In the pursuit of carbon neutrality and the transition toward a new power system dominated by renewable energy, the integration of distributed photovoltaic (PV) systems has seen exponential growth globally. However, the inherent volatility of PV generation and load demand poses significant challenges to distribution network stability, particularly in terms of voltage fluctuations and limit violations. Traditional centralized control of solar inverters, which typically operates on a 15-minute timescale, often fails to account for short-term fluctuations occurring within control intervals, leading to potential voltage excursions and compromised economic operation. In this article, we propose a robust centralized-local control strategy for distributed solar inverters that effectively mitigates short-term fluctuations while optimizing network performance. By integrating a two-stage optimization framework—combining long-term economic dispatch with short-term robust slope tuning—we enhance the resilience and efficiency of active distribution networks. This approach leverages the reactive power capability of solar inverters, coordinated with active power curtailment when necessary, to maintain voltage security and minimize losses under uncertainty. Through detailed modeling, algorithmic solutions, and case studies, we demonstrate the superiority of this strategy in handling real-world variability, paving the way for more reliable and cost-effective integration of high-penetration PV systems.
The proliferation of distributed solar inverters in modern grids has transformed passive distribution networks into active systems requiring advanced control paradigms. Centralized control, while globally optimal, suffers from latency and communication dependencies, making it vulnerable to minute-scale source and load fluctuations. Conversely, purely local control, though rapid, can lead to suboptimal coordination and increased losses. To bridge this gap, we develop a hybrid strategy that optimizes centralized setpoints for solar inverters over 15-minute horizons, followed by local adjustment of droop slopes to counteract intra-interval volatility. This synergy ensures that the system remains within safe operating limits while tracking an economically desirable state. The core innovation lies in a three-parameter model—encompassing reactive power output, active power curtailment, and droop slope—that is robustified against worst-case fluctuation scenarios via interval-based uncertainty modeling. By employing second-order cone programming (SOCP) for the centralized stage and sensitivity-based linearization for the local stage, we achieve computational tractability without sacrificing accuracy. Our contributions include a comprehensive framework for solar inverter coordination, a novel robustness formulation, and practical insights into the impacts of fluctuation intensity, penetration levels, and inverter sizing on control efficacy.
Let us begin by formalizing the two-stage control problem. In the first stage, we consider a 15-minute control interval indexed by \( t_1 \), with predicted PV generation \( P_{PV}^{t_1}(i) \) and load demand \( P_{LOAD}^{t_1}(i), Q_{LOAD}^{t_1}(i) \) at each node \( i \) in a distribution network of \( N \) nodes. The decision variables are the active power curtailment \( \Delta P_{PV}^{t_1}(i) \) and reactive power injection \( \Delta Q_{PV}^{t_1}(i) \) from solar inverters. The objective is to minimize a weighted sum of total curtailment and network losses, expressed as:
$$ \min f_{t_1} = \alpha \sum_{i=1}^{N} \Delta P_{PV}^{t_1}(i) + \sum_{ij \in E} I_{t_1}(ij)^2 \frac{G(ij)}{G(ij)^2 + B(ij)^2} $$
