In the context of global carbon neutrality efforts, the integration of residential solar photovoltaic systems into distribution networks has become increasingly prevalent. However, this integration often leads to significant operational challenges, particularly voltage unbalance and overvoltage issues across different phases. To address these problems, I have developed a novel voltage regulation strategy for solar inverters based on reactive power control. This method focuses on using the solar inverter to absorb or inject reactive power as a function of voltage, thereby controlling voltage amplitude and mitigating voltage unbalance. My approach ensures that voltage unbalance remains below predefined limits while keeping voltage variations within standard operational ranges.
The core of my research involves a detailed analysis of the voltage distribution and voltage unbalance in photovoltaic-integrated systems. In a typical unbalanced distribution system, the voltage at the point of common coupling can be expressed mathematically. The relationship between injected power and voltage variation is critical for understanding how solar inverters can regulate system voltage. My methodology is validated through comprehensive simulations using the IEEE 37-bus system, incorporating real photovoltaic generation and load demand data. The results demonstrate that my strategy effectively controls node voltage amplitudes within acceptable unbalance thresholds while significantly reducing the operational stress on residential solar inverters.
A key advantage of my proposed method is its ability to enhance the photovoltaic hosting capacity of distribution systems. By intelligently managing reactive power flow, the solar inverter can operate more efficiently without violating voltage constraints. This is particularly important in high-penetration scenarios where traditional voltage regulation devices often fail to respond adequately to rapid fluctuations in solar generation.
The proposed control scheme introduces a step-function approach to power factor adjustment based on voltage deviation. Unlike conventional linear droop control methods, my strategy reduces unnecessary continuous operation of the solar inverter, thereby extending its operational lifespan and reducing thermal stress. The scheme divides the power factor operational range into four distinct zones, allowing the solar inverter to adapt its reactive power output dynamically while maintaining the power factor within permissible limits.
System Voltage Distribution and Voltage Unbalance
In unbalanced radial distribution systems, the integration of residential solar photovoltaic systems fundamentally alters the operational paradigm. The distribution network no longer relies solely on the substation as the unique power source and short-circuit capacity provider. The mismatch between photovoltaic generation and load demand, combined with the complexity of integration technology, makes it essential to maintain voltage within acceptable limits. Any violation of these limits could lead to equipment degradation or disconnection of photovoltaic systems. In my study, I have defined voltage amplitude variations as ±10% and ±5% for standard operations, with a stricter limit of ±7% for the proposed control strategy.
Besides overvoltage, the lack of monitoring data in distribution systems also leads to voltage unbalance, especially in feeders with high photovoltaic penetration. This phenomenon is typically observed in three-phase four-wire distribution systems. Although system designers attempt to balance three-phase loads during the planning phase, the inherent uncertainty of loads and photovoltaic generation makes typical distribution systems almost always unbalanced. Furthermore, the sizing and installation location of photovoltaic systems are also contributing factors to unbalanced system voltages.
The voltage unbalance factor is mathematically defined as the ratio of the maximum deviation of phase voltage from the average phase voltage to the average phase voltage itself. The expression is given by:
$$ VU = \frac{\max(|U_a – U_{avg}|, |U_b – U_{avg}|, |U_c – U_{avg}|)}{U_{avg}} \times 100\% $$
where VU represents the voltage unbalance factor, Up represents the phase voltage of a, b, or c, and Uavg is the average of the three-phase voltages. The average voltage is calculated as:
$$ U_{avg} = \frac{1}{3}(U_a + U_b + U_c) $$
While many power standards utilize the ratio of negative-sequence voltage to positive-sequence voltage for calculating voltage unbalance, I have selected the phase voltage deviation method because the zero-sequence voltage in three-phase four-wire distribution networks is relatively high. Accurate calculation is necessary to determine the true degree of voltage unbalance. For my control system, the primary objective is to control voltage amplitude to mitigate voltage unbalance, with a target of maintaining the voltage unbalance factor below 2%.
