In this paper, I present a comprehensive design of a solar inverter control strategy that is capable of operating reliably under three-phase unbalanced grid conditions. The low-voltage distribution network often suffers from unbalanced loads and asymmetrical line parameters, which leads to unbalanced grid voltages. A solar inverter connected to such a grid must therefore be equipped with a robust phase-locked loop (PLL) and a control scheme that can handle negative-sequence components. I first investigate the positive-sequence voltage extraction and phase-locking problem and propose a PLL based on the Second Order Generalized Integrator (SOGI). Then, I analyze the instantaneous power flow of the solar inverter under unbalanced grid voltages and derive current reference calculation methods for different control objectives. Finally, I build a PSCAD/EMTDC simulation model to verify the proposed design. The simulation results demonstrate that the solar inverter maintains balanced grid currents with low total harmonic distortion (THD) even during grid faults, and it recovers quickly after the fault is cleared.
1 Introduction
The application trend of distributed photovoltaic systems is very significant in modern power grids. As a key interface between the photovoltaic array and the grid, the solar inverter must ensure stable power conversion and high power quality under various grid conditions. However, due to the unbalanced parameters of distribution lines and the uneven distribution of single-phase loads, three-phase unbalance is a common phenomenon in low-voltage distribution networks. Consequently, a solar inverter connected to such a grid must be designed with a dedicated control strategy that takes the unbalanced grid voltage into account.
Under unbalanced grid conditions, the grid voltage contains positive-sequence, negative-sequence, and zero-sequence components. For a three-phase three-wire system, the zero-sequence component can be ignored. The presence of the negative-sequence component causes ripple in the active and reactive power at twice the fundamental frequency, and it also induces unbalanced currents in the solar inverter. Therefore, the control strategy of the solar inverter needs to handle both the positive- and negative-sequence components independently.
To ensure the effective operation of a solar inverter under unbalanced grid voltage, two key aspects must be addressed: the phase-locked loop (PLL) and the control strategy. In the literature, several methods have been proposed for phase-locking under unbalanced conditions. In [1], a quarter-cycle delay method based on symmetrical components was designed. However, the introduction of delay limits the dynamic response speed. In [2], a software PLL based on dual synchronous reference frame decoupling was proposed. This method has high steady-state accuracy, but it depends on phase feedback, which may cause large overshoot and long recovery time during grid phase jumps. In [3], a low-pass filter was added before the conventional PLL, but this leads to phase shift and slow response. In [4], an adaptive observer was used for grid phase locking, but the algorithm is computationally intensive and complex.
For the control problem of a solar inverter under unbalanced grid conditions, [5-6] designed a proportional-resonant (PR) controller based on the mathematical model of the inverter in stationary coordinates. The PR controller achieves zero steady-state error for AC signals, but its frequency adaptability is poor; when the grid frequency deviates, the control performance degrades.
In this paper, I propose a SOGI-based PLL for unbalanced grid phase-locking and design current reference calculation methods for different control objectives. The rest of the paper is organized as follows: Section 2 presents the SOGI-based PLL. Section 3 describes the control strategy, including instantaneous power analysis and current reference calculation. Section 4 gives the simulation verification. Section 5 concludes the work.
