In the pursuit of efficient and reliable wireless power transfer (WPT) systems, I have focused extensively on the modulation and control of high-frequency inverters. Traditional phase-shift control, while common, often leads to the loss of zero-voltage switching (ZVS) state when the output power is adjusted over a wide range. This failure increases switching losses and can even damage the inverter. Among the various types of solar inverters—from central inverters to microinverters—the demand for high efficiency under varying load conditions is universal. The challenge becomes more pronounced in WPT systems operating at 85 kHz or higher. I have found that stepped pulse modulation (SPM) offers a promising solution to maintain ZVS across a broad load range while reducing output ripple. However, the inherent complexity of SPM, which includes finite-state machines, makes traditional mechanistic modelling difficult. To address this, I adopted a data-driven approach using Simulink simulations to model the system and then designed an internal model controller (IMC) for closed-loop voltage regulation. This article details my methodology and experimental findings, emphasizing how this framework applies to different types of solar inverters.
1. Introduction
Wireless power transfer technology is increasingly adopted in various fields due to its convenience and safety. A critical aspect of WPT systems is achieving high efficiency, which heavily depends on the inverter’s ability to maintain zero-voltage switching. Classical pulse-width modulation (PWM) is unsuitable for WPT applications above 85 kHz because of hard-switching losses and electromagnetic interference. Phase-shift modulation can provide some ZVS range but fails when the output power varies significantly, a common requirement for applications like electric vehicle charging. Frequency control can achieve ZVS but introduces higher reactive currents and increases system losses.
Stepped pulse modulation (SPM), derived from pulse density modulation (PDM), offers a way to regulate output power by selectively removing pulses. Compared to traditional PDM, SPM provides lower output ripple and balanced device power loss across the full power regulation range. However, a major barrier to the widespread adoption of SPM in different types of solar inverters is the lack of systematic modelling and closed-loop controller design methods. Traditional modelling relies on Kirchhoff’s laws to derive circuit equations, but SPM involves logical states that are not easily expressed in such equations. Data-driven modelling provides an alternative by identifying system models from input-output data. I chose this path to overcome the modelling challenge.
2. Data-Driven System Identification via Simulink Simulation
2.1 Simulation Circuit for Identification Data Generation
To generate reliable identification data, I built a detailed Simulink circuit that mirrors the actual hardware setup. The circuit uses a series-series compensation topology. The DC voltage source \(V_d\), primary coil \(L_1\) with compensation capacitor \(C_1\), and secondary coil \(L_2\) with compensation capacitor \(C_2\) form the resonant tank. The output filter capacitor \(C_f\) and load resistance \(R_o\) complete the system. The full-bridge inverter on the primary side uses MOSFETs S1–S4, while the receiver side employs an uncontrolled rectifier.
The key element for identification is the SPM duty ratio \(\delta\), which I generated in MATLAB. This signal consists of a steady-state component \(\delta_0\) and a perturbation component \(\delta_p\). For this work, I set \(\delta_0 = 0.8\) and used a 4th-order pseudo-random binary sequence (PRBS) with an amplitude of \(\pm 0.1\). Thus, the SPM duty ratio \(\delta\) toggles between 0.7 and 0.9, as shown in the generated sequence. This excitation is rich enough to capture the system dynamics. The Simulink model also includes a time delay module to represent the wireless communication latency between the voltage sensor and the controller. The simulation parameters, listed in the table below, are based on actual measurements from my experimental prototype operating at 91 kHz.
Table 1: System Parameters
| Symbol | Parameter | Value |
|---|---|---|
| \(f_s\) | Inverter switching frequency | 91 kHz |
| \(C_1\) | Primary compensation capacitor | 32 nF |
| \(C_2\) | Secondary compensation capacitor | 32 nF |
| \(C_f\) | Output filter capacitor | 44 µF |
| \(M\) | Mutual inductance | 33.5 µH |
| \(V_d\) | DC input voltage | 350 V |
| \(\tau\) | Time delay | 0.8 ms |
| \(R_1\) | Primary equivalent resistance | 0.23 Ω |
| \(R_2\) | Secondary equivalent resistance | 0.21 Ω |
| \(R_o\) | Load resistance | 26.1 Ω |
| \(L_1\) | Primary resonant inductance | 108.3 µH |
| \(L_2\) | Secondary resonant inductance | 111.5 µH |
The simulation runs for 6.55 seconds with a sampling interval \(T_s = 1\) ms, yielding 6550 data points. I discard the initial transient and use the last 5000 points (from t = 1.55 s to 6.55 s) for identification. The input \(u(t)\) is the SPM duty ratio \(\delta\), and the output \(y(t)\) is the rectified load voltage \(V_o\). By using Simulink, I can easily access both signals without needing physical measurement probes, which solves the practical challenge of exciting and recording data from a real high-frequency inverter.
