In my recent research, I have focused on enhancing the voltage support control for grid-forming inverters in renewable energy systems. The increasing integration of renewable energy sources into microgrids has placed greater demands on the performance of various types of solar inverters. These inverters serve as critical interfaces, converting direct current from solar panels into high-quality alternating current. Among the diverse types of solar inverters, string inverters, microinverters, and hybrid inverters are commonly deployed. Hybrid inverters, such as the 24KW model with battery storage shown in the image below, are particularly interesting for grid-forming applications because they can operate both in grid-tied and islanded modes, providing essential voltage and frequency support.
The effectiveness of these types of solar inverters in a microgrid heavily relies on their control strategies, particularly for grid-forming voltage support. Model Predictive Control (MPC) has emerged as a popular method due to its fast dynamic response and multi-objective optimization capabilities. However, when I examined traditional MPC applied to various types of solar inverters, I discovered a significant vulnerability: its performance is highly sensitive to the accuracy of the inverter’s system parameters, such as filter inductance and capacitance. Parameter mismatches between the model and the actual hardware can lead to substantial voltage prediction errors, degrading the voltage support quality and potentially threatening system stability. To address this critical issue for all types of solar inverters, I proposed a robust predictive control (RPC) strategy. My approach eliminates the dependency on precise model parameters by using an ultra-local model that is updated in real-time, greatly improving the robustness and accuracy of voltage control for grid-forming types of solar inverters.
My analysis begins with a detailed look at the topology and modeling of a typical two-level inverter, which serves as the foundation for many types of solar inverters.
System Topology and Mathematical Model
In my study, I considered a grid-forming inverter with an LC filter. The inverter switches can be controlled to produce different voltage levels. The switching function \( S_j \) (for phase j = a, b, c) determines the state of the switches:
\[
S_j =
\begin{cases}
1, & \text{when S}_{jt} \text{ is on and S}_{jd} \text{ is off}, \\
0, & \text{when S}_{jt} \text{ is off and S}_{jd} \text{ is on}.
\end{cases}
\]
The mathematical model of the inverter in the continuous-time domain is:
\[
L \frac{di_L}{dt} = V_i – V_C,
\]
\[
C \frac{dV_C}{dt} = i_L – i_R.
\]
Where \( L \) is the filter inductance, \( C \) is the filter capacitance, \( V_i \) is the inverter output voltage, \( i_L \) is the inductor current, \( V_C \) is the capacitor voltage (grid-forming voltage), and \( i_R \) is the load current.
Traditional Model Predictive Control (MPC)
Traditional MPC uses a discrete model to predict the future capacitor voltage. Using the forward Euler method for discretization:
\[
i_L(k+1) = i_L(k) + \frac{T_s}{L} (V_i(k+1) – V_C(k)),
\]
\[
V_C(k+1) = V_C(k) + \frac{T_s}{C} (i_L(k+1) – i_R(k)).
\]
From these, I derived the predicted voltage:
\[
V_C(k+1) = V_C(k) + \frac{T_s^2}{LC} (V_i(k+1) – V_C(k)) + \frac{T_s}{C} (i_L(k) – i_R(k)).
\]
A cost function is then used to select the optimal voltage vector from the eight possible voltage vectors:
\[
g = (V_{Cref}(k+1) – V_C(k+1))^2.
\]
This approach works well when the model parameters \( L_0 \) and \( C_0 \), used in the controller, perfectly match the actual physical parameters \( L \) and \( C \). However, parameter mismatches, defined by the parameter error \( P_e = L_0/L = C_0/C \), lead to prediction errors. My analysis shows that when \( P_e \) deviates from 1, the voltage prediction error \( V_{Cerr} \) increases significantly. This is a major drawback for controlling various types of solar inverters in the field, where parameters can drift due to temperature, aging, or manufacturing tolerances.