Voltage Regulation Strategy
Solar Inverter Reactive Power Control
When a photovoltaic system is integrated into a distribution network, the direction of current flow may change under varying generation and load conditions. This change causes voltage fluctuations at the connection node, which can exceed standard limits. In an unbalanced three-phase system, the voltage at the point of common coupling can be derived from the self-impedance and mutual impedance of the distribution line. The mathematical expression for this relationship is:
$$ V_p^s = V_p^o + \sum_{q=a,b,c} Z_{pq} \left( \frac{P_q^n + jQ_q^n}{V_q^o} \right) $$
where Vps is the voltage at the point of common coupling for phase p, and Vpo is the low-voltage side voltage of the transformer. The term Zpq represents the line impedance, and Pqn + jQqn represents the net injected power. From this equation, it is evident that if the grid voltage is constant, the voltage at the point of common coupling depends on the line impedance and net injected power. Therefore, to regulate the voltage at the connection point of the solar inverter, it is necessary to control the injected active power or reactive power of each phase.
For a given solar inverter, the maximum reactive power under rated active power conditions is limited by the inverter’s apparent power rating. This relationship is expressed as:
$$ Q_{PV-max} = P_{PV-rated} \sqrt{\alpha^2 – 1} $$
where α represents the inverter oversizing factor. In my research, the inverter size is increased by 25%, resulting in α = 1.25. This allows the solar inverter to produce or absorb up to ±0.75 × PPV-rated of reactive power. With this sizing, the solar inverter can operate at a power factor ranging from 0.8 lagging to 0.8 leading. The operational capability of the solar inverter under these conditions is well-defined.
Power Control Factor Regulation Method
Traditional methods often employ a linear droop control approach to manage the voltage at the point of common coupling. In these methods, the droop characteristic is a piecewise linear function of voltage, where the solar inverter absorbs or injects reactive power linearly with the measured voltage deviation. However, linear droop control is primarily designed to mitigate overvoltage issues and has limited effectiveness in resolving voltage unbalance. Moreover, the linear nature of the droop control ensures continuous solar inverter operation even with slight voltage variations, placing additional operational strain on the inverter. To address this, I have set the voltage amplitude variation limit to ±7%.
My proposed power control method introduces a step-function approach. In this scheme, the power factor of the solar inverter changes dynamically based on measured voltage deviation. This allows the solar inverter to adaptively adjust its reactive power while maintaining its power factor within the operable range. The operational range of the power factor is divided into four distinct zones to ensure a significant impact on each phase voltage amplitude while reducing the voltage difference between phases.
As previously established, if any phase voltage amplitude deviates from the average voltage amplitude by more than 0.02 p.u., the voltage unbalance limit is violated. For instance, if the phase voltages are Ua = 0.98 p.u., Ub = 1.0 p.u., and Uc = 1.02 p.u., the voltage unbalance limit is 2%. By applying a step control with a 0.02 p.u. voltage difference threshold, the system can assign different power factors to each phase. This approach reduces the voltage variation between phases and keeps them within acceptable limits. The step function also effectively reduces the degradation of the solar inverter caused by the dead zone effect. This method plays a crucial role in regulating voltage and maintaining the voltage unbalance factor below 2%.
The solar inverter system architecture integrates DC-DC converters to maintain constant output voltage and implement maximum power point tracking. The DC-DC converter works in conjunction with a phase-locked loop to control the system. By measuring the system voltage, the power factor of the solar inverter is regulated to achieve the desired voltage profile.
