Modeling and Multi-Objective Optimization of Transcutaneous Energy Transmission Coils Based on Artificial Intelligence

Abstract

This paper proposes a machine learning-based modeling and multi-objective optimization method for transcutaneous energy transfer coils to address the problem that current transcutaneous energy transfer coils with single-objective optimization design methods have difficulty achieving optimal solutions. From modeling to multi-objective optimization design, the whole transcutaneous energy transfer coil process is covered by this approach. This approach models transcutaneous energy transfer coils using the Extreme Learning Machine, and the Gray Wolf Optimization algorithm is used to tune the Extreme Learning Machine’s parameters in order to increase modeling accuracy. The Non-Dominated Sorting Whale Optimization algorithm is utilized for multi-objective optimization of the transcutaneous energy transfer coils, which is based on the established model. Using the optimization of planar helical coils applied in artificial detrusors as an example, a verification analysis was conducted, and the final optimization analysis results were demonstrated. The results indicate that the Gray Wolf Optimization algorithm significantly outperforms the comparison algorithms in tuning the parameters of the Extreme Learning Machine model, and it exhibits good convergence ability and stability. The established transcutaneous energy transfer coil prediction model outperforms the comparative prediction model in terms of evaluation metrics for predicting the three outputs (transmission efficiency, coupling coefficient, and secondary coil diameter), demonstrating excellent prediction performance. The Non-Dominated Sorting Whale Optimization algorithm performs well in the multi-objective optimization process of transcutaneous energy transfer coils, showing excellent results. The Pareto optimal solutions obtained using this algorithm have errors of 3.03%, 0.1%, and 1.7% for transmission efficiency, coupling coefficient, and secondary coil diameter, respectively, when compared to the simulation and experimental calculations. The small errors validate the correctness and effectiveness of the proposed method.

1. Introduction

Implantable medical devices (IMDs) are a type of medical equipment that is surgically implanted or inserted through natural openings in the body to replace, repair, or enhance bodily functions [1]. Such devices play an important role in disease diagnosis and treatment, as well as in monitoring vital signs. Typical IMDs include artificial hearts [2], gastrointestinal stimulators [3], heart pacemakers [4], neurostimulators [5], retinal prosthesis [6], etc. The normal operation of IMDs relies on a stable energy supply, with traditional energy sources mainly being batteries and transcutaneous wiring [7]. Batteries do not depend on external energy sources and can continuously supply energy to IMDs for a long period after implantation. Additionally, the technology is relatively mature and widely applied. Nevertheless, the battery has a limited lifespan and needs to be replaced surgically on a regular basis, which surely raises the patient’s risk of surgery and medical expenses. Furthermore, implanted batteries are frequently big and hefty to satisfy the need for a long-term power source, which somewhat increases bodily intrusion and lowers patient comfort. The transcutaneous wiring power supply method does not require regular replacement and allows IMDs to continuously obtain energy through a wired connection to an external power source. However, this method still has significant limitations. The puncture site for the transcutaneous wiring is highly susceptible to tissue infection risks. Additionally, the wiring somewhat restricts the patient’s normal physiological activities, thereby reducing their quality of life. To break through the bottleneck of traditional energy supply methods, relevant researchers and scholars have conducted in-depth research and active exploration.

With the rise of wireless energy transmission technology, transcutaneous energy transmission (TET) has shown great potential in powering IMDs [8]. The coupling mechanism of TET typically consists of two transcutaneous energy transmission coils (primary coil and secondary coils). Based on the principle of inductive coupling, an alternating current in the external primary coil generates a magnetic field, which in turn induces an induced current in the secondary coil, thereby achieving the transmission of electrical energy. This technology can transmit energy from outside the body to implanted devices inside without any physical connections, effectively avoiding the risk of tissue infection caused by traditional transcutaneous wiring methods and the issues of large sizes and the regular replacement associated with battery-powered methods. TET is currently the most ideal power supply method for IMDs [9]. Although TET has many advantages over traditional power supply methods, it still faces some difficulties and challenges. Since the secondary coil of the TET coupling mechanism needs to be implanted in the patient’s body, its size is limited, which affects the transmission performance of the TET system and, consequently, the stability of the IMD’s power supply. Moreover, due to the influence of tissue conditions and the patient’s daily physiological activities, the relative position and distance between the primary coil outside the body and the secondary coil inside the body may change, thereby reducing the coupling degree between the coils, which in turn affects transmission performance and increases heat loss. In this context, enhancing the transmission performance of the TET system to provide a continuous, stable, and reliable energy supply for IMDs is of great significance.

Relevant researchers and scholars have conducted extensive studies on improving the transmission performance of TET systems [10,11,12,13]. Numerous research results indicate [14,15] that optimizing the design of TET coils can effectively enhance the transmission performance of TET systems. Jow et al. [16] established a theoretical, analytical model of the TET system with the optimization goal of maximizing system transmission efficiency. They proposed a stepwise iterative optimization design method for printed spiral coils (PSC) and conducted example verification. The results showed that under the conditions of working frequencies of 1 MHz and 5 MHz and a coil spacing of 10 mm, transmission efficiencies of 41.2% and 85.8% could be achieved, respectively. Based on theoretical analytical models, similar studies on coil optimization design using a stepwise iterative optimization design method can be found in references [17,18,19,20,21,22,23]. To reduce the power loss of the coil in the implanted artificial heart TET system and avoid thermal damage to the tissue, Leung et al. [24] analyzed the impedance of the entire circuit based on a theoretical analytical model. On this basis, they proposed an optimization design method aimed at minimizing the coil’s power loss. Kim et al. [25] proposed a geometry-constrained coil optimization design method, which uses maximizing transmission efficiency as the optimization objective function. Under strict geometric constraints, the TET coil parameters were optimized, and the design example of the TET coil implanted in a visual prosthesis was verified. The results showed that after optimization of the primary and secondary coils, the experimentally measured transmission efficiency was 52%. To reduce the invasiveness of the implantation process on the human body, Liu et al. [26] proposed a planar spiral coil and performed an in-depth analysis of the coil parameters affecting TET transmission efficiency using the finite element method. They investigated the effects of main and secondary coils of varying diameters on transmission efficiency and conducted analyses to determine the ideal coil construction characteristics. Knecht et al. [27] developed an effective TET system to reduce power loss in the TET coil and implanted electronic components and prevent heat harm to tissues. In addition, to find an optimal combination of the coil parameters, they combined theoretical analytical models with finite element methods to conduct an optimization analysis of the coil structure. References [28,29,30] conducted an optimization design analysis of the TET primary and secondary coils based on experimental methods. To reduce the impact of misalignment between the primary and secondary coils of the TET on transmission efficiency, Wu et al. [8] conducted an optimization design analysis of the primary and secondary coils based on the finite element method, with the maximum coupling coefficient as the optimization objective. The analysis results were verified through the application of an artificial heart example, and a comparative analysis was conducted between the optimized planar coil and the curved coil. The results showed that under conditions of coil misalignment, the curved coil had higher transmission efficiency and lower temperature rise compared to the planar coil. Rodrigues et al. [31] proposed a TET coil optimization design method based on finite element method modeling utilizing genetic algorithm optimization. This method uses the minimization of the difference between the actual output power and the target output power as the optimization objective. Bai et al. [32] also conducted similar research. Based on the theoretical analytical model of the established TET system, they introduced a design factor to comprehensively consider transmission efficiency and transmission power, transforming it into a single-objective problem for analysis. Finally, they used a genetic algorithm to optimize the TET coil parameters. References [33,34] propose a TET coil optimization design method based on finite element method modeling combined with a multi-objective genetic algorithm. This method aims to optimize the TET coil parameters by minimizing the coil size and coil dissipation power in order to provide patients with a more comfortable and reliable TET system.

In conclusion, the following are the primary optimization design approaches currently used for TET coils: analytical, finite element, experimental, analytical modeling in conjunction with a step-by-step iterative optimization design method, analytical or finite element modeling in conjunction with a genetic optimization algorithm, etc. The mathematical model of the TET system developed by the analytical method can thoroughly investigate the effect of each variable on the optimization objective as well as the interaction of the coil structure variables, and the model is highly interpretable. However, the derivation process of the mathematical models is often complex and time-consuming. Additionally, to facilitate computation, mathematical modeling usually requires an assumption of ideal conditions or simplifying physical models, which, to some extent, reduces the model’s accuracy and subsequently affects the optimization results. Moreover, the parasitic internal resistance of the TET coil is influenced by various factors, making it often difficult to establish an accurate mathematical model for solving it, which, to some extent, will also affect the optimization results. The experimental method can only optimize the coil parameters within a relatively limited experimental group. The step-by-step iterative optimization method usually involves optimizing the primary or secondary coils one by one in sequence, making it impossible to achieve synchronous optimization of the coils. The optimization process is relatively cumbersome and inefficient. Moreover, in the existing research on TET coil optimization design, the vast majority are optimized with a single objective, making the solving process relatively simple and the computational efficiency high. However, a single objective may not fully reflect the complexity of the actual application of the TET system, leading to solutions that are not optimal for the actual problem in certain cases. In practical applications, not only should transmission efficiency be considered, but also the size limitations of body implants and the coupling degree of coils, among other requirements, in order to provide patients with a more stable, comfortable, and highly reliable TET system.

To address the aforementioned issues, this study proposes a machine learning-based modeling and multi-objective optimization design method for TET coils. Machine learning-based models do not rely on specific knowledge of the physical principles of the system. Instead, they exhibit extraordinary potential in extracting complex patterns and handling nonlinear relationships. Models based on machine learning possess strong flexibility and adaptability, especially for complex systems with multiple variables. Therefore, using machine learning for modeling is superior to the traditional methods. Furthermore, multi-objective optimization, as opposed to single-objective optimization, can better capture the complexities of the TET system’s real application. By taking into account many aspects, it can identify coil parameter combinations that better fit the features of the real situation. The primary scientific contributions of this work are as follows:

• A method for modeling and optimizing TET coils is proposed, which combines the Extreme Learning Machine (ELM) with the Non-Dominated Sorting Whale Optimization algorithm (NSWOA). The ELM is used to construct a surrogate model for the multi-input and multi-output parameters of the TET coils. Based on this surrogate model, a multi-objective optimization function for the TET coils is constructed, and the NSWOA is then used to perform multi-objective optimization on the TET coils.
• To improve the modeling accuracy of the TET coils, the Gray Wolf Optimizer (GWO) was introduced and applied to the parameter tuning of the ELM model for the TET coils in this study.
• A TET coils simulation platform has been established, and the correctness of the simulation has been verified through experimental measurements. On this basis, the simulation platform was used to collect the data required for TET coil modeling.
• The proposed method was subjected to an in-depth comparative analysis with other methods through a series of evaluation metrics, fully demonstrating its effectiveness and correctness. Additionally, the optimization results obtained using the proposed method were also thoroughly validated.

The remainder of this study is organized as follows. Section 2 introduces the basic background of this research. Section 3 establishes the ELM-based TET coil model, discusses the application of metaheuristic optimization algorithms in tuning the machine learning model parameters, and finally presents the detailed steps of the proposed method’s implementation along with the corresponding flowchart. Section 4 describes the case validation. Section 5 discusses the evaluation metrics for the proposed method and the comparison method in the modeling and optimization process of the TET coil, and the optimization results of the proposed method are demonstrated based on simulations and experiments. Section 6 presents the conclusion and suggests possible future work.

2. Background

This section mainly introduces the composition of the TET system and provides a preliminary discussion on the compensation topology.

2.1. TET System Composition

The TET system mainly consists of two parts: the receiving electronic circuit implanted in the body and the transmitting electronic circuit outside the body. The schematic diagram of its composition principle is shown in Figure 1. The external DC/AC converter converts direct current into alternating current of a certain frequency and transmits it to the primary coil and its compensation topology circuit. Through the alternating magnetic field between the primary coil and the secondary coil, electrical energy is transmitted to the internal secondary coil and its compensation topology circuit. After rectification and filtering by the internal AC/DC converter, it provides continuous and stable direct current to the load (IMDs).

