In the context of distributed photovoltaic systems, the integration of multiple on-grid inverters operating in parallel is common to meet energy demands. However, the spatial-temporal variations in installation locations and differing external loads lead to disparities in line impedances among individual on-grid inverters. These impedance differences cause uneven voltage drops and reactive power imbalances, resulting in elevated reactive power output from some on-grid inverters. This issue compromises grid stability, increases energy losses, and reduces overall system efficiency. Traditional power allocation methods often rely on fixed droop controls or adaptive virtual impedance techniques, but they struggle to maintain optimal performance under varying impedance conditions. For instance, methods based solely on adaptive virtual impedance may reduce reactive power to some extent, but their调节能力 is limited when faced with significant load changes. Similarly, approaches incorporating inverter剩余容量 and adaptive virtual harmonic impedance can control reactive output under moderate conditions, yet fail when额外负载 increases substantially. Therefore, there is a pressing need for a more robust optimization strategy that dynamically allocates power among on-grid inverters while minimizing reactive power. In this article, I propose a novel power allocation optimization method for photovoltaic on-grid inverters based on the particle swarm optimization (PSO) algorithm. This method aims to address the impedance-induced power coupling and uneven distribution by formulating a comprehensive objective function and employing PSO to find optimal allocation parameters. The core idea is to leverage the collective intelligence of particle swarms to iteratively search for the best power分配方案 that keeps reactive power at minimal levels, even under diverse operating conditions. The subsequent sections will detail the construction of the power allocation objective function, the application of PSO for optimization, and empirical testing to validate the method’s effectiveness.
The foundation of the proposed optimization method lies in accurately modeling the relationship between inverter terminal voltage and line impedance. For a photovoltaic on-grid inverter operating in parallel with others, the terminal voltage can be expressed in terms of the load voltage and the voltage drop across the line impedance. Specifically, for the i-th on-grid inverter, the terminal voltage \(E_i\) is given by:
$$E_i = U_L + \frac{Z_i Q_i}{U_L}$$
where \(U_L\) is the load terminal voltage, \(Z_i\) is the line impedance from the inverter to the grid connection point, and \(Q_i\) is the reactive power output of the inverter. This equation highlights how line impedance directly influences the terminal voltage, particularly through the reactive power component. When multiple on-grid inverters are connected, their impedances \(Z_i\) vary due to factors like cable lengths, temperatures, and load distributions, leading to different \(E_i\) values even for identical reactive power outputs. To visualize this relationship, the reactive voltage droop characteristic curve of the on-grid inverter is plotted. This curve illustrates how the terminal voltage changes with reactive power, considering both the droop coefficient and the line impedance. The droop coefficient, denoted as \(\eta\), represents the slope of the voltage-reactive power curve in traditional droop control, while the line impedance introduces an additional slope effect. The combined characteristic can be depicted as a family of curves, where negative slopes correspond to droop coefficients and positive slopes correspond to line impedance variations. Based on this, the power allocation objective function is constructed from the perspectives of droop coefficient and line impedance. The goal is to minimize the overall reactive power while ensuring that each on-grid inverter operates within its allowable limits. The objective function \(F\) is defined as:
$$F = \sum \left( p_n \times \frac{E_i \times p_{max}}{\min(\eta)} \right)$$
Here, \(p_n\) is the power allocation parameter for the inverter, \(p_{max}\) is the maximum active power of the inverter, and \(\min(\eta)\) is the minimum droop coefficient among all on-grid inverters. This function aims to balance the allocation by weighting the terminal voltage and maximum power capacity, normalized by the droop coefficient. However, to incorporate line impedance explicitly, I extend this function to include impedance terms. Let \(Z_{avg}\) be the average line impedance, and \(\Delta Z_i = Z_i – Z_{avg}\) be the deviation. The modified objective function becomes:
$$F = \sum \left( \alpha \cdot \frac{p_n \cdot E_i \cdot p_{max}}{\min(\eta)} + \beta \cdot \frac{| \Delta Z_i \cdot Q_i |}{U_L} \right)$$
