The proliferation of grid connected photovoltaic (PV) systems has significantly altered the landscape of modern power grids. The inherent intermittency of solar energy and the large-scale integration of these systems necessitate stringent requirements for power quality and grid stability. A critical challenge arises during grid faults, particularly voltage sags. If a grid connected inverter disconnects immediately upon detecting a voltage dip, it can exacerbate the fault, hinder grid recovery, and potentially lead to widespread instability. Consequently, grid codes worldwide, including those from entities like the State Grid Corporation of China, mandate that large-scale PV plants possess Low Voltage Ride-Through (LVRT) capability. This requires the grid connected inverter to remain synchronized and operational during specified voltage dips, providing reactive power support to aid grid voltage recovery while managing active power output. The core of fulfilling this mandate lies in devising advanced, robust control strategies for the inverter.
Traditional control paradigms for grid connected inverter systems, such as PI or PR control, face limitations under unbalanced grid conditions typical of LVRT events. PI controllers struggle with negative-sequence currents, while PR controllers can be sensitive to grid frequency deviations. Direct Power Control (DPC) offers fast dynamic response but often suffers from variable switching frequency and high current harmonic distortion. Finite Control Set Model Predictive Control (FCS-MPC) has emerged as a powerful alternative, leveraging the discrete nature of power converters to provide fast, dead-beat control. However, conventional single-vector FCS-MPC can lead to high current harmonics and variable switching frequency. Multi-vector approaches improve performance but often at the cost of high computational burden due to iterative evaluation of numerous vector combinations.
Furthermore, the control system’s reliance on voltage sensors for feedback introduces a point of vulnerability. During severe voltage sags, sensor accuracy can degrade, or sensors may even fail, compromising the entire LVRT strategy. Therefore, sensorless control techniques are highly desirable for enhanced reliability and cost reduction. Virtual flux-based observers are common but susceptible to DC offset and integration drift. Sliding Mode Observers (SMOs) offer robustness but traditional designs often require precise grid frequency information for accurate reconstruction, a parameter that may not be reliable during faults.
To address these intertwined challenges, we propose a comprehensive control scheme for a grid connected inverter. The scheme integrates an improved Multi-Vector Model Predictive Power Control (MV-MPPC) strategy with a frequency-adaptive Sliding Mode Observer (SMO). The MV-MPPC method systematically applies a fixed pair of active voltage vectors and zero vectors within each control cycle, eliminating the need for repetitive online vector selection and evaluation. This approach reduces computational delay, ensures a constant switching frequency through Space Vector Modulation (SVM), and maintains high control accuracy. Simultaneously, the enhanced SMO estimates the grid voltage without physical sensors. A key innovation is a filtering and compensation method that does not rely on a priori knowledge of the grid angular frequency, making it inherently robust to frequency fluctuations during faults. This synergy aims to equip the grid connected inverter with superior LVRT performance: fast and accurate power reference tracking, low grid current harmonics, and high operational reliability.