2.2. Compensating Topology Circuit

In the TET system, in addition to the coupling coils, the compensation topology also has a significant impact on the system’s power transmission efficiency and energy loss. Therefore, it is crucial to reasonably select the compensation topology based on the actual application scenario to minimize reactive power, which is essential for improving power transmission efficiency and reducing energy loss during transmission. According to existing research, the most basic compensation topologies mainly include series–series (SS), parallel–series (PS), series–parallel (SP), and parallel–parallel (PP), with their schematic diagrams shown in Figure 2. In the figure, $L_{\mathrm{P}}$ and $L_{\mathrm{S}}$ represent the inductance of the primary and secondary coils, respectively, while $C_{\mathrm{P}}$ and $C_{\mathrm{S}}$ represent the compensation capacitors of the primary and secondary coils, respectively. When the system is in a fully resonant state, the calculation formulas for the compensation capacitors of the primary and secondary coils for the four compensation topologies are shown in

Table 1. Calculation formula for compensation capacitors.

	Type	${\mathit{C}}_{\mathbf{P}}$	${\mathit{C}}_{\mathbf{S}}$
Parameter
SS		$1/{\omega }^2{L}_{\mathrm{P}}$	$1/{\omega }^2{L}_{\mathrm{S}}$
SP		$1/{\omega }^2{L}_{\mathrm{P}}$	${\displaystyle \frac{1}{{\omega }^2{(L}_{\mathrm{P}}-\frac{M^2}{L_{\mathrm{S}}})}}$
PS		${\displaystyle \frac{R_L^2{L}_{\mathrm{P}}}{{\omega }^2{(L}_{\mathrm{P}}{R}_L^2+M^4{\omega }^2)}}$	$1/{\omega }^2{L}_{\mathrm{S}}$
PP		${\displaystyle \frac{(L_{\mathrm{P}}{L}_{\mathrm{S}}^2-M^2{L}_{\mathrm{S}})L_{\mathrm{S}}^2}{{\omega }^2{{(L}_{\mathrm{P}}{L}_{\mathrm{S}}-M^2)}^2{L}_{\mathrm{S}}^2+M^4{R}_L^2}}$	${\displaystyle \frac{R_L\pm \sqrt{R_L^2-4{\omega }^2{L}_{\mathrm{S}}^2}}{2{\omega }^2{R}_L{L}_{\mathrm{S}}}}$

Note: In the table, $\omega$ is the system’s angular frequency, $M$ is the mutual inductance of the primary and secondary coils, and $R_L$ is the equivalent load.

.

When the TET system provides power to the IMDs, since the secondary coil needs to be implanted in the patient’s body, it is often difficult to ensure that the primary and secondary coils remain in an ideal coupling state. The degree of coupling between the coils usually varies with the patient’s daily physiological activities, leading to changes in mutual inductance between the coils. Table 1 clearly shows that, unlike the other three compensation topologies, the compensation capacitance of the primary and secondary coils in the SS topology is only related to the system’s angular frequency and the coils’ inductance and is independent of the mutual inductance between the primary and secondary coils, as well as the load. In other words, when the coupling degree between coils changes or the load changes, the system can still maintain a resonant state and achieve good transmission performance. It can be seen that SS is more suitable for TET systems compared to the other three compensation topologies.

Based on the aforementioned four basic compensation topologies, various new high-order compensation topologies have also been proposed and applied in various fields of wireless energy transfer. Based on this, this study also selects the high-performance, higher-order compensation topology LCC-S, whose compensation topology circuit is shown in Figure 2e. In the “Results and Discussion” section, a detailed comparative analysis of the impacts of the LCC-S and SS topologies on the TET system will be conducted based on specific application cases.

3. The Proposed Method

The TET coil optimization method proposed in this study includes three parts: (1) Data acquisition. A TET coil simulation model is established based on the finite element method, combined with an equivalent circuit model [35], to obtain input and output data. (2) Establish the GWO-ELM prediction model of the TET coils. Based on the obtained input/output data, use the GWO algorithm to tune the parameters of the ELM model and then establish the TET coils prediction model based on the optimal parameter combination of the ELM. (3) Multi-objective optimization of the TET coils. Based on the established prediction model, a multi-objective optimization function for the TET coils is created, and the NSWOA is used for multi-objective optimization.

3.1. Data Acquisition

In this study, the TET coil’s transmission efficiency, coupling coefficient, and secondary coil diameter are used as output parameters ($\eta$, $k$, and ${\varphi }_{\mathrm{s}}$, respectively). Except for ${\varphi }_{\mathrm{s}}$, $\eta$ and $k$ are not directly related to the structural parameters of the TET coil but rather to the lumped parameters (including the self-inductance and equivalent internal resistance of the primary and secondary coils, as well as mutual inductance, represented as $L_{\mathrm{P}}$, $L_{\mathrm{S}}$, $R_{\mathrm{P}}$, $R_{\mathrm{S}}$, and $M$, respectively). Therefore, this paper first establishes the relationship between the structural parameters of the TET coil and the lumped parameters using the finite element method. Then, based on the equivalent circuit model, establish the relationship between the lumped parameters and the output parameters. Furthermore, research shows that, in addition to the coils’ structural factors, the system’s resonance frequency has a substantial impact on the TET system’s transmission performance. Based on this, this research takes the number of turns, inter-turn spacing, and resonance frequency of the primary and secondary coils as input parameters, denoted as $N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}},$ and $f$.

3.2. TET Coils Model Based on ELM

This study’s TET coils modeling challenge is a multi-input, multi-output problem that requires somewhat sophisticated physical processes and electromagnetic field computations. The classic analytical modeling technique typically begins with building a mathematical model for the coil structural parameters and lumped parameters, followed by establishing a mathematical model for the lumped parameters and output parameters. This is a time-consuming and complicated process. However, general nonlinear fitting methods find it difficult to establish the complex mapping relationship between their inputs and outputs.

As one of the effective machine learning algorithms, ELM can capture complex nonlinear relationships by learning from a large amount of input/output data, and it has many successful applications in high-dimensional nonlinear problems. ELM is a single hidden layer feedforward neural network [36], which randomly selects weights and thresholds from the input layer to the hidden layer. Once randomly generated, they remain unchanged and do not require iterative solving. Then, the weights from the hidden layer to the output layer can be obtained using the Moore–Penrose generalized inverse. Compared to BP neural networks, Random Forest (RF), and support vector machines (SVMs), ELM exhibits advantages such as a faster training speed, fewer parameters with simple implementation, more stable generalization performance, and the ability to effectively avoid local optimum risks by eliminating the need for iterative optimization. Therefore, this paper will use ELM to model the TET coils, thereby constructing the complex mapping relationship between its multiple inputs and multiple outputs.

This study uses $N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}},$ and $f$ as the input parameters for the ELM model, and $\eta , k,$and ${\varphi }_{\mathrm{s}}$ as the output parameters. The ELM model of the TET coil is shown in Figure 3. Assume there are $Q$ sets of the TET coils sampling data ${\left\{({\mathit{X}}_i,{\mathit{Y}}_i)\right\}}_{i=1}^{\mathrm{Q}}$, where the input $X_i={\left[x_{i1},x_{i2},x_{i3},x_{i4},x_{i5}\right]}^{\mathrm{T}}$, $x_{i1},x_{i2}, x_{i3}, x_{i4},$and $x_{i5}$, respectively, represent the five dimensions of the input sample ${\mathit{X}}_i$, namely $N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}},$ and $f$. Output sample $Y_i={\left[y_{i1},y_{i2},y_{i3}\right]}^{\mathrm{T}}$, where $y_{i1},y_{i2}$, and $y_{i3}$ represent the three outputs, namely $\eta , k,$and ${\varphi }_{\mathrm{s}}$, respectively.

$$\sum _{j=1}^m{\mathit{\beta }}_{j}g\left({\mathit{w}}_j{\mathit{X}}_i+b_j\right)={\mathit{Y}}_i,i=\mathrm{1,2},\dots Q$$

(1)

where ${\mathit{w}}_j={\left[w_{j1},w_{j2},w_{j3},w_{j4},w_{j5}\right]}^{\mathrm{T}}$ represents the weights from the input layer to the hidden layer, $b_j$ is the corresponding threshold, ${\mathit{\beta }}_j={\left[{\beta }_{j1},{\beta }_{j2},{\beta }_{j3}\right]}^{\mathrm{T}}$ represents the weights from the hidden layer to the output layer, and $g(·)$ is the activation function.

The matrix form of Equation (1) can be written as:

$$\mathit{Y}=\mathit{H}\mathit{\beta }$$

(2)

where $\mathit{Y}={\left[\begin{array}{c}{\mathit{Y}}_1^{\mathrm{T}}\\ {\mathit{Y}}_2^{\mathrm{T}}\\ \dots \\ {\mathit{Y}}_Q^{\mathrm{T}}\end{array}\right]}_{Q\times 2}$ is the target output matrix, $\mathit{\beta }={\left[\begin{array}{c}{\mathit{\beta }}_1^{\mathrm{T}}\\ {\mathit{\beta }}_2^{\mathrm{T}}\\ \dots \\ {\mathit{\beta }}_m^{\mathrm{T}}\end{array}\right]}_{m\times 2}$, and $\mathit{H}$ is the hidden layer output matrix of the network, which can be expressed as follows:

$$\mathit{H}={\left[\begin{array}{ccc}g({\mathit{w}}_1{\mathit{X}}_1+b_1)& \dots & g({\mathit{w}}_m{\mathit{X}}_1+b_m)\\ ⋮& \dots & ⋮\\ g({\mathit{w}}_1{\mathit{X}}_Q+b_1)& \dots & g({\mathit{w}}_m{\mathit{X}}_Q+b_m)\end{array}\right]}_{Q\times m}$$

(3)

When the number of hidden layer neurons is less than the number of training samples, $\mathit{H}$ is a non-singular matrix, and ELM cannot achieve zero-error approximation to the target. The output weight matrix β can be obtained using the Moore–Penrose (MP) method. According to the generalized inverse theory, the solution for the output weight matrix can be expressed as:

$$\widehat{\mathit{\beta }}={\mathit{H}}^{+}\mathit{Y}$$

(4)

where H+ is the Moore–Penrose generalized inverse of H.

The predictive performance of the TET coils model based on the ELM largely depends on the selection of input weights and thresholds. The input weights and thresholds of the ELM are usually obtained through random generation, and their optimal values are highly uncertain, which, to some extent, reduces the model’s prediction accuracy and stability. In order to achieve satisfactory model prediction accuracy and stability, the input weights and thresholds of the ELM must be adjusted for specific problems. The ELM parameter tuning problem is an NP-hard problem [37], and metaheuristic optimization algorithms have been proven to be effective methods for solving such problems [38]. Therefore, this paper uses metaheuristic optimization algorithms to fine-tune the ELM parameters, aiming to find the optimal parameter combination to improve the prediction accuracy and stability of the TET coil model based on the ELM.

3.3. Metaheuristic Optimization Algorithm and ELM Parameter Tuning

Metaheuristic algorithms are a class of effective general optimization algorithms that may quickly determine the estimated optimal solution by searching the solution space using heuristic techniques. Large-scale, high-dimensional, and difficult optimization issues are best solved with these methods. One of the most common study areas nowadays is the use of metaheuristic optimization approaches to enhance the performance and results of machine learning models. Typical applications include hybrid machine learning combined with beetle antennae search to predict COVID-19 cases [39], an improved sine-cosine algorithm to optimize support vector machines for predicting cryptocurrency values [40], intrusion detection [41,42,43], medical detection and classification [44,45,46,47,48], and so on. Recently, researchers combined metaheuristic algorithms with the ELM model and applied them to real-world problems, with promising results. Examples of applications include using an improved firefly algorithm to optimize the ELM for disease classification detection [49], using an enhanced social network search algorithm to optimize the ELM for phishing website detection [50], detecting river flow [51], predicting groundwater level fluctuations [52], and conducting related research [53,54,55].

