In recent years, the integration of renewable energy sources, particularly through solar inverters, into active distribution networks has become a critical focus for improving grid resilience and efficiency. As a researcher in this field, I have observed that conventional methods for robust optimization often neglect reactive power auxiliary constraints of solar inverters, leading to high costs associated with light abandonment and electricity purchase after distribution network optimization. To address this, we propose a two-stage robust optimal scheduling method that incorporates solar inverters, aiming to minimize total expected costs while enhancing grid robustness against fluctuations in photovoltaic (PV) output and load demand. This article delves into the mathematical modeling, solution approaches, and experimental validation of our method, emphasizing the role of solar inverters in providing reactive power support and voltage regulation. We will use extensive formulas, tables, and analyses to elucidate our approach, ensuring that the keyword ‘solar inverter’ is prominently featured throughout.
The proliferation of distributed generation, especially from PV systems connected via solar inverters, has transformed traditional distribution networks into active systems with bidirectional power flows. However, the inherent variability of PV generation and load demand poses significant challenges for grid stability and economic operation. In my work, I focus on developing robust optimization techniques that can handle these uncertainties effectively. Conventional robust optimization methods, while useful, often fail to account for the reactive power capabilities of solar inverters, which are essential for voltage control and loss reduction. This oversight results in increased penalties for light abandonment and load shedding, thereby raising operational costs. Our proposed method leverages a two-stage robust optimization framework, where the first stage deals with scheduling costs under predicted conditions, and the second stage addresses worst-case scenarios with penalties for deviations. By incorporating constraints related to solar inverter reactive power auxiliary services, we aim to reduce costs and improve grid performance.
To begin, let me outline the core components of our two-stage robust optimization model. The objective is to minimize the total expected cost, which comprises costs from both stages. In the first stage, we consider scheduling costs for PV energy, fluctuating loads, energy storage, grid losses, and electricity purchase. The second stage includes penalty costs for light abandonment and load loss. The mathematical formulation is as follows. The first-stage objective function is denoted as \( C_1 \):
$$C_1 = \sum_{t=1}^{N} \left( U_t + A_t + B_t + R_t + O_t \right)$$
where \( N \) is the scheduling period, \( U_t \) is the PV energy scheduling cost at time \( t \), \( A_t \) is the fluctuating load scheduling cost, \( B_t \) is the energy storage scheduling cost, \( R_t \) is the distribution network loss cost, and \( O_t \) is the electricity purchase cost. These costs can be expressed in detail. For instance, \( U_t \) is given by:
$$U_t = \sum_{t=1}^{N} u_t(p_t)$$
Here, \( p_t \) represents the PV power output at time \( t \), and \( u_t \) is a function defining the cost associated with PV generation. Similarly, other costs are defined with appropriate functions. The second-stage objective function \( C_2 \) focuses on penalties:
$$C_2 = \sum_{t=1}^{N} \left( H_t + K_t \right)$$
where \( H_t \) is the penalty cost for reduced light abandonment, and \( K_t \) is the penalty cost for reduced load loss. The overall robust optimization problem seeks to minimize \( C_1 + C_2 \) under worst-case uncertainty realizations. This formulation ensures that scheduling decisions are resilient to fluctuations in PV output and load, which are critical when integrating solar inverters into the grid.
The constraints play a vital role in shaping the optimization model. For the first stage, we impose constraints based on branch power flow and charging/discharging power limits of the distribution network. The power balance equation is:
$$P = P_1 + P_2 + P_3 – P_4 – P_5$$
where \( P \) is the power flow distribution, \( P_1 \) is the predicted PV power output, \( P_2 \) is the maximum fluctuation difference in active power output, \( P_3 \) and \( P_4 \) are the charging and discharging capabilities of the distribution network, and \( P_5 \) is the load active power. Inequality constraints include limits on PV power output, charging power, and discharging power:
$$P_{\text{min}} \leq P_1 \leq P_{\text{max}}$$
$$S_{\text{min}} \leq S \leq S_{\text{max}}$$
$$F_{\text{min}} \leq F \leq F_{\text{max}}$$
Here, \( S \) and \( F \) represent charging and discharging power, respectively, with subscripts denoting minimum and maximum limits. These constraints ensure operational feasibility within physical network limits. For the second stage, we introduce constraints specific to solar inverter capabilities. A key aspect is the reactive power auxiliary service provided by solar inverters, which helps mitigate voltage issues and reduce losses. The constraint for solar inverter apparent power \( F \) is:
$$F = \frac{J (u_1 – u_2)}{G}$$
where \( J \) is a binary variable for reactive power auxiliary service activation, \( u_1 \) and \( u_2 \) are the upper and lower limits of power factor, and \( G \) is the maximum power at the access node. This constraint leverages the ability of solar inverters to adjust reactive power output, enhancing grid stability. Additionally, we impose constraints on energy storage units and node voltages. For energy storage, the constraints on light abandonment \( D \) and load loss \( V \) are:
$$0 \leq D^a \leq D^{-}$$
$$0 \leq V^b \leq V^{-}$$
where \( D^{-} \) and \( V^{-} \) are maximum allowable losses, and \( a \) and \( b \) are penalty coefficients. Voltage constraints at nodes with solar inverter access are:
$$U_{\text{min}} \leq \sum_{j=1}^{C} U_j \leq U_{\text{max}}$$
with \( C \) being the number of nodes with solar inverters, and \( U_j \) the voltage at node \( j \). These constraints collectively ensure that the optimization model accounts for the reactive power support from solar inverters, which is often overlooked in conventional methods.