Mathematical Model of the Grid Connected Inverter
The topology of a standard two-level three-phase grid connected inverter is considered. The system comprises a DC-link capacitor, three-phase inverter legs, an L filter (inductance $L$ and its inherent resistance $R$), and the point of common coupling to the grid. The dynamic equations in the stationary $\alpha\beta$-frame are given by:
$$
\frac{d i_{g\alpha}}{dt} = -\frac{R}{L} i_{g\alpha} – \frac{1}{L} V_{g\alpha} + \frac{1}{L} U_{g\alpha}
$$
$$
\frac{d i_{g\beta}}{dt} = -\frac{R}{L} i_{g\beta} – \frac{1}{L} V_{g\beta} + \frac{1}{L} U_{g\beta}
$$
where $i_{g\alpha\beta}$ are the grid currents, $V_{g\alpha\beta}$ are the inverter output voltages, and $U_{g\alpha\beta}$ are the grid voltages. Based on the instantaneous power theory, the active ($P$) and reactive ($Q$) power at the inverter output can be expressed as:
$$
P = U_{g\alpha} i_{g\alpha} + U_{g\beta} i_{g\beta}
$$
$$
Q = U_{g\beta} i_{g\alpha} – U_{g\alpha} i_{g\beta}
$$
Differentiating these equations and assuming an ideal grid voltage ($dU_{g\alpha}/dt = -\omega U_{g\beta}$, $dU_{g\beta}/dt = \omega U_{g\alpha}$), the power derivatives are:
$$
\frac{dP}{dt} = -\frac{R}{L}P – \omega Q + \frac{1}{L}(U_{g\alpha}V_{g\alpha} + U_{g\beta}V_{g\beta}) – \frac{R}{L}(U_{g\alpha}i_{g\alpha} + U_{g\beta}i_{g\beta}) – \frac{1}{L}(U_{g\alpha}^2 + U_{g\beta}^2)
$$
$$
\frac{dQ}{dt} = -\frac{R}{L}Q + \omega P + \frac{1}{L}(U_{g\beta}V_{g\alpha} – U_{g\alpha}V_{g\beta}) – \frac{R}{L}(U_{g\beta}i_{g\alpha} – U_{g\alpha}i_{g\beta})
$$
Discretizing the model using forward Euler approximation ($dx/dt \approx (x(k+1)-x(k))/T_s$ with $T_s$ as the sampling period) yields the predictive model for power at the $(k+1)$-th instant:
$$
P(k+1) = P(k) + T_s \left[ -\frac{R}{L}P(k) – \omega Q(k) + \frac{1}{L}(U_{g\alpha}(k)V_{g\alpha}(k) + U_{g\beta}(k)V_{g\beta}(k)) – \frac{R}{L}(U_{g\alpha}(k)i_{g\alpha}(k) + U_{g\beta}(k)i_{g\beta}(k)) – \frac{1}{L}(U_{g\alpha}^2(k) + U_{g\beta}^2(k)) \right]
$$
$$
Q(k+1) = Q(k) + T_s \left[ -\frac{R}{L}Q(k) + \omega P(k) + \frac{1}{L}(U_{g\beta}(k)V_{g\alpha}(k) – U_{g\alpha}(k)V_{g\beta}(k)) – \frac{R}{L}(U_{g\beta}(k)i_{g\alpha}(k) – U_{g\alpha}(k)i_{g\beta}(k)) \right]
$$
This discrete model forms the foundation for the predictive controller, allowing it to forecast the future power state based on present measurements and a candidate inverter voltage vector $V_{g\alpha\beta}(k)$.
Proposed Multi-Vector Model Predictive Power Control (MV-MPPC)
Conventional FCS-MPPC evaluates the cost function for all seven or eight basic voltage vectors of the two-level inverter, selecting the single vector that minimizes the power error for application during the entire next sampling period. This leads to high current ripple and variable switching frequency. Our proposed MV-MPPC strategy synthesizes an optimal reference voltage vector $V_{ref}$ each period using a combination of two adjacent active vectors and the zero vectors.
The core principle is to achieve dead-beat control of power. Assuming the reference power $P^*(k+1)$, $Q^*(k+1)$ is constant over one step, we enforce $P(k+1)=P^*(k)$ and $Q(k+1)=Q^*(k)$. The reference voltage vector $V_{ref}$ needed to achieve this is calculated directly. However, instead of mapping $V_{ref}$ to its sector and calculating dwell times for its two bounding active vectors (a process requiring sector identification and trigonometric calculations each cycle), we propose a fixed-vector-composition approach.
We select a fixed pair of non-adjacent active voltage vectors, for instance, $V_1$ and $V_3$, as a basis. Any reference voltage vector $V_{ref}$ within the hexagon can be uniquely represented by a linear combination of $V_1$, $V_3$, and the zero vector $V_0$:
$$
V_{ref} = \frac{t_1}{T_s} V_1 + \frac{t_3}{T_s} V_3 + \frac{t_0}{T_s} V_0
$$
where $t_1$, $t_3$, and $t_0$ are the dwell times, and $t_0 = T_s – t_1 – t_3$. The relationship between the desired power change and the voltage vectors is established through their power slopes. The slope when applying a vector $V_x$ is:
$$ s_{px} = \frac{1}{L}(e_\alpha u_{x\alpha} + e_\beta u_{x\beta}), \quad s_{qx} = \frac{1}{L}(e_\beta u_{x\alpha} – e_\alpha u_{x\beta}) $$
where $[e_\alpha, e_\beta]^T = [U_{g\alpha}, U_{g\beta}]^T – (R/L)[i_{g\alpha}, i_{g\beta}]^T – (1/L)[U_{g\alpha}^2+U_{g\beta}^2, 0]^T – [-\omega Q, \omega P]^T / ?$ (derived from the power derivative equations). The slopes for the zero vector are $s_{p0}$ and $s_{q0}$.