The following table summarizes the voltage prediction error \( V_{Cerr} \) under different parameter error \( P_e \) for a given reference voltage. This highlights the sensitivity of traditional MPC implemented on standard types of solar inverters.
| Parameter Error (P_e) | Voltage Reference (V) | Prediction Error V_{Cerr} (V) |
|---|---|---|
| 0.5 | 100 | 8.2 |
| 0.8 | 100 | 3.1 |
| 1.0 | 100 | 0.0 |
| 1.2 | 100 | 2.8 |
| 1.5 | 100 | 6.5 |
| 2.0 | 100 | 12.4 |
Proposed Robust Predictive Control (RPC)
To overcome the parameter sensitivity issue for all types of solar inverters, I proposed a robust predictive control (RPC) method based on an ultra-local model. I formulated the voltage dynamics as:
\[
\frac{dV_C}{dt} = \alpha u_i + f.
\]
Where \( \alpha \) is an ultra-local model gain, \( u_i = V_i – V_C \) is the input, and \( f \) is the total disturbance that accounts for all unknown dynamics, parameter variations, and nonlinearities. The terms are defined as:
\[
\alpha = T_s / LC,
\]
\[
u_i = V_i – V_C,
\]
\[
f = (i_L – i_R)/C + f_0.
\]
The discrete form using the forward Euler method is:
\[
V_C(k+1) = V_C(k) + T_s (\alpha u_i(k) + f(k)).
\]
The key contribution of my work is the real-time estimation of \( \alpha \) and \( f \) without requiring knowledge of \( L \) and \( C \). By relating the model at different time steps:
\[
\Delta V_C(k) = V_C(k) – V_C(k-1) = T_s (\alpha u_i(k-1) + f(k-1)),
\]
\[
\Delta V_C(k-1) = T_s (\alpha u_i(k-2) + f(k-2)).
\]
Assuming the disturbance varies slowly, \( f(k-1) \approx f(k-2) \), I derived:
\[
\alpha(k) = \frac{\Delta V_C(k) – \Delta V_C(k-1)}{T_s (u_i(k-1) – u_i(k-2))}.
\]
To prevent calculation failure when \( u_i(k-1) \approx u_i(k-2) \), I improved the formula with a sign function:
\[
\alpha(k) = \frac{\Delta V_C(k) – \Delta V_C(k-1)}{T_s \Delta u_i(k-1) + k_1 \text{sign}(\Delta u_i(k-1))} + \frac{k_1 \text{sign}(\Delta u_i(k-1)) + \hat{\alpha}(k-1)}{T_s \Delta u_i(k-1) + k_1 \text{sign}(\Delta u_i(k-1))}.
\]
High-frequency noise in \( \alpha(k) \) is filtered using a low-pass filter (LPF). The filtered gain, \( \hat{\alpha}(k) \), and the disturbance, \( \hat{f}(k) \), are updated in real-time. This makes the system robust to parameter mismatches.
\[
\hat{\alpha}(k) = B \alpha(k) + (1-B) \hat{\alpha}(k-1),
\]
\[
\hat{f}(k) = C f(k) + (1-C) \hat{f}(k-1).
\]
Finally, using the deadbeat control principle, I set the predicted voltage to equal the reference voltage \( V_{Cref}(k+1) \). The required input \( u_{ref}^i(k) \) is:
\[
u_{ref}^i(k) = \frac{V_{Cref}(k+1) – V_C(k)}{\hat{\alpha} T_s} – \frac{\hat{f}(k)}{\hat{\alpha}}.
\]
The reference inverter voltage for the space vector modulator is then:
\[
V_{ref}^i(k) = u_{ref}^i(k) + V_C(k).
\]
My proposed RPC method ensures that the control performance remains consistent regardless of parameter errors for different types of solar inverters.
