# Relaxation Parameter Optimization in Electrical-to-Mechanical Co-Simulation Based on Time Windowing WR Technique

Md Moktarul Alam, Richard Perdriau, Mohammed Ramdani, Mohsen Koohestani

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Md Moktarul Alam, Richard Perdriau, Mohammed Ramdani, Mohsen Koohestani. Relaxation Parameter Optimization in Electrical-to-Mechanical Co-Simulation Based on Time Windowing WR Technique. IEEE Access, 2025, 13, pp.80775-80785. 10.1109/ACCESS.2025.3567752. [hal-05105459](https://hal.science/hal-05105459v1)

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Submitted on 10 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Received 9 April 2025, accepted 4 May 2025, date of publication 7 May 2025, date of current version 14 May 2025.

Digital Object Identifier 10.1109/ACCESS.2025.3567752

RESEARCH ARTICLE

# Relaxation Parameter Optimization in Electrical-to-Mechanical Co-Simulation Based on Time Windowing WR Technique

MD MOKTARUL ALAM<sup>1,2</sup>, (Student Member, IEEE), RICHARD PERDRIAU<sup>1,3</sup>, (Senior Member, IEEE), MOHAMMED RAMDANI<sup>1,3</sup>, (Senior Member, IEEE), AND MOHSEN KOOHESTANI<sup>1,3</sup>, (Senior Member, IEEE)

<sup>1</sup>École Supérieure d'Electronique de l'Ouest (ESEO), 49107 Angers, France

<sup>2</sup>Institut National des Sciences Appliquées, University of Rennes, 35708 Rennes, France

<sup>3</sup>Institut d'Electronique et des Technologies du numéRique (IETR), 35042 Rennes, France

Corresponding author: Md Moktarul Alam (mdmoktarul.alam@eseo.fr)

This work was supported by French National Research Agency (ANR): Electromagnetic Compatibility Co-Simulation of Complex Electrical Systems (ECOCES Project) under Grant ANR-19-CE05-0016-05.

**ABSTRACT** This paper presents an innovative approach to enhancing the time windowing waveform relaxation (WR) technique in electrical-to-mechanical co-simulation by optimizing relaxation parameters for improved performance. An analytical model is introduced to determine the optimal number of time windows, considering the circuit's dynamic characteristics. Additionally, a genetic algorithm (GA) is applied to refine relaxation parameters (e.g., impedance values), effectively addressing challenges posed by nonlinear device behaviors and improving the accuracy of co-simulation results. In the case study, it is crucial to emphasize that the full system simulation is used exclusively for retrospective validation of the co-simulation, without incorporating its results as inputs or convergence criteria. The proposed method significantly reduces the average error between co-simulation results and the full system output voltage, decreasing it from -2.8 dB to less than -26.2 dB. This demonstrates improved alignment and faster convergence compared to using the WR method. The validation of GA method is then applied to the co-simulation of an electrical (buck converter) and mechanical (DC motor) system, with results compared against the full system. The WR method exhibited significantly lower power efficiency (57.4%) compared to the full system (81.4%), rendering it insufficient. Time windowing WR enhanced power efficiency through simulation segmentation, reaching 74.6%, though it still fell short of the full system. The most advanced approach, time windowing WR with GA optimization, dynamically adjusted relaxation parameters, resulting in a power efficiency of 80.8%, nearly matching the 81.4% recorded in the single-kernel simulation. These findings underscore the effectiveness of integrating time segmentation with adaptive optimization to enhance simulation performance.

**INDEX TERMS** Co-simulation, genetic algorithm, parameter optimization, time windowing technique, waveform relaxation.

## I. INTRODUCTION

Dynamic analyses are essential for the protection and control of electric power systems. Traditionally, these studies have been applied in fields such as communications,

The associate editor coordinating the review of this manuscript and approving it for publication was Sotirios Goudos<sup>1</sup>.
medical electronics, appliance control, and transportation [1]. However, the inclusion of nonlinear components adds complexity. Consequently, modeling and simulation become vital for integrating these intricate system components [2]. It is also crucial to assess electromagnetic disturbance levels during the design phase and continuously refine predictive models. Despite advancements in system-oriented modeling and increased computational power, direct simulations using current tools face challenges due to multiscale issues, high costs, and limited model availability, particularly in the automotive sector. This sector still relies heavily on simulations for complex embedded systems [3], [4], [5].

To address the complexity of modern circuit architectures, numerical simulators must support co-simulation which integrates subsystems with distinct dynamics within a unified framework. Co-simulation is especially effective in complex electrical systems, where varying dynamics can be managed more efficiently by adjusting numerical techniques [6], [7]. It is crucial for maintaining a balance between speed, accuracy, and the diverse requirements of modern design systems (e.g., scalability, accessibility, cross-platform compatibility, maintainability) [3]. Consequently, the waveform relaxation (WR) method offers an alternative by selecting time steps based on the fastest variable's rate of change in conventional simulators, while also highlighting the trade-offs inherent in different simulation methodologies [8]. However, the convergence rate can be sluggish and often requires numerous iterations. The traditional WR method struggles with systems that have strong couplings between subsystems, as tightly linked components may need frequent updates to achieve convergence. To address that, time windowing acceleration is used as an adaptive iterative technique applied in the WR approach [9], [10], [11]. Although determining the optimal segment size remains challenging, time windowing segments are also used to reduce unnecessary computational overhead [12], [13]. An optimization algorithm can be employed to refine the relaxation parameters of subsystems, leading to optimal circuit co-simulation performance [14], [15]. This was achieved using optimization methods such as the genetic algorithm (GA), bayesian algorithm (BA), differential evolution algorithm (DEA), and geometry algorithm (GeA). However, GA is the most prominent and easiest approach for parameter optimization in simulated power systems due to its adaptability, global search capability, and efficiency in handling complex, multi-dimensional problems [11], [16], [17], [18], [19], [20], [21], [22], [23].

All the above methods except GA were previously implemented in [24], [25], and [26]. What makes the GA method more suitable for the intended application is the following:

* GA stands out for its robustness and adaptability in co-simulation-based parameter optimization. Unlike BA optimization, which can struggle with complex or noisy systems, and DEA and GeA which are limited in dynamic and high-dimensional contexts, GA offers

a reliable global search capability that is well-suited to the non-linear, multi-modal nature of co-simulation environments.