2 SOGI-Based Phase-Locked Loop for Unbalanced Grid
Unbalanced grid voltages can be decomposed into positive-sequence, negative-sequence, and zero-sequence components:
$$
v_{abc} = v_{abc}^{+} + v_{abc}^{-} + v_{abc}^{0}
$$
where the superscripts \(+\), \(-\), and \(0\) denote positive, negative, and zero sequences, respectively. For a three-phase three-wire solar inverter system, the zero-sequence component can be neglected. The positive- and negative-sequence components can be obtained using the symmetrical component transformation:
$$
v_{abc}^{+} = T^{+} v_{abc} = \frac{1}{3} \begin{bmatrix} 1 & a & a^2 \\ a^2 & 1 & a \\ a & a^2 & 1 \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix}
$$
$$
v_{abc}^{-} = T^{-} v_{abc} = \frac{1}{3} \begin{bmatrix} 1 & a^2 & a \\ a & 1 & a^2 \\ a^2 & a & 1 \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix}
$$
where \(a = e^{j 2\pi / 3}\) is the complex operator. Applying the Clarke transformation, we obtain the positive- and negative-sequence components in the \(\alpha\beta\) stationary frame:
$$
v_{\alpha\beta}^{+} = T_{\alpha\beta} v_{abc}^{+}, \quad v_{\alpha\beta}^{-} = T_{\alpha\beta} v_{abc}^{-}
$$
$$
T_{\alpha\beta} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}
$$
Through further mathematical manipulation, the positive- and negative-sequence components in the \(\alpha\beta\) frame can be expressed in terms of the original voltage vector and a 90° phase-shifted version:
$$
v_{\alpha\beta}^{+} = \frac{1}{2} \begin{bmatrix} 1 & -q \\ q & q \end{bmatrix} v_{\alpha\beta}
$$
$$
v_{\alpha\beta}^{-} = \frac{1}{2} \begin{bmatrix} 1 & q \\ -q & q \end{bmatrix} v_{\alpha\beta}
$$
where \(q\) is a phase-shift operator that lags the input signal by 90°. This relationship suggests that the positive- and negative-sequence separation can be achieved by orthogonal signal generation. The Second Order Generalized Integrator (SOGI) is an effective way to generate a quadrature signal with the same frequency. The structure of the SOGI is shown in Figure 1.
Based on the SOGI structure, the transfer functions are derived as follows:
$$
D(s) = \frac{v’}{v} = \frac{k \omega’ s}{s^2 + k \omega’ s + \omega’^2}
$$
$$
Q(s) = \frac{q v’}{v} = \frac{k \omega’ s}{s^2 + k \omega’ s + \omega’^2}
$$
In these equations, \(v\) is the input sinusoidal signal, \(\omega’\) is the filter center frequency, and \(k\) is the damping coefficient, which is usually set to \(\sqrt{2}\). When the center frequency of the SOGI matches the input signal frequency, the output \(v’\) has the same amplitude and phase as \(v\), while \(q v’\) has the same amplitude but lags by 90°. Therefore, the SOGI can perfectly serve as a quadrature signal generator for the positive- and negative-sequence separation.
The Bode plots of \(D(s)\) and \(Q(s)\) for different values of \(k\) show that the SOGI has a band-pass characteristic. When \(k\) is small, the bandwidth is narrow and the filtering effect is stronger, but the dynamic response is slower. When \(k\) is larger, the response becomes faster but the filtering performance degrades. In the proposed solar inverter control system, I choose \(k = \sqrt{2}\) as a trade-off between filtering and dynamic performance.
The complete SOGI-based PLL structure is shown in Figure 3. The three-phase grid voltage is first transformed into the \(\alpha\beta\) frame. Then, the SOGI blocks produce two orthogonal signals for each axis, which are used to calculate the positive-sequence component. After that, the positive-sequence voltage is transformed into the synchronous \(dq\) reference frame using the estimated phase angle. A PI controller regulates the \(q\)-axis voltage to zero, which yields the grid angular frequency and phase angle. Since the SOGI blocks have no phase delay at the fundamental frequency, the PLL can achieve fast and accurate phase-locking even under unbalanced grid voltage.
One important advantage of the SOGI-based PLL is that it avoids the computational burden of matrix inversion or complex observers. The SOGI can be easily implemented in a digital signal processor with a simple difference equation. Moreover, the SOGI inherently acts as a band-pass filter, which rejects high-frequency harmonics and dc offset. This property makes the solar inverter PLL robust against grid voltage distortion and measurement noise.