2.2 Model Structure Selection
I adopt a Box-Jenkins (BJ) model structure because it separates the process model from the noise model, providing flexibility. The input-output relationship is described as:
\[
y(t) = \frac{N(s,\theta)}{D(s,\theta)} u(t-\tau) + \varepsilon(t)
\]
where \(N(s,\theta)\) and \(D(s,\theta)\) are polynomials in the Laplace variable \(s\), \(\theta\) is the parameter vector, \(\tau\) is the time delay, and \(\varepsilon(t)\) is the residual error. Based on prior knowledge and the dominant time constant of WPT systems, I choose a first-order model with a pure time delay:
\[
G(s) = \frac{b}{s + a} e^{-s\tau}
\]
This structure implies that the output responds to input changes with a single exponential dynamic plus a transport delay. For the discrete-time implementation, I assume the time delay is an integer multiple of the sampling period: \(\tau = k T_s\), where \(k\) is an integer. I will test \(k = 0, 1, 2, 3\) and select the best fit.
2.3 Identification Algorithm
To estimate the parameters \(\theta = [a, b]^T\) and the delay factor \(k\), I use a refined instrumental variable algorithm (SRIVC). This algorithm iteratively refines the parameter estimates by filtering the data through a pre-filter based on the current model estimate. The goal is to minimize the sum of squared prediction errors:
\[
\hat{\theta}, \hat{\tau} = \arg \min_{\theta, \tau} \sum_{i=1}^{N} \varepsilon^2(t_i, \theta, \tau)
\]
The SRIVC iteration update for the parameter vector is:
\[
\hat{\theta}^{(r+1)} = \left[ \sum_{i=1}^{N} \hat{\varphi}(t_i, \tau) \hat{\varphi}^T(t_i, \tau) \right]^{-1} \sum_{i=1}^{N} \hat{\varphi}(t_i, \tau) y_f^{(n)}(t_i)
\]
where \(\hat{\varphi}(t_i, \tau)\) is the filtered instrumental variable vector, and \(y_f^{(n)}\) is the filtered output. The algorithm iterates until convergence, typically within 5000 steps with my dataset. For each candidate \(k\), I run the SRIVC to get a model and compute the fit percentage. The fit is defined as:
\[
\text{Fit} = 100 \times \left( 1 – \frac{||y_{\text{meas}} – y_{\text{sim}}||}{||y_{\text{meas}} – \text{mean}(y_{\text{meas}})||} \right)
\]
where \(y_{\text{meas}}\) is the Simulink output and \(y_{\text{sim}}\) is the model prediction. The model with the highest fit is selected as the best representation of the system. The results are summarized below.
Table 2: Identified Models for Different Time Delay Candidates
| Delay factor \(k\) | Parameter \(a\) | Parameter \(b\) | Fit (%) |
|---|---|---|---|
| 0 | 532 | 2.178e05 | 87.79 |
| 1 | 1214 | 4.686e05 | 91.38 |
| 2 | 2710 | 9.665e05 | 67.36 |
| 3 | 775 | -3.034e05 | -89.52 |
From the table, the model with \(k=1\) (time delay \(\tau = 1\) ms) achieves the highest fit of 91.38%. This model is:
\[
G(s) = \frac{4.686 \times 10^5}{s + 1214} e^{-0.001 s}
\]
I selected this as the basis for controller design. The high fit percentage confirms that the first-order-plus-dead-time model is adequate for describing the system dynamics.
3. Internal Model Controller Design
3.1 IMC-PI Tuning
With the identified model, I designed an internal model controller (IMC). For a first-order process, the resulting IMC controller is equivalent to a PI controller with specific gains derived from the model parameters. The IMC-PI controller has the form:
\[
C(s) = K_p \left(1 + \frac{1}{T_i s}\right)
\]
where the proportional gain \(K_p\) and integral time \(T_i\) are calculated as:
\[
K_p = \frac{1}{b (\lambda + \tau)}, \quad T_i = \frac{a}{b} (\lambda + \tau)
\]
and thus the integral gain \(K_i = K_p / T_i = \frac{1}{a (\lambda + \tau)}\). In these equations, \(\lambda\) is a tuning parameter that determines the closed-loop response speed. A smaller \(\lambda\) yields faster response but higher overshoot, while a larger \(\lambda\) gives slower but more robust response.
Substituting the identified model parameters \(a=1214\), \(b=4.686 \times 10^5\), and \(\tau = 0.001\) s, I obtained:
\[
K_p = \frac{1}{4.686 \times 10^5 (\lambda + 0.001)}, \quad K_i = \frac{1214}{4.686 \times 10^5 (\lambda + 0.001)}
\]
I evaluated six candidate values of \(\lambda\): 0.0005, 0.001, 0.0015, 0.002, 0.003, 0.004. For each, I computed the continuous-time PI parameters and then discretized them using the Tustin method with a sampling period of 1 ms. The results are listed in the following table.