* GA is well-suited for co-simulation environments due to its ability to handle noise and stochastic behavior without relying on precise models, making it ideal for complex systems. In contrast, BA optimization struggles with discontinuities and randomness, while DEA is limited by its reliance on static data and inability to manage dynamic simulations. GeA, dependent on predefined heuristics, also faces challenges in flexible, unpredictable co-simulation scenarios.

* GA is computationally intensive but highly parallelizable, making it ideal for distributed co-simulation. BA optimization is efficient in low dimensions but becomes costly in higher dimensions. DEA is light on computation but does not scale well for iterative simulations or fine-tuned parameter optimization. GeA works well for low-dimensional problems but struggles with scalability and adaptability in complex, high-dimensional tasks.

This paper introduces a novel optimization methodology for the time windowing WR technique in electrical-to-mechanical co-simulation. The key contributions are as follows:

* A GA is employed to optimize relaxation parameters (e.g., impedance values) in the WR technique. This method addresses nonlinear behavior and significantly improves convergence between subsystems and the full system, reducing the discrepancy.

* By combining time windowing with optimized parameters, the proposed method leads to achieve faster and more accurate convergence in source-to-source circuit co-simulation.

* The approach is validated through full system outputs comparing co-simulation results with two or more kernel simulations, demonstrating significant improvements in accuracy and convergence efficiency.

These advancements position the proposed methodology as a robust tool for co-simulation of complex systems with nonlinear components (e.g., diodes), effectively addressing challenges such as computational overhead, slow convergence, and tightly coupled system dynamics.

The paper is organized as follows: Section II explains the structure of the time windowing WR technique and determines the optimal number of time windows using an analytical model. Section III describes the co-simulation method for source-to-source circuits, where the number of time windows in the time windowing WR technique is determined analytically and compared to the full system. Section IV outlines the optimization method for determining the optimum values of relaxation parameters. Section V presents a case study applying the time windowing technique with optimized parameter values in source-to-source co-simulation, with results compared to the full system. Section VI details the implementation of the GA in the time
windowing WR technique for electrical-to-mechanical co-simulation. Finally, Section VII concludes the paper.

## II. STRUCTURE OF TIME WINDOWING WR CO-SIMULATION METHOD

Ordinary and partial differential equations (ODEs and PDEs) can be solved iteratively using the WR method, which involves discretization, initial guesses, subsystem decomposition, iterative relaxation, convergence checking, and final solution assembly [11], [21]. Solutions are updated at each time step until convergence is achieved. The WR method is applied to solve electric circuits, with the solution's convergence depending on the structure of the subsystems. The convergence error is calculated by simulating each subsystem independently, exchanging interface waveforms at the end of the simulation period [27]. Additionally, time windowing divides simulation time into distinct windows, directing computing resources to active windows, thereby improving simulation accuracy and efficiency. Properly initializing circuit parameters according to the circuit's properties promotes rapid convergence.

Fig. 1 integrates WR, relaxation parameter optimization, and time windowing technique to iteratively solve and optimize complex systems, such as electrical circuits. The WR process divides the system into subsystems, which are solved separately until convergence is achieved. Initially, one subsystem is initialized with an input $X_k^i$ in the current iteration $i$, and its output $X_k^{i+1}$ is recorded. This output is then used as the input for another subsystem, which computes its response $Y_k^{i+1}$, also recorded. Convergence is checked the difference $\Delta X = |X_k^{i+1} - X_k^i|$. If $\Delta X$ is less than a predefined threshold, the process is considered converged. Otherwise, the time windowing technique enhances simulation accuracy by incorporating time-dependent methods. The concept of the optimal number of time windows, $W_f$, is derived from the theory of multiresolution analysis using wavelets [28], [29]. This analytical model of this number of time windows is determined based on the window signal function $f(T)$ and the window size function $W(T)$. At the initial step ($k_0 = 1$), time dependent variables are initialized with $I_L = 0$ and $V_c = 0$. Circuit parameters are then expressed as functions of $T$, such as $f(T) = V_{in}e^{-g(A)T}$, and a window size $W(T) = V_{in}e^{-k_n T}$. The parameter function $g(A)$ is established based on the circuit's output results and the subsystem interface variables $X_n$ and $Y_n$. An analytical model for determining the optimal number of time windows, $W_f$, is derived by integrating $f(T)$ and $W(T)$, as shown in equation (1) [30]. A final convergence check ensures that the simulation results meet the desired accuracy. If the criteria are satisfied, the process ends; otherwise, adjustments are made, and the procedure is repeated to refine the results.

$$W_f = (ceil)(V_{in}e^{-V_{out} \times (ln(X_0/Y_0) + 1) \times k_n T}) \quad (1)$$

The co-simulation results are compared with those of the full system in Sect. III. It is of paramount importance

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Co-simulation methodology developed in the current study using the optimal number of time windows and GA optimization in the time windowing WR method.</figcaption></figure>

to note that the full system simulation is used solely to assess the validity of the co-simulation a posteriori, without incorporating any of its results as inputs for the co-simulation or as convergence criteria. If a difference is still observed, the relaxation parameters are updated using a GA optimization method. The minimum value of the objective function (OF) obtained through this optimization provides the optimum relaxation parameter values. These values are then applied as relaxation parameters in the subsystems and used in the time windowing WR method. As a result, the accuracy is further enhanced by utilizing optimum relaxation parameter values, determined through a GA, in the time windowing WR method for co-simulation.

## III. ANALYSIS OF THE TIME WINDOWING WR CO-SIMULATION

The source-to-source circuit (buck converter) in Fig. 2(a) consists of a switching transistor, an inductor, a capacitor
<figure><img src="image_3.png" /><figcaption>FIGURE 2. (a) Simulink model of the source-to-source operation, illustrating the splitting of the system into (b) subsystem 1 (source) and (c) subsystem 2 (source).</figcaption></figure>

and diodes. In a buck converter, all parameters are calculated using the buck converter equations [31], [32]. Using an input voltage of 4 V, the simulation runs for 0.05 s with an output voltage of 2.4 V. Next, as shown in Figs. 2(b) and 2(c) is split into two subsystems: subsystem 1 (source) and subsystem 2 (source). In both subsystems, the relaxation parameter is incorporated along with the behavioral controller. The relaxation values for each subsystem are calculated using Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL). These relaxation values are then incorporated into each subsystem, and the WR method is applied to facilitate data interfacing between subsystems. The overall procedure is organized and executed using MATLAB® Simulink® software. The entire WR method used to obtain the simulation results is run on a laptop with an Intel® Core™ i7-10850H CPU at 2.70 GHz, 32 GB of RAM, running Microsoft Windows 11 64 bits. The system simulation utilized 18% of the available memory, which increased to 38% during co-simulation.