It is worth noting that although existing research has designed some methods for tuning the ELM’s model parameters, there is still significant room for improvement. According to the no free lunch (NFL) theorem [56], there is no universally applicable method for all optimization problems. Therefore, based on the specific characteristics of the particular problem and the advantages and disadvantages of the algorithms, selecting an appropriate metaheuristic optimization algorithm for model parameter tuning is a prerequisite and important guarantee for achieving optimal results of the prediction model in the specific problem.

3.4. Gray Wolf Optimizer

Inspired by the hunting behavior of gray wolf packs, the Gray Wolf Optimizer (GWO) is a novel swarm intelligence optimization algorithm proposed by Mirjalili et al. [57]. This algorithm simulates the predation behavior of gray wolf packs, achieving optimization based on the cooperative mechanism of the wolf pack. It features a simple structure, requires few adjustable parameters, and is easy to implement. Therefore, this study will use the GWO algorithm to tune the parameters of the ELM-based TET coil model. The mathematical description of the GWO algorithm is as follows:

$$\mathit{D}=\left|\mathit{C}{\mathit{X}}_P\left(t\right)-\mathit{X}\left(t\right)\right|$$

(5)

$$\mathit{X}\left(t+1\right)={\mathit{X}}_P\left(t\right)-\mathit{A}\mathit{D}$$

(6)

$$\left\{\begin{array}{c}A=2a{\mathit{r}}_1-a\\ \mathit{C}=2{\mathit{r}}_2\end{array}\right.$$

(7)

$$a=2\left(1-{\displaystyle \frac{t}{t_{max}}}\right)$$

(8)

where $t$ is the current iteration count, and $t_{max}$ is the maximum iteration count. $\mathit{A}$ and $\mathit{C}$ are both coefficient vectors. ${\mathit{X}}_P\left(t\right)$ is the position vector of the prey, and $\mathit{X}\left(t\right)$ is the position vector of the current gray wolf. $\mathit{D}$ is the distance between the current gray wolf and the prey, and $\mathit{X}\left(t+1\right)$ is the updated position of the gray wolf. ${\mathit{r}}_1$ and ${\mathit{r}}_2$ are random vectors between [0, 1].

$$\left\{\begin{array}{c}{\mathit{D}}_{\alpha }=\left|{\mathit{C}}_1{\mathit{X}}_{\alpha }-\mathit{X}\right|\\ {\mathit{D}}_{\beta }=\left|{\mathit{C}}_2{\mathit{X}}_{\beta }-\mathit{X}\right|\\ {\mathit{D}}_{\delta }=\left|{\mathit{C}}_3{\mathit{X}}_{\delta }-\mathit{X}\right|\end{array}\right.$$

(9)

$$\left\{\begin{array}{c}{\mathit{X}}_1={\mathit{X}}_{\alpha }-{\mathit{A}}_1{\mathit{D}}_{\alpha }\\ {\mathit{X}}_2={\mathit{X}}_{\beta }-{\mathit{A}}_2{\mathit{D}}_{\beta }\\ {\mathit{X}}_3={\mathit{X}}_{\delta }-{\mathit{A}}_3{\mathit{D}}_{\delta }\end{array}\right.$$

(10)

$$\mathit{X}\left(t+1\right)={\displaystyle \frac{{\mathit{X}}_1+{\mathit{X}}_2+{\mathit{X}}_3}{3}}$$

(11)

where ${\mathit{D}}_{\alpha }$, ${\mathit{D}}_{\beta }$, and ${\mathit{D}}_{\delta }$ represent the approximate distances between the current gray wolf and the $\alpha$, $\beta$, and $\delta$ wolves, respectively. ${\mathit{X}}_{\alpha }$, ${\mathit{X}}_{\beta }$, and ${\mathit{X}}_{\delta }$ are the current positions of the $\alpha$, $\beta$, and $\delta$ wolves, respectively. ${\mathit{X}}_1$, ${\mathit{X}}_2$, and ${\mathit{X}}_3$ are the step lengths and directions of the $\omega$ wolf moving towards the $\alpha$, $\beta$, and $\delta$ wolves, respectively. $\mathit{X}\left(t+1\right)$ is the final position of the $\omega$ wolf.

3.5. NSWOA

The NSWOA [58] is an efficient multi-objective optimization algorithm proposed based on the Whale Optimization Algorithm (WOA) by introducing non-dominated sorting strategies and crowding distance calculations. This algorithm simulates the hunting behavior of humpback whales by combining encircling prey, spiral hunting, and random search methods to achieve global exploration of the population. At the same time, it uses a non-dominated sorting strategy to layer the population, evaluate the quality of individuals, and combine crowding distance calculations to ensure the diversity and uniform distribution of the solution set. The WOA algorithm includes the phases of encircling prey, spiral bubble net hunting, and random search, with the mathematical expressions as follows:

3.5.1. Encircling Prey

During the process of the humpback whale surrounding its prey, assuming the current best candidate solution position is the target prey position, the whale will attempt to surround the target prey position. The position update formula is as follows:

$$\overrightarrow{D}=\left|\overrightarrow{C}·\overrightarrow{X^{*}}\left(t\right)-\overrightarrow{X}(t)\right|$$

(12)

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{X^{*}}\left(t\right)-\overrightarrow{A}·\overrightarrow{D}$$

(13)

$$\overrightarrow{A}=2\overrightarrow{a}·\overrightarrow{r}-\overrightarrow{a}$$

(14)

$$\overrightarrow{C}=2·\overrightarrow{r}$$

(15)

where $t$ is the current iteration count, $\overrightarrow{A}$ and $\overrightarrow{C}$ are coefficient vectors, $\overrightarrow{X^{*}}$ is the position vector of the current optimal solution, $\overrightarrow{a}$ decreases linearly from 2 to 0 during the iteration process, and $\overrightarrow{r}$ is a random vector between [0, 1].

3.5.2. Spiral Bubble Net-Feeding Phase

When searching for prey, humpback whales swim around the prey in a shrinking circle while also moving along a spiral path, with their position updates represented by Equation (16). To simultaneously simulate the whale’s shrinking encirclement mechanism and spiral update mechanism, it is assumed that the probabilities of executing these two mechanisms are equal, as represented by Equation (18).

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{D^{’}}·e^{bl}·\cos 2\pi l+\overrightarrow{X^{*}}\left(t\right)$$

(16)

$$\overrightarrow{D^{’}}=\left|\overrightarrow{X^{*}}\left(t\right)-\overrightarrow{X}(t)\right|$$

(17)

$$\overrightarrow{X}\left(t+1\right)=\left\{\begin{array}{c}\overrightarrow{X^{*}}\left(t\right)-\overrightarrow{A}·\overrightarrow{D} if p<0.5\\ \overrightarrow{D^{’}}·e^{bl}·\cos 2\pi l+\overrightarrow{X^{*}}\left(t\right) otherwise\end{array}\right.$$

(18)

where $b$ is a constant representing the spiral shape, $l$ is a random number in the range of [−1, 1], and $\overrightarrow{D^{’}}$ represents the distance from the i-th whale to the prey.

3.5.3. Random Search for Prey

At this stage, the humpback whales will also randomly search for prey. When |$\overrightarrow{A}$| ≥ 1, a random search method is used for position updates; when |$\overrightarrow{A}$| < 1, the whales update their positions based on the current optimal solution. The random search position update formula is as follows:

$$\overrightarrow{D}=\left|\overrightarrow{C}·\overrightarrow{X_{rand}}-\overrightarrow{X}\right|$$

(19)

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{X_{rand}}-\overrightarrow{A}·\overrightarrow{D}$$

(20)

where $\overrightarrow{X_{rand}}$ is the position vector of a randomly selected whale in the current population.

3.6. Multi-Objective Optimization Function for TET Coils

The NSWOA optimization algorithm does not depend on the specific nature of the problem, making it suitable for nonlinear and multi-objective optimization problems. It can effectively handle the complex tasks involved in TET coil optimization. Therefore, this study selects this algorithm as the optimization algorithm for TET coils. Multi-objective optimization models typically include decision variables, constraints, and objective functions. Based on the inputs and outputs of the aforementioned TET coil ELM modeling process, this paper selects $N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}},$ and $f$ as the decision variables for multi-objective optimization, represented by the following equation:

$$V={\left\{N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}}, f\right\}}^T$$

(21)

where $N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}},$ and $f$ represent the number of turns, turn spacing of the primary and secondary coils, and system resonant frequency, respectively. The value ranges of each decision variable are constrained as follows:

$$s.t.\left\{\begin{array}{c}{L}_{N_{\mathrm{P}}}\le N_{\mathrm{P}}\le U_{N_{\mathrm{P}}}\\ L_{N_{\mathrm{s}}}\le N_{\mathrm{S}}\le U_{N_{\mathrm{s}}}\\ L_{d_{\mathrm{P}}}\le d_{\mathrm{P}}\le U_{d_{\mathrm{P}}}\\ L_{d_{\mathrm{P}}}\le d_{\mathrm{S}}\le U_{d_{\mathrm{P}}}\\ L_f\le f\le U_f\end{array}\right.$$

(22)

where $L$ represents the lower limit of each decision variable, and $U$ represents the upper limit. The specific values of each decision variable can be found in the case validation section.

To provide patients with a more stable, comfortable, and highly reliable TET system, it is necessary to consider not only transmission efficiency but also various other requirements, such as the size limitations of body implants and the degree of coupling of the coils. Considering the limitations and complexities in practical applications, this study takes the transmission efficiency ($\eta$), coupling coefficient ($k$), and secondary coil diameter (${\varphi }_{\mathrm{s}}$) as optimization objectives. For TET systems, it is generally desired for $\eta$ and $k$ to be as large as possible and ${\varphi }_{\mathrm{s}}$ to be as small as possible. Therefore, the objective function can be expressed as follows:

$$objective function \left\{\begin{array}{c}\max \left[f_{\eta }\right(N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}}, f\left)\right]\\ \max \left[f_k\left(N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}}, f\right)\right]\\ \min \left[f_{{\varphi }_{\mathrm{s}}}\left(N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}}, f\right)\right]\end{array}\right.$$

(23)

It should be noted that $f_{\eta }$, $f_k$, and $f_{{\varphi }_{\mathrm{s}}}$ do not refer to specific mathematical models but are replaced by the established TET coils ELM model.

3.7. Detailed Steps and Flowchart of the Proposed Method

The TET coils optimization method proposed in this paper is named GWO-ELM-NSWOA. The flowchart of the method is shown in Figure 4, and the detailed steps are as follows:

Step 1: Define the TET coil dataset, which includes 5 input parameters, $N_{\mathrm{P}}, N_{\mathrm{S}}, d_{\mathrm{P}}, d_{\mathrm{S}}, f$, and 3 output variables, $\eta , k, \mathrm{a}\mathrm{n}\mathrm{d} {\varphi }_{\mathrm{s}}$. The input/output sampling dataset is obtained based on the finite element method and equivalent circuit model.

Step 2: Divide the sampled dataset into training and testing sets and perform normalization.

Step 3: Define the range of values for the ELM model parameters (input weights and thresholds) and use the GWO algorithm for parameter tuning. During the parameter tuning process, the average value of the root mean square error of the three output variables is used as the fitness function. Using five-fold cross-validation, the original training set data is randomly and evenly divided into five equal-sized subsets. One subset is sequentially selected as the validation set, while the remaining four subsets are combined to form the training set. Use a set of initial parameters to train on the training set to obtain the corresponding ELM model output weights. Then, based on this set of parameters and the corresponding output weights, establish the TET coils prediction model. Use the validation set to evaluate the fitness function value of this model under this set of parameters, then enter the parameter tuning process, repeat the above steps until the maximum number of iterations is reached, and finally output the optimal parameter combination.

