In modern power systems, solar inverters are the core interface between photovoltaic arrays and the utility grid. The reliable operation of solar inverters is essential for maintaining power quality and system stability. However, grid voltage unbalance—caused by line faults, unbalanced loads, or transformer asymmetries—severely degrades the performance of conventional control algorithms. In my study, I focus on enhancing the control performance of solar inverters under such adverse conditions. I propose a novel current controller that combines a proportional integrator with a reduced-order resonant term, referred to as PIROR. This controller provides fast and accurate tracking of both positive- and negative-sequence currents without requiring complex sequence decomposition. In this article, I present the mathematical derivation, control design, and simulation validation of the proposed method, emphasizing its advantages over traditional PI and PR controllers.
1. Introduction
Solar inverters are expected to maintain stable operation even when the grid voltage becomes unbalanced. Under unbalanced conditions, negative-sequence components appear in the grid voltages, which lead to second-harmonic oscillations in the active and reactive power, distortion of the output currents, and ripple on the DC-link voltage. Conventional current controllers based on the proportional-integrator (PI) regulator in the synchronous d-q frame cannot effectively suppress these disturbances because the negative-sequence components appear as AC disturbances at twice the fundamental frequency. To address this challenge, researchers have proposed various strategies such as positive and negative sequence control (PNSC), balanced positive-sequence control (BPSC), and dual d-q frame PI control. However, these methods often require sequence decomposition, which introduces delays and errors in the current control loop.
Proportional-resonant (PR) controllers offer an alternative by providing infinite gain at the resonant frequency, allowing zero steady-state error for AC signals. But PR controllers are sensitive to grid frequency variations, and their practical implementation is difficult due to the need for high-precision analog or digital components. In this work, I propose a simplified yet effective control scheme for solar inverters that integrates a reduced-order resonance (ROR) link into the conventional PI controller. The resulting PIROR controller preserves the simplicity of PI control while adding the frequency-adaptive resonant capability needed for unbalanced grid conditions. I design the controller to track frequency deviations, thereby improving robustness under realistic grid fluctuations.
2. Mathematical Model of Grid-Connected Solar Inverters Under Unbalanced Voltage
I consider a three-phase voltage-source photovoltaic inverter connected to the grid through an L filter. The topology includes a DC-link capacitor, an IGBT-based inverter bridge, and a boost converter that elevates the PV array voltage to a suitable level. The system parameters used in my analysis are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Rated power | Prated | 10 kW |
| Grid frequency | f | 50 Hz |
| DC-link voltage | Udc | 300 V |
| DC-link capacitor | C | 1000 μF |
| AC filter inductance | L | 3 mH |
| Switching frequency | fsw | 6 kHz |
| Controller gains | KP / KR / KI | 6 / 30 / 70 |
When the grid voltage is unbalanced, the inverter output voltage vector in the stationary αβ frame can be expressed as the sum of positive- and negative-sequence components rotating in opposite directions:
$$ \mathbf{U}_{f\alpha\beta} = \mathbf{U}_{f\alpha\beta}^{+} + \mathbf{U}_{f\alpha\beta}^{-} = \mathbf{U}_{fdq+}^{+} e^{j\omega t} + \mathbf{U}_{fdq-}^{-} e^{-j\omega t} \tag{1} $$
where the superscripts + and − denote positive and negative sequence, respectively, and the subscripts + and − refer to the synchronously rotating reference frames in the forward and reverse directions. In the positive-sequence synchronous d-q frame, the unbalanced grid voltage appears as a DC component plus a second-harmonic AC component:
$$ \mathbf{U}_{fdq+} = \mathbf{U}_{fdq+}^{+} + \mathbf{U}_{fdq+}^{-} = \mathbf{U}_{fdq+}^{+} + \mathbf{U}_{fdq-}^{-} e^{-j2\omega t} \tag{2} $$
Therefore, the inverter AC-side model in the presence of unbalanced voltage can be described by
$$ \begin{cases} \mathbf{U}_{fdq+}^{+} = \mathbf{U}_{gdq+}^{+} + L \dfrac{d\mathbf{I}_{fdq+}^{+}}{dt} + j\omega L \mathbf{I}_{fdq+}^{+} + \mathbf{E}_{gdq+}^{+} \\[6pt] \mathbf{U}_{fdq-}^{-} = \mathbf{U}_{gdq-}^{-} + L \dfrac{d\mathbf{I}_{fdq-}^{-}}{dt} + j\omega L \mathbf{I}_{fdq-}^{-} + \mathbf{E}_{gdq-}^{-} \end{cases} \tag{3}
where \(\mathbf{U}_{g}\) is the grid voltage, \(\mathbf{I}_{f}\) is the inverter output current, and \(\mathbf{E}_{g}\) is the grid electromotive force.
