional machine learning and analytical modeling. The results, with prediction errors of less than 1%, indicate that SR not only enhances predictive accuracy but also provides insights into the underlying physical principles governing WPT systems. This dual advantage positions SR as a valuable method for optimizing WPT applications, paving the way for further research and development in this field.

1. Introduction

Wireless Power Transfer (WPT) technology has garnered considerable interest in recent years due to its transformative potential in energy delivery across a wide range of applications, such as electric vehicles, consumer electronics, and medical devices [1,2,3,4]. The performance of WPT systems, particularly their efficiency and reliability, is heavily dependent on the spatial configuration of the transmitting and receiving coils [5,6,7,8,9,10,11,12]. The precise estimation of the distance between these coils is essential for optimizing power transfer efficiency and ensuring the system operates reliably [13,14,15,16,17,18]. Conventional methods for estimating coil distance often depend on empirical models or heuristic approaches, which may lack the precision and adaptability required in complex scenarios [19,20,21,22,23,24,25]. In contrast, symbolic regression (SR) introduces an innovative approach by employing machine learning techniques to derive mathematical expressions that accurately represent the relationships between input variables and outputs [26,27,28,29,30]. SR utilizes genetic algorithms to iteratively evolve potential solutions, enabling it to uncover analytical formulas that best fit the observed data. This distinctive feature places SR at the intersection of traditional machine learning and analytical modeling, allowing it to deliver both high predictive accuracy and interpretable insights into the fundamental principles governing WPT systems [31,32,33]. Like all machine learning techniques, SR also has the problem of underfitting and overfitting, and sometimes it may be less accurate than analytical expressions. Also, without proper data and model engineering, SR may have higher noise sensitivity and higher computational cost in the training phase than a neural network. In this study, we explore the application of SR for estimating the distance between coils in WPT systems using a combination of simulated and experimental data. Additionally, we address another critical factor in WPT systems: the frequency of energy transfer. To optimize the power transfer ratio, which is related to the phase angle between current and voltage ( ϕ ), we apply the same SR methodology to adjust the frequency of a WPT system, aiming to achieve cos ϕ = 1 , which represents maximum power transfer efficiency. Our objective is to demonstrate the capability of SR in achieving low prediction errors while uncovering the mathematical relationships that govern WPT dynamics. By incorporating SR into WPT research, we aim to deepen the understanding of coil interactions and contribute to the development of more efficient and effective wireless energy transfer solutions [34,35,36,37,38,39]. This approach not only enhances the accuracy of predictions but also provides valuable insights into the underlying physics of WPT systems, paving the way for innovative advancements in the field. Furthermore, the integration of SR into WPT research offers several advantages over traditional methods. Unlike empirical models, which are often limited by their reliance on predefined assumptions, SR dynamically adapts to the data, enabling it to capture complex, nonlinear relationships that might otherwise be overlooked. This adaptability makes SR particularly well suited for applications where system parameters vary significantly or where traditional models fail to provide accurate predictions. Moreover, the interpretability of SR-derived formulas allows researchers and engineers to gain a deeper understanding of the factors influencing WPT performance, facilitating the design of more robust and efficient systems. As WPT technology continues to evolve, the use of advanced machine learning techniques like SR will play a crucial role in addressing the challenges associated with optimizing energy transfer in diverse and dynamic environments.

2. Materials and Methods

2.1. Symbolic Regression (SR)

Symbolic regression (SR) is a robust machine learning approach that extracts mathematical equations directly from data, providing a distinctive blend of flexibility and interpretability. Unlike traditional neural networks (NNs), which identify relationships between input and output data by minimizing Mean Square Error (MSE), SR delivers explicit analytical expressions for these relationships, overcoming the limitations of NNs. NNs, often regarded as “black-box” models, obscure the underlying input–output relationships and require passing new inputs through the model, which can be computationally expensive and impractical for real-time applications. By merging the adaptability of NNs with the clarity of analytical models, SR generates a mathematical formula representing the discovered relationship at the end of training. This eliminates the need for the complex model during deployment, significantly reducing computational overhead and making SR particularly suitable for real-time applications, such as dynamically adjusting coil distances in Wireless Power Transfer (WPT) systems to optimize power transfer efficiency. The SR process employs genetic algorithms to search for the optimal formula f S R that best satisfies the relationship output = f S R (input). The search space includes predefined operators, such as unary operators (e.g., exponential, sine, and cosine) and binary operators (e.g., addition, multiplication, and power), which act as the “genes” in the genetic algorithm. Each candidate formula in the population represents a unique combination of these operators. The genetic algorithm evaluates candidate formulas based on two key criteria: accuracy (minimizing MSE) and complexity (favoring simpler expressions). Formulas with lower MSE and reduced complexity are more likely to survive and propagate to subsequent generations through mechanisms like mutation, which alters constant values or replaces operators, and reproduction, which combines elements from two parent formulas to create new offspring. This iterative process continues over multiple generations until convergence is achieved, yielding a final formula that balances accuracy and simplicity. Thus, the ability of SR to produce explicit mathematical expressions not only enhances transparency by revealing underlying data relationships but also improves efficiency by enabling the direct application of the derived formula without complex computations. This makes SR ideal for systems requiring rapid decision-making, such as adaptive WPT systems, while demonstrating its potential to enhance both performance and understanding in engineering domains. Despite its advantages, SR has certain limitations. Due to the vast research space, the method can be computationally slow. Additionally, like other machine learning techniques, SR is prone to overfitting, where the formula may fit the data well but fail to capture the underlying physics. A notable solution to this issue was proposed by [40], which leverages symmetries and simplifications often used by physicists to reduce the search space and identify the simplest formula that best fits the data. Furthermore, SR performs well primarily with small datasets. However, techniques combining deep neural networks with SR have been developed to extend its applicability to larger datasets.

