With the rapid integration of distributed photovoltaic (DPV) systems into distribution networks, voltage overlimit risks have become a critical challenge due to the inherent randomness and volatility of solar power. Traditional grid-following solar inverters, which rely on phase-locked loops for synchronization, offer limited and slow voltage regulation capabilities, making them insufficient for high-penetration scenarios. In contrast, grid-forming solar inverters, which operate as voltage sources and provide fast, autonomous voltage support, present a promising solution. However, coordinating these diverse solar inverters with conventional devices like on-load tap changers (OLTCs) and shunt capacitor banks (SCBs) remains complex. This paper addresses this issue by proposing a bi-level optimization strategy that leverages the distinct characteristics of grid-forming and grid-following solar inverters to enhance voltage stability and economic efficiency in distribution networks.
Our approach begins with partitioning the distribution network based on comprehensive electrical distance metrics to identify optimal nodes for deploying grid-forming solar inverters. This ensures effective voltage regulation by considering both the sensitivity of nodes to fluctuations and their influence on neighboring nodes. We then calculate the reserve capacity for grid-forming solar inverters to handle short-term voltage fluctuations, ensuring sufficient reactive power margins during extreme scenarios. The core of our strategy involves a day-ahead bi-level optimization model that decouples active and reactive power management. The upper level minimizes operational costs by optimizing active power decisions, such as power purchases, energy storage scheduling, and demand response, while the lower level focuses on reducing voltage deviations by coordinating reactive power outputs from solar inverters and traditional devices. Finally, we employ Fisher’s optimal segmentation method to discretize the actions of OLTCs and SCBs, reducing their switching frequency and prolonging equipment lifespan. Simulation results on an IEEE 33-node system demonstrate that our method significantly improves voltage profiles, reduces operational costs, and minimizes active power losses compared to conventional approaches.
The integration of solar inverters into distribution networks has accelerated with the global shift toward renewable energy. However, the intermittent nature of solar power exacerbates voltage instability, particularly in networks with high DPV penetration. Grid-following solar inverters, while efficient for maximum power point tracking, lack the ability to provide robust voltage support during rapid fluctuations. Grid-forming solar inverters, on the other hand, emulate synchronous generators by regulating voltage and frequency independently, offering superior dynamic performance. Our study explores the synergistic potential of hybrid systems where both types of solar inverters coexist. By formulating a coordinated control framework, we aim to harness the economic benefits of grid-following solar inverters and the stability enhancements of grid-forming solar inverters. This holistic approach not only mitigates voltage violations but also optimizes resource utilization, paving the way for more resilient and cost-effective distribution networks.
Optimal Configuration of Grid-Forming Solar Inverters
Selecting appropriate nodes for deploying grid-forming solar inverters is crucial for maximizing their voltage regulation capabilities. We utilize a clustering-based method to partition the distribution network into regions, ensuring that each region has a grid-forming solar inverter placed at the most influential node. The electrical distance between nodes is defined using a combination of active-power-voltage and reactive-power-voltage sensitivities, which account for the high R/X ratio typical of distribution networks. The comprehensive electrical distance matrix D is computed as follows:
$$d_{ij} = \frac{1}{\sqrt{\gamma_{P,ij}^2 + \gamma_{P,ji}^2 + \gamma_{Q,ij}^2 + \gamma_{Q,ji}^2}}$$
where $\gamma_{P,ij}$ and $\gamma_{Q,ij}$ represent the active and reactive power voltage sensitivities between nodes i and j, respectively. This matrix serves as the similarity measure for the clustering algorithm. We apply the CFSFDP (Clustering by Fast Search and Find of Density Peaks) algorithm to group nodes into clusters, as it effectively handles non-spherical data distributions common in radial networks. Within each cluster, the optimal node for grid-forming solar inverter placement is selected by minimizing the maximum comprehensive voltage sensitivity, ensuring that the chosen node has a strong influence on regional voltage stability. The selection criterion is formulated as:
$$f(i) = \min(L_{ij}) \quad \text{for} \quad i \neq j$$
$$\alpha_m = \max\{f(i)\}$$
where $f(i)$ is the minimum comprehensive sensitivity for node i, and $\alpha_m$ identifies the node with the highest influence in cluster m. This strategy ensures that grid-forming solar inverters are deployed at locations that enhance overall voltage control while considering economic constraints.