3 Control Strategy of Solar Inverter under Unbalanced Grid
3.1 Instantaneous Power Analysis
When the grid voltage is unbalanced, both the voltage and current contain positive- and negative-sequence components. I transform the three-phase voltage and current into the synchronous reference frame using separate positive- and negative-sequence dq transformations. The instantaneous active and reactive powers at the solar inverter output can be expressed as:
$$
P(t) = p_{d}^{+} + p_{q}^{+} + p_{d}^{-} + p_{q}^{-} = p_0 + p_{c2} \cos(2\omega t) + p_{s2} \sin(2\omega t)
$$
$$
Q(t) = q_{d}^{+} + q_{q}^{+} + q_{d}^{-} + q_{q}^{-} = q_0 + q_{c2} \cos(2\omega t) + q_{s2} \sin(2\omega t)
$$
where \(p_0\) and \(q_0\) are the average active and reactive powers, while \(p_{c2}\), \(p_{s2}\), \(q_{c2}\), and \(q_{s2}\) are the coefficients of the double-frequency power ripples. These coefficients depend on the positive- and negative-sequence dq voltages and currents. The matrix relationship can be written as:
$$
\begin{bmatrix} p \\ q \\ p_{c2} \\ p_{s2} \end{bmatrix} = M_e \begin{bmatrix} I_{dq}^{+} \\ I_{dq}^{-} \end{bmatrix}
$$
The matrix \(M_e\) is derived from the instantaneous power theory. The explicit form of \(M_e\) for a solar inverter with an LCL filter is given in the following table:
| Symbol | Expression |
|---|---|
| \(p\) | \(v_{d}^{+} i_{d}^{+} + v_{q}^{+} i_{q}^{+} + v_{d}^{-} i_{d}^{-} + v_{q}^{-} i_{q}^{-}\) |
| \(q\) | \(v_{q}^{+} i_{d}^{+} – v_{d}^{+} i_{q}^{+} + v_{q}^{-} i_{d}^{-} – v_{d}^{-} i_{q}^{-}\) |
| \(p_{c2}\) | \(v_{d}^{+} i_{d}^{-} + v_{q}^{+} i_{q}^{-} + v_{d}^{-} i_{d}^{+} + v_{q}^{-} i_{q}^{+}\) |
| \(p_{s2}\) | \(v_{q}^{+} i_{d}^{-} – v_{d}^{+} i_{q}^{-} – v_{q}^{-} i_{d}^{+} + v_{d}^{-} i_{q}^{+}\) |
| \(q_{c2}\) | \(v_{q}^{+} i_{d}^{-} – v_{d}^{+} i_{q}^{-} + v_{q}^{-} i_{d}^{+} – v_{d}^{-} i_{q}^{+}\) |
| \(q_{s2}\) | \(-v_{d}^{+} i_{d}^{-} – v_{q}^{+} i_{q}^{-} + v_{d}^{-} i_{d}^{+} + v_{q}^{-} i_{q}^{+}\) |
According to the actual control requirements, the current reference can be calculated from the inverse matrix:
$$
\begin{bmatrix} I_{dq}^{+*} \\ I_{dq}^{-*} \end{bmatrix} = M_e^{-1} \begin{bmatrix} p^* \\ q^* \\ p_{c2}^* \\ p_{s2}^* \end{bmatrix}
$$
where the superscript \(*\) denotes the reference value. By choosing different values for the power ripple coefficients, I can achieve different control objectives.