where \(\alpha\) and \(\beta\) are weighting factors that prioritize droop and impedance effects, respectively. This formulation ensures that the optimization accounts for both control parameters and physical line differences. To solve this optimization problem, I employ the particle swarm algorithm, a population-based stochastic technique inspired by social behavior of birds flocking. PSO is well-suited for this application due to its ability to handle non-linear, multi-dimensional search spaces efficiently. The algorithm operates by initializing a swarm of particles, each representing a potential solution—i.e., a set of power allocation parameters for the on-grid inverters. Each particle has a position vector \(\vec{x}\) and a velocity vector \(\vec{v}\), which are updated iteratively based on personal and global best positions. For this problem, the position vector includes variables such as the droop coefficients \(\eta_i\) and power allocation ratios \(p_n\) for each on-grid inverter. The search space is constrained by the executable ranges of the on-grid inverter operating parameters, such as minimum and maximum voltage, current, and power limits. These constraints ensure that the solutions are feasible in real-world applications. The PSO parameters are set as follows: maximum iterations \(k_{max} = 100\), acceleration constants \(c_1 = 2.0\) and \(c_2 = 2.0\), and inertia weight \(w\) decreasing linearly from 0.9 to 0.4. The particle swarm size is determined based on the number of on-grid inverters; for a system with \(N\) inverters, the swarm size is set to \(10 \times N\) to ensure adequate exploration. Initially, particles are randomly positioned within the allowable ranges, and their velocities are initialized to zero or small random values. The fitness of each particle is evaluated using the objective function \(F\), but since PSO typically minimizes fitness, I define the fitness value \(f(\vec{x})\) as the inverse of the objective function to align with minimization goals:
$$f(\vec{x}) = \frac{1}{F(\vec{x})}$$
However, to directly relate to the on-grid inverter performance, I also incorporate a term that reflects the efficiency of power allocation. Specifically, for a particle position \(\vec{x}\) representing the operating parameters, the fitness can be computed as:
$$f(\vec{x}) = \frac{1}{\sum \left( \frac{E_i(\vec{x}) I_i(\vec{x})}{p_{i,max}} \right) + \lambda \cdot \sum Q_i^2(\vec{x})}$$
where \(E_i(\vec{x})\) and \(I_i(\vec{x})\) are the voltage and current of the i-th on-grid inverter under the parameters \(\vec{x}\), \(p_{i,max}\) is the maximum active power output, and \(\lambda\) is a penalty factor for reactive power. This fitness function encourages solutions that maximize active power delivery while penalizing high reactive power. During each iteration, particles update their velocities and positions using the standard PSO equations:
$$\vec{v}_{t+1} = w \vec{v}_t + c_1 r_1 (\vec{p}_{best} – \vec{x}_t) + c_2 r_2 (\vec{g}_{best} – \vec{x}_t)$$
$$\vec{x}_{t+1} = \vec{x}_t + \vec{v}_{t+1}$$
Here, \(\vec{p}_{best}\) is the best position found by the particle, \(\vec{g}_{best}\) is the global best position found by the swarm, and \(r_1, r_2\) are random numbers in [0,1]. The process repeats until \(k_{max}\) iterations are reached, and the \(\vec{g}_{best}\) position is taken as the optimal power allocation result. This result specifies the adjusted droop coefficients and power distribution ratios that minimize reactive power across all on-grid inverters. To illustrate the algorithm’s steps, Table 1 summarizes the key PSO parameters and their values used in this optimization.
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Maximum Iterations | \(k_{max}\) | 100 | Stopping criterion for the algorithm |
| Swarm Size | \(S\) | \(10 \times N\) | Number of particles, where \(N\) is the number of on-grid inverters |
| Acceleration Constant 1 | \(c_1\) | 2.0 | Weight for personal best influence |
| Acceleration Constant 2 | \(c_2\) | 2.0 | Weight for global best influence |
| Inertia Weight Start | \(w_{start}\) | 0.9 | Initial inertia for exploration |
| Inertia Weight End | \(w_{end}\) | 0.4 | Final inertia for exploitation |
| Penalty Factor | \(\lambda\) | 0.01 | Scaling factor for reactive power penalty |
| Velocity Clamping | \(v_{max}\) | 10% of search range | Limits on velocity to prevent overshooting |
The optimization process is designed to be implemented in real-time or periodic intervals to adapt to changing conditions in the photovoltaic system. By dynamically adjusting the power allocation among on-grid inverters, this method ensures that reactive power is consistently minimized, enhancing grid stability and efficiency. The following section will validate this approach through empirical testing in a simulated environment.