Applying the dead-beat condition:
$$ P^*(k) – P(k) = s_{p1}t_1 + s_{p3}t_3 + s_{p0}t_0 $$
$$ Q^*(k) – Q(k) = s_{q1}t_1 + s_{q3}t_3 + s_{q0}t_0 $$
Solving these equations simultaneously with $t_0 = T_s – t_1 – t_3$ yields the dwell times $t_1$ and $t_3$:
$$
t_1 = \frac{ (P_{err})(s_{q3}-s_{q0}) – (Q_{err})(s_{p3}-s_{p0}) }{ (s_{p1}-s_{p0})(s_{q3}-s_{q0}) – (s_{q1}-s_{q0})(s_{p3}-s_{p0}) }
$$
$$
t_3 = \frac{ (P_{err})(s_{q1}-s_{q0}) – (Q_{err})(s_{p1}-s_{p0}) }{ (s_{p3}-s_{p0})(s_{q1}-s_{q0}) – (s_{q3}-s_{q0})(s_{p1}-s_{p0}) }
$$
where $P_{err}=P^*(k)-P(k)$ and $Q_{err}=Q^*(k)-Q(k)$. Once $t_1$ and $t_3$ are computed, $V_{ref}$ is constructed using the fixed vectors $V_1$ and $V_3$. This $V_{ref}$ is then passed to a standard SVM module, which generates the switching signals with a constant switching frequency. This method avoids the iterative search and sector-based calculations, significantly reducing computational load and ensuring a fixed switching pattern, which is crucial for filter design and harmonic performance of the grid connected inverter.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (RMS, phase) | $U_g$ | 220 V |
| Grid Frequency | $f$ | 50 Hz |
| DC-Link Voltage | $V_{dc}$ | 700 V |
| Filter Inductance | $L$ | 1.5 mH |
| Filter Resistance | $R$ | 0.05 Ω |
| Sampling/Switching Frequency | $f_s$ | 10 kHz |
| Rated Grid Current | $I_g$ | 65 A |
Frequency-Adaptive Sliding Mode Observer for Sensorless Operation
To eliminate the voltage sensors, we employ a Sliding Mode Observer (SMO). The SMO structure for the grid current in the $\alpha\beta$ frame is:
$$
\frac{d\hat{i}_{g\alpha\beta}}{dt} = -\frac{R}{L}\hat{i}_{g\alpha\beta} – \frac{1}{L}V_{g\alpha\beta} + \frac{1}{L}\boldsymbol{\Gamma}
$$
where $\hat{i}_{g\alpha\beta}$ is the estimated current, and $\boldsymbol{\Gamma} = [\Gamma_\alpha, \Gamma_\beta]^T$ is the discontinuous switching term defined as $\boldsymbol{\Gamma} = A \cdot \text{sgn}(\mathbf{g})$, with $\mathbf{g} = \hat{i}_{g\alpha\beta} – i_{g\alpha\beta}$ being the current estimation error and $A$ the sliding gain. By defining a Lyapunov function, the gain $A$ can be chosen to ensure the sliding surface $\mathbf{g}=0$ is reached in finite time. Upon reaching the sliding mode ($\mathbf{g}=0, d\mathbf{g}/dt=0$), the equivalent control $\boldsymbol{\Gamma}_{eq}$ is equal to the grid voltage $U_{g\alpha\beta}$.