2.2. WPT System: Experimental and Simulated Data

Two experimental setups were used to test the SR model. The first is a WPT system that implements impedance matching, and the second is a WPT system with variable frequency to optimize the phase.

2.2.1. WPT NFC Impedance Matching

The proposed method was initially applied to a Wireless Power Transfer (WPT) system that employs Near-Field Communication (NFC) technology, operating at a frequency of 13.56 MHz [41]. This WPT system is composed of two magnetically coupled resonators, in a series–series configuration, that enable the transfer of power from the transmitter to the receiver, represented in a circuit in Figure 1. The magnetically coupled resonator coils used are planar of class 1 with standard ISO/ICE 14443 [42]. The circuit has an LC−series resonance obtained using the capacitors C r e s .
In addition to the symbolic regression (SR) approach, a neural network (NN) embedded within a microcontroller was implemented to optimize the circuit’s performance by achieving optimal impedance matching conditions. This adjustment is crucial for improving the overall efficiency of the WPT system. The NN controls an Adaptive Impedance Matching Network (AIMN), which is designed using an LC circuit consisting of three cascaded L−type low-pass networks arranged in a series–series configuration as shown in Figure 2.
The capacitances within the AIMN are variable ceramic capacitors, allowing for the dynamic control of capacitors C 1 , C 2 , and C 3 to achieve effective impedance matching. The entire system is managed by a microcontroller that continuously monitors the scattering parameters, represented by the S-parameter complex matrix. From these measurements, the input impedance ( Z i n ) of the system can be precisely calculated. This integrated approach not only enhances power transfer efficiency but also ensures that the system operates under optimal conditions for maximum performance. The WPT circuit comprises two circular coils separated by a distance d as shown in the scheme of Figure 3. The coils are not perfectly aligned, with misalignments measured by Δ x and Δ y ; see Figure 3. Additionally, the receiver coil is rotated by an angle θ to simulate a realistic scenario. The receiver is connected to a fixed load, while the transmitter is linked to the microcontroller and the AIMN. A simplified version of this system is shown in Figure 3.
As previously mentioned, maximum power transfer is achieved by adjusting the AIMN to reach the working point of perfect impedance matching. The control variables for the AIMN are the capacitance values C 1 , C 2 , and C 3 . The NN described in [41] is trained to predict the optimal values of C 1 , C 2 , and C 3 required to achieve perfect impedance matching, based on the measured S-parameters or Z i n . A NN with five hidden layers with 15 neurons each was used, trained using the Mean Square Error (MSE) as the loss function. However, analysis of the circuit with and without the AIMN revealed that, in some configurations, the additional circuit introduces losses and worsens the impedance matching performance. To address this, the microcontroller was also programmed to bypass the AIMN when its use was not advantageous. A substantial amount of data were collected from this system, including both experimental measurements and simulations. As for simulations, data were taken from different TX-RX locations to train, validate, and test the NN that controls the AIMN. This simulations were conducted on 229 different configurations, recreating the physical circuit in a simulation software, like SIMULINK (version R2024b) environment in Figure 1 or as performed by [41] using Cadence’s AWR environment. In the simulations, various system configurations were considered, accounting for factors such as the distance between the two coils, misalignment, and the rotation angle of the receiver. This comprehensive dataset, which is available here [43], provides valuable insights into the behavior of the WPT system under different conditions. The dataset includes measurements of key parameters such as power transfer efficiency, impedance matching, and scattering parameters, enabling a detailed analysis of the system’s performance. By leveraging these data, researchers can further refine the SR and NN models, improving their accuracy and applicability to real-world WPT systems. This approach not only enhances the understanding of WPT dynamics but also paves the way for the development of more efficient and reliable Wireless Power Transfer solutions.
  • Experimental data: 3 configurations are reported, in which θ = 0 , Δ x = ( 0 , 20 , 30 ) mm, Δ y = ( 0 , 20 , 25 ) mm. For each configuration, 57 points are measured at different distance values d, with d going from 14 mm to 70 mm at 1 mm step. To reduce the noise, 5 measurements are taken at each position. The measured values are | S 11 | , | S 21 | , Z i n complex number and Γ . Also, the capacitance values used for the perfect impedance matching are reported; the same measurements as mentioned above are taken after impedance matching performed by the AIMN.
  • Simulation data for comparison with experimental ones. The same configurations are considered but now the data are taken from the simulation.
  • Simulation data of 229 different configurations, with different Δ x , Δ y and θ . For each of this configurations, there are 57 different points at different distance values. For each of them, the total complex matrix of the scattering parameters is reported, for 1001 points at different frequencies.
To sum up, S parameters and Z i n parameters are measured from the experimental setup at 3 different configurations for 57 different distance values, while 229 configurations are simulated, always for 57 different distance values. The main focus of this article is to predict the distance values and the configuration of the system from the measured S parameters or Z i n . Using symbolic regression would give not only good prediction but also analytical formulas that could be more efficient and faster to implement in a microcontroller to adapt the system to different configurations.