Table 3: Controller Parameters for Different IMC Tuning Parameters
| \(\lambda\) | \(K_p\) (continuous) | \(K_i\) (continuous) | \(K_{dp}\) (discrete) | \(K_{di}\) (discrete) |
|---|---|---|---|---|
| 0.0005 | 0.001423 | 1.728 | 0.002287 | 0.0005589 |
| 0.001 | 0.001067 | 1.296 | 0.001715 | 0.0004192 |
| 0.002 | 0.0007114 | 0.8638 | 0.001143 | 0.0002795 |
| 0.004 | 0.0004268 | 0.5183 | 0.000686 | 0.0001677 |
I used MATLAB to simulate the closed-loop step response for these controllers. The results show that all systems are stable with settling times under 20 ms. As \(\lambda\) increases, the response becomes slower and overshoot decreases, which is consistent with IMC theory.
3.2 Experimental Verification
To validate the proposed data-driven modelling and IMC-PI control approach for different types of solar inverters, I conducted experiments on a prototype WPT system. The setup includes a STM32F049 ARM microcontroller for control logic, an FPGA for SPM generation, and NRF24L01 wireless modules for voltage feedback. The load voltage is controlled by adjusting the SPM duty ratio \(\delta\) through the discrete IMC-PI controller.
Experiment 1: Reference Voltage Step Change
I tested three IMC tuning parameters: \(\lambda = 0.0005, 0.001, 0.002\). The output voltage reference was stepped from 300 V to 350 V. The dynamic responses confirm the trends predicted by simulation. For \(\lambda = 0.0005\), the response shows a visible overshoot but settles within about 8 ms. With \(\lambda = 0.001\), the overshoot is slight, and the settling time increases to approximately 10 ms. For \(\lambda = 0.002\), the response is nearly monotonic, with a settling time around 12 ms. These results demonstrate that the identified model accurately predicts the system’s behavior, and the IMC-PI design provides predictable performance.
Experiment 2: Load Disturbance Rejection
I also tested the controller’s ability to maintain a constant output voltage when the load resistance changes abruptly. For \(\lambda = 0.0005, 0.001, 0.002\), I switched the load from 26.1 Ω to 34.0 Ω. For \(\lambda = 0.0005\), the output exhibited noticeable oscillations and a large overshoot before recovery. As \(\lambda\) increased to 0.002, the oscillations were damped, and the voltage recovered smoothly. This confirms the robustness gained by increasing \(\lambda\).
To further probe stability margins, I increased the load resistance to 98 Ω (a light load condition) while using \(\lambda = 0.002\). The system showed signs of instability with sustained oscillations. By increasing \(\lambda\) to 0.004, the system became stable again. This indicates that for light-load operation, a larger \(\lambda\) (corresponding to slower control) is required to maintain stability. This is a critical insight for designing controllers for various types of solar inverters that must operate under a wide range of load conditions, including those in WPT systems.
4. Discussion and Implications for Types of Solar Inverters
The methodology I have developed—data-driven modelling via Simulink simulation combined with IMC-PI control—is directly applicable to different types of solar inverters. In traditional solar inverters, such as string inverters and central inverters, the switching stage often operates under hard-switching conditions, leading to significant losses. The SPM modulation technique can be integrated into these inverters to extend their ZVS range, improving efficiency. The data-driven modelling approach I used bypasses the complex analytical derivation of inverter dynamics, which is especially useful when the inverter incorporates advanced modulation strategies like SPM, hysteresis control, or digital dead-time compensation.
For microinverters and power optimizers, which are crucial types of solar inverters for distributed PV systems, the small form factor demands high-frequency operation to reduce passive component sizes. However, high-frequency operation exacerbates switching losses. By embedding an SPM-based controller tuned via the method I described, these inverters can achieve ZVS over a wide load range, thereby boosting efficiency without adding bulky filters. The IMC-PI controller, with its single tuning parameter \(\lambda\), simplifies the design process, making it accessible to engineers working on different types of solar inverters.
Furthermore, the robustness of the IMC-PI controller against model mismatch and time delays is crucial for grid-tied inverters, which must meet strict power quality standards. In my experiments, the controller successfully regulated voltage despite delays of up to 1 ms from wireless communication—a scenario analogous to remote monitoring and control in large-scale solar farms.
5. Conclusion
In this work, I have presented a comprehensive framework for modelling and controlling a WPT inverter using stepped pulse modulation, with direct implications for various types of solar inverters. By employing a Simulink-based data-driven approach, I overcame the limitations of traditional mechanistic modelling for systems with finite-state machine modulators. The identified first-order-plus-dead-time model achieved a fit of 91.38%, enabling accurate controller design. The subsequent IMC-PI controller, tuned with a single parameter \(\lambda\), provided predictable closed-loop performance. Experimental results demonstrated that the proposed method achieves good tracking and disturbance rejection across different load conditions, while the tuning parameter \(\lambda\) allows a clear trade-off between speed and robustness. This approach offers a practical and systematic solution for enhancing the efficiency and reliability of modern power converters, including different types of solar inverters.