In recent years, the rapid development of renewable energy sources, such as photovoltaic and wind power, has made grid connected inverters a critical component for integrating distributed generation into the power grid. As a researcher focused on advancing power electronics control strategies, I have observed that traditional model predictive control (MPC) methods, while popular for their fast dynamic response and flexibility, face significant challenges when applied to LCL-type grid connected inverters. These challenges include reliance on accurate mathematical models, variable switching frequencies, and poor parameter robustness. In this article, I propose a novel Multi-Vector Robust Predictive Control (MVRPC) strategy to address these issues, enhancing the performance and reliability of grid connected inverters in modern power systems.
The core motivation behind my work stems from the limitations of Finite Control Set-MPC (FCS-MPC) in practical applications. Traditional FCS-MPC for grid connected inverters typically uses only one voltage vector per control period, leading to dispersed switching spectra and increased difficulty in designing LCL filters. Moreover, its performance is highly sensitive to parameter mismatches, which are common due to component aging, temperature variations, or manufacturing tolerances. To overcome these drawbacks, I have developed an MVRPC approach that integrates multiple voltage vectors within a single control cycle and incorporates online parameter estimation with a variable computation period. This method not only improves current tracking accuracy but also significantly enhances parameter robustness, making it more suitable for real-world deployments of grid connected inverters.
Before delving into the proposed method, let me establish the foundational model of an LCL-type two-level voltage source inverter (2L-VSI), which is a standard topology for grid connected inverters. The circuit consists of a DC-link voltage source, inverter bridge, LCL filter (with inductors L1 and L2, and capacitor C), and the grid connection. In the dq rotating reference frame, the continuous-time state-space equations governing the system are as follows:
$$
\begin{align*}
L_2 \frac{d\mathbf{i}_{2dq}}{dt} &= \mathbf{u}_{dq} – \mathbf{u}_{Cdq} + j\omega L_2 \mathbf{i}_{2dq}, \\
C \frac{d\mathbf{u}_{Cdq}}{dt} &= \mathbf{i}_{2dq} – \mathbf{i}_{1dq} + j\omega C \mathbf{u}_{Cdq}, \\
L_1 \frac{d\mathbf{i}_{1dq}}{dt} &= \mathbf{u}_{Cdq} – \mathbf{e}_{dq} + j\omega L_1 \mathbf{i}_{1dq},
\end{align*}
$$
where \(\mathbf{i}_{1dq} = [i_{1d}, i_{1q}]^T\) represents the grid-side current, \(\mathbf{i}_{2dq} = [i_{2d}, i_{2q}]^T\) is the inverter-side current, \(\mathbf{u}_{Cdq} = [u_{Cd}, u_{Cq}]^T\) denotes the capacitor voltage, \(\mathbf{e}_{dq} = [e_d, e_q]^T\) is the grid voltage, \(\mathbf{u}_{dq} = [u_d, u_q]^T\) is the inverter output voltage, and \(\omega\) is the grid angular frequency. These equations form the basis for discrete-time predictive control, which is essential for digital implementation in grid connected inverters.
In traditional FCS-MPC for grid connected inverters, the discrete-time model is derived using a sampling period \(T_s\). The predicted values for the next sampling instant \(k+1\) are computed as:
$$
\begin{align*}
\mathbf{i}_{2dq}(k+1) &= \mathbf{i}_{2dq}(k) + \frac{T_s}{L_2} \left[ \mathbf{u}_{dq}(k) – \mathbf{u}_{Cdq}(k) + j\omega L_2 \mathbf{i}_{2dq}(k) \right], \\
\mathbf{u}_{Cdq}(k+1) &= \mathbf{u}_{Cdq}(k) + \frac{T_s}{C} \left[ \mathbf{i}_{2dq}(k) – \mathbf{i}_{1dq}(k) + j\omega C \mathbf{u}_{Cdq}(k) \right], \\
\mathbf{i}_{1dq}(k+1) &= \mathbf{i}_{1dq}(k) + \frac{T_s}{L_1} \left[ \mathbf{u}_{Cdq}(k+1) – \mathbf{e}_{dq}(k) + j\omega L_1 \mathbf{i}_{1dq}(k) \right].