To solve the two-stage robust optimization model, we employ a generalized linear decision rule based on affine functions. This approach approximates the uncertainty realizations using auxiliary variables. We define an auxiliary variable \( e \) as:
$$e = f(v) b(v) W E$$
where \( f(v) \) is an affine function of the random variable \( v \) in the uncertainty set, \( b(v) \) is a coupling function, \( E \) is a matching coefficient for variable coupling, and \( W \) is a random value function. Through mixed-integer linear programming, we transform \( e \) into first-stage and second-stage auxiliary variables \( g_1 \) and \( g_2 \):
$$g_1 = \frac{(y_1 – y_2) c p}{h_{\text{max}} – h_{\text{min}}}$$
$$g_2 = q (e – g_1)$$
Here, \( y_1 \) and \( y_2 \) are the reactive and active power adjustment outputs of the solar inverter, \( c \) is a correction coefficient, \( p \) is the light abandonment power, \( q \) is the compensation capacity at the inverter access node, and \( h_{\text{max}} \) and \( h_{\text{min}} \) are the upper and lower limits of compensation capacitance. By iteratively updating these variables, we search for the optimal solution that minimizes the expected cost under worst-case scenarios. This solution method ensures that scheduling decisions are robust and cost-effective, leveraging the flexibility offered by solar inverters.
In our experimental validation, we simulated an active distribution network with a combined heat and power system, integrating six solar inverters to test the proposed method. The network had a PV installed capacity of 1 MW, a combined heat and power unit rated at 1800 kW, and an energy storage system with a capacity of 1600 kWh. The daily forecast error was set at 15%, reflecting real-world variability. The PV output and load curves were modeled as shown in the data, with peak load per unit values of 2.95 for load and 3.02 for PV. To simulate PV fluctuations, we simplified the solar inverter model as a voltage source and current source combination, emphasizing the role of solar inverters in grid integration. The key parameters of the solar inverters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Number of PV units per node | 6 |
| Solar inverter rated capacity | 3300 kW |
| Maximum charging/discharging power | 400 kW |
| Charging efficiency | 0.8 |
| Branch current thermal limit | 600 A |
| Distribution network voltage | 220 V |
| Filter inductance | 5 mH |
| Switching period | 60 μs |
| DC bus capacitance | 2400 μF |
| Total solar inverter capacity | 3800 kW |
| State of charge range | 25% to 85% |
| Solar inverter remaining capacity threshold | 5.5 kVA |
| Discharging efficiency | 1.1 |
| DC bus voltage | 500 V |
| Distribution network frequency | 50 Hz |
| Reactive power compensation range | -110 to 300 kvar |
| Switching frequency | 30 kHz |
| DC boost inductance | 4 mH |
We compared our method with three conventional approaches: adaptive step-size optimization, solar-storage integrated station optimization, and scenario discrimination optimization. The performance was evaluated based on network loss, light abandonment cost, and electricity purchase cost under varying PV fluctuation ranges, which represent different levels of scenario severity. The results are summarized in Table 2, where our method consistently outperforms the others. For instance, the average network loss with our method was 3.84 kW, compared to 4.42 kW, 4.48 kW, and 4.69 kW for the conventional methods. Similarly, the average light abandonment cost was 127 monetary units, significantly lower than 164, 183, and 246 units for the other methods. The electricity purchase cost also saw reductions, with our method averaging 5317 units versus 5404, 5598, and 5689 units. These results highlight the effectiveness of incorporating solar inverter reactive power constraints into the robust optimization framework.
| Method | Average Network Loss (kW) | Average Light Abandonment Cost (monetary units) | Average Electricity Purchase Cost (monetary units) |
|---|---|---|---|
| Our Proposed Method | 3.84 | 127 | 5317 |
| Adaptive Step-Size Optimization | 4.42 | 164 | 5404 |
| Solar-Storage Integrated Station Optimization | 4.48 | 183 | 5598 |
| Scenario Discrimination Optimization | 4.69 | 246 | 5689 |
The integration of solar inverters into active distribution networks not only supports active power injection but also provides crucial reactive power ancillary services. This dual capability is often underutilized in traditional optimization methods. In our approach, by explicitly modeling the reactive power constraints of solar inverters, we enable better voltage regulation and loss reduction. For example, the solar inverter’s apparent power constraint allows it to operate within a defined power factor range, contributing to grid stability. The mathematical formulation ensures that these inverters are dispatched optimally, reducing the need for costly penalties. Moreover, the two-stage structure allows for adaptive decision-making: the first stage sets base schedules, while the second stage adjusts for real-time fluctuations, leveraging the flexibility of solar inverters.