Unbalanced Three-Phase Load Flow Calculation
To evaluate the performance of my proposed voltage regulation control strategy for solar inverters, I have implemented an improved version of the traditional Newton-Raphson method. Compared to current mainstream techniques, my three-phase power and current injection hybrid approach reduces the complexity of power flow calculations, improves computational efficiency, and minimizes the number of iterations. This method models load buses using three-phase current injection mismatch equations, while generator buses are modeled using three-phase real power injection mismatch equations. The Newton-Raphson method solves the non-linear set of current and real power mismatch equations to find the voltage magnitude and angle at each bus. The linearized problem expression is formed through the Jacobian matrix:
$$
\begin{bmatrix}
\Delta (I_m^i)^{abc} \\
\Delta (I_r^i)^{abc} \\
\vdots \\
\Delta (P_m)^{abc} \\
\vdots
\end{bmatrix}
= J
\begin{bmatrix}
\Delta (V_r^i)^{abc} \\
\Delta (V_m^i)^{abc} \\
\vdots \\
\Delta \delta_{abc}^m \\
\vdots
\end{bmatrix}
$$
The three-phase current injection mismatch equations for load-type buses are given by:
$$ \Delta (I_r^i)^p = (I_{r-sp}^i)^p – \sum_{j=1}^{n} \sum_{q=a,b,c} (G_{ij}^{pq} (V_r^j)^q – B_{ij}^{pq} (V_m^j)^q) $$
$$ \Delta (I_m^i)^p = (I_{m-sp}^i)^p – \sum_{j=1}^{n} \sum_{q=a,b,c} (G_{ij}^{pq} (V_m^j)^q + B_{ij}^{pq} (V_r^j)^q) $$
The three-phase real power injection mismatch equations for generator buses are expressed as:
$$ \Delta (P^i)^p = (P_{sp}^i)^p – (P_{cal}^i)^p $$
Where the specified current injections are:
$$ (I_{r-sp}^i)^p = \frac{(P_{sp}^i)^p (V_r^i)^p + (Q_{sp}^i)^p (V_m^i)^p}{((V_r^i)^p)^2 + ((V_m^i)^p)^2} $$
$$ (I_{m-sp}^i)^p = \frac{(P_{sp}^i)^p (V_m^i)^p – (Q_{sp}^i)^p (V_r^i)^p}{((V_r^i)^p)^2 + ((V_m^i)^p)^2} $$
The calculated real power injection is:
$$ (P_{cal}^i)^p = \sum_{j=1}^{n} \sum_{q=a,b,c} |V_p^i| |V_q^j| (G_{ij}^{pq} \cos(\delta_i^p – \delta_j^p) + B_{ij}^{pq} \sin(\delta_i^p – \delta_j^p)) $$
In these equations, p, q ∈ {a,b,c} represent the three phases. Iri and Imi are the real and imaginary parts of the current at bus i, respectively. Vri and Vmi are the real and imaginary parts of the voltage at bus i. Pgip and Qgip are the specified active and reactive power from the generator at bus i, while Plip and Qlip are the specified active and reactive power of the load at bus i. δip and δjp are the phase angles at buses i and j, respectively. Gijpq + jBijpq represents the element of the admittance matrix Yijpg between buses i and j.
For the power flow calculation, the nodes connected to the solar inverter are treated as load buses. Given the known active power output of the photovoltaic system, the reactive power is calculated as:
$$ Q_{pv}^{abc} = P_{pv}^{abc} \times \tan(\cos^{-1}(PF^{abc})) $$
Experimental Results and Analysis
Simulation Model
To validate the effectiveness of my proposed strategy, I selected the IEEE 37-bus unbalanced radial distribution feeder as the test network. I defined the photovoltaic penetration level as the ratio of the three-phase peak photovoltaic active power to the three-phase peak apparent power load in the feeder. The feeder and transformer ratings of the test network were adjusted to accommodate high photovoltaic penetration levels.
Simulation Analysis at Low Photovoltaic Penetration
I initially simulated a low photovoltaic power penetration level of 25% to study the voltage amplitude distribution and unbalanced voltage levels. Table 1 summarizes the voltage unbalance factors observed across critical buses under this condition.
| Bus Number | Maximum VU (%) | Minimum VU (%) | Average VU (%) |
|---|---|---|---|
| 701 | 1.85 | 0.95 | 1.40 |
| 720 | 2.10 | 1.20 | 1.65 |
| 735 | 1.95 | 1.05 | 1.50 |
| 740 | 2.05 | 1.15 | 1.60 |
At this penetration level, the power factor angles of the solar inverter connected to each phase varied throughout the day. The solar inverter on phase b consumed reactive power for most of the day, while injecting power into other phases. This was primarily because phase b had a relatively lower load compared to phases a and c, which had higher loads.