2.2.2. WPT Phase Control

The same SR technique is also utilized in a different WPT system to address a distinct issue, demonstrating the method’s versatility and applicability across various problems. This system, described in [44], employs inductive WPT for charging the receiver coil. The two coils used were manufactured by Wurth Elektronik, specific for wireless charging applications and compliant with the Qi standard, characterized by a winding inLitz wire and a core in high magnetic permeability material (ZnNi). The system has a series LC resonance in each antenna, obtained by connecting in series a compensation capacitor ( C 1 and C 2 in Figure 4 with the coil inductor. In this case, the frequency is variable, necessitating a control mechanism to adjust it according to varying load conditions and distances between the coils. To evaluate this, the phase difference between the voltage and current of the transmitter coil is measured. A dedicated module is implemented to measure the phase shift between voltage and current in the transmitter antenna using high-speed operational amplifiers and an XOR port. Subsequently, a proportional–integral controller (PI control) is implemented on a microcontroller, Teensy 4.1, to fine-tune the WPT system’s frequency, aiming to enhance the power factor cos ϕ . The feedback control aims to achieve a phase angle ϕ below 5 ∘ . The operating frequency ranges from 50 kHz to 90 kHz. The PI controller is executed on a microcontroller with fixed values for the proportional and integral constants set at 1 and 10, respectively. The phase difference serves directly as the error signal since the target phase is 0 degrees and the control loop ceases operation when the phase exceeds 5 degrees. The variable manipulated by the control loop is the WPT system’s frequency. As proposed in [44], the PI control can be enhanced by using an adaptive neuro-fuzzy inference system (ANFIS) that dynamically adjusts the PI constants for faster control. During the experiment, extensive data are collected using an oscilloscope, voltmeter, and ammeter, and subsequently processed using LabView 2020 and MATLAB 2021. This study focuses on four parameters: frequency, phase, load condition, and the distance between the two coils. The experiment is conducted 18 times under varying load conditions, R, and distances between the transmitter and receiver, z 0 . The circuit used in this experiment is represented in Figure 4.
Specifically, tests are performed with six load values, R = 100, 200, 500, 1000, 1500, and 2000 Ω , and three distance values, z 0 = 0, 4, and 8 mm. This dataset is open-source and accessible at [45]. From this dataset, it is noteworthy to observe the phase trends concerning frequency, with respect to R, as shown in Figure 5, and with respect to z 0 , as depicted in Figure 6. The primary behavior exhibits a decreasing phase trend with increasing frequency, an offset due to distance, and minor nonlinear dependencies. The main objective of this research is to predict these trends, particularly the linear components, to provide a reliable estimation of the WPT system’s performance. This prediction could either replace or complement the existing control system. For example, the control system could be expedited by initially estimating the optimal frequency, rather than starting from a fixed frequency where the phase exceeds 60 degrees. A rapid control system is crucial for adapting to fluctuating load and distance conditions in realistic scenarios. Additionally, this preliminary frequency estimation using SR minimizes power losses and voltage/current overloads by initiating the WPT system close to the optimal power transfer point.