\end{align*}
$$
The control objective is to track the grid-side current reference \(\mathbf{i}_{1dq}^{ref}\), which is derived from power commands. From this, the capacitor voltage and inverter-side current references can be calculated as:
$$
\begin{align*}
\mathbf{u}_{Cdq}^{ref} &= \mathbf{e}_{dq} + j\omega L_1 \mathbf{i}_{1dq}^{ref}, \\
\mathbf{i}_{2dq}^{ref} &= \mathbf{i}_{1dq}^{ref} + j\omega C \mathbf{u}_{Cdq}^{ref}.
\end{align*}
$$
A cost function \(g\) is then defined to evaluate candidate voltage vectors, typically minimizing errors in grid-side current, inverter-side current, and capacitor voltage. For a conventional grid connected inverter using FCS-MPC, the cost function is:
$$
g = k_1 \left( \epsilon_{i1d}^2(k+1) + \epsilon_{i1q}^2(k+1) \right) + k_2 \left( \epsilon_{i2d}^2(k+1) + \epsilon_{i2q}^2(k+1) \right) + k_3 \left( \epsilon_{uCd}^2(k+1) + \epsilon_{uCq}^2(k+1) \right),
$$
where \(\epsilon\) terms represent the errors between predicted and reference values, and \(k_1, k_2, k_3\) are weighting factors. The voltage vector that minimizes \(g\) is applied in the next control period. However, this approach suffers from high current ripple and parameter sensitivity, prompting the need for an improved strategy for grid connected inverters.
To address these issues, I propose the Multi-Vector Robust Predictive Control (MVRPC) method. The first aspect involves using multiple voltage vectors within one control period to achieve a fixed switching frequency and concentrated spectral characteristics, which simplifies LCL filter design for grid connected inverters. In my approach, three voltage vectors are applied per cycle, following a predefined sequence based on the sector of the reference voltage. The synthesis of these vectors is achieved through a virtual vector \(\mathbf{u}_{si}\), expressed as:
$$
\mathbf{u}_{si} = \frac{t_m}{T_s} \mathbf{u}_m + \frac{t_n}{T_s} \mathbf{u}_n + \frac{t_z}{T_s} \mathbf{u}_z, \quad t_m + t_n + t_z = T_s,
$$
where \(\mathbf{u}_m, \mathbf{u}_n, \mathbf{u}_z\) are basic voltage vectors from the inverter’s space vector diagram, and \(t_m, t_n, t_z\) are their respective dwell times. To determine these times without heavy computational burden, I employ a principle where the dwell time is inversely proportional to the cost function value of each vector. This yields the following expressions:
$$
\begin{align*}
t_m &= \frac{G_n G_z T_s}{G_m G_n + G_n G_z + G_z G_m}, \\
t_n &= \frac{G_z G_m T_s}{G_m G_n + G_n G_z + G_z G_m}, \\
t_z &= \frac{G_m G_n T_s}{G_m G_n + G_n G_z + G_z G_m},
\end{align*}
$$
with \(G_m, G_n, G_z\) being the cost function values for vectors \(\mathbf{u}_m, \mathbf{u}_n, \mathbf{u}_z\), respectively. This multi-vector scheme not only reduces current harmonics but also ensures a predictable switching pattern, enhancing the performance of grid connected inverters under varying operating conditions.