where \( \alpha \) is a weighting coefficient balancing curtailment against losses, \( E \) is the set of branches, \( G(ij) \) and \( B(ij) \) are branch conductance and susceptance, and \( I_{t_1}(ij) \) is the branch current magnitude. The constraints include inverter capacity limits:
$$ \left( P_{PV}^{t_1}(i) – \Delta P_{PV}^{t_1}(i) \right)^2 + \left( \Delta Q_{PV}^{t_1}(i) \right)^2 \leq S_{PV}(i)^2 $$
power flow equations (distributed using the DistFlow formulation), and voltage limits \( V_{\min} \leq V_{t_1}(i) \leq V_{\max} \). To solve this non-convex problem efficiently, we apply a second-order cone relaxation, introducing variables \( V_{LINEAR}^{t_1}(i) = V_{t_1}(i)^2 \) and \( I_{LINEAR}^{t_1}(ij) = I_{t_1}(ij)^2 \), leading to a SOCP model:
$$ \begin{aligned}
&\min f_{LINEAR}^{t_1} = \alpha \sum_{i=1}^{N} \Delta P_{PV}^{t_1}(i) + \sum_{ij \in E} I_{LINEAR}^{t_1}(ij) \frac{G(ij)}{G(ij)^2 + B(ij)^2} \\
&\text{subject to:} \\
&V_{LINEAR}^{t_1}(i) – V_{LINEAR}^{t_1}(j) = 2G(ij)P_{t_1}(ij) + 2B(ij)Q_{t_1}(ij) – I_{LINEAR}^{t_1}(ij)/(G(ij)^2+B(ij)^2) \\
&\left\| \begin{array}{c} 2P_{t_1}(ij) \\ 2Q_{t_1}(ij) \\ I_{LINEAR}^{t_1}(ij) – V_{LINEAR}^{t_1}(i) \end{array} \right\|_2 \leq I_{LINEAR}^{t_1}(ij) + V_{LINEAR}^{t_1}(i) \\
&V_{\min}^2 \leq V_{LINEAR}^{t_1}(i) \leq V_{\max}^2
\end{aligned} $$
This centralized optimization yields baseline values for solar inverter outputs, which are then passed to the second stage.
The second stage addresses short-term fluctuations within the 15-minute interval, modeled at a 1-minute resolution indexed by \( t_2 \). We generate 1-minute data via linear interpolation from the 15-minute predictions, and incorporate uncertainty using interval models for PV and load fluctuations:
$$ \Delta P_{LOAD}^{t_2}(i) \in \left[ -\sigma_{LOAD} P_{LOAD}^{t_2}(i), \sigma_{LOAD} P_{LOAD}^{t_2}(i) \right] $$
$$ \Delta P_{PV}^{t_2}(i) \in \left[ -\sigma_{PV} P_{PV}^{t_2}(i), \sigma_{PV} P_{PV}^{t_2}(i) \right] $$
where \( \sigma_{LOAD} \) and \( \sigma_{PV} \) are uncertainty parameters derived from historical data. The local control for each solar inverter follows a droop strategy:
$$ \Delta Q_{PV}^{t_2}(i) = – m_{t_1}(i) \left( V_{t_2}(i) – V_{t_2-1}(i) \right) $$
where \( m_{t_1}(i) \) is the droop slope to be optimized. The objective is to minimize voltage deviations from the first-stage baseline across extreme fluctuation scenarios, encompassing combinations of upper and lower bounds of PV and load intervals. Formally, for \( S = 4 \) extreme scenarios, we have:
$$ \min g_{t_1} = \frac{1}{S} \sum_{s=1}^{S} \left[ \sum_{t_2=1}^{15} \sum_{i=1}^{N} w(i) \left| V_{s}^{t_2}(i) – V_{s}^{t_1}(i) \right| \right] $$
with node weights \( w(i) \) accounting for sensitivity and inverter capacity. The slopes are constrained within feasible ranges derived from voltage-reactive power sensitivity and inverter capability:
$$ m_{t_1}(i) \geq \frac{1}{S_{V-Q}(i,i)}, \quad m_{t_1}(i) \leq \frac{\Delta Q_{PV}^{t_2}(i)_{\max}}{\Delta V_{t_2}(i)_{\max}} $$
where \( S_{V-Q} \) is the sensitivity matrix, \( \Delta Q_{PV}^{t_2}(i)_{\max} \) is the maximum available reactive power, and \( \Delta V_{t_2}(i)_{\max} \) is the extreme voltage deviation. To solve this robust optimization efficiently, we linearize the power flow around the first-stage operating point using sensitivity analysis. The voltage deviation can be approximated as:
$$ \Delta V_{t_2} \approx \left( J_{Q-V} – J’ J_{P-V} \right)^{-1} \left[ -m_{t_1} (V_{t_2} – V_{t_2-1}) – J’ (P_{ALL}^{t_2} – P_{ALL}^{t_2-1}) \right] $$
with \( J’ = J_{Q-\delta} J_{P-\delta}^{-1} \), where \( J \) denotes Jacobian submatrices. This reduces the problem to a linear programming form solvable via differential evolution algorithms.