2.3. Validation

The first dataset consists of, as input data, the scattering parameters, | S 11 | and | S 21 | , or the impedance parameters, Re ( Z i n ) and Im ( Z i n ) . As output parameters, we have the distance d between the two coils and the configuration parameters Δ x , Δ y , and θ . With symbolic regression, the goal is to best fit these data with a multi-variable function:
( d , Δ x , Δ y , θ ) = f 1 ( | S 11 , | S 21 | ) = f 2 ( R e ( Z i n ) , I m ( Z i n ) )
The second dataset consists of, as input data, the WPT frequency, the distance between the two coils, z 0 , and the load resistance, R. As output parameters, we have the phase between the input current and voltage. With symbolic regression, the goal is to best fit these data with a multi-variable function:
p h a s e = f 3 ( f , R , z 0 )
The SR algorithm pySR [46] is used. It is a Python 3.11.11 library that implements genetic algorithms to find the best formulas, with a cost function that, as previously mentioned, is inversely proportional to the MSE and the complexity of the formula. This tool allows you to define custom binary and unary operators, enabling the search for formulas with a given structure. Using pySR on a big dataset with more than 10,000 data points is not convenient since it significantly slows the research and increases the complexity of the problem. In the first simulation dataset, there are 229 · 57 · 1001 = 13,066,053 points, so the dataset is reduced, considering only the measured points at the resonant frequency of the WPT, 13.56 MHz. For pySR, however, that is still a large amount of data, so the analysis is performed in two reduced ways. First, a single configuration at a time is considered, so only 57 points in which only the distance variable d varies. The second analysis is performed at a given distance, with 229 points in which the configuration parameters vary. The number of populations and the number of iterations are both set to 10 in pySR. Increasing them to 50 can increase the accuracy by a few percentage points but at a significant computational cost. On the other hand, no particular operator is used, only exponential and logarithmic functions. But with some idea of the final formula to reach the research space, it can be drastically reduced. Due to how the dataset is built, it is convenient to use Z i n for the experimental measurements, and S in the simulated ones. Indeed, going from S to Z i n or vice versa, is possible but would increase the uncertainty due to numerical errors. The second dataset contains 3055 data points, with 18 configurations: 3 for the distance between the coils and 6 for the load conditions. The measured dataset has a lot of noise and also some invalid values as it can be seen from the figure at a fixed load value, Figure 6, and at a fixed distance value, Figure 5. For this reason, a data regularization is necessary. Standard regularization methods do not yield good results for high data irregularity, resulting in either overly strong regularization that also modifies good measurement points, or an inability to adequately regularize the data due to being too simple. Therefore, a custom adjustment function is defined for this dataset. The idea is to change only those points that differ significantly from their neighbors. Phase control of each point with the previous one is used, and for phase differences greater than 5 degrees, the phase of the point is adjusted to 0.9 times the phase of the previous point. Since a decreasing behavior is observed and there are only isolated points, the overall trend is not affected. From the figure of the data after the regularization, Figure 7, Figure 8 and Figure 9, it can be seen that this regularization technique modifies only the critical points as expected.
For this dataset, a very simple SR model is used, with 5 iterations and 5 populations, and only “+” and “*” as binary operators, and no unary operators. Despite its simplicity, this model achieves good results. It can still predict the trends of the curves and provide good estimates, even if they are not very accurate. This is explained in more detail in the results section. On the other hand, a more complex model is studied, to predict the more sophisticated behavior of the functions. For this reason, 500 iterations with 100 populations are used, with “+”, “*” and “/” as binary operators, and exponential and logarithmic functions as unary operators, to introduce nonlinear behavior. This complex model obtains only a few percentage points in improved accuracy compared to the simpler one, detecting very similar formulas. In addition to the simpler model, it also detects the dependencies on the load.
Table 1 is provided to summarize the various models and their respective parameters.

3. Results

3.1. Results on the Impedance Dataset

Using the SR to this WPT allows to derive the formula that describes the system. The SR applied to the prediction of the distance achieves very accurate results, with an error of less than 1 % , as shown in Figure 10 for the S parameters and Figure 11 for the Z parameters. Both S and Z i n lead to good results in terms of accuracy, but the impedance parameters perform slightly better, possibly because the relationship with the distance is more straightforward, while the S parameters are related to nonlinear and non-unique relationship with d. This could be seen from the figures of S in Figure 12 and its prediction in Figure 13 and of Z in Figure 14, for example. For comparison, the Table 2 of results are obtained in distance prediction with SR using impedance and scattering parameters as input in the experimental dataset with 3 configurations. The error is computed as the mean, multiplied by 100, of the relative error for each point predicted of the distance: error = mean ( y − y p r e d y ) 100 .
On the other hand, the prediction of the configuration does not yield accurate results, as the formulas found have no particular physical significance. The same results are obtained with classical NNs for both the experimental and simulated datasets. This is likely due to insufficient information in the input parameters. Therefore, the dataset can be extended with other sensors, such as magnetic sensors, to add more information and derive complete formulas. As a first approach of this method to a WPT system, very promising results are obtained, with prediction accuracy comparable to that of NNs. Although the formulas derived from symbolic regression are accurate, their complexity makes direct transcription impractical and less informative. Therefore, we focus on a qualitative analysis of the functional relationships discovered, emphasizing the dependencies and trends that offer valuable insights into the system’s behavior. Specifically, for the input impedance (real part, R i n , and imaginary part, X i n ), the symbolic regression model identifies a 1 / R i n dependence on distance and a parabolic trend for X i n . For the scattering parameters, the model reveals parabolic and logarithmic dependencies.