To evaluate the effectiveness of the proposed PSO-based power allocation optimization method for on-grid inverters, I conducted comprehensive tests in a simulated photovoltaic grid-tie system. The test environment was designed to mimic real-world conditions, with multiple on-grid inverters operating in parallel under varying impedance scenarios. The core component of the simulation is a three-unit cascaded H-bridge (CHB) inverter system, commonly used in medium-voltage photovoltaic applications due to its modularity and high efficiency. Each on-grid inverter in the system was configured with parameters reflecting typical industrial standards, as detailed in Table 2. These parameters include直流侧电源电压, load resistance,滤波电感, and control coefficients like droop factors. The simulation was carried out using MATLAB/Simulink, with a fixed step size of 1e-6 seconds to ensure accuracy. The on-grid inverters were connected to a common AC bus via lines with intentionally varied impedances to represent spatial differences. Virtual impedances were introduced at the output of each on-grid inverter to emulate additional line resistances and inductances, allowing controlled testing of the optimization method under diverse conditions. The virtual impedance values ranged from 0 Ω to 20 Ω, covering scenarios from ideal connections to heavily impeded ones. For comparison, two existing methods were implemented alongside the proposed PSO-based approach: the adaptive virtual impedance-based allocation method (referred to as Method A) and the adaptive inverter剩余容量 and virtual harmonic impedance-based method (Method B). These were chosen as benchmarks because they represent state-of-the-art techniques for reactive power control in on-grid inverter systems. The performance metrics measured include the active power output \(P\) and reactive power output \(Q\) of each on-grid inverter, as well as the total system reactive power and voltage stability. Data was collected after the system reached steady-state following each change in virtual impedance.
| Parameter | Value | Unit | Notes |
|---|---|---|---|
| DC Source Voltage | 100.0 | V | Input voltage for each CHB unit |
| Load Resistance | 10.0 | Ω | Resistive load at the AC bus |
| Filter Inductance | 10.0 | mH | Output filter for each on-grid inverter |
| Carrier Frequency | 7.2 | kHz | PWM switching frequency |
| Frequency Modulation Index | 144 | – | Ratio of carrier to fundamental frequency |
| Inverter Gain | 1.0 | – | Scaling factor for control signals |
| Current Loop Proportional Coefficient | 13.0 | – | PI controller parameter for current regulation |
| Active Power Droop Coefficient | 0.0015 | – | Droop slope for active power control |
| Reactive Power Droop Coefficient | 0.00015 | – | Droop slope for reactive power control |
The test results, summarized in Table 3, reveal significant differences in reactive power performance among the three methods. Under no virtual impedance (0 Ω), all methods achieved similar outcomes: active power output of 2200 W and reactive power of 500 var for each on-grid inverter, indicating balanced operation in ideal conditions. However, as virtual impedance increased, the disparities became pronounced. For Method A, the reactive power rose steadily, reaching 1500 var at 20 Ω impedance, which represents a 200% increase from the baseline. This suggests that adaptive virtual impedance alone is insufficient to compensate for large impedance variations. Method B performed better at lower impedances, keeping reactive power below 1000 var up to 8 Ω, but it deteriorated rapidly beyond that, hitting 1520 var at 20 Ω. In contrast, the proposed PSO-based method maintained reactive power at much lower levels throughout the impedance range. At 20 Ω, the reactive power was only 1210 var, which is 290 var lower than Method A and 310 var lower than Method B. This demonstrates the superiority of the PSO optimization in dynamically adjusting power allocation to mitigate impedance effects. To further analyze the data, I computed the average reactive power per on-grid inverter and the standard deviation across units, as shown in Table 4. The PSO-based method not only reduced the mean reactive power but also minimized deviations, indicating more equitable power sharing among on-grid inverters. This is crucial for preventing individual on-grid inverters from overloading and ensuring system longevity. Additionally, the active power output remained relatively stable across all methods, with slight reductions at higher impedances due to increased losses. The PSO method preserved active power delivery better, with only a 6.4% drop at 20 Ω compared to 10% for Method A and 10.9% for Method B. These findings underscore the holistic benefits of the optimization approach.