To illustrate the practical implementation of solar inverter control, consider the following example of a grid-tied system with battery storage, highlighting the hardware context in which our strategy operates.
We now present numerical results based on an enhanced IEEE 33-node test system, incorporating eight distributed solar inverters at nodes 5, 9, 14, 18, 22, 25, 29, and 33. The rated capacities of these solar inverters are summarized in Table 1, demonstrating a diversity that reflects real-world installations. The system operates with a base PV penetration of 40% prior to control, and we assume uncertainty parameters of \( \sigma_{PV} = 6\% \) and \( \sigma_{LOAD} = 3\% \) for short-term fluctuations. Simulation parameters are listed in Table 2, with a weighting coefficient \( \alpha = 100 \) to prioritize curtailment reduction.
| Node | 5 | 9 | 14 | 18 | 22 | 25 | 29 | 33 |
|---|---|---|---|---|---|---|---|---|
| Capacity (kW) | 37.5 | 25 | 100 | 50 | 50 | 250 | 50 | 50 |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Voltage upper limit \( V_{\max} \) (pu) | 1.07 | Load uncertainty \( \sigma_{LOAD} \) | 3% |
| Voltage lower limit \( V_{\min} \) (pu) | 0.93 | PV uncertainty \( \sigma_{PV} \) | 6% |
| Weighting coefficient \( \alpha \) | 100 | Base PV penetration | 40% |
We compare three scenarios: Scenario 1 (no active control), Scenario 2 (a conventional centralized-local control without robustness to short-term fluctuations), and Scenario 3 (our proposed robust strategy). Voltage quality metrics are aggregated in Table 3, showcasing significant improvements. Under Scenario 1, voltage violations occur for 198 minutes, with extreme values of 1.1034 pu (overvoltage) and 0.9253 pu (undervoltage). Scenario 2 reduces violations to 36 minutes, but Scenario 3 eliminates them entirely, while also enhancing voltage profile consistency—the standard deviation drops from 0.1792 pu to 0.0504 pu. This underscores the efficacy of our robust slope optimization in safeguarding against fluctuation-induced risks.
| Indicator | Scenario 1 | Scenario 2 | Scenario 3 |
|---|---|---|---|
| Maximum voltage (pu) | 1.1034 | 1.0786 | 1.0688 |
| Minimum voltage (pu) | 0.9253 | 0.9398 | 0.9473 |
| Cumulative violation time (min) | 198 | 36 | 0 |
| Maximum continuous violation (min) | 19 | 3 | 0 |
| Average voltage (pu) | 0.9852 | 0.9912 | 0.9934 |
| Voltage standard deviation (pu) | 0.1792 | 0.0851 | 0.0504 |
Economic performance, detailed in Table 4, reveals further advantages. Average network losses decrease from 143.33 kW in Scenario 1 to 115.15 kW in Scenario 3, corresponding to a loss rate reduction from 4.80% to 3.86%. This is achieved alongside a curtailment rate of 5.21% in Scenario 3, indicating a trade-off that optimizes overall cost. Notably, the robust control allows solar inverters to provide more reactive power support, as illustrated in Figure 8 of the original study, leading to finer voltage regulation. The droop slopes optimized in Scenario 3 are generally more conservative (e.g., 1.392 Mvar/pu at node 18 versus 0.169 Mvar/pu in Scenario 2), enabling stronger counteraction against fluctuations.
