Integrating distributed renewable energy sources into the main grid relies heavily on the grid connected inverter, which serves as the critical power conversion interface. A paramount requirement for these inverters is to maintain high-quality power output and stable operation under various grid disturbances. Among these, unbalanced grid voltages, often caused by faults or unequal single-phase loads, present a significant challenge. Under such conditions, the power delivered by a standard grid connected inverter exhibits severe double-frequency (2ω) oscillations. More critically, inherent power coupling can degrade the transient stability of the inverter when power references change, potentially endangering the broader power system’s reliability. This work addresses the persistent issue of power coupling in Model Predictive Direct Power Control (MPDPC) for grid connected inverters operating with unbalanced grids. We propose a novel decoupling strategy based on dynamic feed-forward power compensation, which effectively suppresses oscillatory power components and decouples the active and reactive power dynamics during transients.
1. Introduction and Problem Statement
The proliferation of distributed generation has made the grid connected inverter a ubiquitous component in modern power systems. Its primary function is to convert DC power from sources like photovoltaics or batteries into AC power synchronized with the utility grid. However, the ideal condition of a perfectly balanced three-phase grid is often not met in practice. Voltage imbalances introduce negative-sequence components, leading to non-sinusoidal current injection and oscillatory power flow from the grid connected inverter. These oscillations, pulsating at twice the fundamental grid frequency (2ω), can cause overheating, torque pulsations in nearby motors, and interfere with protective relays.
Several control strategies have been developed to manage grid connected inverter operation under imbalance. Traditional methods include Voltage-Oriented Control (VOC), Virtual Synchronous Generator (VSG) control, and Direct Power Control (DPC). DPC, in particular, is favored for its fast dynamic response due to the direct regulation of instantaneous active (P) and reactive (Q) power without internal current loops. The integration of DPC with Finite-Control-Set Model Predictive Control (FCS-MPC) gives rise to Model Predictive Direct Power Control (MPDPC). MPDPC leverages a system model to predict future behavior over a horizon and selects the optimal switching state that minimizes a predefined cost function. This approach offers several advantages, including the ability to handle multiple constraints and objectives, such as switching frequency reduction or common-mode voltage mitigation.
Nevertheless, a critical shortcoming of conventional MPDPC under unbalanced grids is the coupling between active and reactive power loops during transient events, such as a step change in power reference. This coupling manifests as a significant disturbance in one power component when the reference of the other is changed, impairing the grid connected inverter‘s transient performance and stability. Existing decoupling techniques, like virtual power or virtual impedance methods, are primarily designed for balanced systems or exhibit limited performance under deep imbalance. This paper introduces a comprehensive solution. First, we employ a newly defined instantaneous reactive power within the MPDPC framework. This definition inherently eliminates the 2ω oscillations from both active and the newly defined reactive power without requiring complex positive- and negative-sequence extraction. Second, and most innovatively, we design a dynamic feed-forward power compensation scheme. This scheme actively decouples the power dynamics by injecting compensating signals into the power reference paths, dramatically improving the transient response of the grid connected inverter when power commands change under unbalanced voltage conditions.
2. Theoretical Foundation: Power Definitions under Unbalance
The analysis begins with the standard model of a three-phase two-level grid connected inverter with an L-type filter. In the stationary αβ-reference frame, the system dynamics are given by:
$$ \mathbf{u} = L \frac{d\mathbf{i}}{dt} + R\mathbf{i} + \mathbf{e} $$
where $\mathbf{u}$, $\mathbf{i}$, and $\mathbf{e}$ are the inverter output voltage, grid current, and grid voltage vectors, respectively, and L and R are the filter inductance and resistance.