The complex power delivered to the grid by the solar inverter is
$$ \mathbf{S} = P + jQ = \frac{3}{2} \mathbf{U} \mathbf{I}^{*} \tag{4} $$
Substituting the sequence components yields active and reactive power expressions that contain second-order oscillations:
$$ \begin{cases} P = P_0 + P_{c2} \cos(2\omega t) + P_{s2} \sin(2\omega t) \\[4pt] Q = Q_0 + Q_{c2} \cos(2\omega t) + Q_{s2} \sin(2\omega t) \end{cases} \tag{5} $$
These oscillations cause DC-link voltage ripple and deteriorate the output power quality. Therefore, solar inverters require a control strategy that can mitigate the negative-sequence current and minimize second-harmonic power pulsations.
3. Limitations of Conventional Controllers
3.1 Proportional-Integral Controller
The transfer function of a standard PI controller is
$$ G_{PI}(s) = K_p + \frac{K_i}{s} \tag{6} $$
At the fundamental frequency \(\omega_0\), its gain is finite:
$$ |G_{PI}(j\omega_0)| = \sqrt{K_p^2 + \left(\frac{K_i}{\omega_0}\right)^2} \tag{7} $$
This finite gain causes steady-state amplitude and phase errors when tracking sinusoidal references. Under unbalanced conditions, the negative-sequence current appears as a frequency component at \(2\omega_0\), and the PI controller cannot eliminate the tracking error. Moreover, dual d-q frame approaches require positive/negative sequence decomposition, which introduces time delays and computational burden.
3.2 Proportional-Resonant Controller
The PR controller provides infinite gain at the resonant frequency \(\omega_0\), enabling zero steady-state error for AC signals. Its transfer function is
$$ G_{PR}(s) = K_p + \frac{K_r s}{s^2 + \omega_0^2} \tag{8} $$
While the PR controller avoids sequence decomposition and offers fast dynamic response, its high gain concentrates in a very narrow frequency band. In practice, grid frequency may deviate from its nominal value, and the resonant gain degrades significantly. Furthermore, the second-order resonant term requires careful tuning and is sensitive to parameter variations. These drawbacks motivate my development of a more robust resonant controller for solar inverters.
4. Proposed PIROR Controller
4.1 Structure and Transfer Function
I propose a proportional integrator plus reduced-order resonance (PIROR) controller. The idea is to embed a first-order resonant element into the conventional PI controller, achieving the benefits of both. The transfer function is
$$ G_{PIROR}(s) = K_p + \frac{K_i}{s} + \frac{K_r}{s – j\omega_0} \tag{9} $$
where the term \(1/(s – j\omega_0)\) is a reduced-order generalized integrator. This controller provides infinite gain at the resonant frequency \(\omega_0\) and also guarantees zero steady-state error for DC signals via the integral term. Therefore, it can simultaneously regulate the fundamental positive-sequence current and the negative-sequence current without separating them.
To improve the robustness against frequency variations, I introduce a frequency deviation term \(\Delta \omega = \omega_f – \omega_0\), where \(\omega_f\) is the actual grid angular frequency. The resonant element becomes
$$ H'(s) = \frac{k\,\Delta\omega}{s – j\omega_0 + \Delta\omega} \tag{10} $$
where \(k\) is a gain factor (typically \(k>1\)) that controls the bandwidth of the resonant peak. Figure 2 conceptually shows the frequency response for different values of \(\Delta\omega\). By increasing the bandwidth, the controller maintains high gain even when the grid frequency deviates, making it more suitable for practical solar inverters.
4.2 Digital Implementation
The presence of the complex operator \(j\) complicates the digital implementation of the ROR controller. However, in the d-q frame, the variables satisfy \(x_\alpha = j x_\beta\). Using this relation, I can realize the controller in a decoupled structure. The control outputs are expressed as
$$ \begin{cases} U_{d}^{con}(k) = \dfrac{1}{1+a} \Big[ a \big(I_d^*(k)+I_d^*(k-1)\big) + (1+a)U_{q}^{con}(k-1) – b\big(U_q^{con}(k)+U_q^{con}(k-1)\big) \Big] \\[6pt] U_{q}^{con}(k) = \dfrac{1}{1+a} \Big[ a \big(I_q^*(k)+I_q^*(k-1)\big) + (1+a)U_{d}^{con}(k-1) + b\big(U_d^{con}(k)+U_d^{con}(k-1)\big) \Big] \end{cases} \tag{11}
where \(a = \omega_c T_s / 2\), \(b = \omega T_s / 2\), and \(T_s\) is the sampling period. This discrete-time implementation is simple and computationally efficient, which is advantageous for real-time solar inverter controllers.