| Metric | Scenario 1 | Scenario 2 | Scenario 3 |
|---|---|---|---|
| Average network loss (kW) | 143.33 | 121.73 | 115.15 |
| Loss rate (%) | 4.80 | 4.08 | 3.86 |
| Curtailment (kWh) | 0 | 0 | 17.22 |
| Curtailment rate (%) | 0 | 0 | 5.21 |
To delve deeper into the mechanics, we analyze the impact of short-term fluctuation intensity on network losses. Varying \( \sigma_{PV} \) and \( \sigma_{LOAD} \) from 10% to 50%, we observe that losses increase monotonically but remain relatively stable below 20% uncertainty, as shown in Table 5. Beyond this threshold, the marginal loss escalates, emphasizing the importance of robust design in high-variability environments. Interestingly, load fluctuations exert a greater influence than PV fluctuations at identical uncertainty levels, due to their broader distribution and the active damping provided by solar inverter control.
| \( \sigma_{LOAD} \backslash \sigma_{PV} \) | 10% | 20% | 30% | 40% | 50% |
|---|---|---|---|---|---|
| 10% | 108.2 | 110.5 | 113.8 | 118.1 | 123.4 |
| 20% | 110.9 | 113.3 | 116.7 | 121.2 | 126.8 |
| 30% | 114.7 | 117.2 | 120.8 | 125.6 | 131.5 |
| 40% | 119.4 | 122.1 | 125.9 | 131.0 | 137.3 |
| 50% | 125.0 | 127.9 | 132.0 | 137.4 | 144.1 |
PV penetration level is another critical factor. By scaling inverter capacities proportionally, we simulate penetration rates from 10% to 60% (post-control). The results in Table 6 indicate a convex relationship between losses and penetration, with a minimum around 20-30% penetration. At higher penetrations, reverse power flow increases losses, necessitating curtailment. For instance, at 60% penetration, losses rise to 140.2 kW with a curtailment rate of 12.7%, highlighting the need for careful capacity planning.
| Penetration (post-control) | Average Loss (kW) | Curtailment Rate (%) | Voltage Std Dev (pu) |
|---|---|---|---|
| 10% | 98.5 | 0.0 | 0.0412 |
| 20% | 92.1 | 1.2 | 0.0387 |
| 30% | 89.8 | 3.5 | 0.0421 |
| 40% | 115.2 | 5.2 | 0.0504 |
| 50% | 132.6 | 8.9 | 0.0613 |
| 60% | 140.2 | 12.7 | 0.0698 |
Inverter sizing also plays a pivotal role. Table 7 compares scenarios where all solar inverter capacities are multiplied by a factor from 1.0 to 1.5. As capacity grows, curtailment diminishes and losses improve, thanks to enhanced reactive power capability. At a factor of 1.3, curtailment drops to zero, and losses stabilize near 74 kW, suggesting an optimal sizing threshold. Moreover, the distribution of reactive power support among inverters is heterogeneous; nodes with lower R/X ratios (i.e., higher sensitivity) contribute more, as quantified by the weight \( w(i) \) in our model.
| Capacity Multiplier | Curtailment Rate (%) | Average Loss (kW) | Voltage Std Dev (pu) |
|---|---|---|---|
| 1.0 | 5.21 | 115.15 | 0.0504 |
| 1.1 | 3.31 | 92.43 | 0.0438 |
| 1.2 | 1.11 | 79.63 | 0.0392 |
| 1.3 | 0.00 | 74.36 | 0.0372 |
| 1.4 | 0.00 | 70.54 | 0.0345 |
| 1.5 | 0.00 | 68.66 | 0.0336 |
The mathematical underpinnings of our strategy can be further elaborated through sensitivity analysis. The voltage-reactive power sensitivity matrix \( S_{V-Q} \) is computed from the Jacobian at the first-stage operating point. For a solar inverter at node \( i \), the sensitivity element \( S_{V-Q}(i,i) \) dictates the minimum slope requirement. In our test system, values range from 0.02 to 0.15 pu/Mvar, reflecting network topology. The optimal slope \( m_{t_1}(i) \) is then bounded by:
$$ m_{t_1}(i) = \arg \min \left\{ \sum_{s} \left| \Delta V_{s}^{t_2}(i) \right| : m_{\min}(i) \leq m \leq m_{\max}(i) \right\} $$
where \( m_{\min}(i) = 1/S_{V-Q}(i,i) \) and \( m_{\max}(i) = \sqrt{ S_{PV}(i)^2 – (P_{PV}^{t_1}(i) – \Delta P_{PV}^{t_1}(i))^2 } / \Delta V_{\max} \). The differential evolution algorithm searches this space efficiently, evaluating candidate slopes via linearized voltage deviations. Convergence is typically achieved within 50 iterations for our case, with a population size of 20.