Under balanced conditions, the standard instantaneous active power P and reactive power Q are defined as:
$$ P = \frac{3}{2} (i_{\alpha}e_{\alpha} + i_{\beta}e_{\beta}), \quad Q = \frac{3}{2} (i_{\alpha}e_{\beta} – i_{\beta}e_{\alpha}) $$
Under unbalanced grid voltages, both voltage and current vectors contain positive- and negative-sequence components:
$$ \mathbf{e} = \mathbf{e}^+_{\alpha\beta} + \mathbf{e}^-_{\alpha\beta} = \mathbf{e}^+_{dq}e^{j\omega t} + \mathbf{e}^-_{dq}e^{-j\omega t} $$
$$ \mathbf{i} = \mathbf{i}^+_{\alpha\beta} + \mathbf{i}^-_{\alpha\beta} = \mathbf{i}^+_{dq}e^{j\omega t} + \mathbf{i}^-_{dq}e^{-j\omega t} $$
Substituting these into the standard power definitions yields power expressions containing a constant term and 2ω oscillatory terms:
$$ P = P_0 + P_{c2}\cos(2\omega t) + P_{s2}\sin(2\omega t) $$
$$ Q = Q_0 + Q_{c2}\cos(2\omega t) + Q_{s2}\sin(2\omega t) $$
The oscillation coefficients ($P_{c2}, P_{s2}, Q_{c2}, Q_{s2}$) depend on the interaction between positive- and negative-sequence components of voltage and current. Suppressing these oscillations to achieve constant power delivery is a key control objective for a grid connected inverter.
To simplify control and ensure sinusoidal grid currents, a newly defined reactive power $Q_{nov}$ is adopted. It is formulated using a quarter-period delayed version of the grid voltage vector, $\mathbf{e}’$:
$$ \mathbf{e}’ = -j\mathbf{e}^+_{\alpha\beta} + j\mathbf{e}^-_{\alpha\beta} $$
$$ Q_{nov} = \frac{3}{2} \text{Re}(\mathbf{i}^* \cdot \mathbf{e}’) $$
The power expressions then become:
$$ P = P_0 + P_{c2}\cos(2\omega t) + P_{s2}\sin(2\omega t) $$
$$ Q_{nov} = Q_{nov0} + Q_{nov_{c2}}\cos(2\omega t) + Q_{nov_{s2}}\sin(2\omega t) $$
Crucially, analysis shows that $P_{c2} = Q_{nov_{s2}}$ and $P_{s2} = -Q_{nov_{c2}}$. Therefore, the control objective of setting $P_{c2}=0$ and $P_{s2}=0$ automatically forces $Q_{nov_{c2}}=0$ and $Q_{nov_{s2}}=0$. This means that by actively controlling P and $Q_{nov}$ to be constant, we simultaneously eliminate the 2ω oscillations from both power quantities and, as a result, force the grid currents to be balanced and sinusoidal. This forms the first pillar of our proposed control strategy for the grid connected inverter.
3. Model Predictive Direct Power Control (MPDPC) Formulation
MPDPC for the grid connected inverter is designed to directly regulate the instantaneous active power P and the newly defined reactive power $Q_{nov}$. The control algorithm operates in discrete time with a sampling period $T_s$.