The WR approach, as shown in Figs. 2(b) and 2(c), uses X (the output voltage of subsystem 2) and Y (the initial value for the voltage controller) as interface variables to assess convergence. This process continues until system convergence is achieved. If convergence is not reached, the time windowing strategy is applied, with initial values for inductor current ($I_L$) and capacitor voltage ($V_C$) serving as interface variables. The optimal number of time windows is

<figure><img src="image_4.png" /><figcaption>FIGURE 3. Comparison between the full system and co-simulation using (a) the initial number of time windows in WR and (b) extracted from the analytical model of the time window.</figcaption></figure>

determined based on the system's nonlinear behavior, with the converged values of $I_L$ and $V_C$ from one window used as initial conditions for the next, ensuring an accurate system response across multiple number of time windows.

The number of time windows is determined using Equation (1), with the values of $X_0$ and $Y_0$ taken from the WR method, resulting in a value of 6. This number of time windows is then applied in the time windowing WR technique. The co-simulation results are subsequently compared with the full system simulation results, as shown in Fig. 3. Fig. 3(a) presents the comparison using WR method, while Fig. 3(b) shows the comparison with the optimal number of time window 6, as determined from equation (1).
Fig. 3(a) illustrates that while convergence is achieved at the final point of the simulation, a difference of up to 40% is observed at the initial and middle points compared to the full system. Additionally, the co-simulation result does not align with the trend of the full system simulation and diverges from it. After applying the optimal number of time window in the time windowing WR technique, the co-simulation results are compared with the full system in Fig. 3(b). This comparison shows that while the co-simulation output follows the same trend as the full simulation, a difference of up to 15% persists due to the relaxation parameters calculated using the *KVL* and *KCL* methods. Note that the relaxation parameters calculated using *KVL* and *KCL* are based on linear approximations, which may not fully capture the system's complexity. In contrast, co-simulation models incorporate nonlinearities and dynamic interactions between components, such as the behavior of diodes and transistors, resulting in differences in the relaxation parameters. Additionally, co-simulation accounts for the coupling and feedback between subsystems, further contributing to discrepancies between the parameters calculated by *KVL* and *KCL* and those used in co-simulation. The optimum relaxation parameter is required to ensure that the co-simulation accurately reflects the behavior of the full system. By accurately capturing the system's nonlinearities, dynamic interactions, and coupling effects, the co-simulation will yield results that closely match the real-world performance of the full system, enhancing the reliability and validity of the simulation. The next section discusses optimization techniques to address this challenge.

## IV. PROPOSED OPTIMIZATION OF WAVEFORM PARAMETERS

The integration of the time windowing WR technique with the conventional optimization process, as shown in Fig. 4, enhances the optimization of the relaxation parameters. The *GA* is used to determine the optimum relaxation parameters in the time windowing WR method due to its ability to optimize complex, nonlinear, and multidimensional problems. It handles multiple parameters simultaneously, improving overall simulation performance. The global search capability of the *GA* helps avoid local optima, ensuring the optimal solution for the relaxation parameters. Its adaptability to complex systems allows it to adjust to varying circuit configurations and nonlinear behaviors. Additionally, *GA* improve efficiency in parameter tuning, reducing the need for exhaustive manual tuning by converging to the optimal solution over multiple generations [11], [33], [34], [35].

The *GA* method starts by evaluating the *OF* values at the initial vertices, ranking these $n + 1$ points from best (smallest *OF*) to worst (largest *OF*). This forms a triangular simplex for a two-variable optimization problem, as illustrated in Fig. 4. In this simplex, one vertex (*H*) corresponds to the highest *OF* value, one vertex (*L*) to the lowest, and one vertex (*M*) holds an intermediate value. In degenerate cases, where *M* and *L* have the same value, it is arbitrary and labeled *M* or *L* [11], [33].

<figure><img src="image_5.png" /><figcaption>(a) Reflection</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) Outside contraction</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d) Inside contraction</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(e) Shrinking</figcaption></figure>

**FIGURE 4.** The objective function values (*OF*) of the simplex vertices determine the basic optimisation strategy used in numerous *GA* operations, where *OF(L)*< *OF(M)* < *OF(H)*. (a) Reflection: shifting the least favourable vertex (*H*) towards the minimum point (*L*).(b) Expansion: executing if *OF(R)* < *OF(L)*.(c) Outside contraction: if *OF(L)* < *OF(R)* < *OF(M)*.(d) Inside contraction: if *OF(R)* > *OF(M)*.(e) Shrinking: carried out in the case where *OF(OC or IC)* > *OF(H)*.

Fig. 4(a) illustrates how the *GA* algorithm eliminates the vertex with the largest *OF* by reflecting it over the midpoint of the other two vertices to reduce the value. The *OF* at the reflected point (*R*) is then evaluated. If the new vertex (*R*) has a smaller *OF* than the current best vertex (*L*), the *GA* algorithm recognizes the direction as promising and expands by multiplying the reflected length by a factor $\alpha > 1$, as shown in Fig. 4(b). The factor $\alpha$ acts as a scaling factor, controlling the rate of change applied to the solution during mutation or crossover operations [11].

If the *OF* at *R* falls between *L* and *M*, an external contraction is performed, as shown in Fig. 4(c), by multiplying the reflected length by a factor $\beta < 1$. The factor $\beta$ controls the influence each parent has in creating offspring, especially in parent selection or crossover operations. In certain crossover methods (e.g., linear crossover), $\beta$ scales of two parents
<figure><img src="image_9.png" /><figcaption>FIGURE 5. GA based variation of the OF to optimize parameter values.</figcaption></figure>

are combined. Fig. 4(d) illustrates the need for a more significant internal contraction when R's OF is greater than M [36].