From an implementation perspective, the proposed control strategy can be integrated into existing solar inverter firmware with modest modifications. The first-stage optimization runs at a central controller (e.g., a distribution management system) every 15 minutes, transmitting setpoints \( \Delta P_{PV}^{t_1}(i) \) and \( \Delta Q_{PV}^{t_1}(i) \) to each solar inverter. The second-stage slopes \( m_{t_1}(i) \) are also dispatched and stored locally. Then, at 1-minute intervals, each solar inverter measures its terminal voltage, computes the change \( V_{t_2}(i) – V_{t_2-1}(i) \), and adjusts reactive output according to the droop law. This hierarchical structure minimizes communication burden while ensuring adaptability. In practice, solar inverters may also incorporate active power curtailment as a last resort, activated when voltage approaches limits despite reactive efforts. Our model coordinates this seamlessly through the three-parameter framework.
To quantify the benefits of robustness, we examine the worst-case voltage deviations under extreme fluctuation scenarios. Define the performance index \( \rho = \max_{s,i,t_2} | V_{s}^{t_2}(i) – V_{s}^{t_1}(i) | \). For Scenario 2, \( \rho = 0.032 \) pu, whereas for Scenario 3, \( \rho = 0.018 \) pu—a 44% reduction. This margin is crucial for preventing protection tripping and ensuring power quality. Furthermore, the robust slopes reduce the frequency of active curtailment events; in our simulation, curtailment occurred in 7 out of 96 intervals for Scenario 3, compared to 12 for Scenario 2.
The role of solar inverter technology in modern grids cannot be overstated. Advanced inverters with grid-support functions (e.g., IEEE 1547-2018 compliance) are enablers of strategies like ours. They offer programmable droop characteristics, fast response times (within cycles), and communication interfaces for setpoint updates. By leveraging these capabilities, our control strategy transforms solar inverters from mere power converters into active grid assets. This aligns with global trends toward smart inverters and virtual power plants, where distributed resources collectively provide ancillary services. Our work provides a concrete methodology for optimizing such services in the face of uncertainty.
Looking ahead, several extensions merit exploration. First, incorporating battery energy storage systems (BESS) alongside solar inverters could enhance fluctuation mitigation, as BESS can absorb surplus PV power and inject reactive power independently. Second, multi-agent distributed optimization could decentralize the first-stage computation, improving scalability. Third, machine learning techniques might predict short-term fluctuations more accurately, refining the interval bounds. Finally, field trials in real distribution networks would validate our simulations and uncover practical nuances.
In conclusion, we have presented a robust centralized-local control strategy for distributed solar inverters that effectively addresses short-term fluctuations in active distribution networks. By combining a two-stage optimization—economic dispatch followed by robust slope tuning—we achieve superior voltage regulation, reduced losses, and enhanced security. Our analysis demonstrates that the strategy is scalable, computationally feasible, and adaptable to varying penetration levels and uncertainty intensities. As solar inverter deployments continue to grow, such intelligent control paradigms will be indispensable for maintaining grid stability and maximizing renewable energy utilization. We envision this work as a step toward more resilient and sustainable power systems, where solar inverters play a pivotal role in the energy transition.