3.1 Predictive Model
From the system dynamic equation, the derivative of the current is:
$$ \frac{d\mathbf{i}}{dt} = \frac{1}{L} (\mathbf{u} – R\mathbf{i} – \mathbf{e}) $$
The derivatives of the power terms, considering the definitions of P and $Q_{nov}$, are derived as:
$$ \frac{dP}{dt} = -\frac{R}{L}P – \omega Q_{nov} + \frac{3}{2L}(\mathbf{e} \cdot \mathbf{u} – |\mathbf{e}|^2) $$
$$ \frac{dQ_{nov}}{dt} = -\frac{R}{L}Q_{nov} + \omega P + \frac{3}{2L}[(\mathbf{u} – \mathbf{e}) \cdot \mathbf{e}’] $$
Using a forward Euler discretization, the power predictions for the next sampling instant (k+1) based on measurements at instant (k) are obtained:
$$ P^{k+1} = P^k – \frac{R T_s}{L}P^k – \omega T_s Q_{nov}^k + \frac{3T_s}{2L}(\mathbf{e}^k \cdot \mathbf{u}^k – |\mathbf{e}^k|^2) $$
$$ Q_{nov}^{k+1} = Q_{nov}^k – \frac{R T_s}{L}Q_{nov}^k + \omega T_s P^k + \frac{3T_s}{2L}[(\mathbf{u}^k – \mathbf{e}^k) \cdot \mathbf{e}’^k] $$
3.2 Cost Function and Control Law
The core of the FCS-MPC algorithm is to evaluate all possible voltage vectors $\mathbf{u}^k$ generated by the eight switching states of the inverter. For each candidate vector, the future powers $P^{k+1}$ and $Q_{nov}^{k+1}$ are predicted using the equations above. These predictions are then compared to their reference values ($P_{ref}$, $Q_{ref}$) via a cost function J, typically chosen as the sum of squared errors:
$$ J = (P_{ref} – P^{k+1})^2 + (Q_{ref} – Q_{nov}^{k+1})^2 $$
The voltage vector that minimizes this cost function is selected and applied during the next sampling period. This process is repeated every $T_s$ seconds. This MPDPC structure inherently handles the objective of constant P and $Q_{nov}$ under unbalanced grids when $Q_{ref}$ is set for $Q_{nov}$.
| Switching State (SaSbSc) | Voltage Vector $\mathbf{u}^k$ ($\alpha\beta$) | Description |
|---|---|---|
| 000 | 0 | Zero Vector V0 |
| 111 | 0 | Zero Vector V7 |
| 100 | $\frac{2}{3}U_{dc}$ | Active Vector V1 |
| 110 | $\frac{1}{3}U_{dc} + j\frac{\sqrt{3}}{3}U_{dc}$ | Active Vector V2 |
| 010 | $-\frac{1}{3}U_{dc} + j\frac{\sqrt{3}}{3}U_{dc}$ | Active Vector V3 |
| 011 | $-\frac{2}{3}U_{dc}$ | Active Vector V4 |
| 001 | $-\frac{1}{3}U_{dc} – j\frac{\sqrt{3}}{3}U_{dc}$ | Active Vector V5 |
| 101 | $\frac{1}{3}U_{dc} – j\frac{\sqrt{3}}{3}U_{dc}$ | Active Vector V6 |
4. Proposed Feed-Forward Power Decoupling Strategy
While the MPDPC with $Q_{nov}$ successfully suppresses steady-state power oscillations, the transient coupling issue remains. The single cost function J treats both power errors equally. When $P_{ref}$ changes abruptly, the term $(P_{ref} – P^{k+1})^2$ dominates J. The controller’s primary effort becomes correcting P, often at the expense of allowing a temporary but significant deviation in $Q_{nov}$, and vice-versa. This cross-coupling can be analyzed through the relationship between power errors and current deviations.
Let the nominal current at an operating point be $\mathbf{i}_n$. A change in power reference causes a current deviation $\Delta \mathbf{i}$. The resulting power deviations can be approximated by the Jacobian matrix $\mathbf{J}$:
$$ \begin{bmatrix} \Delta P \\ \Delta Q_{nov} \end{bmatrix} = \mathbf{J} \begin{bmatrix} \Delta i_{\alpha} \\ \Delta i_{\beta} \end{bmatrix} = \begin{bmatrix} e_{\alpha} & e_{\beta} \\ e’_{\alpha} & e’_{\beta} \end{bmatrix} \begin{bmatrix} \Delta i_{\alpha} \\ \Delta i_{\beta} \end{bmatrix} $$
The current deviation required to achieve a certain power change is:
$$ \begin{bmatrix} \Delta i_{\alpha} \\ \Delta i_{\beta} \end{bmatrix} = \mathbf{J}^{-1} \begin{bmatrix} \Delta P \\ \Delta Q_{nov} \end{bmatrix} = \begin{bmatrix} H_{P_{i\alpha}} & H_{Q_{i\alpha}} \\ H_{P_{i\beta}} & H_{Q_{i\beta}} \end{bmatrix} \begin{bmatrix} e_P \\ e_Q \end{bmatrix} $$
where $e_P = P_{ref} – P$ and $e_Q = Q_{ref} – Q_{nov}$ are the instantaneous power errors, and $H_{ij}$ are elements of the inverse Jacobian.