If the value at the contracted point (IC or OC) is smaller than at the worst point (H), the dimensions of the GA simplex are reduced. As shown in Fig. 4(e), this is done by shrinking the simplex lengths by a factor $γ < 1$ towards the smallest point (L). The factor $γ$ serves as a selection pressure in algorithms, controlling the preference for better individuals over worse ones during the selection process, thereby influencing the rate of convergence towards an optimal solution. By replacing the ‘centroid’ of the $n-1$ smallest-valued vertices with the ‘midpoint’ of the two smallest-valued vertices, this process can be extended to any simplex dimension $n$.

## V. CO-SIMULATION RESULTS FOR THE PROPOSED OPTIMIZATION OF TIME WINDOWING WR

This section presents and analyzes the results of the co-simulation using the proposed genetic optimization method to determine the optimum relaxation parameter values, replacing those initially obtained using the KVL and KCL methods. The OF values of 1.2 and 2.8, obtained from the GA optimization method, are illustrated in Fig. 5 and detailed in Table 1, which presents their optimum and average values. The updated relaxation parameter values from Table 1 are applied in Figs. 2(a) and 2(b). These optimized relaxation values are then used in the time windowing WR technique to refine the process, significantly improving both the accuracy and efficiency of the co-simulation compared to full system simulation.

$$ Error = 20\log\left|\frac{V_{\text{fullsystem}}^{i+1} - V_{\text{co-simulation}}^{i+1}}{V_{\text{fullsystem}}^{i+1}}\right| \quad (2) $$

The initial relaxation parameter values, calculated using the KVL and KCL methods and listed in Table 1, are used in the genetic optimization method. The first two parameters, L and C, are assigned to subsystem 1, connected to the

<figure><img src="image_10.png" /><figcaption>FIGURE 6. Comparison of relative error between full system and co-simulation using WR, time windowing WR, time windowing WR with average values and GA optimized time windowing WR.</figcaption></figure>

current controller, while the third parameter, R, is assigned to subsystem 2, connected to the voltage controller. The comparison in Fig. 6, using (2), shows that the WR method results in an average error of -2.8 dB. After applying the time windowing method, the average error improves to less than -14.8 dB. The GA optimization method generates a new vector of optimized relaxation parameters (R', L', and C'). The stopping criterion for the GA is set to a convergence threshold of 0.001, based on the circuit characteristics [22], [27] in the OF. It takes approximately 120 minutes to reach the minimum OF value, which defines the optimum relaxation parameter values: R' = 9.9 Ω, L' = 1.5 mH, and C' = 12 μF. These optimized values are then applied to subsystem 1 and subsystem 2, replacing the previous relaxation values.

<table><caption>TABLE 1. The initial parameter estimations for the simplex algorithm and the associated optimization outcomes.</caption><thead><tr><th rowspan="2">Specification</th><th colspan="3">Relaxation Parameters</th><th rowspan="2">OF</th><th rowspan="2">Elapsed Time (s)</th></tr><tr><th>R (Ω)</th><th>L (mH)</th><th>C (μF)</th></tr></thead><tbody><tr><td>KVL and KCL<br>(Initial guess)</td><td>7</td><td>0.8</td><td>5</td><td>23.6</td><td>33.6</td></tr><tr><td>Guess 1</td><td>10</td><td>1</td><td>10</td><td>2.8</td><td>30.2</td></tr><tr><td>Guess 2</td><td>13</td><td>2</td><td>15</td><td>7.4</td><td>31.8</td></tr><tr><td>Worst</td><td>18.4</td><td>12</td><td>0.7</td><td>27.6</td><td>42.4</td></tr><tr><td>Optimum</td><td>9.9</td><td>1.5</td><td>12</td><td>1.2</td><td>29.7</td></tr></tbody></table>

Fig. 7(a) compares the co-simulation and the full system using the average relaxation parameter values from Guess 1 in Table 1: $R_{avg} = 10$ Ω, $L_{avg} = 1$ mH and $C_{avg} = 10$ μF, within the time-windowing WR technique, where the OF value is 2.8. These results can be compared to Fig. 3 where the initial KVL and KCL values were used. In this case, Fig. 6 shows that the average error in output results between the co-simulation and the full system is approximately -22.6 dB. It is noted that using a relaxation parameter corresponding to an objective function value of 2.8 reduces the difference from -16.8 dB to -22.6 dB through the time windowing technique. This highlights the crucial role
<figure><img src="image_11.png" /><figcaption>FIGURE 7. Comparison of the full system and co-simulation output voltages using time windowing WR technique: (a) average relaxation values and (b) relaxation values proposed by genetic optimization method.</figcaption></figure>

of accurate relaxation parameter values in minimizing the difference.

The elapsed times shown in Table 1 represent the duration required by the optimization algorithm, when applied within the time windowing WR method, to reach the best solution for each set of parameters. Generally, the closer the initial guess to the optimal solution, the shorter the optimization time, with the optimal guess resulting in the fastest convergence. The algorithm converges to the same final values regardless of the initial guess. Therefore, the primary consideration for the cost function is the elapsed time, as the resulting waveform remains unchanged. Notably, the elapsed time using the optimal parameters in the time windowing WR method is 29.8 s, which is nearly identical to the 29.4 s observed for the full system. This demonstrates that co-simulation using time windowing WR with GA optimized parameters achieves comparable performance to the full system in terms of computational efficiency and accuracy.

A comparison between the full system and the co-simulation using the time windowing WR method with the proposed genetic optimization method for relaxation parameter values is shown in Fig. 7(b). As shown in Fig. 5, the last three OF values are nearly identical. The GA yields an optimal OF value of 1.2, determining the optimized relaxation parameter values listed in Table 1. These values are then applied in the time windowing WR technique and the resulting co-simulation output is compared with the full system. The full system simulation takes 15 s, while the co-simulation converges after 12 iterations with an elapsed time of 17 s, representing an 11.8% increase. Nonetheless, the co-simulation output closely matches the full system, with an average error of less than -26.2 dB.