3.2 Current Reference Calculation for Different Control Objectives
There are three typical control objectives for a solar inverter operating under unbalanced grid voltage:
| Control Objective | Description | Power Ripple Settings |
|---|---|---|
| 1. Suppress active power ripple | The instantaneous active power is kept constant, so \(p_{c2}^* = 0\) and \(p_{s2}^* = 0\). | \(p_{c2}^*=0, p_{s2}^*=0\) |
| 2. Suppress reactive power ripple | The instantaneous reactive power is kept constant, so \(q_{c2}^* = 0\) and \(q_{s2}^* = 0\). | \(q_{c2}^*=0, q_{s2}^*=0\) |
| 3. Suppress negative-sequence current | The grid current is balanced, so \(I_{dq}^{-*} = 0\). | \(i_{d}^{-*}=0, i_{q}^{-*}=0\) |
For the negative-sequence current suppression objective, the current references are simply:
$$
i_{d}^{+*} = \frac{p^* v_{d}^{+} + q^* v_{q}^{+}}{(v_{d}^{+})^2 + (v_{q}^{+})^2}, \quad i_{q}^{+*} = \frac{p^* v_{q}^{+} – q^* v_{d}^{+}}{(v_{d}^{+})^2 + (v_{q}^{+})^2}
$$
$$
i_{d}^{-*} = 0, \quad i_{q}^{-*} = 0
$$
For the active power ripple suppression objective, the positive- and negative-sequence current references are more involved. They can be obtained by solving the linear system in the equation above. The resulting expressions involve both the positive- and negative-sequence voltages. I do not present the full algebraic expansion here because it is lengthy; instead, the key idea is to set the double-frequency active power coefficients to zero. The same method applies to the reactive power ripple suppression objective.
In practice, the choice of control objective depends on the grid code and the application scenario. For example, if the solar inverter is connected to a weak grid with a high impedance, suppressing the active power ripple can help avoid dc-link voltage oscillation. If the grid is stiff and the main concern is current quality, suppressing the negative-sequence current is preferable. In this paper, I focus on the negative-sequence current suppression objective, because it is the most straightforward and results in balanced grid currents.
3.3 Dual Synchronous Reference Frame Control Structure
To implement the proposed current reference calculation, I use a dual synchronous reference frame (SRF) control structure. The positive-sequence current is regulated in the positive dq frame, while the negative-sequence current is regulated in the negative dq frame. The control block diagram is shown in Figure 4
The control system consists of two decoupled current controllers. The current controller in each frame is a PI regulator with feedforward compensation of the cross-coupling terms and the grid voltage. The output of the PI regulators is transformed back to the \(\alpha\beta\) frame, and then the sum of the positive- and negative-sequence switching signals is sent to the PWM modulator. The modulation signals are given by:
$$
v_{d}^{c+} = v_{d}^{+} – \omega L i_{q}^{+} + \Delta v_{d}^{+}
$$
$$
v_{q}^{c+} = v_{q}^{+} + \omega L i_{d}^{+} + \Delta v_{q}^{+}
$$
$$
v_{d}^{c-} = v_{d}^{-} + \omega L i_{q}^{-} + \Delta v_{d}^{-}
$$
$$
v_{q}^{c-} = v_{q}^{-} – \omega L i_{d}^{-} + \Delta v_{q}^{-}
$$
where \(\Delta v_{d}^{+}\), \(\Delta v_{q}^{+}\), \(\Delta v_{d}^{-}\), and \(\Delta v_{q}^{-}\) are the outputs of the PI current controllers. The parameters of the PI controllers are designed based on the LCL filter transfer function. In my design, the current controller gains are \(K_p = 8\) and \(K_i = 10\). The voltage controller gains are \(K_p = 0.5\) and \(K_i = 0.08\). These values are tuned to achieve a fast dynamic response while maintaining sufficient stability margin.
One challenge in the dual SRF control structure is that the positive- and negative-sequence current loops can interact with each other if the phase-locked loop is not accurate. However, with the SOGI-based PLL described in Section 2, the positive-sequence phase angle is extracted precisely, so the positive- and negative-sequence components are well decoupled. This makes the dual SRF control structure effective for the solar inverter.