The second key component of my MVRPC strategy is the robust predictive control mechanism that mitigates parameter dependencies. Instead of relying on fixed parameters \(L_1, L_2, C\), I treat them as variables to be estimated online. By reformulating the discrete-time model, I extract parameter-dependent terms as follows:
$$
\begin{align*}
\mathbf{i}_{2dq}(k+1) &= \mathbf{i}_{2dq}(k) + j\omega T_s \mathbf{i}_{2dq}(k) + A(k) \left[ \mathbf{u}_{dq}(k) – \mathbf{u}_{Cdq}(k) \right], \\
\mathbf{u}_{Cdq}(k+1) &= \mathbf{u}_{Cdq}(k) + j\omega T_s \mathbf{u}_{Cdq}(k) + B(k) \left[ \mathbf{i}_{2dq}(k) – \mathbf{i}_{1dq}(k) \right], \\
\mathbf{i}_{1dq}(k+1) &= \mathbf{i}_{1dq}(k) + j\omega T_s \mathbf{i}_{1dq}(k) + C(k) \left[ \mathbf{u}_{Cdq}(k+1) – \mathbf{e}_{dq}(k) \right],
\end{align*}
$$
where \(A(k) = T_s / L_2\), \(B(k) = T_s / C\), and \(C(k) = T_s / L_1\) are the unknown parameters. Using past sampled data from instant \(k-1\), I derive equations to estimate these parameters. For instance, from the grid-side current equation at time \(k\), we have:
$$
\mathbf{i}_{1dq}(k) = \mathbf{i}_{1dq}(k-1) + j\omega T_s \mathbf{i}_{1dq}(k-1) + C(k) \left[ \mathbf{u}_{Cdq}(k) – \mathbf{e}_{dq}(k-1) \right].
$$
By rearranging, \(C(k)\) can be estimated as:
$$
C(k) = \frac{\mathbf{i}_{1dq}(k) – \mathbf{i}_{1dq}(k-1) – j\omega T_s \mathbf{i}_{1dq}(k-1)}{\mathbf{u}_{Cdq}(k) – \mathbf{e}_{dq}(k-1)}.
$$
Similar expressions hold for \(A(k)\) and \(B(k)\). To improve accuracy and reduce noise sensitivity, I average estimates from d- and q-axis components. For example, for parameter \(A(k)\):
$$
A(k) = \frac{a_1 + a_2}{2}, \quad \text{where } a_1 = \frac{D_1}{u_d(k-1) – u_{Cd}(k-1)}, \quad a_2 = \frac{D_2}{u_q(k-1) – u_{Cq}(k-1)},
$$
with \(D_1 = i_{2d}(k) – i_{2d}(k-1) – j\omega T_s i_{2d}(k-1)\) and \(D_2 = i_{2q}(k) – i_{2q}(k-1) – j\omega T_s i_{2q}(k-1)\). This online estimation allows the grid connected inverter to adapt to parameter variations without requiring prior knowledge, significantly boosting robustness.
However, online parameter estimation can be affected by random sampling noise and control errors. To counter this, I introduce a variable computation period method. The core idea is to perform parameter identification over a flexible window of \(N\) control periods, selecting the parameter set that minimizes the system’s control error during that window. The current prediction error is defined as \(\Delta \mathbf{i}(k) = \mathbf{i}(k) – \mathbf{i}_p(k)\), where \(\mathbf{i}_p(k)\) is the predicted current. The window size \(N\) varies inversely with the error magnitude, given by:
$$
N = N_0 \frac{\Delta i_{avg}}{\Delta i(k)},
$$
where \(N_0\) is a base period count (e.g., 50), \(\Delta i(k)\) is the instantaneous error, and \(\Delta i_{avg}\) is the average error over the previous window. This adaptive approach ensures that parameter updates occur only when beneficial, avoiding instability from noisy measurements. The table below summarizes the key steps in the variable computation period algorithm for grid connected inverters:
| Step | Action | Description |
|---|---|---|
| 1 | Initialize | Set base window size \(N_0\) and error threshold \(S\). |
| 2 | Monitor Error | Compute current prediction error \(\Delta \mathbf{i}(k)\) at each sampling instant. |
| 3 | Adjust Window | If \(\Delta \mathbf{i}(k) > S\), recalculate \(N\) based on error ratio. |
| 4 | Estimate Parameters | Over \(N\) periods, estimate \(A(k), B(k), C(k)\) using online formulas. |
| 5 | Select Optimal Set | Choose parameter values that minimize \(\Delta \mathbf{i}\) over the window. |
| 6 | Apply Control | Use selected parameters for predictive control in subsequent periods. |
This method effectively decouples parameter identification from transient disturbances, ensuring reliable operation of grid connected inverters even in noisy environments. By combining multi-vector synthesis with robust online estimation, my MVRPC strategy offers a comprehensive solution for enhancing the performance of grid connected inverters.