This significant reduction in discrepancy highlights that the optimization of relaxation parameters in the time windowing WR technique effectively represents the full system with high accuracy. It demonstrates that applying the GA for parameter tuning within this method leads to more precise and reliable co-simulation results.

## VI. VALIDATION OF THE PROPOSED OPTIMIZATION FOR ELECTRICAL-TO-MECHANICAL CO-SIMULATION

For the proof-of-concept, the GA optimization approach is applied to enhance the co-simulation between the

electrical and mechanical subsystems. The converter-to-DC motor circuit, shown in Fig. 8(a), includes a switching transistor, an inductor, a capacitor, diodes, and a voltage controller. The equivalent of DC motor is represented by equations (3) and (4). In these equations, (i + 1) denotes the iteration number, and k represents the optimal number of time windows.

$$V(T) = L\left(\frac{di}{dT}\right)_k^{i+1} + R(T) + k_e\omega_k^{i+1}(T) \quad (3)$$

where $V(t)$ is the input voltage supplied to the DC motor, $L$ is the inductance of the motor armature, and $\frac{di}{dT}$ represents the rate of change of current ($A/s$), indicating whether the current in the motor armature circuit is increasing or decreasing. $R$ is the resistance of the motor armature ($\Omega$), $k_e$ is the back EMF constant ($Vs/rad$), and $\omega(T)$ is the angular velocity of the motor's rotor (rad/s).

$$J\left(\frac{d\omega(T)}{dT}\right)_k^{i+1} + B\omega_k^{i+1}(T) = T_q i(T)_k^{i+1} \quad (4)$$

$J$ is the moment of inertia of the motor's rotor ($kgm^2$), $\frac{d\omega(T)}{dT}$ represents the angular acceleration ($rad/s^2$), $B$ is the viscous damping coefficient ($Nms/rad$), $T_q$ is the torque constant ($Nm/A$), and $i(T)$ is the armature current ($A$).

In this setup, the subsystems operate on different kernels: subsystem 1 represents the electrical circuit (a buck converter), while subsystem 2 corresponds to the mechanical equivalent circuit of a DC motor, based on Equations (3) and (4), specifically the DC-Micromotors 1218.SXR model from MATLAB® [37]. An input DC voltage of 15 V is applied, which is sufficient to start the motor within a total simulation time of 1 s. In subsystem 1, the parameters of the buck converter are determined using its governing equations [38], whereas in subsystem 2, the parameter values are calculated based on the equations governing the behavior of the DC motor [39], [40]. In both subsystems, the orange color highlights the parameter initialization and relaxation parameters, which are integrated alongside the behavioral controller. The relaxation values for each subsystem are computed using KVL and KCL. These computed relaxation values are then embedded into their respective subsystems. Finally, the WR method is applied to facilitate seamless data exchange and interaction between the two subsystems.

In Figs. 8(b) and 8(c), the WR algorithm is applied, demonstrating the interaction between the subsystems. In subsystem 1, the internal parameters of the inductor and capacitor are initialized with values $IL1$ and $X$, respectively. The output $X$ from subsystem 1 is used as the input to subsystem 2. A relaxation parameter, $R1$ and $L1$ is introduced in subsystem 1, with calculated values of 12 $\Omega$ and 32 $\mu H$, respectively. Similarly, in subsystem 2, the inductor and the integration of motor speed are initialized with $IL2$ and Speed. The relaxation parameter $R2$ connected to the current controller in subsystem 2, provides feedback to Subsystem 1, represented as $Y$. The convergence error is dependent on the boundary condition of subsystem 1 and subsystem 2. The convergence error is calculated based on [30], which depends
<figure><img src="image_12.png" /><figcaption>FIGURE 8. (a) Simulink model of the electrical-to-mechanical operation, illustrating the splitting of the system into (b) subsystem 1 (buck converter) and (c) subsystem 2 (DC motor).</figcaption></figure>

on the boundary conditions of each subsystem. At each iteration, it is updated and advances to the next iteration until the convergence criteria are met.

Time windowing WR, applied using equation (1), determines the number of time windows, segmenting the simulation into smaller intervals. Following this, the converter and DC motor powers are compared, with the results shown in Fig. 9. To improve both error reduction and efficiency between the converter and the DC motor, the GA optimization method is applied to determine the optimized values of the relaxation parameters, as shown in the Table 2.

In Table 2, the initial relaxation parameter values, calculated using the KVL and KCL methods, are utilized in the GA optimization method. Subsystem 1, connected to the voltage controller, is assigned the parameters $R1$ and $L1$, while subsystem 2, connected to the current

controller is assigned $R2$. The genetic optimization method generates a new set of optimized relaxation parameters ($R1'$, $L1'$, and $R2'$), with the stopping criterion defined by a convergence threshold based on circuit characteristics (e.g., 10% [22], [27]).

After approximately 4 hours of optimization in MATLAB using the GA method, the optimum OF value of 2.2 is achieved, yielding the optimized relaxation parameters: $R1' = 18 \, \Omega$, $L1' = 368 \, \mu H$, and $R2' = 1.28 \, \Omega$. These values are then applied to subsystem 1 and subsystem 2, replacing the initial relaxation parameters. Their effectiveness is validated using the time windowing WR method, as shown in Fig. 9 and Fig. 10.

Fig. 9 illustrates the comparison of relative error between the full system mechanical power and the co-simulation mechanical power, calculated using the formula in equation 2.
<table><caption>**TABLE 2. Optimization of relaxation parameters using GA for electrical-to-mechanical co-simulation.**</caption><thead><tr><th rowspan="2">Specification</th><th colspan="3">Relaxation parameters</th><th rowspan="2">OF</th></tr><tr><th>R1 (Ω)</th><th>L1 (μH)</th><th>R2 (Ω)</th></tr></thead><tbody><tr><td>KVL and KCL<br>(Initial guess)</td><td>12</td><td>32</td><td>1.2</td><td>26.8</td></tr><tr><td>Guess 1</td><td>5</td><td>10</td><td>0.5</td><td>4.8</td></tr><tr><td>Guess 2</td><td>20</td><td>400</td><td>2</td><td>22.3</td></tr><tr><td>Worst</td><td>27</td><td>800</td><td>1.8</td><td>38.4</td></tr><tr><td>Optimum</td><td>18</td><td>368</td><td>1.28</td><td>2.2</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**FIGURE 9.** Comparison of relative mechanical power error between full system and co-simulation using WR, time windowing WR and GA optimized time windowing WR.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**FIGURE 10.** Comparison of power efficiency between full system and co-simulation using WR, time windowing WR and GA optimized time windowing WR.</figcaption></figure>