The coupling effect stems from the fact that a change in $P_{ref}$ (i.e., a non-zero $e_P$) produces not only $\Delta i_P$ but also, through the off-diagonal terms of $\mathbf{J}^{-1}$, a component $\Delta i_Q$. This $\Delta i_Q$ directly perturbs $Q_{nov}$. The core idea of the proposed decoupling strategy is to anticipate and cancel this effect using a feed-forward compensation signal added to the power references.
We design compensation powers $P_{com}$ and $Q_{com}$ as functions of the power errors and system variables:
$$ P_{com} = H_{Q_{i\alpha}} e_{\alpha} e_Q + k_P e_P e_Q $$
$$ Q_{com} = H_{P_{i\alpha}} e’_{\alpha} e_P + k_Q e_P e_Q $$
Here, $k_P$ and $k_Q$ are tunable compensation gain coefficients. The modified power references ($P_{ref}^*$, $Q_{ref}^*$) fed into the MPDPC cost function become:
$$ P_{ref}^* = P_{ref} + P_{com} $$
$$ Q_{ref}^* = Q_{ref} + Q_{com} $$
The first term in each compensation equation ($H_{Q_{i\alpha}} e_{\alpha} e_Q$ and $H_{P_{i\alpha}} e’_{\alpha} e_P$) provides the primary decoupling action derived from the system model. The second term ($k e_P e_Q$) adds a stabilizing correction that further fine-tunes the transient response. When a step change in $P_{ref}$ occurs, $e_P$ becomes large, causing $Q_{com}$ to generate a counter-acting signal that modifies the $Q_{ref}$ input to the MPDPC. This proactive adjustment guides the controller to act in a way that minimizes the induced disturbance in $Q_{nov}$. This strategy significantly enhances the transient stability and dynamic performance of the grid connected inverter without altering the fundamental MPDPC stability properties.
| Control Feature | Conventional MPDPC (Standard Q) | Proposed MPDPC (New Qnov) | Proposed MPDPC with Decoupling |
|---|---|---|---|
| Sequence Extraction | Required for oscillation suppression | Not Required | Not Required |
| Steady-State 2ω Oscillations | Can be suppressed with proper control | Fully Suppressed | Fully Suppressed |
| Grid Current THD under Unbalance | High (Non-sinusoidal) | Low (Sinusoidal) | Low (Sinusoidal) |
| Transient Power Coupling | Significant | Significant | Effectively Eliminated |
| Control Complexity | Moderate to High | Moderate | Moderate (adds simple feed-forward) |
5. Simulation Verification and Discussion
The effectiveness of the proposed decoupling strategy for the grid connected inverter is validated through detailed simulations in MATLAB/Simulink. The system parameters used are listed in the table below. A three-phase grid voltage imbalance is created by setting the phase voltages to 0.8, 0.9, and 1.0 per unit of the nominal 100V RMS value.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link Voltage | $U_{dc}$ | 520 V |
| DC-link Capacitance | $C_{dc}$ | 1000 µF |
| Filter Inductance | $L$ | 10 mH |
| Filter Resistance | $R$ | 0.1 Ω |
| Grid Voltage (Nominal) | $e_{abc}$ | 100 V RMS |
| Grid Frequency | $f$/$\omega$ | 50 Hz / 100$\pi$ rad/s |
| Sampling Frequency | $f_s$ | 40 kHz |
| Compensation Gains | $k_P$, $k_Q$ | -0.01, -1/2400 |
5.1 Coupling Phenomenon and Basic Decoupling
First, the inherent coupling in the basic MPDPC (using $Q_{nov}$) is demonstrated. At t=0.1s, the active power reference $P_{ref}$ is stepped from 0 W to 2000 W while $Q_{ref}$ is kept at 0 var. Without decoupling, the transient causes a significant coupling spike in $Q_{nov}$ exceeding 150 var. Conversely, when $Q_{ref}$ is stepped to 1500 var, a coupling spike of about 240 W appears in P. After activating the proposed feed-forward power compensation, these coupling spikes are virtually eliminated in both test scenarios, confirming the core decoupling functionality.
