With the increasing integration of distributed photovoltaic (PV) systems into low-voltage distribution networks, voltage violations have become a critical challenge due to reverse power flow and fluctuating generation. Traditional voltage regulation methods, such as installing reactive compensation devices or curtailing PV output, often suffer from limited effectiveness, high costs, or reduced resource utilization. In this context, solar inverters and energy storage systems (ESS) offer promising solutions due to their fast response and declining costs. However, existing approaches often fail to fully coordinate the reactive power control of solar inverters and the active power control of ESS, leading to suboptimal performance. To address this, we propose a novel grouped coordination strategy based on consensus algorithms, which prioritizes devices with higher voltage-cost sensitivity (FU-C) to enhance economic efficiency and voltage stability.
The core of our strategy lies in analyzing the voltage-cost sensitivity (FU-C), which quantifies the economic effectiveness of voltage regulation devices. For a solar inverter at node j affecting node i, FU-C is defined as $$F_{U-C,PV,ij} = \frac{\Delta U_{ij}}{C_{PV,j}} = \frac{S_{U-Q,ij} \Delta Q_{PV,j}}{c_{PV}} = \frac{S_{U-Q,ij}}{c_{PV}}$$ where $\Delta U_{ij}$ is the voltage change at node i due to power adjustment at node j, $C_{PV,j}$ is the cost of reactive power regulation for the solar inverter, $S_{U-Q,ij}$ is the voltage-reactive power sensitivity, $\Delta Q_{PV,j}$ is the reactive power change, and $c_{PV}$ is the unit cost (approximately 0.067 USD/kvar·h). Similarly, for an ESS at node j, $$F_{U-C,ESS,ij} = \frac{\Delta U_{ij}}{C_{ESS,j}} = \frac{S_{U-P,ij} \Delta P_{ESS,j}}{c_{ESS}} = \frac{S_{U-P,ij}}{c_{ESS}}$$ where $S_{U-P,ij}$ is the voltage-active power sensitivity, $\Delta P_{ESS,j}$ is the active power change, and $c_{ESS}$ is the unit cost (0.6-1.0 USD/kWh). By comparing these, we find that for the same node, the ratio $$\frac{F_{U-C,ESS,ij}}{F_{U-C,PV,ij}} = \frac{S_{U-P,ij}}{S_{U-Q,ij}} \cdot \frac{c_{PV}}{c_{ESS}} = \frac{R}{X} \cdot \frac{c_{PV}}{c_{ESS}}$$ is typically less than 1 in low-voltage networks (where R/X ranges from 1.14 to 5.56 and $c_{PV}$ is 0.067-0.112 times $c_{ESS}$), indicating that solar inverters are more economical for voltage regulation. Thus, our strategy prioritizes reactive power control from solar inverters before utilizing active power from ESS.
Furthermore, we group regulation devices based on their FU-C values relative to critical nodes. Devices located at or downstream of a critical node have higher FU-C due to greater voltage sensitivity. For instance, in an IEEE 14-node network, we form two groups: Group GV1 (nodes 3, 4, 5, 8) and Group GV2 (nodes 7, 9, 13, 14), with GV2 having higher FU-C for the critical node (e.g., node 14). The control framework proceeds in two stages: first, solar inverters adjust reactive power, and if insufficient, ESS adjusts active power. Within each group, consensus algorithms ensure balanced resource utilization. For solar inverters, the consensus variable is reactive power utilization rate $\mu$, updated as $$\mu_{GVi,j}(k+1) = \sum_{m=1}^{N_P} \beta_{PV,GVi,jm} \mu_{GVi,m}(k) + d_{PV,GVi,j} \lambda_1 (\mu_{GVi,j}(k) – \mu_{ref,GVi,n}(k))$$ where $\beta_{PV,GVi,jm}$ is the weight based on communication links, $d_{PV,GVi,j}$ indicates connection to the leader node, and $\lambda_1$ is the iteration step. The reactive power output is $Q_{GVi,j} = \mu_{GVi,j} Q_{max,GVi,j}$, subject to power factor constraints (e.g., $\cos \phi_i \in [0.9,1.0] \cup [-1.0,-0.9]$). For ESS, the consensus variable is the state of charge (SOC) change $\Delta S$, updated as $$\Delta S_{GVi,j}(k+1) = \sum_{m=1}^{N_b} \beta_{ESS,GVi,jm} \Delta S_{GVi,m}(k) + d_{ESS,GVi,j} \lambda_2 (\Delta S_{GVi,j}(k) – \Delta S_{ref,GVi,n}(k))$$ with active power output $P_{ESS,i,j} = \Delta S_{GVi,j}(k+1) S_{ESS,i,j} / (\eta \Delta t)$, constrained by SOC limits (e.g., 20% to 80%) and rated power. Inter-group coordination ensures that groups with higher FU-C are activated first, reducing unnecessary costs.