Step 4: Construct the TET coils prediction model based on GWO-ELM under the optimal parameter combination and evaluate the model based on the test set data, with the evaluation metrics being $RMSE$, $R^2$, and $MAE$.

Step 5: Define the range of decision variable values using $\mathrm{m}\mathrm{a}\mathrm{x}{f}_{\eta }$, $\mathrm{m}\mathrm{a}\mathrm{x}{f}_k$, and $\mathrm{m}\mathrm{i}\mathrm{n}{f}_{{\varphi }_{\mathrm{s}}}$ as the multi-objective optimization functions. Based on the GWO-ELM prediction model of the TET coils, perform multi-objective optimization on the TET coils and output the Pareto solution set after reaching the maximum number of iterations.

4. Case Validation

To verify the effectiveness of the aforementioned method, this study uses the planar helical coil applied in an artificial detrusor energy supply as an example for validation. Compared to solenoid coils, planar spiral coils have the characteristics of being small in volume and thin in thickness, making them more suitable for implantation in narrow spaces such as subcutaneous areas. They are widely used in powering applications for IMDs. Therefore, this paper conducts an optimization analysis using planar spiral coils as an example.

4.1. Range of Decision Variable Values

According to the design requirements of the TET coil size parameters and the limitations of the implantation size, the value ranges of each decision variable have been determined, and Equation (22) can be specifically expressed as:

$$s.t.\left\{\begin{array}{c}5\le N_{\mathrm{P}}\le 31\\ 5\le N_{\mathrm{S}}\le 25\\ 0 \mathrm{mm}\le d_{\mathrm{P}}\le 4 \mathrm{mm}\\ 0 \mathrm{mm}\le d_{\mathrm{S}}\le 2 \mathrm{mm}\\ 100 \mathrm{kHz}\le f\le 400 \mathrm{kHz}\end{array}\right.$$

(24)

where $N_{\mathrm{P}}$ is set to seven levels: 5, 10, 15, 18, 20, 25, and 31. $N_{\mathrm{S}}$ is set to five levels: 5, 10, 15, 20, and 25. $d_{\mathrm{P}}$ is set at intervals of 0.5 mm, from 0 mm to 4 mm, with a total of nine levels. $d_{\mathrm{S}}$ is set at intervals of 0.25 mm, from 0 mm to 2 mm, with a total of nine levels. $f$ is set at intervals of 50 kHz, with a total of seven levels. To ensure the comprehensiveness and reliability of the experimental results, as well as to establish a reliable and accurate TET coil model, this study conducted a full-factorial experimental design, requiring the collection of a total of 19,845 sets of data.

4.2. Construction and Verification of the TET Coils Simulation Platform

This study establishes a finite element simulation model of the TET coils based on the ANSYS Maxwell 2023R1 to obtain the relationship between the coil structural parameters and the lumped parameters. Due to the symmetry of the planar helical coil structure, a simplified two-dimensional model is used for modeling to improve the efficiency of the simulation solution. Before the formal modeling analysis, in order to verify the accuracy of the simulation solution, an analysis is conducted using a planar helical coil with an inner diameter of 80 mm, an outer diameter of 120 mm, and 10 turns as an example. The physical object and the two-dimensional simulation model of this coil are shown in Figure 5. The coil is wound with Litz wire, and the operating frequency is set to 10 kHz. The inductance ($L$), internal resistance ($R$), and quality factor ($Q$) of the coil are calculated based on simulations and experiments, and a comparison is made. The simulation analysis type is set to Eddy current, the coil structure parameters and frequency are set as described above, the boundary conditions are set to Balloon, the excitation is set to Winding, the type is current, the magnitude is 3.5 A, the coil type is Stranded, and the z-axis is the symmetry axis. The experimental instrument used for measurement is the Tonghui TL2812D digital bridge tester (Manufacturer: China Changzhou Tonghui Electronics Co., LTD). The simulation and experimental results of the sample coil are shown in Figure 6, and the summarized results are shown in

Table 2. Summary of the experimental and simulation results, for example, the coil.

Result Category	Measurement Parameters
	$R$ (mΩ)	$L$ (uH)	$Q$
Simulation	34.52	15.24	27.7
Experiment	37.2	16.22	27.4
Simulation error	7.2%	6%	1.1%

. It is not difficult to see that the simulation and experimental measurement results are quite consistent, with a maximum error of only 7.2%, mainly due to model rounding errors. This indicates that the simulation solution has high accuracy and meets the computational requirements. The TET coils simulation model established in this study is shown in Figure 7. The parameters of the TET coils are set according to Equation (24). The inner radius of both the primary and secondary coils is set to 5 mm, the wire diameter of the primary coil is set to 2 mm, and the wire diameter of the secondary coil is set to 1 mm, both using Litz wire. The axial distance between the two coils is set to 15 mm (the distance between the implanted secondary coil and the primary coil). The other simulation settings are the same as those of the aforementioned example coil settings.

This paper is based on the equivalent circuit model derived in reference [36], using Equations (25) and (26) to establish the relationship between the lumped parameters and $\eta$, $k$, with ${\varphi }_{\mathrm{s}}$ determined by Equation (27).

$$\eta ={\displaystyle \frac{{\omega }^2{M}^2{R}_L}{({\omega }^2{M}^2+R_{\mathrm{p}}\left(R_{\mathrm{L}}+R_{\mathrm{s}}\right))(R_{\mathrm{L}}+R_{\mathrm{s}})}}$$

(25)

$$k={\displaystyle \frac{M}{\sqrt{L_{\mathrm{p}}{L}_{\mathrm{s}}}}}$$

(26)

$${\varphi }_{\mathrm{s}}=2\ast (r_{\mathrm{i}\mathrm{n}}+(N_{\mathrm{s}}-1)\ast (D_{\mathrm{S}}+d_{\mathrm{s}}\left)\right)$$

(27)

where ω is the system’s angular frequency, $\omega =2\pi f$, $M$ is the mutual inductance of the two coils, $R_{\mathrm{p}}$ and $R_{\mathrm{s}}$ are the internal resistances of the primary and secondary coils, respectively, $R_L$ is the equivalent internal resistance of the load and $R_L$ ≈ 0.81$R_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$ [36]. In this study, the load of the artificial detrusor is approximately 5Ω, $R_L$ is set to 4Ω, $r_{\mathrm{i}\mathrm{n}}$ is the inner radius of the coil, and $D_{\mathrm{S}}$ is the wire diameter of the secondary coil, set to 1 mm.

5. Results and Discussion

The evaluation criteria for the modeling and multi-objective optimization of the TET coils are first introduced in this section. Second, it investigates and discusses the GWO algorithm’s optimization performance during the parameter tuning process of the TET coils prediction model; this is followed by an examination and discussion of the prediction performance of the TET coils GWO-ELM prediction model. Finally, the performance of the NSWOA in the multi-objective optimization process of TET coils is evaluated and discussed, the optimal Pareto solution set for the TET coils is obtained, and the optimization results are simulated and experimentally validated.

5.1. Outcome Evaluation Indexes

The evaluation of model performance is an important component of machine learning and data analysis. By selecting the appropriate evaluation metrics, the performance and accuracy of the model can be effectively measured.

In order to accurately evaluate the performance of the TET coil model established in this paper, a set of standard evaluation metrics was used. These metrics include the mean absolute error ($MAE$), root mean squared error ($RMSE$), and coefficient of determination ($R^2$). Among them, the $RMSE$ can intuitively reflect the difference between the model’s predicted values and the actual values. The smaller the $RMSE$, the higher the model’s accuracy. $R^2$ is used to evaluate the goodness of fit of the model. The closer this value is to 1, the higher the model’s degree of fit and the better the model’s predictive performance. $MAE$ is the average of the absolute differences between the predicted values and actual values, and it is relatively less sensitive to outliers. The mathematical expressions for the above three evaluation metrics are as follows:

$$MAE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|{{(S}_O)}_i-{{(S}_C)}_i\right|$$

(28)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{({{(S}_O)}_i-{{(S}_C)}_i)}^2}$$

(29)

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^N{({{(S}_O)}_i-{{(S}_C)}_i)}^2}{\sum _{i=1}^N{({{(S}_O)}_i-\overline{S})}^2}}$$

(30)

where ${{(S}_O)}_i$ is the actual value of the data point, ${{(S}_C)}_i$ is the predicted value, $\overline{S}$ is the average value, and $N$ is the number of data points.

Evaluation metrics for multi-objective optimization: This study selects four evaluation metrics, namely Inverted Generational Distance ($IGD$), Hypervolume ($HV$), Spread, and Spacing ($SP$). Among them, the $IGD$ indicator calculates the average Euclidean distance from each point on the true Pareto front to the nearest point in the non-dominated solution set obtained by the algorithm. The smaller the value, the better the overall performance of the algorithm. $HV$ is used to quantify the hypervolume covered by the solution set generated by the optimization method in the goal space. The larger its value, the better the quality of the solution set, meaning the solution set better covers the objective space, reflecting the convergence and diversity of the solution set. The $SP$ indicator is used to measure the uniformity of the distribution of the solution set generated by the algorithm. The smaller the SP value, the more uniform the distribution of the solution set. Spread is an indicator used to measure the uniformity and coverage of the solution set in the objective space. The smaller the value, the more uniform the distribution of the solution set and the broader the coverage. The mathematical expression is as follows:

$$SP=\sqrt{{\displaystyle \frac{1}{n-1}}{\sum }_{i=1}^n{(\overline{d}-d_i)}^2}$$

(31)

$$IGD={\displaystyle \frac{\sqrt{\sum _{i=1}^{\left|Z\right|}{d}_i^2}}{\left|Z\right|}}$$

(32)

$$HV=\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\left({\bigcup }_{i=1}^n{v}_i\right)$$

(33)

$$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}={\displaystyle \frac{\sum _{i=1}^{n-1}\left|d_i-\overline{d}\right|+\sum _{j=1}^{m}d(f_j,\mathrm{P}\mathrm{F})}{\sum _{i=1}^{n-1}\left|d_i-\overline{d}\right|+m·\overline{d}}}$$

(34)

where $d_i$ is the Euclidean distance between the adjacent solutions, $\overline{d}$ is the average of all $d_i$, and $d(f_j,\mathrm{P}\mathrm{F})$ is the minimum distance from the solution set $f_j$ to the true Pareto front. $n$ is the number of solutions in the solution set, m is the number of objective functions, and $v_i$ is the hypercube formed by the solution set and reference points.

5.2. Analysis of the Optimization Performance of the GWO Algorithm in the Parameter Tuning Process of the TET Coils ELM Prediction Model

This subsection uses the GWO algorithm for parameter tuning of the ELM model and establishes the GWO-ELM prediction model for the TET coils. To verify the optimization performance of the GWO algorithm in the ELM parameter tuning process, as well as the prediction performance of the established TET coil GWO-ELM prediction model, this paper also establishes TET coil prediction models using PSO, CFOA, DE, and HOA algorithms for the ELM parameter tuning, defined as PSO-ELM, CFOA-ELM, DE-ELM, and HOA-ELM, respectively. The parameter settings of the five algorithms are shown in

Table 3. Parameter setting of each algorithm.

Algorithm	Parameter	Value
PSO	$c_1$ $c_2$ $w$	1.2 1.5 0.7
DE	$beta$ $pCR$	[0.2, 0.8] 0.2
GWO	$r1$ $r2$	[0, 1] [0, 1]
HOACFOA	$SF$ $theta$ $per$	[0, 50] [0, 1] $[3, 4]$

. By splitting the dataset into three distinct ratios of the training set to test set—7:3, 8:2, and 9:1—we were able to further investigate the effects of the dataset partition ratios on the TET coil prediction model.