In practice, $\boldsymbol{\Gamma}$ is a high-frequency switching signal. A Low-Pass Filter (LPF) is used to extract $\boldsymbol{\Gamma}_{eq}$ (i.e., the grid voltage estimate):
$$
\hat{U}_{g\alpha\beta}^{filt} = \frac{\omega_c}{s + \omega_c} \boldsymbol{\Gamma}
$$
where $\omega_c$ is the LPF cutoff frequency. However, this filtering introduces both amplitude attenuation and phase lag at the fundamental frequency $\omega$, distorting the estimate. Traditional compensation requires precise knowledge of $\omega$:
$$
\hat{U}_{g\alpha\beta}^{comp} = \hat{U}_{g\alpha\beta}^{filt} \cdot \frac{j\omega + \omega_c}{\omega_c}
$$
This dependency on $\omega$ is problematic during LVRT as grid frequency may fluctuate.
We propose a novel compensation method independent of $\omega$. The key idea is to process the already filtered signal $\hat{U}_{g\alpha\beta}^{filt}$ through a second, identical LPF:
$$
\hat{U}_{g\alpha\beta}^{filt2} = \frac{\omega_c}{s + \omega_c} \hat{U}_{g\alpha\beta}^{filt}
$$
Both filtered signals suffer the same distortion characteristics relative to the original $\boldsymbol{\Gamma}_{eq}$. Let the original signal have amplitude $M$ and phase $\phi$. The first filtered output has amplitude $M_1$ and phase $\phi_1$, and the second has $M_2$ and $\phi_2$. The attenuation ratio $\Delta M$ and phase shift difference $\Delta \phi$ between the two filtered signals are:
$$
\Delta M = \frac{M_1}{M_2}, \quad \Delta \phi = \phi_1 – \phi_2
$$
These ratios are functions of the LPF characteristics and $\omega$, but crucially, they can be measured directly from the two filtered signals without knowing $\omega$. Using $\Delta M$ and $\Delta \phi$, the original amplitude $M$ and phase $\phi$ can be reconstructed geometrically, effectively compensating for the LPF distortion. The final compensated grid voltage estimate is calculated as:
$$
\hat{U}_{g\alpha} = \sqrt{ (\hat{U}_{g\alpha}^{filt2} \cdot \Delta M \cos(\Delta\phi))^2 + (\hat{U}_{g\beta}^{filt2} \cdot \Delta M \sin(\Delta\phi))^2 } \cdot \cos(\angle(…))
$$
$$
\hat{U}_{g\beta} = \sqrt{ (\hat{U}_{g\alpha}^{filt2} \cdot \Delta M \cos(\Delta\phi))^2 + (\hat{U}_{g\beta}^{filt2} \cdot \Delta M \sin(\Delta\phi))^2 } \cdot \sin(\angle(…))
$$
This estimated voltage $\hat{U}_{g\alpha\beta}$ is then fed into the MV-MPPC predictive model, closing the loop without physical voltage sensors and maintaining accuracy even during grid frequency variations.
Simulation Analysis and Performance Evaluation
The proposed integrated control scheme for the grid connected inverter was simulated in MATLAB/Simulink using the parameters in Table 1. The performance was evaluated under LVRT conditions and compared with conventional DPC.
LVRT Under Balanced Voltage Sag: A 50% balanced voltage dip was applied at t=0.05s, lasting until t=0.15s. The proposed MV-MPPC controller demonstrated excellent stability. The grid current quickly adapted to the new voltage level while maintaining sinusoidal shape and synchronism. The active power reference was adjusted according to LVRT requirements, and reactive current was injected as per grid code mandates, showcasing the controller’s fast dynamic response and robustness.
LVRT Under Unbalanced Voltage Sag: A single-phase (phase C) voltage dip of 50% was applied. Despite the severe unbalance, the MV-MPPC strategy effectively regulated the power and maintained controlled grid currents without excessive distortion or DC offset, proving its capability under non-ideal fault conditions.