4 Simulation and Verification
To verify the correctness of the proposed control strategy, I built a three-level solar inverter model using the electromagnetic transient simulation software PSCAD/EMTDC. The simulation parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Grid voltage peak (V) | 220 |
| DC-link capacitors (\(\mu F\)) | \(C_1 = C_2 = 300\) |
| System rated power (kW) | 10 |
| LCL filter – grid-side inductor (mH) | \(L_1 = 0.05\) |
| LCL filter – inverter-side inductor (mH) | \(L_2 = 0.45\) |
| Filter capacitor (\(\mu F\)) | \(C = 10\) |
| Damping resistor (\(\Omega\)) | \(R = 1\) |
| Current controller gains | \(K_p = 8\), \(K_i = 10\) |
| Voltage controller gains | \(K_p = 0.5\), \(K_i = 0.08\) |
In the simulation, the grid A phase experiences a ground fault at \(t = 0.2\) s, and the voltage drops to 50% of its original value. The fault is cleared at \(t = 0.25\) s, and the voltage returns to normal. The three-phase grid voltage waveform is shown in Figure 5. The A-phase voltage dip creates a pronounced imbalance in the three-phase voltage system.
First, I simulated the conventional control strategy, which uses a standard single SRF PLL and a positive-sequence current controller only. The resulting grid currents and the THD analysis of the A-phase current are shown in Figure 6. It can be observed that during the fault period, the solar inverter output currents become three-phase asymmetrical, and the THD increases significantly, far exceeding the grid code requirements. The conventional controller cannot maintain balanced currents because it does not have the ability to regulate the negative-sequence current. The negative-sequence voltage produces a large negative-sequence current, which distorts the current waveform and causes double-frequency power oscillations.
Next, I simulated the proposed control strategy based on the SOGI PLL and the dual SRF control with negative-sequence current suppression. The resulting grid currents and the THD analysis of the A-phase current are shown in Figure 7. From the results, it is clear that after the A-phase voltage sag, the solar inverter grid currents still maintain good symmetry. With the action of the dual SRF controllers, the THD of the grid current remains around 3%, which is well within the acceptable limit. Moreover, when the fault is cleared, the solar inverter quickly returns to its normal operating state without significant overshoot or oscillation.
The simulation results confirm that the proposed control strategy can effectively handle grid voltage imbalance. The SOGI-based PLL is able to extract the positive-sequence phase angle accurately, and the dual SRF controllers successfully suppress the negative-sequence current. The dynamic performance of the solar inverter is also satisfactory: the current control reacts quickly to the fault condition and recovers smoothly when the fault clears.
I also performed additional simulations for the other two control objectives (active power ripple suppression and reactive power ripple suppression). For brevity, I do not present all waveforms here; however, the results are consistent with the theoretical analysis. The proposed current reference calculation method is flexible and can be easily adapted to different control requirements by changing the desired power ripple coefficients.
5 Conclusion
In this paper, I have designed a solar inverter control strategy that is suitable for three-phase unbalanced grid conditions. The main contributions are summarized as follows:
(1) I designed a SOGI-based PLL that can quickly and accurately lock the grid phase under unbalanced voltage conditions. The SOGI generates orthogonal signals for positive- and negative-sequence separation, and the PLL achieves good filtering performance with a fast dynamic response.
(2) I designed current reference calculation methods for different control objectives, including suppressing active power ripple, suppressing reactive power ripple, and suppressing negative-sequence current. These methods are based on the instantaneous power theory and are expressed as a unified matrix equation.
(3) I established a PSCAD/EMTDC simulation model and verified the correctness of the proposed methods. The simulation results show that the solar inverter can maintain balanced grid currents with low THD during grid faults, and it recovers quickly after the fault is cleared.
The proposed control strategy is practical and effective for solar inverter applications in low-voltage distribution networks where three-phase unbalance is common. The use of a SOGI-based PLL reduces the computational burden compared to adaptive observer methods, and the dual SRF control provides a straightforward way to handle the negative-sequence component. Future work will focus on extending the method to more complex grid fault scenarios, such as voltage swells and phase jumps, and on optimizing the controller parameters for different grid impedance conditions.
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