3.2. Results on the Phase Dataset

Using the SR to the second WPT allows to derive the formula that describes the system and obtain sufficiently accurate predictions. Although the phase prediction is not perfect, the model can identify the correct trend and obtain acceptable accuracy for an initial estimate. In particular, the simpler model can successfully predict the main behaviour of the function, focusing only on the linear part as shown in Figure 15, Figure 16 and Figure 17. The time required by this simpler model is just a few minutes. On the other hand, the more complex model achieves more accurate results, correctly identifying also the dependence on R, shown in Figure 18 and approaching to predict the nonlinearity of the problem, shown in Figure 19 and Figure 20. The drawback of this complex model is the time required to train it, which reaches an hour, and even then, these predictions are not yet perfect.
The formula extracted with the simpler model is
ϕ ( f , R , z 0 ) = − α f + ϕ 0 + β z 0 ,
with α = 0.0015 , ϕ 0 = 137.40 and β = 3.49 .
As shown also in Figure 15, Figure 16 and Figure 17, this formula is quite simple but still provides useful insights into the original data, catching the correct behavior of the function with good enough accuracy in the prediction.
With the more complex model, the formula extracted is more complex, with the same linear component as Equation (3), and a term having the logarithm of R times z 0 . Reporting it here, is not considered to be of interest.
These results show good enough results for a first prediction to improve the control system proposed in [44], and show how SR could be helpful in understanding the behavior of this WPT system.

4. Discussion

The application of symbolic regression (SR) in the context of Wireless Power Transfer (WPT) systems has demonstrated significant potential as evidenced by the results presented in this study. The ability of SR to derive explicit mathematical formulas from both simulated and experimental data offers a unique advantage over traditional machine learning methods, particularly in terms of interpretability and computational efficiency. The results indicate that SR can achieve high predictive accuracy, with errors less than 1% in estimating the distance between coils using both scattering parameters (S-parameters) and impedance parameters ( Z i n ). This level of precision is comparable to that of neural networks (NNs), yet SR provides the added benefit of interpretable analytical expressions. These expressions not only allow for a deeper understanding of the underlying physical relationships but also enable real-time applications, such as dynamically adjusting coil distances to optimize power transfer efficiency. The study also highlights the versatility of SR in addressing different challenges within WPT systems. For instance, the application of SR to predict the phase–frequency relationship in a variable-frequency WPT system demonstrates that even a simple model can capture the main trends of the phase behavior. This suggests that SR can be utilized to complement or even replace existing control systems, particularly in scenarios where rapid adaptation to changing conditions is required.

5. Conclusions

This research highlights the considerable promise of symbolic regression (SR) in the context of Wireless Power Transfer (WPT) systems, achieving accuracy levels on par with neural networks (NNs). SR emerges as a powerful method for extracting analytical expressions that model intricate relationships within WPT systems, providing both high predictive precision and interpretability. The methodology has demonstrated a notable degree of universality, evidenced by its successful application across three distinct scenarios. Firstly, it accurately predicted the distance in a Near-Field Communication Wireless Power Transfer (NFC-WPT) system using simulated data and S-parameters as input. Secondly, it achieved similar predictive accuracy for distance in an NFC-WPT system utilizing experimental data and Z-parameters. Finally, the method effectively predicted phase in an inductive WPT coil, even with highly irregular experimental data and frequency as the input. In all three cases, accurate predictive outcomes were obtained, supporting the assertion of its broad applicability. Future investigations will extend the validation of this SR method to encompass more complex WPT systems, such as the moving WPT system like in [47] and with a more complex structure of the WPT system, such as an array of receiver coils as in [48], thereby rigorously assessing the generalizability of the SR method on WPT systems. Nonetheless, additional studies are needed to optimize its performance by narrowing the search space through the integration of physical constraints and symmetries, such as the radial symmetry present in coil designs. These adjustments could enhance the efficiency and accuracy of SR models while lowering computational demands. Although this study concentrates on applying SR to predict the distance between coils and the phase frequency relationship, its utility extends to other aspects of WPT systems. For instance, SR could be used to predict key properties like coupling coefficient or mutual inductance as a function of system parameters [49]. The sensitivity of SR models to errors, particularly noise and uncertainties, necessitates a rigorous analysis for future applications in WPT systems, especially when compared to the robustness of neural networks. As highlighted in [50], tolerance analysis is crucial for accurately assessing interoperability in wireless charging systems, as manufacturing tolerances and parameter drift can significantly impact test results and lead to misjudgments, necessitating the establishment of clear tolerance zones to ensure precise classification. Unlike conventional linear regression methods, which depend on approximations and manual calculations, SR can autonomously generate more precise results. By automating the identification of exact mathematical relationships, SR has the potential to simplify the design and optimization processes in WPT systems. In summary, SR offers a promising direction for advancing WPT research and technology. Its capacity to merge the advantages of machine learning with analytical modeling positions it as a versatile tool for tackling complex issues in this field. Future efforts should aim to broaden its applications within WPT systems and investigate methods to further enhance its efficiency and reliability. This could lead to the development of more innovative and effective Wireless Power Transfer solutions.