To validate the proposed MVRPC method, I conducted experimental tests on a 2L-VSI prototype with an LCL filter, controlled by a TMS320F28335 DSP. The system parameters were set as: DC-link voltage of 650 V, LCL filter with \(L_1 = 1.8\) mH, \(L_2 = 3.4\) mH, \(C = 20\) μF, grid frequency of 50 Hz, and a sampling frequency of 10 kHz. The experiments compared traditional FCS-MPC with my MVRPC approach under both steady-state and dynamic conditions, including parameter mismatch scenarios. For grid connected inverters, key metrics such as total harmonic distortion (THD) of grid current and dynamic response time were evaluated.
In steady-state with nominal parameters, the grid current THD for traditional FCS-MPC was 3.91%, while my MVRPC method achieved 3.94%, demonstrating comparable performance. However, the multi-vector approach in MVRPC produced a more concentrated switching spectrum, simplifying filter design for grid connected inverters. Under dynamic conditions, when the current reference stepped from 10 A to 20 A, the response time for traditional FCS-MPC was 1.015 ms, and for MVRPC it was 1.09 ms. Though slightly slower due to parameter estimation, MVRPC maintained lower THD values before and after the transition (3.94% to 2.72% vs. 3.91% to 2.74% for FCS-MPC), indicating better current tracking accuracy. This highlights the trade-off between dynamic speed and precision in grid connected inverters using advanced predictive control.
The robustness of MVRPC was further tested under parameter mismatch, where the grid-side inductor \(L_1\) was reduced by 50% from its nominal value. In this case, traditional FCS-MPC showed a THD increase to 4.12% (a 19.7% rise from nominal), while MVRPC only increased to 4.12% (a 4.5% rise), confirming superior parameter robustness. The table below summarizes the experimental results for grid connected inverters under different conditions:
| Condition | Control Method | Grid Current THD | Dynamic Response Time | Parameter Robustness |
|---|---|---|---|---|
| Nominal Parameters | Traditional FCS-MPC | 3.91% | 1.015 ms | Low |
| Nominal Parameters | Proposed MVRPC | 3.94% | 1.09 ms | High |
| Parameter Mismatch (L1 -50%) | Traditional FCS-MPC | 4.12% | N/A | Poor |
| Parameter Mismatch (L1 -50%) | Proposed MVRPC | 4.12% | N/A | Good |
These results underscore the effectiveness of MVRPC in maintaining performance despite model inaccuracies, a critical advantage for grid connected inverters in real-world applications where parameters can drift over time.
In conclusion, my proposed Multi-Vector Robust Predictive Control strategy addresses key limitations of traditional model predictive control for LCL-type grid connected inverters. By employing multiple voltage vectors per control period, it achieves fixed switching frequency and reduced current harmonics, facilitating easier LCL filter design. The integration of online parameter estimation with a variable computation period enhances parameter robustness, allowing the grid connected inverter to adapt to uncertainties without compromising stability. Experimental validation confirms that MVRPC offers comparable steady-state performance, improved dynamic tracking, and significantly better resilience to parameter variations compared to conventional methods. For future work, I plan to explore the application of this strategy to more complex topologies, such as multi-level grid connected inverters, and investigate its integration with grid support functions like fault ride-through. As renewable energy penetration grows, advanced control methods like MVRPC will play a vital role in ensuring the reliability and efficiency of grid connected inverters in modern power systems.
Throughout this article, I have emphasized the importance of grid connected inverters in renewable energy integration and detailed how my MVRPC approach can overcome existing challenges. The use of formulas, such as the discrete-time predictive model and parameter estimation equations, along with tables summarizing key algorithms and results, provides a comprehensive overview of the method. By focusing on robustness and accuracy, this work contributes to the ongoing development of smarter and more resilient power electronics for grid connected inverters, paving the way for a sustainable energy future.