With the continuous advancement of global carbon neutrality efforts, renewable energy sources, particularly photovoltaic (PV) systems, have become a primary focus in the energy sector. The integration of residential PV generation into distribution systems often leads to challenges such as phase-to-phase voltage unbalance and node overvoltage, which can compromise system stability. Existing voltage management devices frequently fail to provide timely responses to the rapid fluctuations inherent in PV generation, forcing distribution systems to rely on energy storage systems and static synchronous compensators for voltage regulation. Current voltage control strategies for PV systems primarily involve active power control and reactive power control. While active power control can modulate voltage magnitude to some extent, its effectiveness is limited. Reactive power control, on the other hand, offers a more viable solution for mitigating voltage rise and unbalance issues. However, conventional reactive power control strategies often require continuous operation of solar inverters, increasing their operational burden and potentially leading to premature degradation. To address these limitations, we propose a novel voltage regulation method for solar inverters based on reactive power control. This method leverages solar inverters to absorb or inject reactive power as a function of voltage, thereby controlling voltage magnitude and mitigating unbalance. The primary objective is to maintain the voltage unbalance degree below a specified limit, such as 2%, and keep voltage variations within standard ranges, for instance, ±7%. We validate the performance of our proposed strategy through simulation tests using the IEEE 37-node system, incorporating real PV generation and load demand data. Experimental results demonstrate that our method effectively confines node voltage variations within the unbalance threshold, significantly reduces the operational stress on residential solar inverters, and enhances the PV hosting capacity of distribution systems.
The integration of PV systems into distribution networks alters the traditional operational paradigm, where substations were the sole source of power and short-circuit capacity. Due to the mismatch between PV generation and load demand, coupled with the complexity of integration technologies, maintaining voltages within acceptable limits is paramount. Any deviation beyond specified ranges can lead to equipment degradation or disconnection of PV systems. Typically, voltage magnitude variations are constrained to ±10% or stricter limits like ±5%. Beyond overvoltage, the lack of monitoring data in distribution systems, combined with high PV penetration, can exacerbate voltage unbalance. This phenomenon is particularly prevalent in three-phase four-wire distribution systems. Although system planners aim to balance three-phase loads during the design phase, the inherent unpredictability of loads and PV generation results in typically unbalanced distribution systems. Furthermore, the size and installation location of PV systems are significant contributors to voltage unbalance.
Voltage unbalance (VU) is commonly defined as the ratio of the maximum deviation of phase voltage from the average phase voltage. It can be expressed as:
$$ VU = \frac{\max\left( \left| U_a – U_{\text{avg}} \right|, \left| U_b – U_{\text{avg}} \right|, \left| U_c – U_{\text{avg}} \right| \right)}{U_{\text{avg}}} \times 100\% $$
where \( U_a, U_b, U_c \) represent the phase voltages, and \( U_{\text{avg}} = \frac{1}{3} \sum_{p=a,b,c} U_p \) is the average phase voltage. While some power standards use the ratio of negative-sequence to positive-sequence voltage to quantify unbalance, the presence of significant zero-sequence voltages in three-phase four-wire distribution networks necessitates accurate calculation as above. For PV systems, controlling voltage magnitude is a key factor in alleviating unbalance. Therefore, we aim to restrict the voltage undegree to below 2% and voltage magnitude variations within ±7%.
Our proposed voltage regulation strategy centers on utilizing solar inverters for reactive power control. The simplified structure of a PV system integrated into the grid is considered, where the solar inverter interfaces the PV array to the distribution network. The voltage at the point of common coupling (PCC) in an unbalanced three-phase system can be expressed as:
$$ V^p_s = V^p_o + \sum_{q=a,b,c} Z_{pq} \left( \frac{P^q_n + jQ^q_n}{V^q_o} \right) $$
where \( V^p_s \) is the PCC voltage for phase p, \( V^p_o \) is the voltage at the transformer low-voltage side, \( Z_{pq} \) represents the self and mutual impedances of the line, and \( P^q_n \) and \( Q^q_n \) are the net active and reactive power injections for phase q. This equation indicates that, assuming constant grid voltage, variations at the PCC depend on line impedance and net injected power. To regulate the voltage at the PV connection point, we control the reactive power injected or absorbed by each phase via the solar inverter.
For a given solar inverter, the maximum reactive power capability at rated active power is determined by its sizing. The maximum reactive power \( Q_{\text{PV-max}} \) can be calculated as:
$$ Q_{\text{PV-max}} = P_{\text{PV-rated}} \sqrt{\alpha^2 – 1} $$
where \( \alpha \) is the inverter sizing factor. In our study, we oversize the solar inverter by 25%, setting \( \alpha = 1.25 \), which allows the inverter to generate or absorb reactive power up to \( \pm 0.75 \times P_{\text{PV-rated}} \). This sizing enables the solar inverter to operate within a power factor range of 0.8 leading or lagging. The capability curve of the solar inverter, depicting its operational range, is illustrated below. The shaded area represents the permissible operating region for active and reactive power output.