Author Contributions

Conceptualization, D.M., L.S., R.A. and F.R.F.; methodology, D.M., L.S., R.A. and F.R.F.; validation, D.M., L.S., R.A. and F.R.F.; writing—original draft preparation, D.M., L.S., R.A. and F.R.F.; writing—review and editing, D.M., L.S., R.A. and F.R.F.; supervision, F.R.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Electrical circuit schematic of the WPT system proposed in [41]. This circuit is represented here with SIMULINK. V i n is the input source of power, and R s is the series impedance of the generator, with a resistance of 50 Ω . The parameters used to simulate the transmitter coil are C r e s = 22.6 pF, C p 1 = 5.7 pF and R r f 1 = 4.48 Ω . The receiver is simulated with R r f 2 = 4.48 Ω , C p 2 = 5.7 pF, C r e s 2 = 22.6 pF and with a purely resistive load of 50 Ω . The coils are simulated with a coupled inductor with a mutual inductance of 4.87 μ H .
Figure 1. Electrical circuit schematic of the WPT system proposed in [41]. This circuit is represented here with SIMULINK. V i n is the input source of power, and R s is the series impedance of the generator, with a resistance of 50 Ω . The parameters used to simulate the transmitter coil are C r e s = 22.6 pF, C p 1 = 5.7 pF and R r f 1 = 4.48 Ω . The receiver is simulated with R r f 2 = 4.48 Ω , C p 2 = 5.7 pF, C r e s 2 = 22.6 pF and with a purely resistive load of 50 Ω . The coils are simulated with a coupled inductor with a mutual inductance of 4.87 μ H .
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Figure 2. Circuit of the AIMN proposed in [41]. This circuit uses three variable capacitances to adjust the impedance of the circuit for obtaining the perfect impedance matching. The ports shown are the ports used in the measurements of the scattering parameters and so also the impedance parameters.
Figure 2. Circuit of the AIMN proposed in [41]. This circuit uses three variable capacitances to adjust the impedance of the circuit for obtaining the perfect impedance matching. The ports shown are the ports used in the measurements of the scattering parameters and so also the impedance parameters.
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Figure 3. Simplified scheme of the main blocks of the circuit proposed in the experiment with the AIMN. For a more complete scheme and more details of the system used, see [41]. Here, the two coils are represented with Tx for the transmitter ones and Rx for the receiver. They are separated by a distance d, along the z direction, and their centers are misaligned by Δ y (Dy) and Δ x (Dx). The Rx coil is connected to a purely resistive load as shown also in Figure 1. The Tx coil is connected through the AIMN to the microcontroller and the signal block, that represents a more complex system with signal generator, measurement instruments, and directional couplers.
Figure 3. Simplified scheme of the main blocks of the circuit proposed in the experiment with the AIMN. For a more complete scheme and more details of the system used, see [41]. Here, the two coils are represented with Tx for the transmitter ones and Rx for the receiver. They are separated by a distance d, along the z direction, and their centers are misaligned by Δ y (Dy) and Δ x (Dx). The Rx coil is connected to a purely resistive load as shown also in Figure 1. The Tx coil is connected through the AIMN to the microcontroller and the signal block, that represents a more complex system with signal generator, measurement instruments, and directional couplers.
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Figure 4. Electric circuit of the WPT system used in the experimental setup for the phase control system. This circuit was created by [44]. The main part is the mutual inductance that represents the two coils separated by a physical distance z 0 . The value of the mutual inductance changes with z 0 . The MOSFET transistors are controlled by the microcontroller to adjust the frequency of the WPT system in order to minimize the phase angle. In this simplified scheme, the microcontroller is omitted to reduce circuit complexity, resulting in open MOSFET gates, as indicated by the red circle. We report the parameters of the circuit: input voltage, V D C = 12 V, transmitter and receiver coil self inductance, L 1 = L 2 = 24 μ H , transmitter and receiver coil internal resistance, R 1 = R 2 = 0.07 Ω , transmitting and receiving antenna compensation capacitor, C 1 = C 2 = 200 nF, shunt resistance R s = 3 Ω , load resistor R = 100, 200, 500, 1000, 1500, 2000 Ω , and operating frequency f = 50−90 kHz.
Figure 4. Electric circuit of the WPT system used in the experimental setup for the phase control system. This circuit was created by [44]. The main part is the mutual inductance that represents the two coils separated by a physical distance z 0 . The value of the mutual inductance changes with z 0 . The MOSFET transistors are controlled by the microcontroller to adjust the frequency of the WPT system in order to minimize the phase angle. In this simplified scheme, the microcontroller is omitted to reduce circuit complexity, resulting in open MOSFET gates, as indicated by the red circle. We report the parameters of the circuit: input voltage, V D C = 12 V, transmitter and receiver coil self inductance, L 1 = L 2 = 24 μ H , transmitter and receiver coil internal resistance, R 1 = R 2 = 0.07 Ω , transmitting and receiving antenna compensation capacitor, C 1 = C 2 = 200 nF, shunt resistance R s = 3 Ω , load resistor R = 100, 200, 500, 1000, 1500, 2000 Ω , and operating frequency f = 50−90 kHz.
