With the increasing integration of high-penetration distributed photovoltaic systems into low-voltage distribution networks, overvoltage issues at grid connection points have become a critical challenge. This article explores an overvoltage suppression strategy for two-stage single-phase neutral-point-clamped (NPC) three-level solar inverters using finite control set model predictive control (FCS-MPC). The approach leverages an improved particle swarm optimization (PSO) algorithm to track the maximum power point (MPP) of photovoltaic arrays, serving as the active power reference. By analyzing the root causes of overvoltage in distribution networks with solar inverters and applying reactive power voltage regulation theory, the reactive power reference for the solar inverter is derived. Through a designed objective function, the solar inverter outputs time-varying active and reactive power, addressing overvoltage at the source. Simulations in Simulink validate the strategy, demonstrating reduced steady-state oscillations and tracking errors compared to traditional methods, along with effective voltage regulation within safe limits.
The proliferation of solar inverters in low-voltage grids has led to complex power flow dynamics, increasing the risk of voltage violations. Solar inverters, as key components in photovoltaic systems, can actively participate in grid voltage control. When photovoltaic penetration exceeds 30%, solar inverters can replace traditional voltage regulation devices like capacitors. This study focuses on enhancing the performance of solar inverters through advanced control strategies, specifically addressing overvoltage by coordinating active and reactive power outputs. The two-stage single-phase NPC solar inverter structure is employed, combining Boost converter-based MPPT control with FCS-MPC for grid-side management. The improved PSO algorithm optimizes MPPT tracking, while FCS-MPC enables precise power control, ensuring grid stability.
Overvoltage in distribution networks with high solar inverter penetration primarily occurs during periods of high irradiance and low load demand. This results in reverse power flow, elevating voltages beyond permissible limits. The reactive power capability of solar inverters is harnessed to mitigate this issue. The maximum reactive power capacity of a solar inverter is given by:
$$Q_{max}^{pv} = \pm \sqrt{S_{inv}^2 – P_{pv}^2}$$
where \(S_{inv}\) is the solar inverter’s rated capacity, and \(P_{pv}\) is the active power from the photovoltaic array. For a distribution line with impedance \(r_i + jx_i\) and node voltages \(V_i\) and \(V_{i+1}\), the voltage drop \(\Delta V\) is expressed as:
$$\Delta V = V_{i+1} – V_i = -\frac{P_i r_i + Q_i x_i}{V_1}$$
When overvoltage occurs (\(\Delta V > \Delta V_{limit}\)), the required reactive power reference \(Q_{PV,ref}\) for each solar inverter in a system with \(n\) inverters is derived as:
$$Q_{PV,ref} = A – B – \sigma P_{PV}$$
where \(A = \frac{Q_T}{n}\), \(B = \frac{\Delta V \times V_1 + P_T r_i}{n x_i}\), \(P_{PV} = \frac{P_i}{n}\), and \(\sigma = \frac{r_i}{x_i}\). This formulation allows solar inverters to inject reactive power dynamically, counteracting overvoltage.
For MPPT control, traditional methods like perturb and observe (P&O) and incremental conductance (InC) suffer from fixed step sizes, leading to oscillations and slow tracking. The improved PSO algorithm addresses these limitations by adaptively adjusting parameters. The velocity and position update equations are:
$$v_{i}^{k+1} = \omega v_{i}^{k} + c_1 r_1 (P_{ibest}^{k} – x_{i}^{k}) + c_2 r_2 (P_{gbest}^{k} – x_{i}^{k})$$
$$x_{i}^{k+1} = x_{i}^{k} + v_{i}^{k+1}$$
Here, \(\omega\) is the inertia weight, \(c_1\) and \(c_2\) are learning factors, and \(r_1\), \(r_2\) are random numbers in (0,1). Adaptive adjustments are made as:
$$\omega = \omega_{max} \times \exp\left(\log\left(\frac{\omega_{min}}{\omega_{max}}\right) \times \rho\right)$$
$$c_1 = c_{1max} – (c_{1max} – c_{1min}) \times \rho$$
$$c_2 = c_{2max} – (c_{2max} – c_{2min}) \times \rho$$
where \(\rho = \frac{k}{k_{max}}\) is the system state factor. The algorithm includes restart functionality based on power changes (\(\Delta P > 0.1\)), enhancing tracking under varying environmental conditions. Comparative analysis shows the improved PSO algorithm’s superiority in reducing steady-state error and oscillations, as summarized in Table 1.
| Metric | P&O Method | InC Method | Improved PSO |
|---|---|---|---|
| Range Error (W) | 30/26/33 | 19/18/22 | 8/12/12 |
| Average Power (W) | 4785/5997/7192.5 | 4781.5/5998/7195 | 4792/6001/7200 |
| Steady-State Error (%) | 0.187/0.100/0.104 | 0.261/0.083/0.069 | 0.041/0.033/0 |
The FCS-MPC strategy for the solar inverter involves modeling the single-phase NPC three-level structure. The inverter outputs three states: P, O, and N, corresponding to switch configurations. Using virtual voltage vectors in \(\alpha\beta\) coordinates, the output states are defined in Table 2.