The visual representation above illustrates a typical solar inverter system integrated with battery storage, highlighting the hardware involved in such deployments. In our simulation, similar setups were modeled to assess performance. The solar inverter parameters, as listed in Table 1, were derived from real-world specifications to ensure practical relevance. The optimization model’s effectiveness stems from its ability to handle uncertainties in PV output and load demand, which are inherent in networks with high renewable penetration. By using robust optimization, we account for the worst-case scenarios without relying on precise probability distributions, which are often difficult to obtain. This is particularly important for solar inverters, as their output can vary rapidly due to weather conditions.
To further elaborate on the mathematical details, let me discuss the uncertainty set formulation. We define an uncertainty set \( \mathcal{U} \) that contains all possible realizations of PV output and load demand fluctuations. This set is constructed based on historical data and forecast errors. For instance, if the predicted PV output at time \( t \) is \( \hat{P}_t \), the actual output \( P_t \) lies within \( [\hat{P}_t – \Delta_t, \hat{P}_t + \Delta_t] \), where \( \Delta_t \) is the maximum deviation. Similarly, load demand uncertainties are modeled. The robust optimization problem then becomes:
$$\min_{x} \max_{u \in \mathcal{U}} \left( C_1(x, u) + C_2(x, u) \right)$$
where \( x \) represents the first-stage decision variables (e.g., scheduling of solar inverter output, storage dispatch), and \( u \) represents the uncertain parameters. This min-max formulation ensures that the solution is feasible for all uncertainty realizations within \( \mathcal{U} \). The inclusion of solar inverter constraints modifies the feasible region, allowing for more resilient schedules. For example, the reactive power output \( Q_t \) of a solar inverter at time \( t \) is constrained by its apparent power rating \( S_{\text{max}} \) and active power output \( P_t \):
$$Q_t^2 \leq S_{\text{max}}^2 – P_t^2$$
This constraint ensures that the solar inverter operates within its capability curve, providing both active and reactive power support as needed. In our model, this is incorporated into the second-stage constraints to enhance voltage stability and reduce losses.
In the experimental setup, we varied the PV fluctuation range from 0.1 to 0.7 per unit to simulate increasingly severe scenarios. The network loss, light abandonment cost, and electricity purchase cost were recorded for each method. Our method showed a slower increase in these metrics with growing fluctuation ranges, indicating better robustness. For instance, when the PV fluctuation range was 0.7, the network loss for our method was 5.2 kW, compared to 6.0 kW or higher for conventional methods. This demonstrates the advantage of our approach in extreme conditions, where solar inverter reactive power support becomes crucial. The reduction in light abandonment cost is directly linked to the optimal utilization of PV generation through solar inverters, minimizing curtailment even during high variability.
Another key aspect is the computational efficiency of our solution method. The use of affine decision rules and mixed-integer linear programming allows for scalable implementation. We solved the optimization problem using MATLAB/Simulink, with simulations run on a standard computing platform. The solution time for a 24-hour scheduling period was within acceptable limits for day-ahead scheduling, making it practical for real-world applications. The iterative process converges to a robust solution by progressively refining the auxiliary variables, as described earlier. This efficiency is essential for networks with multiple solar inverters, where coordination is required to achieve global optimality.
Looking ahead, there are several directions for future research. While our method effectively incorporates solar inverter constraints, it could be extended to include more detailed models of inverter dynamics, such as harmonic distortion or fault ride-through capabilities. Additionally, integrating demand response mechanisms with solar inverter dispatch could further enhance grid flexibility. We also plan to explore distributionally robust optimization approaches that use Wasserstein distances to better capture uncertainty distributions, potentially improving cost savings. However, the current work establishes a solid foundation for leveraging solar inverters in robust grid scheduling.
In conclusion, our two-stage robust optimal scheduling method for centralized solar inverter integration in active distribution networks addresses the limitations of conventional approaches by explicitly accounting for reactive power auxiliary constraints. Through mathematical modeling and simulation, we have demonstrated significant reductions in network loss, light abandonment cost, and electricity purchase cost under various fluctuation scenarios. The key innovation lies in the holistic consideration of solar inverter capabilities, enabling both active and reactive power support for grid stability. As renewable energy penetration grows, such methods will be vital for ensuring economic and reliable operation. We believe that this work contributes to the broader goal of building resilient smart grids, where solar inverters play a pivotal role in balancing supply and demand amidst uncertainty.