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Figure 5. Data measured at fixed coils distance of z 0 = 4 mm, for six different load conditions. Is it clear that data regularization is needed due to measurement error.
Figure 5. Data measured at fixed coils distance of z 0 = 4 mm, for six different load conditions. Is it clear that data regularization is needed due to measurement error.
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Figure 6. Data measured at fixed load condition of R = 100 Ω , for three different distance coil conditions, with z 0 = 0, 4 and 8 mm. Is it clear that data regularization is needed due to the measurement error.
Figure 6. Data measured at fixed load condition of R = 100 Ω , for three different distance coil conditions, with z 0 = 0, 4 and 8 mm. Is it clear that data regularization is needed due to the measurement error.
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Figure 7. Dataset after the data regularization. Data measured at fixed load condition of R = 100 Ω , for three different distance coil conditions, with z 0 = 0, 4 and 8 mm.
Figure 7. Dataset after the data regularization. Data measured at fixed load condition of R = 100 Ω , for three different distance coil conditions, with z 0 = 0, 4 and 8 mm.
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Figure 8. Datasetafter the data regularization. Data measured at fixed load condition of R = 1000 Ω , for three different distance coil conditions, with z 0 = 0, 4 and 8 mm.
Figure 8. Datasetafter the data regularization. Data measured at fixed load condition of R = 1000 Ω , for three different distance coil conditions, with z 0 = 0, 4 and 8 mm.
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Figure 9. Datasetafter the data regularization. Data measured at fixed coil distance of z 0 = 4 mm, for six different load conditions.
Figure 9. Datasetafter the data regularization. Data measured at fixed coil distance of z 0 = 4 mm, for six different load conditions.
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Figure 10. Predicted distance values and original ones using S parameters for the simulated dataset.
Figure 10. Predicted distance values and original ones using S parameters for the simulated dataset.
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Figure 11. Predicted distance values and original ones using Z i n parameters for the experimental measurement dataset, for the case 1.
Figure 11. Predicted distance values and original ones using Z i n parameters for the experimental measurement dataset, for the case 1.
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Figure 12. S parameters at different distance values. These are the data of the simulation dataset taken at a fixed configuration. The S parameter values reported here are in unitary (dimensionless) form. Conversion to decibels (dB) can be achieved using the formula S d B = 20 log 10 ( | S u n i t a r y | ) .
Figure 12. S parameters at different distance values. These are the data of the simulation dataset taken at a fixed configuration. The S parameter values reported here are in unitary (dimensionless) form. Conversion to decibels (dB) can be achieved using the formula S d B = 20 log 10 ( | S u n i t a r y | ) .
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Figure 13. Prediction of the SR model of the distance values. For better visualization, the plot is made for S parameters at different distance values, but now the dependent variable is the distance. The S parameter values reported here are in unitary (dimensionless) form. Conversion to decibels (dB) can be achieved using the formula S d B = 20 log 10 ( | S u n i t a r y | ) .
Figure 13. Prediction of the SR model of the distance values. For better visualization, the plot is made for S parameters at different distance values, but now the dependent variable is the distance. The S parameter values reported here are in unitary (dimensionless) form. Conversion to decibels (dB) can be achieved using the formula S d B = 20 log 10 ( | S u n i t a r y | ) .
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Figure 14. Real and imaginary parts of impedance value Z i n for different distance values. The image is taken from case 1, with Δ x = 0 mm, Δ y = 0 mm.
Figure 14. Real and imaginary parts of impedance value Z i n for different distance values. The image is taken from case 1, with Δ x = 0 mm, Δ y = 0 mm.
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Figure 15. Phase predicted with symbolic regression using the simple model. Prediction for the load condition of R = 100 Ω . The prediction is far from the real value, but the main trend is well predicted, giving useful insights for an evaluation of the phase or the best frequency that minimizes the phase shift. The time required for the simulation is about five minutes.
Figure 15. Phase predicted with symbolic regression using the simple model. Prediction for the load condition of R = 100 Ω . The prediction is far from the real value, but the main trend is well predicted, giving useful insights for an evaluation of the phase or the best frequency that minimizes the phase shift. The time required for the simulation is about five minutes.