The simulation environment consists of the following: MATLAB R2023b software, a Win10 64-bit operating system, an Intel^® Core™ i7-9750H CPU, a base frequency of 2.60 GHz, and 16 GB of RAM. In the process of parameter tuning of the ELM model using various optimization algorithms, the objective function (fitness function) for the tuning process is defined as the average value of the root mean square error of the output variables $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$. The experimental parameters were uniformly set as follows: the maximum number of iterations was 200, and the population size was set to 30. Each optimization algorithm was run independently 15 times to obtain the optimal value (best), worst value (worst), mean value (mean), median value (median), standard deviation (std), and variance (var) of the objective function during the ELM parameter tuning process. The statistical results are shown in

Table 4. The statistical results of the evaluation metrics for the objective function of ELM parameter tuning using various optimization algorithms.

Train: Test	Metrics	CFOA-ELM	DE-ELM	GWO-ELM	PSO-ELM	HOA-ELM
7:3	Best	3.47 × 10−2	3.45 × 10−2	2.22 × 10−2	2.40 × 10−2	2.94 × 10−2
	Worst	4.30 × 10−2	3.78 × 10−2	2.33 × 10−2	2.58 × 10−2	3.50 × 10−2
	Mean	3.97 × 10−2	3.66 × 10−2	2.28 × 10−2	2.47 × 10−2	3.21 × 10−2
	Median	4.01 × 10−2	3.68 × 10−2	2.29 × 10−2	2.46 × 10−2	3.18 × 10−2
	Std	2.48 × 10−3	1.01 × 10−3	3.36 × 10−4	5.19 × 10−4	1.31 × 10−3
	Var	6.16 × 10−6	1.01 × 10−6	1.13 × 10−7	2.69 × 10−7	1.72 × 10−6
8:2	Best	3.79 × 10−2	3.64 × 10−2	2.26 × 10−2	2.49 × 10−2	2.85 × 10−2
	Worst	4.19 × 10−2	3.71 × 10−2	2.35 × 10−2	2.64 × 10−2	3.71 × 10−2
	Mean	3.98 × 10−2	3.68 × 10−2	2.33 × 10−2	2.53 × 10−2	3.24 × 10−2
	Median	3.93 × 10−2	3.69 × 10−2	2.35 × 10−2	2.50 × 10−2	3.17 × 10−2
	Std	1.71 × 10−3	3.25 × 10−4	3.35 × 10−4	5.64 × 10−4	2.99 × 10−3
	Var	2.94 × 10−6	1.06 × 10−7	1.12 × 10−7	3.18 × 10−7	8.93 × 10−6
9:1	Best	3.62 × 10−2	3.57 × 10−2	2.14 × 10−2	2.26 × 10−2	2.75 × 10−2
	Worst	4.50 × 10−2	3.81 × 10−2	2.39 × 10−2	2.75 × 10−2	3.67 × 10−2
	Mean	4.10 × 10−2	3.69 × 10−2	2.26 × 10−2	2.44 × 10−2	3.12 × 10−2
	Median	4.12 × 10−2	3.68 × 10−2	2.23 × 10−2	2.42 × 10−2	3.12 × 10−2
	Std	2.38 × 10−3	6.69 × 10−4	6.85 × 10−4	1.09 × 10−3	2.28 × 10−3
	Var	5.68 × 10−6	4.48 × 10−7	4.69 × 10−7	1.19 × 10−6	5.20 × 10−6

and Figure 8 and Figure 9.

Table 4 shows the statistical results of the evaluation metrics for the objective function of ELM parameter tuning using various optimization algorithms. From the table, it can be seen that when the training set and test set are in a 7:3 ratio, all evaluation metrics of the GWO-ELM objective function are superior to those of other comparative algorithms. When the dataset is divided in a ratio of 8:2 and 9:1, the std and var metrics of DE-ELM outperform other methods, followed by GWO-ELM. However, GWO-ELM’s other four evaluation metrics are globally optimal, and the overall quality of GWO-ELM’s objective function value is significantly better than that of DE-ELM. Moreover, under the three dataset partitioning ratios, the optimal values of the best, mean, and median evaluation metrics of the GWO-ELM objective function were all achieved at a partitioning ratio of 9:1. The above conclusion is also demonstrated in the box plot statistical results of the objective functions for the parameter tuning of various algorithms in Figure 8. From the figure, it is evident that under the three dataset partitioning ratios, the box width of the GWO-ELM objective function box plot is narrow and positioned at the bottom, with the objective function values being relatively concentrated and stable. This further indicates that the overall quality of the GWO-ELM objective function values is significantly superior to that of other algorithms. Figure 9 shows the convergence graph of the objective function for the parameter tuning of various algorithms. From the convergence graph, it can be seen that under the three dataset partition ratios, GWO-ELM exhibits good convergence performance. Compared to the comparison algorithms, it converges faster, and the final convergence value of the objective function is also globally optimal.

Based on the above analysis, to fully demonstrate the optimization performance of GWO in the parameter tuning process of ELM, a Wilcoxon rank-sum test was further conducted on the parameter tuning processes of GWO and the comparison algorithms. The significance level was defined as 0.05, and the statistical results are shown in

Table 5. Wilcoxon rank-sum test statistical results of the parameter tuning process for each algorithm.

Train: Test	GWO-ELM vs. CFOA-ELM	GWO-ELM vs. DE-ELM	GWO-ELM vs. PSO-ELM	GWO-ELM vs. HOA-ELM
7:3	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5
8:2	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5
9:1	3.05 × 10−5	3.05 × 10−5	1.53 × 10−4	3.05 × 10−5

. From the table, it can be seen that all results are less than 0.05, and based on the above analysis, it can be fully demonstrated that during the parameter tuning process of the TET coil ELM model, the optimization performance of GWO is significantly superior to that of the comparison algorithms.

5.3. The Prediction Performance of the TET Coil GWO-ELM Prediction Model

The above analysis fully demonstrates the excellent optimization performance of GWO in the parameter tuning process of the TET coil ELM model. Next, the predictive performance of the established TET coil GWO-ELM model will be analyzed. Based on the aforementioned comparison methods, a TET coil prediction model, established by ELM without parameter tuning (where input weights and thresholds are obtained through random generation), will be further added for comparison. The test sets with three different dataset partition ratios will be used to perform a predictive statistical analysis on each prediction model, and the statistical results are shown in

Table 6. Evaluation metrics for prediction results of each model (Train:Test = 7:3).

Output	Metrics	CFOA	DE-ELM	GWO-ELM	PSO-ELM	HOA-ELM	ELM
		-ELM
$\eta$	$RMSE$	5.00 × 10−2	4.90 × 10−2	2.72 × 10−2	2.94 × 10−2	3.49 × 10−2	8.66 × 10−2
	$R^2$	9.40 × 10−1	9.39 × 10−1	9.83 × 10−1	9.79 × 10−1	9.70 × 10−1	8.25 × 10−1
	$MAE$	3.88 × 10−2	3.71 × 10−2	1.96 × 10−2	2.21 × 10−2	2.63 × 10−2	6.53 × 10−2
$k$	$RMSE$	5.12 × 10−1	5.98 × 10−1	1.82 × 10−1	3.64 × 10−1	6.67 × 10−1	2.13
	$R^2$	0.99978	0.99970	0.99997	0.99989	0.99963	0.99620
	$MAE$	3.84 × 10−1	4.32 × 10−1	1.36 × 10−1	2.56 × 10−1	4.82 × 10−1	1.58
${\varphi }_{\mathrm{s}}$	$RMSE$	2.78 × 10−2	2.85 × 10−2	2.16 × 10−2	2.28 × 10−2	2.69 × 10−2	3.04 × 10−2
	$R^2$	9.61 × 10−1	9.60 × 10−1	9.78 × 10−1	9.75 × 10−1	9.64 × 10−1	9.53 × 10−1
	$MAE$	2.14 × 10−2	2.20 × 10−2	1.60 × 10−2	1.67 × 10−2	1.92 × 10−2	2.31 × 10−2

,

Table 7. Evaluation metrics for prediction results of each model (Train:Test = 8:2).

Output	Metrics	CFOA	DE-ELM	GWO-ELM	PSO-ELM	HOA-ELM	ELM
		-ELM
$\eta$	$RMSE$	5.64 × 10−2	4.61 × 10−2	2.82 × 10−2	3.13 × 10−2	3.80 × 10−2	7.63 × 10−2
	$R^2$	9.23 × 10−1	9.48 × 10−1	9.81 × 10−1	9.76 × 10−1	9.63 × 10−1	8.65 × 10−1
	$MAE$	4.43 × 10−2	3.59 × 10−2	2.00 × 10−2	2.35 × 10−2	2.81 × 10−2	6.19 × 10−2
$k$	$RMSE$	1.15	1.35	1.27 × 10−1	2.58 × 10−1	4.27 × 10−1	1.65
	$R^2$	0.9988	0.99847	0.99999	0.99994	0.99985	0.9977
	$MAE$	8.20 × 10−1	9.87 × 10−1	9.94 × 10−2	2.01 × 10−1	3.14 × 10−1	1.23
${\varphi }_{\mathrm{s}}$	$RMSE$	2.75 × 10−2	2.95 × 10−2	2.18 × 10−2	2.34 × 10−2	2.56 × 10−2	4.41 × 10−2
	$R^2$	9.61 × 10−1	9.54 × 10−1	9.77 × 10−1	9.73 × 10−1	9.67 × 10−1	9.06 × 10−1
	$MAE$	2.08 × 10−2	2.25 × 10−2	1.67 × 10−2	1.71 × 10−2	1.99 × 10−2	3.49 × 10−2

and

Table 8. Evaluation metrics for prediction results of each model (Train:Test = 9:1).

Output	Metrics	CFOA	DE-ELM	GWO-ELM	PSO-ELM	HOA-ELM	ELM
		-ELM
$\eta$	$RMSE$	5.42 × 10−2	4.91 × 10−2	2.46 × 10−2	2.97 × 10−2	3.10 × 10−2	7.66 × 10−2
	$R^2$	9.22 × 10−1	9.33 × 10−1	9.83 × 10−1	9.74 × 10−1	9.72 × 10−1	8.51 × 10−1
	$MAE$	4.20 × 10−2	3.82 × 10−2	1.73 × 10−2	2.26 × 10−2	2.33 × 10−2	6.14 × 10−2
k	$RMSE$	9.82 × 10−1	6.87 × 10−1	1.18 × 10−1	2.07 × 10−1	4.29 × 10−1	1.60
	$R^2$	0.99919	0.99961	0.99999	0.99996	0.99985	0.9978
	$MAE$	7.11 × 10−1	5.03 × 10−1	9.34 × 10−2	1.54 × 10−1	3.10 × 10−1	1.14
${\varphi }_{\mathrm{s}}$	$RMSE$	2.58 × 10−2	2.84 × 10−2	2.04 × 10−2	1.92 × 10−2	2.54 × 10−2	5.10 × 10−2
	$R^2$	9.65 × 10−1	9.59 × 10−1	9.80 × 10−1	9.81 × 10−1	9.67 × 10−1	8.69 × 10−1
	$MAE$	1.98 × 10−2	2.13 × 10−2	1.53 × 10−2	1.43 × 10−2	1.92 × 10−2	3.95 × 10−2

and Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

Table 6, Table 7 and Table 8 show the evaluation metrics for the prediction results of various models when the dataset is divided into training and test sets in the ratios of 7:3, 8:2, and 9:1, respectively. The optimal values for each metric are indicated in bold. From the table, it can be seen that the TET coil GWO-ELM model outperforms the comparison prediction model in all prediction performance evaluation metrics for the three outputs when the training set/test set ratio is 7:3 and 8:2. When the training set/test set ratio is 9:1, the GWO-ELM model’s $\eta$ and $k$ evaluation metrics are all optimal, and the evaluation metrics of ${\varphi }_{\mathrm{s}}$ are second only to PSO-ELM, outperforming the other prediction models. The GWO-ELM model has $R^2$ greater than 0.95 for all three outputs under different dataset partition ratios. Furthermore, across three different dataset partition ratios, all evaluation measures for the three outputs of the GWO-ELM prediction model obtained optimum values when the training set/test set ratio was 9:1. The fundamental reason is that increasing the amount of the training set improves the prediction model’s fit.