To validate our approach, we simulate an IEEE 14-node low-voltage distribution network with distributed solar inverters and ESS using PSCAD and MATLAB. The network has a rated voltage of 380 V, line impedance of (0.602 + j0.232) Ω/km, and solar inverter capacities ranging from 10 kVA to 12 kVA, paired with ESS capacities of 8-10 kWh and rated powers of 1.6-2.0 kW. PV output profiles vary over time, causing voltage violations at the critical node. We compare our grouped coordination strategy (S3) with two alternatives: S1 (solar inverter reactive control only) and S2 (ESS active control only). The voltage sensitivity matrix $S_{ij}$ is derived from line parameters: $$S_{ij} = \begin{cases} \left[ \frac{1}{U_0} \sum_{n=1}^{i} R_n \quad \frac{1}{U_0} \sum_{n=1}^{i} X_n \right] & i \leq j \\ \left[ \frac{1}{U_0} \sum_{n=1}^{j} R_n \quad \frac{1}{U_0} \sum_{n=1}^{j} X_n \right] & i > j \end{cases}$$ where $U_0$ is the transformer outlet voltage. Parameters for consensus algorithms are set as $\lambda_1 = 0.35$ and $\lambda_2 = 0.40$, with tuning factors $\alpha = 1$ and $b = 1$ for leader node updates.
Simulation results demonstrate that S1 fails to fully mitigate voltage violations due to limited reactive capacity and power factor constraints, while S2 requires excessive ESS capacity and incurs high costs. In contrast, S3 successfully maintains voltages within limits (0.95-1.05 pu) by coordinating solar inverters and ESS. Specifically, S3 uses 87.31 kvar·h of reactive power from solar inverters and only 8.03 kWh from ESS, whereas S2 requires 50.81 kWh from ESS. This translates to a 84.2% reduction in ESS capacity usage for S3 compared to S2. The economic benefits are evident in the regulation costs: S3 totals 10.67 USD (5.85 USD for solar inverters and 4.82 USD for ESS), while S2 costs 30.49 USD. Additionally, we compare S3 with a global consensus strategy (S4) from literature, which uses 93.60 kvar·h from solar inverters and 11.38 kWh from ESS, costing 13.20 USD. Thus, S3 reduces costs by 19.2% and ESS usage by 29.4% compared to S4, highlighting the advantage of grouped coordination.
| Strategy | PV Reactive Usage (kvar·h) | ESS Active Usage (kWh) | Total Cost (USD) |
|---|---|---|---|
| S1 (PV Only) | 87.31 | 0.00 | 5.85 |
| S2 (ESS Only) | 0.00 | 50.81 | 30.49 |
| S3 (Proposed) | 87.31 | 8.03 | 10.67 |
| S4 (Global Consensus) | 93.60 | 11.38 | 13.20 |
The consistency of solar inverters in responding to voltage deviations is further analyzed through iterative updates. For example, the leader node in GV2 updates its reactive utilization reference as $$\mu_{ref,GV2,n}(k+1) = \mu_{ref,GV2,n}(k) – \alpha [U_n(k+1) – U_{max}]$$ for overvoltage conditions, ensuring convergence to the setpoint. Similarly, ESS SOC changes are computed as $$\Delta S_{ref,GV2,n}(k+1) = \Delta S_{ref,GV2,n}(k) – b [U_n(k+1) – U_{max}]$$ The weights $\beta$ in consensus algorithms are normalized sums of communication links, promoting equitable burden sharing. This approach not only enhances voltage stability but also optimizes the lifetime of solar inverters and ESS by preventing excessive usage.
In conclusion, our grouped coordination strategy based on consensus algorithms effectively addresses voltage violations in low-voltage distribution networks with high PV penetration. By leveraging the economic superiority of solar inverters for initial reactive control and sequentially activating ESS, we minimize costs and resource usage. The integration of FU-C based grouping and distributed control ensures scalability and robustness, making it suitable for real-world applications. Future work could explore dynamic grouping adaptations and hybrid consensus models for larger networks.
The mathematical formulation of voltage sensitivities plays a key role in optimizing the control strategy. For instance, the sensitivity of voltage at node i to power changes at node j is encapsulated in the matrix $$S_{ij} = \frac{1}{U_0} \begin{bmatrix} \sum_{n=1}^{i} R_n & \sum_{n=1}^{i} X_n \\ \sum_{n=1}^{j} R_n & \sum_{n=1}^{j} X_n \end{bmatrix}$$ which influences the FU-C calculations. This allows us to dynamically prioritize solar inverters in regions with higher sensitivity, ensuring efficient voltage regulation. Moreover, the consensus algorithm parameters are tuned to balance convergence speed and accuracy, as shown in empirical tests where $\alpha$ and $b$ values of 1.0 achieve optimal iterations (e.g., 208 for solar inverters and 173 for ESS) with high precision (errors below $10^{-3}$).
| Parameter | Value | Iterations | Control Accuracy |
|---|---|---|---|
| $\alpha$ | 0.1 | 684 | 6.15e-5 |
| $\alpha$ | 1.0 | 208 | 6.58e-5 |
| $\alpha$ | 10.0 | 202 | 1.65e-3 |
| $b$ | 0.1 | 231 | 1.69e-4 |
| $b$ | 1.0 | 173 | 6.32e-3 |
| $b$ | 10.0 | 154 | 1.52e-2 |
Overall, the proposed strategy demonstrates significant improvements in managing voltage issues through coordinated use of solar inverters and energy storage. The emphasis on solar inverters as primary regulators not only capitalizes on their cost-effectiveness but also aligns with the trend of increasing solar integrations in modern grids. By incorporating consensus-based distributed control, we ensure that all devices contribute optimally, enhancing the overall resilience and efficiency of low-voltage distribution networks.