(using 10log instead of 20log). Initially, the WR method models the interaction between the converter and the DC motor, resulting in an average error of approximately -3.4 dB after 28 iterations, with a CPU elapsed time of 38 s. This error reflects the inherent power mismatch between the two systems under standard conditions. Subsequently, applying the time windowing WR method with six optimal time windows reduces an average error to around -9.6 dB after 42 iterations and a CPU elapsed time of 30 s. This improvement demonstrates that dividing the simulation into smaller

time segments enhances the accuracy of power calculations. Finally, applying the same number of time windows in the WR time windowing with the GA method to adjust the relaxation parameters achieves a significant reduction in average error between the electrical and mechanical powers to approximately -17.6 dB, highlighting the effectiveness of GA-based optimization in fine-tuning system parameters for improved precision and efficiency.

In Fig. 10, the WR method, which divides the system into subsystems and iteratively solves them, stabilizes over time to a much lower power efficiency (57.4 %) compared to the full system (81.4 %) and, therefore, is not satisfactory at all. Time windowing WR improves this by using equation (1) to determine the optimal six time windows, breaking the simulation into smaller segments, but remains also rather unfaithful to the full system simulation, with 74.6 % computed power efficiency. The most advanced method, time windowing WR with GA optimization, quickly and consistently reaches a sensible power efficiency figure by dynamically optimizing relaxation parameters, minimizing error propagation, and ensuring smoother transitions. Importantly, GA optimization in the time windowing WR technique estimates power efficiency to 80.8 %, to be compared with the 81.4 % computed with the single-kernel simulation. Even if these results are specific to the considered scenario, they underscore the effectiveness of combining time segmentation with adaptive optimization to enhance simulation performance.

The proposed optimization method can be applied to enhance the overall efficiency of an electric vehicle (EV) powertrain. By optimizing converter parameters and motor control strategies under varying load and speed conditions, the system can significantly improve energy utilization and extend battery life. Given the dynamic constraints and nonlinear nature of such systems, they are well-suited for heuristic-based optimization approaches.

## VII. CONCLUSION

This research highlights the significant advancements made in electrical-to-mechanical co-simulation through the optimization of the WR time windowing methodology. By addressing the challenges of nonlinear behavior and disturbances inherent in power electronic systems, the study demonstrates the effectiveness of an analytical model to optimize the number of time windows, thus improving simulation accuracy and performance. During the implementation in buck converter co-simulation, a 4 V input voltage was applied, resulting in a 2.4 V output in both simulation and co-simulation modes. It is important to highlight that the full system simulation is solely used for retrospective validation of the co-simulation, with no results being used as inputs or convergence criteria. When using the initial number of time windows, a comparison between WR results and full-system simulations revealed an average error of up to -2.8 dB. This error was then reduced to -16.8 dB after optimizing the number of time windows using the developed analytical model. To further minimize the gap, a GA was employed
to optimize relaxation parameters, ultimately reducing the average error to less than -26.2 dB. To validate the GA optimization, the time windowing WR method was applied to the co-simulation between the electrical (buck converter) and mechanical (DC motor) systems, using an input voltage of 15 V, which was sufficient to start the motor within a total simulation time of 1 s. The WR method resulted in significantly lower power efficiency (57.4%) compared to the full system (81.4%), rendering it insufficient. Time windowing WR improved this by segmenting the simulation, achieving a power efficiency of 74.6%, though it still lagged behind the full system. The most advanced approach, time windowing WR with GA optimization, dynamically adjusted relaxation parameters to achieve a power efficiency of 80.8%, closely approaching the 81.4% observed in the single-kernel simulation. These findings highlighted the effectiveness of combining time segmentation with adaptive optimization to enhance simulation performance.

## ACKNOWLEDGMENT

Special thanks to Prof. P. Besnier (HDR, Directeur de Recherche, CNRS) INSA, Rennes, for his guidance and expertise in contributing to this article.

## REFERENCES

[1] J. G. Kassakian, D. J. Perreault, G. C. Verghese, and M. F. Schlecht, *Principles of Power Electronics*. Cambridge, U.K.: Cambridge Univ. Press, 2023.

[2] A. Bhargava, D. Pommerenke, K. Kam, X. Chang, F. Centola, C.-W. Lam, and R. Steinfeld, "EMI prediction in switched power supplies by full-wave and non-linear circuit co-simulation," in *Proc. IEEE Int. Symp. Electromagn. Compat.*, Aug. 2009, pp. 41–46.

[3] A. M. Gole, S. Filizadeh, and P. L. Wilson, "Inclusion of robustness into design using optimization-enabled transient simulation," *IEEE Trans. Power Del.*, vol. 20, no. 3, pp. 1991–1997, Jul. 2005.

[4] M. Segura, T. Poggi, and R. Barcena, "A generic interface for X-in-the-Loop simulations based on distributed co-simulation protocol," *IEEE Access*, vol. 11, pp. 5578–5595, 2023.

[5] Y. Liu, R. Huang, W. Du, A. Singhal, and Z. Huang, "Highly-scalable transmission and distribution dynamic co-simulation with 10,000+ grid-following and grid-forming inverters," *IEEE Trans. Power Del.*, vol. 39, no. 1, pp. 578–590, Feb. 2024.

[6] S. Sicklinger, V. Belsky, B. E. Engelmann, H. Elmqvist, H. Olsson, R. Wüchner, and K. Bletzinger, "Interface Jacobian-based Co-simulation," *Int. J. Numer. Methods Eng.*, vol. 98, no. 6, pp. 418–444, Mar. 2014.

[7] C. Lohmann, J. Dünnebacke, and S. Turek, "Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equation," *J. Numer. Math.*, vol. 30, no. 3, pp. 173–207, Sep. 2022.

[8] B. Rüth, B. Uekermann, M. Mehl, P. Birken, A. Monge, and H.-J. Bungartz, "Quasi-Newton waveform iteration for partitioned surface-coupled multiphysics applications," *Int. J. Numer. Methods Eng.*, vol. 122, no. 19, pp. 5236–5257, Oct. 2021.