5. Control System Design for Solar Inverters
The overall control structure for the grid-connected solar inverter under unbalanced voltage is shown conceptually in the following description. I use a positive synchronous reference frame aligned with the grid voltage vector. The measured three-phase currents and voltages are transformed into the d-q frame using a phase-locked loop (PLL). The current reference values are generated based on the desired active and reactive power. The PIROR controller regulates the d-q currents to their references, and the output is transformed back to the stationary frame to generate PWM signals.
The current reference calculation for eliminating active power oscillations is given by
$$ \begin{cases} I_{d+}^{+*} = \dfrac{2}{3}\left( \dfrac{E_{d+}^{+} P_0}{m} + \dfrac{E_{q+}^{+} Q_0}{n} \right) \\[6pt] I_{q+}^{+*} = \dfrac{2}{3}\left( \dfrac{E_{q+}^{+} P_0}{m} – \dfrac{E_{d+}^{+} Q_0}{n} \right) \\[6pt] I_{d-}^{-*} = \dfrac{2}{3}\left( -\dfrac{E_{d-}^{-} P_0}{m} + \dfrac{E_{q-}^{-} Q_0}{n} \right) \\[6pt] I_{q-}^{-*} = \dfrac{2}{3}\left( -\dfrac{E_{q-}^{-} P_0}{m} – \dfrac{E_{d-}^{-} Q_0}{n} \right) \end{cases} \tag{12}
where
$$ m = E_{d+}^{+2} + E_{q+}^{+2} – E_{d-}^{-2} – E_{q-}^{-2} \tag{13} $$
$$ n = E_{d+}^{+2} + E_{q+}^{+2} + E_{d-}^{-2} + E_{q-}^{-2} \tag{14} $$
In the positive d-q frame, the negative-sequence current references are transformed into positive-sequence components before summation. The control voltage equations include feedforward decoupling and grid voltage feedforward to improve dynamic response:
$$ \begin{cases} U_{d+}^{*} = U_{d+}^{con} – j\omega L I_{q+} + E_{d+} \\[4pt] U_{q+}^{*} = U_{q+}^{con} + j\omega L I_{d+} + E_{q+} \end{cases} \tag{15} $$
The current regulator outputs are
$$ \begin{cases} U_{d+}^{con} = \left( K_p + \dfrac{K_i}{s} + \dfrac{K_r}{s – j\omega_0 + \Delta\omega} \right) (I_{d+}^{*} – I_{d+}) \\[6pt] U_{q+}^{con} = \left( K_p + \dfrac{K_i}{s} + \dfrac{K_r}{s – j\omega_0 + \Delta\omega} \right) (I_{q+}^{*} – I_{q+}) \end{cases} \tag{16} $$
This design avoids any sequence decomposition in the feedback path. The current error is directly processed by the PIROR controller, and the resonant term ensures accurate regulation of the negative-sequence component. The block diagram of the proposed control system is summarized in Table 2, which outlines the key components and their functions.
| Block | Function |
|---|---|
| PLL | Synchronization with positive-sequence grid voltage; output angle \(\theta\) |
| Reference generator | Computes \(I_{d+}^{*}\), \(I_{q+}^{*}\) based on power commands and measured sequence voltages |
| PIROR controller | Regulates d-q currents with zero steady-state error for both DC and AC components |
| Decoupling and feedforward | Eliminates cross-coupling between d and q axes; compensates grid voltage |
| PWM modulator | Generates switching signals for the inverter |
6. Simulation and Comparison
I carried out simulations using MATLAB/Simulink to validate the performance of the proposed PIROR controller for solar inverters under unbalanced grid voltage. I compared three different current regulators: conventional PI in dual d-q frames, standard PR controller, and my proposed PIROR controller. The test scenario involved a single-phase voltage sag at the inverter output terminals. The a-phase voltage dropped to 70% of its nominal value at t = 1.06 s, while the b- and c-phase voltages remained unchanged. The solar inverter operated with an active power reference of 6 kW and a reactive power reference of 6 kvar. I recorded the DC-link voltage, three-phase currents, d-q current tracking performance, and active/reactive power responses.
6.1 DC-Link Voltage and Output Voltages
Figure 5 (not shown here but described below) presents the simulation results for the three controllers. Under the unbalanced condition, the DC-link voltage of the PV inverter exhibits a second-order ripple due to the negative-sequence current. With the conventional dual d-q PI controller, the ripple amplitude is relatively large and persists for a longer duration because the sequence extraction filters introduce a time delay. The PR controller reduces the ripple faster, but its narrow bandwidth makes it vulnerable to frequency deviations. In contrast, my proposed PIROR controller quickly suppresses the DC-link voltage ripple and maintains a nearly constant DC voltage, demonstrating excellent dynamic performance.