Experimental Validation
I validated my proposed RPC method on a three-phase inverter experimental platform. I compared traditional MPC, traditional Space Vector MPC (SV-MPC), and my proposed RPC method. The inverter parameters used are:
| Parameter | Value |
|---|---|
| DC Link Voltage Vdc | 300 V |
| Reference Voltage VCref | 50 V, 100 V |
| Filter Inductance L | 1.7 mH |
| Filter Capacitance C | 15 μF |
| Load Resistance R | 10 Ω |
| Control Period Ts | 50 μs |
Steady-State Performance
In steady-state, I compared the Total Harmonic Distortion (THD) of the output voltage. Traditional MPC had a THD of 5.63%. Traditional SV-MPC improved this to 1.82%. My proposed RPC achieved an even lower THD of 1.32%, demonstrating superior steady-state performance compared to other control methods for types of solar inverters.
The following table summarizes the steady-state performance comparison for the different types of solar inverters control strategies.
| Control Method | THD (%) |
|---|---|
| Traditional MPC | 5.63 |
| Traditional SV-MPC | 1.82 |
| Proposed RPC | 1.32 |
Dynamic Performance
For dynamic performance, I tested the response when the voltage reference stepped from 50 V to 100 V. Traditional MPC was fast with a response time of 0.12 ms, but had a large voltage error. Traditional SV-MPC was slower at 0.36 ms. My proposed RPC achieved a fast response time of 0.2 ms while maintaining a low voltage error, balancing speed and accuracy effectively for grid-forming types of solar inverters.
| Control Method | Response Time (ms) | Voltage Error |
|---|---|---|
| Traditional MPC | 0.12 | High |
| Traditional SV-MPC | 0.36 | Medium |
| Proposed RPC | 0.20 | Low |
Robustness Against Parameter Mismatch
This is the most critical test for my proposed method. I introduced a parameter mismatch where the model parameters \( L_0 \) and \( C_0 \) differed from the actual values \( L \) and \( C \). Traditional MPC and SV-MPC both exhibited significant voltage errors due to the mismatch. In contrast, my proposed RPC method maintained the desired voltage with high accuracy, completely unaffected by the parameter mismatches. This is because RPC estimates the ultra-local model parameters in real-time, making it inherently robust. This result is crucial for the reliable operation of all types of solar inverters in the field.
The following table shows the voltage prediction error for different control methods under a 50% parameter mismatch (L0/L = 1.5, C0/C = 1.5).
| Control Method | Parameter Error (P_e) | Voltage Prediction Error V_{Cerr} (V) |
|---|---|---|
| Traditional MPC | 1.5 | 7.8 |
| Traditional SV-MPC | 1.5 | 4.1 |
| Proposed RPC | 1.5 | 0.3 |
Implementation Process of the Proposed RPC
The implementation of my robust predictive control for types of solar inverters involves the following steps:
1. Sample capacitor voltages V_C(k), V_C(k-1), and V_C(k-2).
2. Calculate the ultra-local model gain \(\alpha(k)\) and disturbance \(f(k)\) using the formulas developed above.
3. Filter \(\alpha(k)\) and \(f(k)\) using a low-pass filter to obtain \(\hat{\alpha}(k)\) and \(\hat{f}(k)\).
4. Predict the k+1 capacitor voltage using the ultra-local model.
5. Calculate the required reference voltage V_ref^i(k) using the deadbeat principle, which eliminates the influence of system parameters.
6. Apply the reference voltage via space vector modulation (SVM) to generate the switching signals for the inverter.
Conclusion
In my work, I have successfully developed and validated a robust predictive control strategy for grid-forming voltage support in various types of solar inverters. My method, based on a real-time updated ultra-local model, completely eliminates the detrimental effects of system parameter mismatches that plague traditional model predictive control. Experimental results confirm that my proposed RPC method not only achieves lower steady-state THD (1.32% compared to 5.63% for traditional MPC) but also provides a good balance between dynamic response and voltage accuracy. Most importantly, it demonstrates superior robustness, maintaining excellent performance even when the model parameters are incorrect. This makes my proposed RPC method a highly reliable and effective solution for enhancing the voltage support capability of modern types of solar inverters used in complex microgrid environments. The ability to operate independently of precise parameter knowledge is a significant advantage for practical deployment, where component values can vary. This work contributes to making grid-forming inverters more resilient and efficient, which is essential for the widespread adoption of renewable energy systems.