5.2 Performance under Different Operating Points
To evaluate robustness, the decoupling performance is tested with different levels of active power reference. Three step changes are applied: to 1200 W, 1600 W, and 2000 W. The amplitude of the resulting $Q_{nov}$ coupling spike in the uncompensated system increases proportionally with the step size (70 var, 90 var, 130 var). With the decoupling strategy active, the coupling is reduced to near zero in all three cases. Similarly, when reactive power steps are applied, the induced P coupling is also effectively removed, demonstrating the strategy’s efficacy across the operating range of the grid connected inverter.
5.3 Performance under Different Voltage Imbalance Depths
The strategy is tested under varying severity of voltage imbalance by reducing the amplitude of phase-A voltage to 0.9, 0.8, 0.7, and 0.6 per unit. In all cases, the uncompensated MPDPC shows significant transient coupling (e.g., ~170 var coupling for a 2000W P step at 0.8 p.u. imbalance). The proposed compensation scheme successfully suppresses this coupling to minimal levels regardless of the imbalance depth, proving its adaptability to different grid conditions.
5.4 Benefits of the New Reactive Power Definition
A critical comparison is made between using standard Q and the new $Q_{nov}$ within the MPDPC framework under the same unbalanced grid. With standard Q control (and no oscillation suppression), the grid currents become highly distorted and non-sinusoidal, with a Total Harmonic Distortion (THD) of 7.51%. In contrast, using $Q_{nov}$ as the control variable forces the currents to remain sinusoidal (THD of 1.70%) while naturally eliminating the 2ω oscillations from P. This clearly establishes the superiority of $Q_{nov}$ for maintaining power quality in a grid connected inverter under unbalanced voltage conditions.
| Test Scenario | Uncompensated Coupling Magnitude | Compensated Coupling Magnitude | Improvement |
|---|---|---|---|
| $P_{ref}$ step 0→2000W (Q coupling) | ~150 var | ~0 var | ~100% |
| $Q_{ref}$ step 0→1500var (P coupling) | ~240 W | ~0 W | ~100% |
| Va = 0.7 p.u., $Q_{ref}$ step | ~460 W | ~0 W | ~100% |
| Different P step levels (1200, 1600, 2000W) | Proportional increase (70-130 var) | Consistently near 0 var | Effective across range |
6. Conclusion
This work has presented a comprehensive solution to enhance the performance of a grid connected inverter under unbalanced grid voltage conditions. The integration of a newly defined instantaneous reactive power ($Q_{nov}$) within a Model Predictive Direct Power Control (MPDPC) framework provides a foundational improvement. It eliminates the need for complex sequence decomposition, suppresses the inherent double-frequency power oscillations, and ensures the injection of sinusoidal currents into the grid.
The primary contribution, however, is the novel dynamic feed-forward power compensation strategy designed to address the transient power coupling problem in MPDPC. By analyzing the relationship between power errors and current deviations, we derived compensation signals that are added to the power references. This proactive measure effectively decouples the active and reactive power dynamics during reference changes. Extensive simulation studies have conclusively demonstrated the strategy’s effectiveness. It robustly eliminates transient coupling spikes across different operating points and under varying degrees of grid voltage imbalance, significantly boosting the transient stability of the grid connected inverter.
The proposed method offers a practical and effective enhancement for MPDPC, improving the reliability and power quality of renewable energy integration systems facing non-ideal grid conditions. Future work will focus on experimental validation using a laboratory-scale prototype and extending the strategy to other inverter topologies, such as three-level or modular multilevel grid connected inverters.