[9] J. A. White, A. Sangiovanni-Vincentelli, F. Odeh, and A. Ruehli, "Wave-form relaxation: Theory and practice," Electron. Res. Lab., Berkeley, CA, USA, Tech. Rep., 1985.

[10] T. Wack, A. Schröder, and G. Deerberg, "Towards the stability of waveform relaxation in distributed process Co-simulation," *Chem. Ingenieur Technik*, vol. 94, no. 10, pp. 1426–1437, Oct. 2022.

[11] M. M. Goulkhah, "Waveform relaxation based hardware-in-the-loop simulation," Ph.D. Dissertation, Dept. of Elect. and Comp. Eng., Univ. Manitoba, Winnipeg, MB, Canada, 2014.

[12] M. A. Mamun, S. Paudyal, and S. Kamalasadan, "Computationally efficient dynamic simulation of integrated transmission and distribution systems," *IEEE Trans. Power Syst.*, vol. 39, no. 3, pp. 4865–4880, May 2024.

[13] T. Zhang, H. Bao, P. Gu, D. Ding, D. H. Werner, and R. Chen, "An arbitrary high-order DGTD method with local time-stepping for nonlinear field-circuit cosimulation," *IEEE Trans. Antennas Propag.*, vol. 70, no. 1, pp. 526–535, Jan. 2022.

[14] H. H. Zhang, P. P. Wang, L. J. Jiang, W. E. I. Sha, M. S. Tong, Y. Liu, W. J. Wu, and G. M. Shi, "Parallel higher order DGTD and FETD for transient electromagnetic-circuital-thermal co-simulation," *IEEE Trans. Microw. Theory Techn.*, vol. 70, no. 6, pp. 2935–2947, Jun. 2022.

[15] F. Plumier, P. Aristidou, C. Geuzaine, and T. Van Cutsem, "Co-simulation of electromagnetic transients and phasor models: A relaxation approach," *IEEE Trans. Power Del.*, vol. 31, no. 5, pp. 2360–2369, Oct. 2016.

[16] Y. Yang, Y. Ge, Z. J. Wang, and Y. Kang, "An automated electro-thermal-mechanical co-simulation methodology based on PSpice-MATLAB-COMSOL for SiC power module design," in *Proc. IEEE Workshop Wide Bandgap Power Devices Appl. Asia*, Aug. 2021, pp. 499–503.

[17] C. Wang, Y. Lian, H. Ling, H. Pan, and Y. Zhang, "Multiphysics field co-simulation research based on electric drive system for a specific type of car," *Automot. Innov.*, vol. 2, no. 3, pp. 223–230, Sep. 2019.

[18] J. Zheng, Z. Zhao, Y. Zeng, S. Ji, and L. Yuan, "An event-driven parallel acceleration real-time simulation for power electronic systems without simulation distortion in circuit partitioning," *IEEE Trans. Power Electron.*, vol. 37, no. 12, pp. 15626–15640, Dec. 2022.

[19] M. Benedikt and E. Drenth, "Relaxing stiff system integration by smoothing techniques for non-iterative co-simulation," in *Proc. IUTAM Symp. Solver-Coupling Co-Simul.*, Darmstadt, Germany, Cham, Switzerland: Springer, Sep. 2019, pp. 1–25.

[20] W. Ahmed, S. Hussain, A. M. Khan, and R. Ali, "Analysis of efficient FPGA based pid controller for DC–DC buck boost converter using hardware co-simulation setup," *Sukkur IBA J. Emerg. Technol.*, vol. 4, no. 2, pp. 33–39, Nov. 2021.

[21] M. K. Al-Nussairi and R. Bayindir, "Co-simulation of fuzzy logic control for a DC–DC buck converter in cascade system," in *Artificial Intelligence and Evolutionary Computations in Engineering Systems*. Cham, Switzerland: Springer, 2020, pp. 561–569.

[22] A. B. Diallo, C. Vollaire, A. Breard, M. Bensetti, and L. Pichon, "Co-simulation of circuit/circuit type solvers for EMC applications using a new relaxation method," in *Proc. Int. Symp. Electromagn. Compat.*, Sep. 2022, pp. 366–371.

[23] R. Acharya, F. A. Mohiyaddin, A. Potocnik, K. De Greve, B. Govoreanu, I. P. Radu, G. Gielen, and F. Catthoor, "Circuit models for the co-simulation of superconducting quantum computing systems," in *Proc. Design, Autom. Test Eur. Conf. Exhib. (DATE)*, Feb. 2021, pp. 968–973.

[24] H. M. Torun, M. Swaminathan, A. Kavungal Davis, and M. L. F. Bellaredj, "A global Bayesian optimization algorithm and its application to integrated system design," *IEEE Trans. Very Large Scale Integr. (VLSI) Syst.*, vol. 26, no. 4, pp. 792–802, Apr. 2018.

[25] R. Wagle, L. N. H. Pham, G. Tricarico, P. Sharma, J. L. Rueda, and F. Gonzalez-Longatt, "Co-simulation-based optimal reactive power control in smart distribution network," *Electr. Eng.*, vol. 106, no. 3, pp. 2391–2405, Jun. 2024.

[26] E. O. Inci, W. Desmet, and J. Croes, "System-level co-simulation of mechatronic systems for design space exploration," KU Leuven, Belgium, Tech. Rep., 2023.

[27] M. M. Alam, M. Koohestani, M. Ramdani, and R. Perdriau, "Circuit-to-Circuit cosimulation for closed-loop electrical systems using waveform relaxation," *IEEE Trans. Power Electron.*, vol. 40, no. 6, pp. 8553–8565, Jun. 2025.

[28] S. G. Mallat, "A theory for multiresolution signal decomposition: The wavelet representation," *IEEE Trans. Pattern Anal. Mach. Intell.*, vol. 11, no. 7, pp. 674–693, Jul. 1989.

[29] J. R. Hartman, "Networked control system co-simulation for co-design: Theory and experiments," Cleveland, OH, USA, Tech. Rep., 2004.