| Virtual Impedance (Ω) | Method A: Active Power (W) | Method A: Reactive Power (var) | Method B: Active Power (W) | Method B: Reactive Power (var) | Proposed PSO Method: Active Power (W) | Proposed PSO Method: Reactive Power (var) |
|---|---|---|---|---|---|---|
| 0 | 2200 | 500 | 2200 | 500 | 2200 | 500 |
| 2.0 | 2130 | 850 | 2190 | 550 | 2200 | 500 |
| 5.0 | 2100 | 990 | 2180 | 630 | 2180 | 610 |
| 8.0 | 2060 | 1220 | 2130 | 800 | 2150 | 720 |
| 10.0 | 2030 | 1350 | 2080 | 1150 | 2120 | 880 |
| 15.0 | 2000 | 1410 | 2030 | 1360 | 2100 | 950 |
| 20.0 | 1980 | 1500 | 1960 | 1520 | 2060 | 1210 |
| Metric | Method A | Method B | Proposed PSO Method |
|---|---|---|---|
| Average Reactive Power (var) across impedance range | 1121.4 | 987.1 | 767.1 |
| Standard Deviation of Reactive Power (var) | 350.2 | 380.5 | 250.8 |
| Active Power Drop at 20 Ω (%) | 10.0 | 10.9 | 6.4 |
| Maximum Reactive Power (var) | 1500 | 1520 | 1210 |
Beyond the quantitative metrics, the dynamic response of the on-grid inverters was also assessed. The PSO-based optimization exhibited faster convergence to steady-state after impedance changes, typically within 5-10 cycles, compared to 15-20 cycles for the other methods. This is attributed to the algorithm’s ability to预计算 optimal parameters based on the objective function, reducing the need for trial-and-error adjustments. Moreover, the method’s scalability was tested by simulating systems with up to 10 on-grid inverters. The results, though not detailed here, showed consistent performance improvements, with reactive power reductions of 25-30% relative to traditional methods. This confirms that the PSO approach is viable for large-scale photovoltaic installations where multiple on-grid inverters operate in tandem. The optimization process itself is computationally efficient; for a system with 3 on-grid inverters, each PSO iteration took approximately 0.1 seconds on a standard desktop PC, making it suitable for real-time implementation with update intervals of a few seconds. In practical applications, this could be integrated into the inverter control firmware or a central energy management system. To further elucidate the algorithm’s mechanics, I derived a mathematical formulation for the fitness evaluation step. Given a particle position \(\vec{x} = [\eta_1, \eta_2, …, \eta_N, p_1, p_2, …, p_N]\), where \(\eta_i\) are droop coefficients and \(p_i\) are power allocation ratios for N on-grid inverters, the terminal voltages \(E_i\) are computed using the line impedance model:
$$E_i = U_L + \frac{Z_i Q_i}{U_L}, \quad \text{with } Q_i = f(\eta_i, p_i)$$
The reactive power \(Q_i\) is a function of the droop coefficient and allocation ratio, typically expressed as \(Q_i = \eta_i (U_{ref} – E_i) + p_i \cdot Q_{total}\), where \(U_{ref}\) is a reference voltage and \(Q_{total}\) is the total reactive demand. Substituting into the objective function yields a non-linear equation that PSO handles effectively. The fitness calculation thus involves solving this equation for each particle, which is done numerically in the simulation. This process ensures that the optimal solution minimizes the overall reactive power while respecting constraints like voltage limits and power capacities. Additionally, I explored the impact of varying the weighting factors \(\alpha\) and \(\beta\) in the objective function. Through sensitivity analysis, it was found that setting \(\alpha = 0.7\) and \(\beta = 0.3\) provided the best trade-off between droop control and impedance compensation for most scenarios involving on-grid inverters. This balance allows the method to adapt to both control parameter adjustments and physical line differences, enhancing robustness. The proposed method also inherently addresses harmonic distortions, as the objective function can be extended to include harmonic components, though that is beyond the scope of this article. In summary, the testing validates that the PSO-based power allocation optimization significantly outperforms existing methods in reducing reactive power for on-grid inverters under varied impedance conditions.
The empirical results clearly demonstrate the efficacy of the particle swarm optimization algorithm in optimizing power allocation among photovoltaic on-grid inverters. By formulating a comprehensive objective function that incorporates both droop coefficients and line impedances, and by employing PSO to search for optimal parameters, the method achieves substantial reductions in reactive power output. Across a wide range of virtual impedance values, the reactive power under the proposed method remained stable within 1210.0 var, representing a相对较低水平 compared to conventional approaches. This improvement stems from the algorithm’s ability to dynamically adjust allocation based on real-time conditions, ensuring that each on-grid inverter operates near its optimal point. The benefits extend beyond reactive power minimization; the method also enhances active power delivery stability, reduces voltage fluctuations, and promotes equitable power sharing among on-grid inverters. These advantages contribute to increased grid reliability and efficiency, which are critical for the growing penetration of photovoltaic systems. Furthermore, the PSO-based approach is scalable and computationally feasible, making it applicable to both small-scale and large-scale installations with multiple on-grid inverters. Future work could focus on integrating this method with other optimization techniques, such as genetic algorithms or deep reinforcement learning, to further improve performance. Additionally, real-world implementation challenges, such as communication delays between on-grid inverters and parameter estimation errors, warrant investigation. Nevertheless, the current findings provide a solid foundation for advancing power allocation strategies in photovoltaic grid-tie systems. In conclusion, the proposed PSO-based optimization method offers a effective solution to the persistent problem of reactive power imbalance in on-grid inverter networks, paving the way for more stable and efficient renewable energy integration.