The aforementioned study shows that the GWO-ELM prediction model surpasses the comparison prediction models in terms of overall predictive performance, with more accurate prediction outcomes, greater prediction accuracy, and obtaining the global optimal value at a training set/test set ratio of 9:1.

Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 provide a statistical comparison of the TET coil prediction models’ predictions of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ across three dataset partition ratios. Figure 10a, Figure 11a, Figure 12a, Figure 13a, Figure 14a, Figure 15a, Figure 16a, Figure 17a and Figure 18a show the comparison between the predicted and observed values of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ under three different dataset partition ratios. From those figures, it can be initially observed that the predicted values of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ from each prediction model exhibit similar distribution patterns to the observed values. Further quantitative analyses of the errors between the predicted and observed values for each model were conducted. Figure 10b, Figure 11b, Figure 12b, Figure 13b, Figure 14b, Figure 15b, Figure 16b, Figure 17b and Figure 18b show the error curves of the predicted and observed values of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ under three different dataset partition ratios. From those figures, it can be initially observed that, compared to the comparative prediction models, the actual error of the TET coil GWO-ELM prediction model is relatively small, with most data points’ errors fluctuating around the reference line (zero-error line). Figure 10c, Figure 11c, Figure 12c, Figure 13c, Figure 14c, Figure 15c, Figure 16c, Figure 17c and Figure 18c show the prediction error violin plots of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ under three different dataset partition ratios. The violin plots combine the characteristics of box plots and kernel density plots. The width and area of the contours in different intervals represent the concentration and probability density of the data in those intervals, allowing for a more intuitive understanding of the actual prediction error distribution of each prediction model. From these graphs, it is evident that the violin plot of the prediction errors of the GWO-ELM prediction model near the zero-error line has a relatively large contour width and area, indicating that the prediction errors of the GWO-ELM model are more concentrated and have a higher probability density in this region. This further demonstrates that, compared to other prediction models, the GWO-ELM model has the smallest overall prediction error and the best prediction quality. Figure 10d–i, Figure 11d–i, Figure 12d–i, Figure 13d–i, Figure 14d–i, Figure 15d–i, Figure 16d–i, Figure 17d–i and Figure 18d–i show the linear fitting graphs of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ for each prediction model under three different dataset partition ratios. From these figures, it can be seen that when the training set to test set ratio of the dataset is 7:3 or 8:2, the GWO-ELM prediction model fits $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ the best, with the fitted R² value closest to 1; PSO-ELM is second; CFOA-ELM, DE-ELM, and HOA-ELM follow. The TET coil ELM prediction model without parameter tuning performs relatively poorly. When the ratio of the training set to the test set is 9:1, the GWO-ELM prediction model still outperforms other comparative prediction models in fitting $\eta$ and $k$. Although PSO-ELM slightly outperforms GWO-ELM in fitting ${\varphi }_{\mathrm{s}}$, the overall prediction quality of GWO-ELM is superior to that of PSO-ELM.

In summary, it can be further stated that the TET coil GWO-ELM prediction model has good stability and high prediction accuracy. Moreover, based on the above analysis, when the ratio of the training set to the test set is 9:1, the prediction performance of the GWO-ELM model is globally optimal. As a result, the GWO-ELM prediction model trained with a training-to-test set ratio of 9:1 is chosen for the following phase in the multi-objective optimization analysis of TET coils.

5.4. TET Coils Multi-Objective Optimization Results

The aforementioned analysis has fully demonstrated the optimization performance of the GWO algorithm in the parameter tuning process of the ELM model, as well as the prediction performance of the TET coil GWO-ELM prediction model. This subsection will perform multi-objective optimization on the TET coils based on the established GWO-ELM prediction model. For details related to multi-objective optimization, please refer to the proposed methods section of this paper. To analyze the optimization performance of the NSWOA in the multi-objective optimization process of the TET coils, this study also selected the MOPSO, MOMVO, MSSA, and MODA multi-objective optimization algorithms for comparative validation. The relevant parameters of each algorithm were set according to the parameters in their published papers. The simulation environment is as follows: Win10 64-bit operating system, Intel (R) Core (TM) i7-9750H CPU, base frequency 2.60 GHz, 16 GB RAM, and software MATLAB R2023b.

The experimental parameters were uniformly set as follows: the maximum number of iterations was 100, the population size was set to 100, the archive size was set to 100, and the experiment was run independently 15 times. This paper selects four multi-objective optimization evaluation metrics: IGD, HV, Spread, and SP. Detailed discussions on each metric can be found in Section 4 of this article. The analysis obtained the best, worst, mean, median, standard deviation, and variance values of the evaluation metrics for each multi-objective optimization algorithm. The statistical results are shown in

Table 9. Statistical results of evaluation metrics for various multi-objective optimization algorithms.

Metrics		Algorithm
		MOPSO	MOMVO	NSWOA	MSSA	MODA
IGD	Best	5.94 × 10−1	8.57 × 10−1	3.90 × 10−1	1.20	1.50
	Worst	1.42	4.70	5.84 × 10−1	5.52	4.92
	Mean	1.00	1.86	4.59 × 10−1	2.73	2.40
	Median	9.91 × 10−1	1.47	4.56 × 10−1	2.27	2.47
	Std	1.85 × 10−1	1.13	4.51 × 10−2	1.37	8.63 × 10−1
	Var	3.42 × 10−2	1.27	2.04 × 10−3	1.87	7.45 × 10−1
HV	Best	5.92 × 10−1	6.91 × 10−1	5.97 × 10−1	6.91 × 10−1	6.71 × 10−1
	Worst	5.79 × 10−1	5.48 × 10−1	5.92 × 10−1	5.12 × 10−1	5.48 × 10−1
	Mean	5.85 × 10−1	6.11 × 10−1	5.95 × 10−1	5.89 × 10−1	5.76 × 10−1
	Median	5.87 × 10−1	5.99 × 10−1	5.96 × 10−1	5.78 × 10−1	5.69 × 10−1
	Std	4.31 × 10−3	4.22 × 10−2	1.60 × 10−3	4.02 × 10−2	2.80 × 10−2
	Var	1.86 × 10−5	1.78 × 10−3	2.57 × 10−6	1.61 × 10−3	7.82 × 10−4
SP	Best	9.24 × 10−1	8.59 × 10−1	6.84 × 10−1	9.19 × 10−1	6.43 × 10−1
	Worst	1.60	2.47	9.05 × 10−1	4.55	6.21
	Mean	1.17	1.54	7.83 × 10−1	2.00	2.94
	Median	1.12	1.55	7.97 × 10−1	1.51	3.00
	Std	1.73 × 10−1	4.47 × 10−1	6.90 × 10−2	1.11	1.29
	Var	3.00 × 10−2	1.99 × 10−1	4.77 × 10−3	1.23	1.67
Spread	Best	9.03 × 10−1	1.07	6.45 × 10−1	1.18	1.35
	Worst	1.14	1.32	8.48 × 10−1	1.68	1.85
	Mean	1.04	1.22	7.41 × 10−1	1.43	1.68
	Median	1.03	1.25	7.29 × 10−1	1.43	1.73
	Std	6.38 × 10−2	7.77 × 10−2	5.66 × 10−2	1.52 × 10−1	1.29 × 10−1
	Var	4.07 × 10−3	6.04 × 10−3	3.21 × 10−3	2.32 × 10−2	1.66 × 10−2

and

Table 10. Wilcoxon rank-sum test statistical results of the evaluation metrics for various multi-objective optimization algorithms.

Metrics	NSWOA vs. MOPSO	NSWOA vs. MOMVO	NSWOA vs. MSSA	NSWOA vs. MODA
IGD	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5
HV	3.05 × 10−5	8.2 × 10−1	1.4 × 10−1	4.18 × 10−3
SP	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5	6.10 × 10−5
Spread	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5	3.05 × 10−5

and Figure 19.

Table 9 shows the statistical results of the evaluation metrics for various multi-objective optimization algorithms, with the bolded data representing the optimal values for each metric. It is evident from the table that the NSWOA performs quite well in the multi-objective optimization process of the TET coils, with the majority of metrics surpassing the comparison methods and demonstrating good stability. Figure 19 shows the box plots of the evaluation metrics for various multi-objective optimization algorithms. From the figure, it can be seen that in the IGD, Spread, and SP evaluation metrics, the box plot for the NSWOA is narrow and located at the bottom, indicating that the overall quality of the statistical results for these metrics is high and the stability is good. However, the HV indicator works differently; the greater the indicator, the better the optimization performance of the multi-objective optimization method. Although the ideal value of the HV indicator for the NSWOA is not globally optimal, its box plot is the smallest, suggesting the highest level of stability in its solutions.

Based on the above analysis, to fully demonstrate the optimization performance of the NSWOA in the multi-objective optimization process of the TET coils, a Wilcoxon rank-sum test was further conducted on the optimization evaluation metrics of the NSWOA and the comparison algorithms, with a significance level defined as 0.05. The statistical results are shown in Table 10. From the table, it can be seen that except for the HV evaluation indicators, where the results of the NSWOA vs. MOMVO and the NSWOA vs. MSSA are greater than 0.05, all other results are less than 0.05, indicating that the NSWOA significantly outperforms the comparison algorithms in the TET coils optimization process.

In summary, the NSWOA shows good statistical results for most evaluation metrics, with high overall solution quality and good stability. Therefore, this study uses the Pareto solution set of the TET coils optimized by the NSWOA as the final optimization analysis result.

The Pareto solution set of the NSWOA is shown in Figure 20. All points on the Pareto front are optimal values. Decision-makers need to choose an appropriate plan based on the actual situation. In most cases, the best plan is a compromise plan.

Table 11. Candidate solutions and corresponding decision variables.

Candidate Solution	Decision Variables					Optimization Objective
	${\mathit{N}}_{\mathbf{P}}$	${\mathit{N}}_{\mathbf{S}}$	${\mathit{d}}_{\mathbf{P}}$ (mm)	${\mathit{d}}_{\mathbf{S}}$(mm)	$\mathit{f}$ (kHz)	$\mathit{\eta }$	$\mathit{k}$	${\mathit{\varphi }}_{\mathbf{s}}$ (mm)
1	5.00	5.00	1.66	0.00	341.53	0.54	0.09	17.50
2	25.56	15.46	0.09	1.79	129.56	1.00	0.44	90.45
3	31.00	25.00	0.15	2.00	400.00	0.95	0.60	154.07
4	10.63	21.42	1.62	0.00	281.05	0.99	0.29	50.85

lists four sets of candidate solutions and their corresponding decision variable values. Among them, candidate solution 1, candidate solution 2, and candidate solution 3 are the extreme solutions of the approximate solution set obtained by the NSWOA on the three objectives, representing the maximum preference on the three objectives, achieving the optimal value on only one objective. Candidate solution 4 is a compromise solution that balances the values of the three objectives. Therefore, this paper chooses candidate solution 4 as the final optimization result.