[30] M. M. Alam, M. Koohestani, M. Ramdani, and R. Perdriau, "Optimization using time windowing technique for closed loop circuit co-simulation using wr method," *IEEE Open J. Ind. Appl.*, 2025.

[31] D. Schelle and J. Castorena, "Buck-converter design demystified," *Power Electron. Technol.*, vol. 2006, pp. 1–6, Jun. 2006.
[32] P. Cheng, M. Vasic, O. García, J. Á. Oliver, P. Alou, and J. A. Cobos, "Minimum time control for multiphase buck converter: Analysis and application," *IEEE Trans. Power Electron.*, vol. 29, no. 2, pp. 958-967, Feb. 2014.

[33] Y. Li, H. Hu, and Q. Wang, "Non-dominated sorting genetic algorithm II (NSGA2)-based parameter optimization of the MSMGWB model used in remote infrared sensing prediction for hot combustion gas plume," *Remote Sens.*, vol. 16, no. 17, p. 3116, Aug. 2024.

[34] A. Konak, D. W. Coit, and A. E. Smith, "Multi-objective optimization using genetic algorithms: A tutorial," *Rel. Eng. Syst. Saf.*, vol. 91, no. 9, pp. 992-1007, Sep. 2006.

[35] T. Hamiti, C. Gerada, and M. Rottach, "Weight optimisation of a surface mount permanent magnet synchronous motor using genetic algorithms and a combined electromagnetic-thermal co-simulation environment," in *Proc. IEEE Energy Convers. Congr. Expo.*, Sep. 2011, pp. 1536-1540.

[36] D. F. Cook, C. T. Ragsdale, and R. L. Major, "Combining a neural network with a genetic algorithm for process parameter optimization," *Eng. Appl. Artif. Intell.*, vol. 13, no. 4, pp. 391-396, Aug. 2000.

[37] K. Ghlam and M. Oukli, "Performance analysis of rotary electromagnetic micromotors across different size and weight scales," *J. Européen des Syst. Autom.*, vol. 56, no. 3, pp. 459-463, Jun. 2023.

[38] R. Kaur and N. Kaur, "Mathematical modelling of buck converter," *Int. J. Recent Innov. Trends Comput. Commun.*, vol. 2, no. 5, pp. 1226-1229, 2014.

[39] J. Yao, Z. Jiao, and D. Ma, "Adaptive robust control of DC motors with extended state observer," *IEEE Trans. Ind. Electron.*, vol. 61, no. 7, pp. 3630-3637, Jul. 2014.

[40] W. P. Aung, "Analysis on modeling and simulink of DC motor and its driving system used for wheeled mobile robot," *World Acad. Sci., Eng. Technol.*, vol. 1, no. 8, pp. 1141-1148, Aug. 2007.

<figure><img src="image_15.png" /></figure>

**MD MOKTARUL ALAM** (Student Member, IEEE) received the B.Sc. degree in electrical and electronic engineering in Bangladesh, in 2016, and the master's degree in electronic engineering from International Islamic University, Malaysia, in 2019. He is currently pursuing the Doctor of Philosophy degree with INSA, Rennes.

From 2017 to 2019, he was a Research Assistant and a Teaching Assistant with International Islamic University. He is a Research Assistant with the Faculty of Electronic Engineering, ESEO, Angers, France. He is involved in Electromagnetic Compatibility Research Projects. His main research interests include the domain of propagation, antenna, EMC, and co-simulation. He also has good research experience in the area of propagation, electromagnetic, and antennas. He has published several research papers in national/international journals and conferences.

<figure><img src="image_16.png" /></figure>

**RICHARD PERDRIAU** (Senior Member, IEEE) was born in Angers, France, in 1971. He received the Engineering degree in electronics and computer science from ESEO, Angers, in 1992, the Ph.D. degree in applied sciences from the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, in 2004, and the Habilitation à diriger des recherches" (Accreditation degree) from the Université de Rennes 1, France, in 2012.

From 1992 to 2012, he was an Assistant, then Associate Professor with ESEO in the fields of microelectronics and embedded systems. In 2013, he was promoted to a Full Professor. His research interests include EMC of integrated circuits, mixed-signal hardware description languages, and integrated circuit design. He is an Active Member of IEC SC47A. He served as the Vice-Chair and the Technical Program Co-Chair for EMC Europe 2017. Moreover, he is the Technical Program Chair of EMC Europe 2025. He is the current Secretary of the IEEE France Section.

<figure><img src="image_17.png" /></figure>

**MOHAMMED RAMDANI** (Senior Member, IEEE) was born in Oujda, Morocco. He received the Ph.D. degree in microelectronics from the Université Paul Sabatier, Toulouse, France, in 1989, and the "Habilitation à diriger des recherches" (Accreditation degree) from Université de Rennes 1, France, in 2004.

He is currently a Professor with ESEO, an engineering school located in Angers, France. His research focuses on the electromagnetic compatibility (EMC) of integrated circuits and integrated circuit design. He has contributed extensively to the field, authoring several book chapters on EMC of ICs and publishing numerous peer-reviewed scientific articles.

Prof. Ramdani is an Active Member of the International Steering Committee (ISC) of EMC Europe. He also served as the General Chair for EMC Europe 2017, which took place in Angers.

<figure><img src="image_18.png" /></figure>

**MOHSEN KOOHESTANI** (Senior Member, IEEE) received the joint Ph.D. degree (Hons.) in electromagnetics from the École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, and the Universidade de Lisboa (ULISBOA), Lisbon, Portugal, in 2014, and the Higher Doctorate degree (H.D.R.) from the Université de Rennes, Rennes, France, in 2023.

From 2014 to 2018, he was a Senior Research Fellow with the Institut d'Electronique et des Technologies du numérique (IETR), Université de Rennes 1. Since 2018, he has been an Associate Professor with the École Supérieure d'Électronique de l'Ouest (ESEO), Angers, France, where he was promoted to a Full Professor, in 2024. He is currently an Associate Researcher with IETR, UMR CNRS 6164. He has authored over 80 peer-reviewed scientific articles. His main research interests include the domain of antennas, microwaves, EMC, and biomedical engineering.

Prof. Koohestani is an Official Member of AFNOR, the IEC Standardization Working Group (WG2 and WG9), in the SC47A French Subcommittee on EMC for integrated circuits.

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