Reserve Capacity Calculation for Grid-Forming Solar Inverters
To address short-term voltage fluctuations caused by uncertainties in solar generation and load demand, we compute the reserve capacity for grid-forming solar inverters. This reserve represents the reactive power margin reserved for extreme scenarios, ensuring that the solar inverters can respond rapidly to voltage deviations without compromising day-ahead schedules. The calculation involves three steps: first, we solve a day-ahead optimization model to determine baseline operational plans; second, we model uncertainties in photovoltaic output and load using normal distributions with variances set to 10% of forecasted values; and third, we optimize for extreme voltage deviations under these uncertainties to derive the required reserve capacities. The reserve capacity $Q_{\gamma,t}$ for a grid-forming solar inverter at node $\gamma$ and time t is given by:
$$Q_{\gamma,t} = \max\left\{ |Q_{\text{PVGW2,max},\gamma,i,t} – Q_{\text{PVGW},\gamma,t}|, |Q_{\text{PVGW2,min},\gamma,i,t} – Q_{\text{PVGW},\gamma,t}| \right\}$$
where $Q_{\text{PVGW2,max},\gamma,i,t}$ and $Q_{\text{PVGW2,min},\gamma,i,t}$ are the reactive power outputs under maximum and minimum voltage extremes, respectively, and $Q_{\text{PVGW},\gamma,t}$ is the day-ahead reactive power output. This approach ensures that grid-forming solar inverters maintain sufficient flexibility to handle real-time disturbances, thereby enhancing system reliability.
Bi-Level Optimization Model for Active and Reactive Power
Our bi-level optimization model decouples active and reactive power decisions to improve computational efficiency and solution quality. The upper level focuses on minimizing day-ahead operational costs, while the lower level aims to reduce voltage deviations through reactive power coordination. This structure allows for iterative feedback between the two levels, ensuring that active and reactive power decisions are mutually consistent.
Upper-Level Active Power Optimization
The objective of the upper-level model is to minimize total operational costs, including power purchase expenses, revenue from power sales, penalties for photovoltaic curtailment, energy storage costs, and demand response compensation. The objective function is formulated as:
$$\min F_{\text{total}} = f_{\text{buy}} – f_{\text{sell}} + f_{\text{PV}} + f_{\text{ESS}} + f_{\text{DR}}$$
where each cost component is defined as follows:
$$f_{\text{buy}} = \sum_{t=1}^{24} c_{\text{buy},t} P_{\text{grid},t}$$
$$f_{\text{sell}} = \sum_{t=1}^{24} c_{\text{sell},t} P_{\text{sell},t}$$
$$f_{\text{PV}} = \sum_{t=1}^{24} \sum_{g=1}^{n_{\text{PV}}} a_q P_{q,g,t}$$
$$f_{\text{ESS}} = \sum_{t=1}^{24} \sum_{e=1}^{n_{\text{ESS}}} a_{\text{ESS},e} P_{\text{ESS},e,t}$$
$$f_{\text{DR}} = \sum_{t=1}^{24} \sum_{d=1}^{n_{\text{DR}}} a_{\text{DR}} P_{\text{DR},d,t}$$
Constraints include power flow equations based on the DistFlow model, power balance constraints, energy storage operational limits, and bounds on decision variables such as photovoltaic curtailment and demand response. The power flow constraints are expressed as:
$$\sum_{i \in u(j)} (P_{ij,t} – I_{ij,t}^2 r_{ij}) = \sum_{k \in v(j)} P_{jk,t} + P_{j,t}$$
$$\sum_{i \in u(j)} (Q_{ij,t} – I_{ij,t}^2 x_{ij}) = \sum_{k \in v(j)} Q_{jk,t} + Q_{j,t}$$
$$U_{i,t}^2 – U_{j,t}^2 = 2(r_{ij} P_{ij,t} + x_{ij} Q_{ij,t}) – (r_{ij}^2 + x_{ij}^2) I_{ij,t}^2$$
$$I_{ij,t}^2 U_{i,t}^2 = P_{ij,t}^2 + Q_{ij,t}^2$$
These constraints ensure physical feasibility while accounting for losses and voltage drops across the network.