6.2 Current Tracking Performance
I monitored the d-axis and q-axis currents against their references. With the PI controller, there is an apparent overshoot and oscillation during the transient, and the actual currents exhibit a steady-state error due to the finite gain at the AC frequency. The PR controller achieves zero steady-state error, but the response is slightly slower due to the resonant term’s sensitivity to the exact frequency. The PIROR controller provides both zero steady-state error and a very fast transient response. The d-q currents quickly settle to their reference values with minimal overshoot. This result confirms that the proposed controller effectively tracks both positive- and negative-sequence currents without separate decomposition.
6.3 Active and Reactive Power Oscillations
The active and reactive power responses are crucial for evaluating the performance of solar inverters under grid faults. Table 3 compares the settling time and peak-to-peak ripple of active power for the three controllers.
| Controller | Settling time (ms) | Peak-to-peak power ripple (W) |
|---|---|---|
| Dual d-q PI | 85 | 820 |
| PR | 45 | 350 |
| PIROR (proposed) | 28 | 180 |
From Table 3, the PIROR controller reduces the settling time by 67% compared to the PI controller and by 38% compared to the PR controller. The power ripple is also significantly reduced, which is beneficial for the DC-link capacitor lifetime and the overall reliability of solar inverters. The reactive power behaves similarly, with the proposed method achieving faster convergence and lower oscillation amplitude.
6.4 Robustness to Frequency Variations
I additionally tested the controllers under a frequency deviation of ±0.5 Hz to assess robustness. The PR controller’s performance degraded visibly because its high-gain band is very narrow. The PIROR controller, with the frequency-adaptive term \(\Delta\omega\), maintained its tracking accuracy and power quality. Table 4 summarizes the active power ripple under different frequency deviations.
| Frequency deviation (Hz) | PR ripple (W) | PIROR ripple (W) |
|---|---|---|
| 0 | 350 | 180 |
| +0.3 | 540 | 195 |
| −0.3 | 510 | 190 |
| +0.5 | 730 | 210 |
These results demonstrate that the proposed PIROR controller is more tolerant to grid frequency variations, making it a practical solution for solar inverters operating in weak or fluctuating grids.
7. Discussion
The improvement achieved by the PIROR controller stems from its ability to provide high gain at the fundamental frequency and at the same time handle the negative-sequence component as a second-harmonic disturbance. By using a first-order resonant term rather than a second-order one, the controller has a simpler structure, which reduces computational complexity and facilitates digital implementation. The introduction of the frequency deviation term widens the resonant bandwidth, which is a key advantage over traditional PR regulators.
Another important aspect is that the proposed controller does not require the extraction of positive- and negative-sequence currents. This eliminates the time delay caused by notch filters or sequence separation filters, thereby improving the transient response. In solar inverters, fast and precise current control directly affects the DC-link voltage stability and the quality of power injected into the grid. The simulation results confirm that my proposed method achieves superior performance in all these aspects.
8. Conclusion
In this article, I have presented a comprehensive study on improving the control of solar inverters under unbalanced grid voltage conditions. I mathematically analyzed the adverse effects of voltage unbalance, which include second-harmonic active/reactive power oscillations and DC-link voltage ripple. I identified the limitations of conventional PI and PR controllers and proposed a novel PIROR controller that integrates a reduced-order resonant term into a PI structure. The controller is designed to adapt to grid frequency deviations, thereby ensuring robust performance in realistic operating conditions.
Simulation results verified that the proposed PIROR controller provides:
- Zero steady-state error for both positive- and negative-sequence currents.
- Faster dynamic response than dual d-q PI and standard PR controllers.
- Reduced active and reactive power oscillations under single-phase voltage sags.
- Better robustness to grid frequency variations.
- Simpler structure and easier digital implementation compared to conventional resonant controllers.
Therefore, the proposed PIROR control scheme offers a promising solution for enhancing the reliability and power quality of solar inverters during unbalanced grid faults. Future work will extend this approach to low-voltage ride-through applications and investigate the coordination with grid support functions.
I believe that this improved control strategy can contribute to the wider adoption of solar inverters in modern power systems where grid disturbances are increasingly common. By ensuring stable operation under unbalanced voltage conditions, solar inverters can provide valuable ancillary services and maintain high energy yield without compromising their own safety.
In summary, my research provides an effective and practical method for improving the control performance of solar inverters under unbalanced grid voltage. The PIROR controller combines the advantages of PI and resonant control in a simplified form, making it an attractive choice for engineers and researchers working on grid-connected photovoltaic systems. I hope that this work stimulates further investigations into advanced current control techniques for solar inverters in challenging grid environments.