Harmonic Performance under Distorted Grid: To evaluate steady-state harmonic rejection, the grid voltage was deliberately distorted by injecting 15% 3rd harmonic in phase A, 15% 5th in phase B, and 20% 7th in phase C. The Total Harmonic Distortion (THD) of the grid current was analyzed. The conventional DPC method yielded a current THD of 7.06%. In contrast, the proposed MV-MPPC strategy, with its fixed switching frequency and SVM modulation, achieved a significantly lower THD of 3.82%. This highlights a major advantage for power quality in the grid connected inverter.
Observer Performance under Frequency Deviation: The proposed SMO was tested against a traditional frequency-dependent SMO. With grid frequency fixed at 50 Hz, both observers performed accurately. However, when the grid frequency shifted to 45 Hz (simulating a severe fault condition), the traditional observer exhibited a clear phase error (~5 ms lead) due to using incorrect compensation parameters. The proposed frequency-adaptive observer maintained accurate amplitude and phase estimation, validating its robustness and eliminating the need for a PLL or precise frequency measurement during faults.
| Performance Metric | Conventional DPC | Proposed MV-MPPC with SMO |
|---|---|---|
| Switching Frequency | Variable, High | Constant (10 kHz) |
| Current THD (Distorted Grid) | 7.06% | 3.82% |
| Dynamic Response to Power Step | Fast | Very Fast (~50% faster settling time) |
| LVRT Current Limiting | Effective | Effective, with smoother current |
| Sensor Requirement | Voltage & Current Sensors | Current Sensors Only |
| Robustness to Grid Frequency Change | N/A (relies on sensors) | High (Frequency-adaptive SMO) |
| Computational Burden | Moderate | Low-Moderate (Fixed vector calculation) |
Experimental Validation
A laboratory-scale prototype was built using a TMS320F28335 DSP controller to validate the simulation findings. The same system parameters were used.
Steady-State Performance: Figure 11 (referenced from conceptual waveforms) shows the steady-state grid phase voltage and current under both DPC and the proposed MV-MPPC. The current waveform under MV-MPPC is visibly smoother with lower ripple, corroborating the lower THD observed in simulation.
Dynamic LVRT Performance: A 50% voltage sag was applied using a transformer tap change. Both controllers successfully limited the current surge within 1.1 pu as required. However, the current waveform from the MV-MPPC-based grid connected inverter exhibited less transient distortion and settled more gracefully, as seen in the conceptual Figure 12.
Reference Tracking Speed: A critical test involved a step change in active power reference from 0.25 kW to 1.4 kW. The proposed MV-MPPC strategy achieved reference tracking in approximately half the time required by a traditional multi-vector predictive controller that performs sector-based vector selection. This demonstrates the computational advantage of the fixed-vector approach, leading to faster control action—a paramount feature for LVRT where conditions change rapidly.
Conclusion
This work presents a comprehensive and robust control solution to enhance the Low Voltage Ride-Through capability of a grid connected inverter. The proposed scheme synergistically combines an improved Multi-Vector Model Predictive Power Control (MV-MPPC) with a frequency-adaptive Sliding Mode Observer (SMO).
The MV-MPPC strategy fundamentally improves upon traditional methods by employing a fixed pair of active voltage vectors ($V_1$, $V_3$) to synthesize the optimal reference voltage every control cycle. This eliminates the iterative search and sector identification processes, reducing computational delay—a critical factor for fast LVRT response. The subsequent use of Space Vector Modulation ensures a constant switching frequency, which directly translates to lower output current harmonic distortion (THD reduced from 7.06% to 3.82% in tests) and simplifies filter design for the grid connected inverter.
Complementing this, the novel SMO design provides accurate grid voltage estimation without physical sensors, enhancing system reliability and reducing cost. Its distinctive filtering and compensation method does not require prior knowledge of the grid angular frequency, making it inherently robust to frequency fluctuations that commonly occur during network faults. This ensures the control loop receives accurate feedback even during challenging LVRT events.
Simulation and experimental results consistently validate the effectiveness of the proposed approach. The controller maintains stable operation, provides fast dynamic power response, injects appropriate reactive current during voltage sags, and yields high-quality grid currents under both balanced and unbalanced fault conditions. Therefore, this integrated control strategy represents a significant step forward in developing highly reliable, efficient, and grid-code-compliant grid connected inverter systems for modern renewable energy integration.