5.5. Verification of the Multi-Objective Optimization Results of the TET Coils

To verify the effectiveness of the multi-objective optimization analysis results of the TET coil calculated using the method proposed in this paper (candidate solution 4 in Table 10), the decision variables corresponding to candidate solution 4 in Table 10 were rounded (i.e., $N_{\mathrm{P}}$, $N_{\mathrm{S}}$, $d_{\mathrm{P}}$, $d_{\mathrm{S}}$, and $f$ were set to 11, 21, 1.62 mm, 0 mm, and 281 kHz, respectively). Subsequently, a simulation model was established in Maxwell based on the aforementioned coil structure parameters, and the remaining simulation settings were consistent with the previous case validation. The simulation calculations yielded $k$ and ${\varphi }_{\mathrm{s}}$ as 0.2897 and 50 mm, respectively, with calculation result errors of 0.1% and 7.57% compared to the results corresponding to candidate solution 4 in Table 10. Further simulation calculations yielded the self-inductance of the two coils, $L_{\mathrm{P}}$ and $L_{\mathrm{S}}$, as 4.143 × 10⁻⁶ H and 10.934 × 10⁻⁶ H, respectively, and the parasitic resistances $R_{\mathrm{P}}$ and $R_{\mathrm{S}}$ as 34 mΩ and 156 mΩ, respectively. Based on the above calculation results, $\eta$ can be calculated using Equation (35) [59].

$$\eta ={\displaystyle \frac{k^2{Q}_1{Q}_2}{{(1+\sqrt{1+k^2{Q}_1{Q}_2})}^2}}$$

(35)

$$Q_1=\omega L_{\mathrm{P}}/R_{\mathrm{P}}$$

(36)

$$Q_2=\omega L_{\mathrm{S}}/R_{\mathrm{S}}$$

(37)

$$\omega =2\pi f$$

(38)

where $\omega$ is the angular frequency, and $Q_1$ and $Q_2$ are the quality factors of the primary and secondary coils, respectively.

Using Equation (35), the theoretically calculated $\eta$ is 0.96, with a calculation error of 3.03% compared to the result corresponding to candidate solution 4 in Table 10. In summary, the errors between the $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ obtained from the simulations and theoretical calculations, and the corresponding values calculated using the method proposed in this paper are 3.03%, 0.1%, and 1.7%, respectively. The small errors demonstrate the effectiveness and correctness of the proposed method.

5.6. Comparison and Verification of the Anti-Misalignment Performance of the TET System Composed of the Optimal Coil Structure and Two Compensation Topologies

The aforementioned analysis has demonstrated the effectiveness and correctness of the proposed method. This subsection will analyze the anti-misalignment performance of the TET system constructed with the optimized TET coil structure and the two compensation topologies (S-S and LCC-S) discussed in Section 2—using the Simulink simulation software (MATLAB R2023b)—and select the compensation topology more suitable for the IMDs power supply.

Mutual inductance is a measure of the magnetic coupling between two coils, reflecting the intensity of interaction between the coils. When the relative positions of the TET coils are misaligned, it will lead to corresponding changes in mutual inductance between the coils. Therefore, for the convenience of research and analysis, mutual inductance can be used to characterize the positional misalignment of the coils. Taking into account the actual application scenarios of the TET coils, such as coil position shifts caused by patients’ physiological activities, individual differences, and changes in body posture, as well as previous related research [7], this study sets the coil’s axial spacing range to 15–23 mm and the deflection angle to 0–9° for simulation analysis. This model mimics the coil position changes in real-world circumstances, resulting in mutual inductance values ranging from 1.0 × 10⁻⁶ H to 1.95 × 10⁻⁶ H.

The simulation circuit of the TET system based on the S-S and LCC-S compensation topology is shown in Figure 21. The DC/AC conversion uses a full-bridge inverter composed of four Metal-Oxide-Semiconductor Field-Effect Transistors (MOSFETs), with the MOSFET switching states controlled by PWM signals. The operating frequency is set to the optimized coil resonant frequency of 281 kHz, and the AC/DC conversion uses a bridge rectifier. The relevant simulation parameter settings are shown in

Table 12. Circuit simulation parameter settings.

Parameters	Explanation	Values
$L_{\mathrm{p}}$	Inductance of the primary coil	4.143 × 10−6 H
$L_{\mathrm{s}}$	Inductance of the secondary coil	10.394 × 10−6 H
$L_{\mathrm{q}}$	Inductance of LCC-S topology component	1.6 × 10−6 H
$C_{\mathrm{p}}$(LCC-S)	LCC-S topology primary tuning capacitor	1.26 × 10−7 F
$C_{\mathrm{p}}$(S-S)	S-S topology primary tuning capacitor	7.74 × 10−8 F
$C_{\mathrm{s}}$(LCC-S)	LCC-S topology secondary tuning capacitor	2.93 × 10−8 F
$C_{\mathrm{s}}$(S-S)	S-S topology secondary tuning capacitor	2.93 × 10−8 F
$C_{\mathrm{q}}$	$L_{\mathrm{q}}$ tuning capacitor	2 × 10−7 F
$M$	Mutual inductance of coils	1 × 10−6 H~1.95 × 10−6 H
$R_{\mathrm{p}}$	Primary coil resistance	34 mΩ
$R_{\mathrm{s}}$	Secondary coil resistance	156 mΩ
$C_{\mathrm{f}}$	Filter capacitor	4.7 × 10−6 F
$R_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$	Load resistance	5 Ω
$U_{\mathrm{D}\mathrm{C}}$	DC voltage source	15 V

.

Figure 22 shows the variations in electrical parameters of the TET system with coils mutual inductance. Among them, Figure 22a shows the variation of TET system transmission efficiency and coil transmission efficiency with mutual inductance. The system transmission efficiency is the ratio of the system output power to the input power, while the coil transmission efficiency is the ratio of the power of the secondary coil (including compensation topology) to the power of the primary coil (including compensation topology). The figure shows that when the coils are ideally coupled (i.e., the axial distance between the coils is 15 mm, and they are perfectly aligned with a mutual inductance of approximately 1.95 × 10⁻⁶ H), the transmission efficiency of the TET system based on the two topologies is approximately equal, as is the coil transmission efficiency. However, when the coupling degree between the coils decreases and the mutual inductance decreases, the system transmission efficiency of the two topologies shows a downward trend. Given the range of mutual inductance variation, the coil transmission efficiency of the S-S topology fluctuates less compared to LCC-S, while the fluctuation in system transmission efficiency is smaller for LCC-S. Figure 22b shows the variations in the input AC voltage and load output voltage with mutual inductance. $U_{\mathrm{i}\mathrm{n}}$ is the AC voltage obtained through DC/AC conversion and $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the voltage across the load terminals. It is not difficult to see from the diagram that the $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$ of the two topologies changes in opposite trends with mutual inductance. The $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$ of S-S increases as mutual inductance decreases, while LCC-S behaves oppositely. This is determined by the inherent characteristics of the topologies, and similar conclusions can be drawn from existing research [60]. Given the range of mutual inductance variation, the fluctuation amplitude of $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$ for S-S is approximately 13 V, while the fluctuation for LCC-S is relatively smaller, with an amplitude of about 8 V. Compared to $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$, the $U_{\mathrm{i}\mathrm{n}}$ of the two topologies is relatively less affected by the mutual inductance changes and remains approximately constant. Figure 22c shows the variations in the primary and secondary trunk currents with mutual inductance. $I_{\mathrm{i}\mathrm{n}}$ and $I_{\mathrm{o}\mathrm{u}\mathrm{t}}$ are the primary and secondary trunk currents, respectively. As shown in the figure, LCC-S’s $I_{\mathrm{i}\mathrm{n}}$ and $I_{\mathrm{o}\mathrm{u}\mathrm{t}}$ decrease with the reduction of mutual inductance, while S-S shows the opposite trend. Given the range of mutual inductance variations, the fluctuation amplitudes of $I_{\mathrm{i}\mathrm{n}}$ and $I_{\mathrm{o}\mathrm{u}\mathrm{t}}$ for LCC-S are 3 A and 1.8 A, respectively, whereas for S-S, the fluctuation amplitudes of $I_{\mathrm{i}\mathrm{n}}$ and $I_{\mathrm{o}\mathrm{u}\mathrm{t}}$ are relatively larger, approximately 11 A and 2.7 A, respectively. Figure 22d shows the variations in system output power with mutual inductance. From the figure, it is not difficult to see that the system output power of the two topologies exhibits opposite variation patterns. The output power of the S-S system increases as mutual inductance decreases, with a fluctuation amplitude of approximately 116 W, while the output power of the LCC-S system decreases as mutual inductance decreases, with a fluctuation amplitude of approximately 42 W.

In summary, when the coils are in an ideal coupling state, the two topologies exhibit similar output characteristics. In certain application scenarios where the coupling degree of the coils remains unchanged, the S-S topology is simpler and more advantageous. In the IMD’s power supply application discussed in this study, coil misalignment is inevitable. Through the above analysis, it is evident that under the same conditions, the electrical parameters of the TET system composed of the LCC-S topology exhibit smaller fluctuations compared to the S-S topology, indicating its stronger anti-misalignment performance. Moreover, when the misalignment between the coils is too large (i.e., when mutual inductance is low), the $I_{\mathrm{i}\mathrm{n}}$, $I_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$ based on the S-S topology increase sharply, which can easily damage the electronic components of the circuit and cause severe heating, thereby endangering patient safety. In contrast, the $I_{\mathrm{i}\mathrm{n}}$, $I_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and $U_{\mathrm{o}\mathrm{u}\mathrm{t}}$ based on the LCC-S topology gradually decreases with the reduction of coil mutual inductance, effectively avoiding the aforementioned safety hazards. Therefore, the LCC-S topology is more suitable for powering applications in IMDs.

5.7. Verification of Electromagnetic Safety of Optimal Coil Structure

To verify whether the optimal coil structure obtained by the proposed method meets the safety requirements for electromagnetic radiation, this section will conduct an electromagnetic safety analysis of the optimal coil structure based on HFSS. Usually, the SAR (the specific absorption rate) [61] is used as an electromagnetic safety assessment index. This index is defined as the electromagnetic power absorbed by a unit mass of human tissue, and its differential expression is:

$$SAR={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\mathrm{d}W}{\mathrm{d}m}}\right)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\mathrm{d}W}{\rho \mathrm{d}V}}\right)$$

(39)

In the formula, $W$ is the power of the electromagnetic radiation absorbed by the tissue, measured in watts (W). $m$, $\rho$, and $V$ represent the mass, density, and volume of the tissue exposed to electromagnetic radiation, measured in kilograms (Kg), kilograms per cubic meter (Kg/${\mathrm{m}}^3$), and cubic meters (${\mathrm{m}}^3$), respectively. The SAR is measured in watts per kilogram (W/Kg). The SAR is divided into a local SAR and an average SAR. Since the TET coils used to power IMDs typically generate electromagnetic radiation only in a localized area of the human body, using a local SAR to characterize the electromagnetic radiation produced by the TET coils is more accurate. The local SAR can be expressed as:

$$SAR={\displaystyle \frac{\sigma {\left|E\right|}^2}{2\rho }}$$

(40)

where $\sigma$ is the tissue conductivity, and $E$ is the root mean square value of the local electric field strength.

Regarding the SAR safety limit standards, there are mainly two types. One is the IEEE1528SAR-200x standard, formulated by the International Commission on Non-Ionizing Radiation Protection (ICNIRP) [62], which stipulates that the SAR of biological tissues in an uncontrolled electromagnetic environment should be less than 2 W/kg. The other is the ANSIEEE standard, established by the American National Standards Institute (ANSI) [63], which stipulates that the SAR of biological tissues should not exceed 1.6 W/kg.

The electromagnetic safety simulation analysis model is shown in Figure 23, where the human body model references the high-precision electromagnetic model of the Chinese digitally visualized human body created by Zhang et al. [64]. The basic parameters of the human body model are a height of 180 cm and a mass of 65 kg. Due to the application of the artificial detrusor, the TET coils act on the abdominal area of the human body. Therefore, to reduce the computational load, the model has been appropriately simplified, using the torso part of the human body model for analysis. The coil is set according to the optimal coil structure parameters obtained through optimization, with a frequency set to 281 kHz. The coil is placed on the abdomen of the human body, and the radiation boundary conditions are set.