Lower-Level Reactive Power Optimization
The lower-level model minimizes the average daily voltage deviation across all nodes, formulated as:
$$\min f_p = \frac{1}{24n} \sum_{t=1}^{24} \sum_{i=1}^{n} |U_{i,t} – U_{i}^{\text{ref}}|$$
Reactive power balance is maintained through:
$$Q_{i,t} = Q_{\text{PVG},i,t} + Q_{\text{PVGW},i,t} + Q_{\text{CB},i,t} – Q_{\text{load},i,t} + Q_{\text{DR},i,t}$$
where $Q_{\text{PVG},i,t}$ and $Q_{\text{PVGW},i,t}$ denote reactive power outputs from grid-following and grid-forming solar inverters, respectively. Constraints include capacity limits for solar inverters, operational bounds for SCBs and OLTCs, and voltage security limits. For solar inverters, the reactive power capability is constrained by:
$$-Q_{\text{PV,max},i,t \leq Q_{\text{PV},i,t} \leq Q_{\text{PV,max},i,t$$
Here, $Q_{\text{PV,max},i,t}$ is derived from the apparent power capacity of the solar inverters, considering active power outputs and reserve capacities. The iterative process between upper and lower levels continues until convergence, ensuring optimal coordination between active and reactive power resources.
Dynamic Reactive Power Optimization Using Fisher’s Optimal Segmentation
To handle the discrete nature of OLTCs and SCBs, we apply Fisher’s optimal segmentation method to determine optimal switching times and tap positions. This approach reduces the action frequency of these devices, mitigating wear and tear. The method involves partitioning the day into segments where device settings remain constant, minimizing the loss function defined as the sum of squared deviations within each segment. For a sequence of settings ${x_1, x_2, \ldots, x_{24}}$, the segment diameter for a segment from index v to h is computed as:
$$Z(v,h) = \sum_{a=v}^{h} [x_a – \bar{x}(v,h)]^2$$
where $\bar{x}(v,h)$ is the segment mean. The optimal segmentation is found by recursively minimizing the loss function across possible partitions, subject to constraints on the maximum number of actions. This results in a practical schedule that balances voltage control with device longevity.
Simulation Results and Analysis
We validate our proposed strategy using the IEEE 33-node system, incorporating multiple grid-following and grid-forming solar inverters. The simulation considers typical daily profiles for solar generation and load demand, with uncertainties modeled as normal distributions. The following table summarizes the key parameters used in the simulation:
| Parameter | Value |
|---|---|
| Number of grid-following solar inverters | 6 |
| Number of grid-forming solar inverters | 3 |
| OLTC tap range (p.u.) | 0.95–1.05 |
| SCB capacity (kvar) | 80 × 5 |
| Maximum OLTC actions per day | 12 |
| Maximum SCB actions per day | 16 |
The optimization results demonstrate significant improvements in voltage profiles and economic metrics. Compared to traditional methods, our approach reduces the average daily voltage deviation by over 9% and active power losses by approximately 8%. The following table compares the performance of five different schemes, highlighting the superiority of our bi-level optimization with dynamic reactive power control:
| Scheme | Daily Operational Cost ($) | Average Voltage Deviation (p.u.) | Daily Active Loss (kWh) |
|---|---|---|---|
| Scheme 1 (Grid-following only) | 16,303.85 | 0.01473 | 1,525.26 |
| Scheme 2 (Hybrid with reference node selection) | 15,576.29 | 0.01368 | 1,419.53 |
| Scheme 3 (Hybrid with proposed node selection) | 15,366.83 | 0.01331 | 1,399.07 |
| Scheme 4 (Bi-level optimization without segmentation) | 14,788.17 | 0.01235 | 1,340.39 |
| Scheme 5 (Bi-level optimization with segmentation) | 14,896.01 | 0.01246 | 1,361.17 |
Our method effectively reduces voltage violations across all time periods, particularly during peak solar generation hours when overvoltages are common. The reserve capacity calculation ensures that grid-forming solar inverters can respond to extreme fluctuations, as illustrated by the following formula for reactive power capability:
$$Q_{\text{PVGW,max},i,t} = \sqrt{S_{\text{PVGW},i}^2 – (P_{\text{PVGW,max},i,t} – P_{q\text{GW},i,t})^2} – |Q_{\text{PVGWst},i,t}|$$
where $S_{\text{PVGW},i}$ is the rated capacity of the grid-forming solar inverter, and $Q_{\text{PVGWst},i,t}$ is the reserve capacity. This ensures that the solar inverters operate within their limits while providing necessary voltage support.
Conclusion
In this paper, we have developed a comprehensive strategy for coordinating grid-forming and grid-following solar inverters in distribution networks. Our approach combines optimal node selection, reserve capacity allocation, bi-level active-reactive power optimization, and dynamic device scheduling to enhance voltage stability and economic performance. The results confirm that grid-forming solar inverters play a crucial role in mitigating voltage violations, especially in high-penetration scenarios. By decoupling active and reactive power optimization, we achieve better resource utilization and cost savings. Future work will focus on multi-time-scale optimization to further leverage the fast response capabilities of grid-forming solar inverters in real-time voltage control.