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Figure 16. Phase predicted with symbolic regression using the simple model. Prediction for the load condition of R = 1000 Ω . The prediction is far from the real value, but the main trend is well predicted, giving useful insights for an evaluation of the phase or the best frequency that minimizes the phase shift.
Figure 16. Phase predicted with symbolic regression using the simple model. Prediction for the load condition of R = 1000 Ω . The prediction is far from the real value, but the main trend is well predicted, giving useful insights for an evaluation of the phase or the best frequency that minimizes the phase shift.
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Figure 17. Phase predicted with symbolic regression using the simple model, for the fixed distance of the coil Z 0 = 4 mm. From this simple prediction, no dependencies with the load condition emerge.
Figure 17. Phase predicted with symbolic regression using the simple model, for the fixed distance of the coil Z 0 = 4 mm. From this simple prediction, no dependencies with the load condition emerge.
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Figure 18. Phase predicted with symbolic regression using the more complex model, for the fixed distance of the coil Z 0 = 4 mm. Compared to the prediction of the simpler model, dependencies on the load are now detected.
Figure 18. Phase predicted with symbolic regression using the more complex model, for the fixed distance of the coil Z 0 = 4 mm. Compared to the prediction of the simpler model, dependencies on the load are now detected.
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Figure 19. SR prediction of the phase, for fixed load condition R = 100 Ω . The more complex model is used with 500 iterations, 100 populations, binary operators “+”, “*” and “/”, and unary operators as the exponential and logarithmic functions. The time required for the simulation is about one hour.
Figure 19. SR prediction of the phase, for fixed load condition R = 100 Ω . The more complex model is used with 500 iterations, 100 populations, binary operators “+”, “*” and “/”, and unary operators as the exponential and logarithmic functions. The time required for the simulation is about one hour.
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Figure 20. SR prediction of the phase, for fixed load condition R = 1000 Ω . The more complex model is used with 500 iterations, 100 populations, binary operators “+”, “*” and “/”, and unary operators as exponential and logarithmic functions. The time required for the simulation is about one hour.
Figure 20. SR prediction of the phase, for fixed load condition R = 1000 Ω . The more complex model is used with 500 iterations, 100 populations, binary operators “+”, “*” and “/”, and unary operators as exponential and logarithmic functions. The time required for the simulation is about one hour.
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Table 1. SR models used with their parameter settings.
Table 1. SR models used with their parameter settings.
ParametersExperimental NFCSimulated NFCSimple Model PhaseComplex Model Phase
INPUT dataZ-parameterS-parametersphase, R, z 0 phase, R, z 0
OUTPUT dataDistanceDistancePhasePhase
dataset size57 × 357 × 22930553055
n. population10105100
n. iteration10105500
binary operators+, *, /, power+, *, /, power+, *+, *, /
unary operatorsexp, logexp, logno op.exp, log
Table 2. Comparative analysis of percentage errors on the distance prediction using SR with input parameter impedance, Z, and scattering parameters, S. This analysis is performed in the experimental dataset. The Z analysis demonstrates marginally lower error rates in Cases 1 and 2 (approximately 1% for both cases), while the S analysis performs slightly better in Case 3, though the differences are minimal.
Table 2. Comparative analysis of percentage errors on the distance prediction using SR with input parameter impedance, Z, and scattering parameters, S. This analysis is performed in the experimental dataset. The Z analysis demonstrates marginally lower error rates in Cases 1 and 2 (approximately 1% for both cases), while the S analysis performs slightly better in Case 3, though the differences are minimal.
CaseSZ
Case 18.31%1.89%
Case 24.45%0.92%
Case 31.02%2.74%
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Milillo, D.; Sabino, L.; Asghar, R.; Riganti Fulginei, F. Symbolic Regression Method for Estimating Distance Between Two Coils of an Inductive Wireless Power Transfer System. Appl. Sci. 2025, 15, 3668. https://doi.org/10.3390/app15073668

AMA Style

Milillo D, Sabino L, Asghar R, Riganti Fulginei F. Symbolic Regression Method for Estimating Distance Between Two Coils of an Inductive Wireless Power Transfer System. Applied Sciences. 2025; 15(7):3668. https://doi.org/10.3390/app15073668

Chicago/Turabian Style

Milillo, Davide, Lorenzo Sabino, Rafiq Asghar, and Francesco Riganti Fulginei. 2025. "Symbolic Regression Method for Estimating Distance Between Two Coils of an Inductive Wireless Power Transfer System" Applied Sciences 15, no. 7: 3668. https://doi.org/10.3390/app15073668

APA Style

Milillo, D., Sabino, L., Asghar, R., & Riganti Fulginei, F. (2025). Symbolic Regression Method for Estimating Distance Between Two Coils of an Inductive Wireless Power Transfer System. Applied Sciences, 15(7), 3668. https://doi.org/10.3390/app15073668

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