| State | Switch State [Sa1 Sa2 Sb1 Sb2] | Output Level | \(u_{\alpha}\) | \(u_{\beta}\) |
|---|---|---|---|---|
| 0 | [1 1 1 1] | 0 | 0 | \(U_{dc}\) |
| 1 | [1 1 0 1] | \(\frac{1}{2}U_{dc}\) | \(\frac{1}{2}U_{dc}\) | \(\frac{\sqrt{3}}{2}U_{dc}\) |
| 2 | [1 1 0 0] | \(U_{dc}\) | \(U_{dc}\) | 0 |
| 3 | [0 1 1 1] | \(-\frac{1}{2}U_{dc}\) | \(-\frac{1}{2}U_{dc}\) | \(\frac{\sqrt{3}}{2}U_{dc}\) |
| 4 | [0 1 0 1] | 0 | 0 | 0 |
| 5 | [0 1 0 0] | \(\frac{1}{2}U_{dc}\) | \(\frac{1}{2}U_{dc}\) | \(-\frac{\sqrt{3}}{2}U_{dc}\) |
| 6 | [0 0 1 1] | \(-U_{dc}\) | \(-U_{dc}\) | 0 |
| 7 | [0 0 0 1] | \(-\frac{1}{2}U_{dc}\) | \(-\frac{1}{2}U_{dc}\) | \(-\frac{\sqrt{3}}{2}U_{dc}\) |
| 8 | [0 0 0 0] | 0 | 0 | \(-U_{dc}\) |
The discrete-time model of the solar inverter in \(\alpha\beta\) coordinates is:
$$i_{s\alpha}(k+1) = i_{s\alpha}(k) + \frac{T_s}{L} [u_{\alpha}(k) – e_{\alpha}(k) – R i_{s\alpha}(k)]$$
$$i_{s\beta}(k+1) = i_{s\beta}(k) + \frac{T_s}{L} [u_{\beta}(k) – e_{\beta}(k) – R i_{s\beta}(k)]$$
where \(T_s\) is the sampling period, \(L\) is the filter inductance, and \(R\) is the resistance. Power predictions are obtained using second-order generalized integrator (SOGI) transformations:
$$P(k+1) = \frac{1}{2} \left( e_{\alpha}(k) i_{s\alpha}(k+1) + e_{\beta}(k) i_{s\beta}(k+1) \right)$$
$$Q(k+1) = \frac{1}{2} \left( e_{\beta}(k) i_{s\alpha}(k+1) – e_{\alpha}(k) i_{s\beta}(k+1) \right)$$
For DC-link capacitor voltage balance in the solar inverter, the model is:
$$u_{C1}(k+1) = u_{C1}(k) + \frac{T_s}{C_1} i_{C1}(k)$$
$$u_{C2}(k+1) = u_{C2}(k) + \frac{T_s}{C_2} i_{C2}(k)$$
with currents \(i_{C1}(k)\) and \(i_{C2}(k)\) derived from switch states and grid current \(i_s(k)\). The cost function for FCS-MPC is:
$$J(k) = |P_{ref}(k) – P(k+1)| + |Q_{ref}(k) – Q(k+1)| + \lambda |u_{C1}(k+1) – u_{C2}(k+1)|$$
where \(\lambda\) is a weighting factor. This enables the solar inverter to track active and reactive power references while maintaining DC-link balance.
Simulation results validate the proposed strategy. The solar inverter system parameters are listed in Table 3.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(C_{b1}\) | 2.2 μF | \(C_1, C_2\) | 1080 μF |
| \(C_{b2}\) | 22 μF | Filter Inductance \(L\) | 50 mH |
| \(L_b\) | 5 mH | Grid Voltage \(e\) | 220 V |
| Sampling Frequency | 20 kHz | Grid Frequency \(f\) | 50 Hz |
| Weighting Factor \(\lambda\) | 5 | PWM Frequency | 20 kHz |
Under varying irradiance (1000 W/m² to 1400 W/m²), the improved PSO algorithm achieves MPPT with minimal oscillation, as shown in Figure 5 (waveforms at 25°C). Without overvoltage control, the grid voltage exceeds 235.4 V during high irradiance periods. With reactive power control, the solar inverter outputs \(Q_{ref}\) as per Equation (8), regulating voltage within safe limits. The power tracking waveform in Figure 8 demonstrates the solar inverter’s ability to inject up to -900 var of reactive power during overvoltage, while maintaining active power output at MPP.
In conclusion, the integration of improved PSO-based MPPT and FCS-MPC in solar inverters effectively suppresses overvoltage in low-voltage distribution networks. The solar inverter’s dynamic reactive power support, combined with accurate active power tracking, ensures grid stability. Future work could explore multi-objective optimization for solar inverters in larger networks, enhancing their role in smart grid applications.