The simulation analysis results are shown in Figure 24 and Figure 25. Figure 24 is the SAR cloud map. It is not difficult to see that the closer the tissue is to the coil, the larger the SAR value. The maximum SAR value is 0.00046 W/Kg, located in the abdominal skin tissue, which is far below the safety limits set by IEEE1528SAR-200x and ANSIEEE. This indicates that the electromagnetic radiation produced by the optimized coil structure proposed in this paper meets the electromagnetic safety standards. In addition, the distribution pattern of the SAR along the negative direction of the x-axis (towards the interior of the human body) was further extracted, as shown in Figure 25. From the figure, it can be seen that the SAR distribution shows a decreasing trend towards the interior of the human body. Additionally, this study further analyzed the electric field distribution of the coil, as illustrated in Figure 26. The results show that the maximum electric field strength is 0.348 V/m, which is significantly lower than the safety limits specified by the ICNIRP. In summary, both the SAR and the electric field strength are below the safety thresholds. This demonstrates that the electromagnetic radiation dose received by the human body within the electromagnetic field generated by this coil fully complies with the established electromagnetic radiation safety standards.

5.8. Analysis of the Performance Verification of Artificial Detrusor Powered by TET

The aforementioned analysis performed theoretical and simulation verifications on the optimal coil structure obtained through the optimization of the method proposed in this paper, chose the LCC-S compensation topology that is better suited for powering IMDs, and investigated the optimal coil’s electromagnetic safety. This subsection will design a complete TET system prototype based on the aforementioned optimal coil structure and LCC-S compensation topology to power the artificial detrusor (driven by shape memory alloy (SMA) spring) and conduct a practical verification. By analyzing the driving performance of the artificial detrusor, the actual power supply effect of the TET system prototype will be verified.

An artificial detrusor is an implanted device that helps people with neurogenic bladders perform voluntary urination. Its primary role is to replace the injured natural detrusor, providing the contraction force needed to compress the bladder, thereby enabling the patient to urinate independently. The volume of urine expelled is an important evaluation indicator for measuring the driving performance of an artificial detrusor. Previous research has shown that the driving voltage of the SMA spring-driven artificial detrusor, which is involved in this study, has a significant impact on its driving performance. If the driving voltage is too low, the driving effect of the artificial detrusor will be poor and unable to effectively assist patients in urination. Conversely, if the driving voltage is too high, it will cause an excessive temperature rise in the SMA spring, leading to thermal damage to the tissue. The research results indicate that setting the driving voltage to 8 V is most suitable, as it ensures a certain amount of urine output while also keeping the temperature rise of the SMA spring low, thereby reducing its impact on human tissue.

As seen in Figure 1, the TET system is primarily composed of a DC/AC converter, primary and secondary (transmission and receiving) coils, and compensatory topology AC/DC converters. The primary and secondary coil structures, as well as the compensatory topology components, are determined using the aforementioned methodology. The external DC/AC converter uses a full-bridge inverter circuit, which consists of four MOSFETs. The system operating frequency (MOSFET switching frequency) determined in this study is 281 kHz. The switching time of the MOSFET is relatively short, which can easily lead to excessive voltage change rate du/dt and current change rate di/dt during the switching process, thereby increasing the losses during the on-off state switching process. Additionally, it may directly damage the switching device, leading to system failure. Based on this, this study employs a dual RCD snubber circuit to reduce the switching losses of power electronic switching devices, increase the circuit’s ability to withstand short-term overloads, and thus improve the switching devices’ reliability during inverter circuit operation. The on-off control of the MOSFET in this study is conducted using pulse width modulation (PWM), with the control module employing the STM32F103C8T6 microcontroller, primarily to generate two complementary PWM signals with dead time. However, the PWM control signal generated by the microcontroller is relatively weak and insufficient to enable the MOSFET to achieve rapid turn-on and turn-off. Therefore, it is necessary to amplify the PWM control signal so that it can quickly turn on and off the MOSFET. This study chooses the IR2110 chip for driving. In addition, to prevent the PWM control signals from being interfered with, two 6N137M high-speed optocouplers are used for isolation. The internal AC/DC converter uses a bridge rectifier DB307S, and after rectification, a filter capacitor is used for filtering. According to the aforementioned anti-misalignment verification analysis in Figure 22, the output load-end voltage based on the LCC-S topology varies with the degree of coil coupling, with a range of approximately 9–17 V. As previously mentioned, the optimal driving voltage for the SMA spring-driven artificial detrusor in this study is 8 V. Therefore, based on this requirement, this paper uses the TPSM63604RDLR buck chip to stabilize the output voltage at 8 V. A physical prototype of the TET system was built, and the artificial detrusor experimental platform powered by the TET system is shown in Figure 27. It mainly consists of the TET system, an artificial detrusor prosthesis, and SMA springs.

The TET system includes an external primary coil and a transmission module (composed of a control module, opto-isolation circuit, drive circuit, inverter circuit, and absorption buffer circuit), as well as a secondary coil and a receiving module (composed of a secondary coil compensation topology and rectification, filtering, and voltage regulation circuits). The specific execution procedure may be broadly summarized as follows: utilizing a syringe to gently inject water into a simulated bladder to replicate the process of human bladder filling, with a total injection volume of 300 mL (the usual bladder capacity of an adult is around 300–500 mL). The external transmission module transforms the DC power into AC power with a frequency of 281 kHz after the injection is finished, and the three-way valve’s injection side is closed. The AC power is then sent to the primary coil and its compensation topology circuit. Through the alternating magnetic field between the primary coil and the secondary coil, electrical energy can be transmitted to the secondary coil within the body and its compensation topology circuit. After rectification, filtering, and voltage regulation by the internal receiving module, it can provide a continuous and stable power supply for the artificial detrusor. The SMA spring fitted on the artificial detrusor prosthesis undergoes a phase change and heats up under electrical excitation, generating a restoring force that causes it to contract. Under the influence of this restoring force, the prosthesis rotates inward around the support frame, “closing” and compressing the bladder, providing the necessary pressure for bladder urination. As the pressure gradually increases, the liquid level in the syringe on the upper left rises, allowing the volume of the expelled liquid to be read. The diagram of the driving process of the artificial detrusor is shown in Figure 28.

During the experiment, the relative position between the primary and secondary coils was adjusted to simulate the positional changes between the coils in actual applications. The power-on time was set to 30 s, and the coil position adjustment range was referenced from the aforementioned anti-misalignment performance comparison verification section, set to an axial distance of 15 mm–23 mm and a primary coil deflection of 0°–9°. The emptying rate $∆\mathit{V}$ of the artificial detrusor (the ratio of the volume of liquid expelled to the total volume of the injected simulated bladder) was used as the evaluation index for its driving performance to verify the actual power supply effect of the TET system. The analysis results are shown in Figure 29.

Figure 29a shows the curve of $∆V$ changing with the coil position. From the figure, it is not difficult to see that $∆V$ decreases as the axial distance of the coil increases, and it also shows a downward trend with the increase in the deflection angle. Within the entire range of coil position variation, the change in $∆V$ is relatively small. When the axial spacing is 23 mm and the deflection angle is 9°, the maximum change rate is only 7.3%. The main reason for this is that within the range of coil position variation, the voltage after filtering is greater than 8 V, so it can be stabilized around 8 V after stepping down. This indicates that the TET system, based on the optimal coil structure and LCC-S compensation topology, can still maintain a good driving performance of the artificial detrusor even when the coil position is offset. Figure 29b shows the curve of $∆V$ changing with the coil energization time. From the figure, it can be seen that $∆V$ increases with the increase in energization time. The maximum $∆V$ can be reached after approximately 16 s of energization, after which it will no longer change, laying the foundation for determining the optimal energization time.

6. Conclusions

The majority of current research on TET system coil optimization design focuses on single-objective optimization. In certain situations, single-objective optimization produces solutions that are not the best fit for the real issue because it is unable to accurately capture the complexity of the TET system’s actual application. In practical applications, not only should transmission efficiency be considered, but also the size limitations of body implants and the degree of coil coupling, among other requirements, in order to provide patients with a more stable, comfortable, and highly reliable TET system. This study proposes a machine learning-based modeling and multi-objective optimization method for TET coils, covering the complete process from modeling to multi-objective optimization design. Verification analysis was conducted using a planar helical coil applied in an artificial detrusor as an example, and the final optimization analysis results were validated. The following conclusions can be drawn:

(1) The GWO algorithm significantly outperforms the comparison algorithms in terms of optimization performance during the parameter tuning process of the ELM model for the TET coils, and it also exhibits good convergence capability and stability. The TET coil prediction model based on GWO-ELM achieves globally optimal prediction performance when the training set to test set ratio is 9:1. The $RMSE$, $R^2$, and $MAE$ for predicting $\eta$ are 2.46 × 10⁻², 9.83 × 10⁻¹, and 1.73 × 10⁻², respectively. For predicting the $k$, the $RMSE$, $R^2$, and $MAE$ are 1.18 × 10⁻¹, 0.99999, and 9.34 × 10⁻², respectively. For predicting the ${\varphi }_{\mathrm{s}}$, the $RMSE$, $R^2$, and $MAE$ are 2.04 × 10⁻², 9.80 × 10⁻¹, and 1.53 × 10⁻², respectively. All these values are superior to the comparison prediction models, demonstrating excellent prediction performance.

(2) Based on the GWO-ELM prediction model, a multi-objective optimization function for the TET coil was established. The NSWOA algorithm showed superior optimization performance during the multi-objective optimization process of the TET coil, outperforming most evaluation metrics compared to the benchmark algorithms. Using this algorithm, the Pareto solution set for the TET coil was obtained, with the selected optimal solutions for $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ being 0.99, 0.29, and 50.85 mm, respectively. Based on the simulation and theoretical analysis, the values of $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ are 0.96, 0.2897, and 50 mm, respectively. The errors between these values and the corresponding values calculated by the proposed method are 3.03%, 0.1%, and 1.7%, respectively. The small errors verify the correctness and effectiveness of the proposed method.

(3) Based on the optimized optimal coil, the anti-offset performance of the TET systems created by the optimal coil with S-S and LCC-S compensation topologies was investigated further. The findings reveal that the TET system composed of the optimum coil and LCC-S has superior anti-misalignment performance, making it more suited for powering IMDs. Furthermore, the electromagnetic safety of the optimal coil was further analyzed, with its maximum SAR value being 0.00046 W/Kg, which is far below the safety limit. This indicates that the electromagnetic radiation generated by the optimal coil structure, optimized by the proposed method, meets the electromagnetic safety standards. Finally, based on the optimal coil structure and LCC-S compensation topology, a complete TET system prototype was designed to power the SMA spring-driven artificial bladder, and an example verification was conducted. The findings show that, with a coil axis spacing of 15–23 mm and a coil offset range of 0°–9°, the physical prototype of the transcutaneous energy transfer system, which was designed using the best coil structure and LCC-S compensation topology, can still deliver a steady energy supply to the artificial detrusor while preserving good driving performance.

The method proposed in this paper provides theoretical support for the optimal design of TET coils and is applicable not only to planar coils but also to TET coils of other structural forms. Although the approach suggested in this research has shown good optimization results for planar coils, there are certain restrictions. This study focuses on optimizing the TET coil’s $\eta$, $k$, and ${\varphi }_{\mathrm{s}}$ parameters. It may be expanded to include more optimization targets to better reflect the complexity of the real use of TET coils, resulting in a better balance of physiological needs and safety constraints. The next step in this research plan is to conduct in vivo animal experiments to validate the designed TET coils.