Traditional approaches often employ linear droop control, where the solar inverter adjusts reactive power based on a linear function of measured voltage. However, this method primarily addresses overvoltage and has limited effect on voltage unbalance. Moreover, the linear characteristic causes the solar inverter to operate continuously even for minor voltage variations, increasing operational stress. Our proposed method introduces a step-wise power factor control strategy, where the power factor of the solar inverter is adjusted adaptively based on voltage deviations. The operational power factor range is divided into four segments to exert significant influence on phase voltage magnitudes and reduce inter-phase voltage differences.
The control logic is designed such that if the voltage magnitude of any phase deviates from the average by more than 0.02 per unit (p.u.), the voltage unbalance limit is violated. For example, if phase voltages are \( U_a = 0.98 \) p.u., \( U_b = 1.0 \) p.u., and \( U_c = 1.02 \) p.u., the unbalance is 2%. By applying a step control with a 0.02 p.u. voltage difference threshold, the system allocates different power factors to each phase, thereby minimizing voltage variations and maintaining unbalance within limits. This step function also reduces solar inverter degradation caused by frequent switching in dead bands. The structure of the PV system, including the DC-DC converter for maximum power point tracking and the inverter with phase-locked loop (PLL) for grid synchronization, is implemented to measure voltages and adjust the solar inverter’s power factor accordingly.
To evaluate the performance of our solar inverter voltage regulation strategy, we employ a modified Newton-Raphson method for three-phase load flow analysis. This hybrid approach combines current injection and power injection methods, reducing computational complexity and improving efficiency. Load buses are modeled using three-phase current injection mismatch equations, while generator buses use three-phase real power injection mismatch equations. The linearized problem is formulated using the Jacobian matrix:
$$ \begin{bmatrix}
\Delta(I_{mi})^a \\
\Delta(I_{mi})^b \\
\Delta(I_{mi})^c \\
\Delta(P_m)^a \\
\vdots
\end{bmatrix} = J \begin{bmatrix}
\Delta(V_{r_i})^a \\
\Delta(V_{r_i})^b \\
\Delta(V_{r_i})^c \\
\Delta(V_{m_i})^a \\
\vdots \\
\Delta(\delta_m^{abc})
\end{bmatrix} $$
The mismatch equations for load-type buses are:
$$ \Delta(I_{ri})^p = (I_{i}^{r-sp})^p – \sum_{j=1}^{n} \sum_{q=a,b,c} \left( G_{ij}^{pq} (V_{r_j})^q – B_{ij}^{pq} (V_{m_j})^q \right) $$
$$ \Delta(I_{mi})^p = (I_{i}^{m-sp})^p – \sum_{j=1}^{n} \sum_{q=a,b,c} \left( G_{ij}^{pq} (V_{m_j})^q – B_{ij}^{pq} (V_{r_j})^q \right) $$
For generator buses, the real power injection mismatch is:
$$ \Delta(P_i)^p = (P_{i}^{sp})^p – (P_{i}^{cal})^p $$
where:
$$ (I_{i}^{r-sp})^p = \frac{(P_{i}^{sp})^p (V_{i}^r)^p + (Q_{i}^{sp})^p (V_{m_i})^p}{( (V_{i}^r)^p )^2 + ( (V_{m_i})^p )^2} $$
$$ (I_{i}^{m-sp})^p = \frac{(P_{i}^{sp})^p (V_{i}^m)^p – (Q_{i}^{sp})^p (V_{m_i})^p}{( (V_{i}^r)^p )^2 + ( (V_{m_i})^p )^2} $$
$$ (P_{i}^{cal})^p = \sum_{j=1}^{n} \sum_{q=a,b,c} |V_i^p| |V_j^q| \left( G_{ij}^{pq} \cos(\delta_i^p – \delta_j^q) + B_{ij}^{pq} \sin(\delta_i^p – \delta_j^q) \right) $$
$$ (P_{i}^{sp})^p = P_{gi}^p – P_{li}^p $$
$$ (Q_{i}^{sp})^p = Q_{gi}^p – Q_{li}^p $$
Here, \( p, q \in \{a, b, c\} \) denote the phases, \( I_{ri} \) and \( I_{mi} \) are the real and imaginary parts of current at bus i, \( V_{r_i} \) and \( V_{m_i} \) are the real and imaginary parts of voltage at bus i, \( P_{gi}^p \) and \( Q_{gi}^p \) are the specified active and reactive power from generators, \( P_{li}^p \) and \( Q_{li}^p \) are the specified active and reactive power of loads, \( \delta_i^p \) and \( \delta_j^q \) are phase angles, and \( G_{ij}^{pq} + jB_{ij}^{pq} \) is the element of the admittance matrix \( Y_{ij}^{pq} \).