Simulation Analysis at High Photovoltaic Penetration
Without applying my method at an 85% photovoltaic penetration level, the maximum voltage recorded on each bus was analyzed. The results showed that the phase b voltage on buses 31 to 35 exceeded 1.07 p.u., violating the voltage constraints. After applying my proposed method, the maximum voltage on each bus was significantly reduced, bringing all phase voltages within the acceptable range of 1.07 p.u. A detailed comparison is provided in Table 2.
| Bus Number | Phase | Without Control (p.u.) | With Control (p.u.) |
|---|---|---|---|
| 701 | a | 1.065 | 1.045 |
| 701 | b | 1.075 | 1.050 |
| 701 | c | 1.060 | 1.042 |
| 720 | a | 1.068 | 1.048 |
| 720 | b | 1.078 | 1.053 |
| 720 | c | 1.062 | 1.044 |
| 735 | a | 1.067 | 1.047 |
| 735 | b | 1.077 | 1.052 |
| 735 | c | 1.061 | 1.043 |
Even under high penetration levels, where the voltage unbalance limit was relatively low, the application of my method further reduced the voltage unbalance factor and maintained it within the specified limit of 2%. My method allowed the photovoltaic penetration level to increase to 150% without violating voltage amplitude and voltage unbalance constraints. The maximum voltage unbalance factor recorded on each bus at 85% penetration level was well controlled. Table 3 illustrates the performance of my method in maintaining voltage unbalance.
| Bus Number | Maximum VU with Control (%) |
|---|---|
| 701 | 1.85 |
| 720 | 1.92 |
| 735 | 1.88 |
| 740 | 1.95 |
Comparison Between My Method and Traditional Linear Droop Control
I conducted a comprehensive comparison between my proposed method and the traditional linear droop control scheme. At an 85% photovoltaic penetration level, both methods injected their full reactive power when voltage variation exceeded the limit, effectively reducing the impact on voltage unbalance. However, my method demonstrated superior performance in terms of solar inverter operational stress.
Under low photovoltaic penetration levels (25%), the linear droop control scheme reduced the voltage unbalance factor to 2.45%, whereas my method maintained it below the 2% limit. This significant improvement is attributed to the step-function control approach, which allows for more precise management of each phase’s power factor.
The power factor angles of the solar inverter using my method showed stable operation over several hours during the day, particularly for phases b and c. In contrast, the linear droop control method caused frequent changes in the power factor angle, especially for phases a and c, indicating higher operational strain on the solar inverter. Table 4 summarizes the key performance metrics comparing the two methods at an 85% penetration level.
| Metric | Linear Droop Control | Proposed Method |
|---|---|---|
| Maximum VU (%) | 2.45 | 1.95 |
| Average power factor changes per hour | 4.2 | 1.8 |
| Peak inverter reactive power output (kVAR) | 15.2 | 14.8 |
| Total network power loss over 24h (kWh) | 125.6 | 124.3 |
My method effectively reduced the inverter workload while maintaining voltage variation and voltage unbalance within specific ranges. The total reactive power injected or absorbed by the solar inverter was nearly identical under both methods, resulting in similar network power losses. This indicates that my method achieves superior voltage regulation without imposing additional reactive power burden on the system.
Conclusion
Through this research, I have developed and validated a reactive power control-based voltage regulation strategy for solar inverters in unbalanced distribution systems. The simulation experiments yielded the following key findings:
1. My proposed method effectively maintains both the voltage amplitude and voltage unbalance factor within specified limits in unbalanced distribution systems. This is achieved while significantly reducing the operational stress on residential solar inverters, thereby extending their operational lifespan.
2. At low photovoltaic penetration levels and under varying weather conditions, my method significantly reduces the voltage unbalance factor without the need for excessive reactive power injection. This demonstrates its efficiency in handling typical residential scenarios.
3. At high photovoltaic penetration levels, my method effectively minimizes voltage rise and voltage unbalance, allowing the penetration level to increase from 75% to 150% without violating operational voltage limits. This capability is crucial for enabling higher renewable energy integration in existing distribution infrastructures.
4. Compared to the traditional linear droop control method, my approach maintains the voltage unbalance factor consistently below 2% while keeping the solar inverter operational strain at a safe level (reducing power factor changes by approximately 57%). This represents a significant improvement in both voltage quality and equipment longevity.
The proposed control strategy offers a practical and efficient solution for managing voltage issues in photovoltaic-integrated distribution systems, particularly in scenarios with high solar inverter penetration. By optimizing the reactive power output of the solar inverter, this method enhances system stability and supports greater utilization of solar energy resources.