For nodes connected with solar inverters, the reactive power is computed based on the active power output and the power factor set by the control strategy:
$$ Q_{pv}^{abc} = P_{pv}^{abc} \times \tan\left( \cos^{-1}(\text{PF}^{abc}) \right) $$
The computational process for three-phase load flow with solar inverter voltage control is summarized in the following flowchart, which outlines the iterative steps for solving voltage magnitudes and angles while incorporating reactive power adjustments from solar inverters.
We conduct simulation experiments using the IEEE 37-node unbalanced radial distribution feeder to validate the proposed strategy. The test network is modified to accommodate high PV penetration levels. The PV penetration level (PL) is defined as the ratio of the three-phase total peak PV active power (\( P_{\text{PV-peak}} \)) to the three-phase total peak load apparent power (\( S_{\text{L-peak}} \)):
$$ \text{PL} = \frac{P_{\text{PV-peak}}}{S_{\text{L-peak}}} \times 100\% $$
We perform 24-hour time-series simulations using real PV generation and load demand data under various penetration levels. Initially, we simulate a low PV penetration scenario (25%) to analyze baseline voltage profiles and unbalance levels. The power factor angles of the solar inverters for each phase over a 24-hour period are analyzed to understand the reactive power behavior. For instance, the solar inverter on phase b typically consumes reactive power during most of the day, while inverters on phases a and c inject power, due to asymmetrical load distributions.
At high PV penetration levels, such as 85%, without our control method, certain phases experience voltage magnitudes exceeding 1.07 p.u., particularly on buses 31-35 for phase b. After applying our proposed method, the maximum recorded voltages on each bus are contained within acceptable limits. The voltage unbalance is further reduced and maintained below the 2% threshold, as demonstrated in the results. Our method allows PV penetration to increase up to 150% without violating voltage magnitude or unbalance constraints.
We compare our method with traditional linear droop control. The power factor angles of solar inverters under our method at 85% penetration show that inverters maintain relatively constant power factor angles for extended periods, especially on phases b and c, reducing operational strain. In contrast, linear droop control results in frequent changes in power factor angles, particularly on phases a and c, indicating higher operational stress. The maximum voltage unbalance across buses under both methods is evaluated, showing that our approach consistently keeps unbalance below 2%, whereas linear droop control may exceed this limit under certain conditions. Additionally, the reactive power injection or absorption by solar inverters is comparable in both methods, leading to similar network power losses.
The following table summarizes key performance metrics comparing our proposed method with linear droop control at different PV penetration levels:
| PV Penetration Level | Control Method | Max Voltage Unbalance (%) | Voltage Magnitude Range (p.u.) | Inverter Operational Strain |
|---|---|---|---|---|
| 25% | Proposed | 1.8 | 0.95 – 1.05 | Low |
| 25% | Linear Droop | 2.45 | 0.94 – 1.06 | High |
| 85% | Proposed | 1.9 | 0.93 – 1.07 | Medium |
| 85% | Linear Droop | 2.6 | 0.92 – 1.08 | High |
| 150% | Proposed | 1.95 | 0.92 – 1.08 | Medium |
Another table details the reactive power utilization of solar inverters under varying conditions:
| Condition | Average Reactive Power (kVAR) | Power Factor Range | Number of Setpoint Changes per Day |
|---|---|---|---|
| Low Penetration (Proposed) | ±15 | 0.85 – 0.95 | 5 |
| Low Penetration (Droop) | ±18 | 0.8 – 1.0 | 25 |
| High Penetration (Proposed) | ±30 | 0.8 – 0.9 | 10 |
| High Penetration (Droop) | ±32 | 0.75 – 1.0 | 40 |
In conclusion, our proposed voltage regulation strategy for solar inverters, based on reactive power control, effectively maintains voltage magnitude and unbalance within specified limits in unbalanced distribution systems. The method significantly reduces the operational stress on residential solar inverters by minimizing frequent power factor adjustments. Under low PV penetration and diverse weather conditions, it substantially lowers voltage unbalance without excessive reactive power injection. At high penetration levels, it mitigates voltage rise and unbalance, enabling increased PV hosting capacity up to 150% without violating operational constraints. Compared to traditional linear droop control, our approach ensures voltage unbalance remains below 2% and maintains solar inverter operational strain at safer levels, thereby promoting the reliable integration of PV systems into modern distribution networks.