Accurate modeling of grid-connected solar inverters is crucial for analyzing power system fault characteristics under large-scale photovoltaic integration. However, existing simulation models often deviate from actual solar inverter dynamics due to unknown control parameters, stemming from proprietary manufacturer data or operational environmental changes. To address this, parameter identification techniques are employed to construct precise solar inverter models. This study proposes a stepwise identification strategy using an improved chaos genetic particle swarm optimization (CGAPSO) algorithm, leveraging real-world data from a 1 MW solar power plant. The solar inverter’s operational phases—normal, fault-tolerant, and recovery—are analyzed to classify parameters by sensitivity using mathematical perturbation methods. High-sensitivity parameters are prioritized in identification to enhance accuracy, while initial parameter ranges are derived from measurable external characteristics. Comparative results demonstrate the superiority of CGAPSO over synchronous identification, with errors below 1.1% for all parameters. The identified model closely matches actual solar inverter output, validating its effectiveness in replicating real-world behavior.
The proliferation of solar energy integration has heightened the need for reliable solar inverter models. Solar inverters convert DC power from photovoltaic arrays to AC power synchronized with the grid, and their control strategies—such as dual-loop control during normal operation and low-voltage ride-through (LVRT) during faults—dictate system behavior. Traditional models often inaccurately represent solar inverter dynamics due to unverified control parameters, leading to erroneous fault analysis. This work focuses on identifying key control parameters, including proportional-integral (PI) gains and LVRT coefficients, to bridge this gap. By categorizing parameters based on sensitivity and employing a structured identification approach, the proposed method ensures high-fidelity modeling of solar inverters.
Solar inverter control systems typically utilize a dual-loop structure: an outer voltage loop and an inner current loop. The mathematical model for a grid-connected solar inverter in the dq-frame is expressed as:
$$ U_q(s) = \left( K_{iP} + \frac{K_{iI}}{s} \right) \left( -I_q(s) + \omega L I_d(s) \right) $$
$$ U_d(s) = \left( K_{iP} + \frac{K_{iI}}{s} \right) \left( K_{uP} (U_{dcref} – U_{dc}) + K_{uI} \int_0^{t_1} (U_{dcref} – U_{dc}) \, dt – I_d(s) \right) – \omega L I_q(s) + E_d(s) $$
Here, \( U_d \), \( U_q \), \( I_d \), and \( I_q \) represent dq-axis voltages and currents at the inverter output, \( E_d \) is the filtered grid voltage, \( \omega \) is the angular frequency, \( K_{iP} \) and \( K_{iI} \) are current loop PI parameters, \( K_{uP} \) and \( K_{uI} \) are voltage loop PI parameters, \( U_{dc} \) is the DC-link voltage, and \( U_{dcref} \) is its reference. During grid faults, solar inverters switch to LVRT mode, where reactive current support is activated based on voltage sag depth:
$$ I_{q\_fault}^* = \begin{cases}
0, & U_{pcc} > 0.9 \, \text{p.u.} \\
K_q (0.9 – U_{pcc}) I_n, & 0.2 \, \text{p.u.} \leq U_{pcc} \leq 0.9 \, \text{p.u.} \\
1.1 I_n, & U_{pcc} < 0.2 \, \text{p.u.}
\end{cases} $$
The active current reference during faults is:
$$ I_{d\_fault}^* = \min \left( I_{dref\_normal}, \sqrt{I_{n,\max}^2 – (I_{q\_fault}^*)^2} \right) $$
Post-fault recovery involves a linear active current ramp:
$$ I_{d\_recovery}^* = (t – t_1) K_{dp} + I_{d\_fault} $$
$$ I_{q\_recovery}^* = 0 $$
where \( K_q \) is the reactive support coefficient, \( K_{dp} \) is the active recovery coefficient, and \( I_n \) is the rated current. These equations define the parameters for identification: \( K_{uP} \), \( K_{uI} \), \( K_{iP} \), \( K_{iI} \) in normal operation; \( K_q \), \( K_{iP} \), \( K_{iI} \) during faults; and \( K_{dp} \), \( K_{iP} \), \( K_{iI} \) during recovery.
Mathematical perturbation analysis reveals parameter sensitivity by examining their impact on the current loop PI output. For normal operation, the d-axis PI output is:
$$ Y_{PI.d} = K_{iP} \left( K_{uP} (U_{dcref} – U_{dc}) + K_{uI} \int_0^{t_1} (U_{dcref} – U_{dc}) \, dt – I_d \right) + K_{iI} \int_0^{t_1} \left( K_{uP} (U_{dcref} – U_{dc}) + K_{uI} \int_0^{t_1} (U_{dcref} – U_{dc}) \, dt – I_d \right) dt $$
Deviations in high-sensitivity parameters (e.g., \( K_{uP} \), \( K_{uI} \)) cause significant output errors due to integral action, whereas low-sensitivity parameters (e.g., \( K_{iP} \), \( K_{iI} \)) induce minor changes. Similarly, during faults, \( K_q \) dominates the output deviation. Thus, parameters are classified as:
- High-sensitivity: \( K_{uP} \), \( K_{uI} \), \( K_q \), \( K_{dp} \)
- Low-sensitivity: \( K_{iP} \), \( K_{iI} \)
Synchronous identification often fails for low-sensitivity parameters due to dominance by high-sensitivity ones. A stepwise approach is proposed: first identify high-sensitivity parameters with low-sensitivity ones fixed at empirical values, then refine low-sensitivity parameters. This reduces interference and improves accuracy.
Initial parameter ranges are derived from operational data to guide identification. For voltage loop parameters, the relationship between current and voltage deviations is:
$$ \frac{\Delta I_d}{\Delta U_{dc}} = K_{uP} + K_{uI} \Delta t $$
Using measured \( \Delta I_d \) and \( \Delta U_{dc} \) during disturbances, \( K_{uP} \) and \( K_{uI} \) ranges are estimated. For current loop parameters, the system order (first or second) is determined by comparing fitness values under different ranges. The fitness function is:
$$ \text{Obj. Fun} = \frac{1}{n} \sum_{j=1}^{n} | I_{d\_true,j} – I_{d\_sim,j} | $$
where \( n \) is data length. LVRT parameters \( K_q \) and \( K_{dp} \) are bounded using fault current profiles.
The improved CGAPSO algorithm combines genetic operations with chaotic perturbations. Initially, genetic algorithm-based particle swarm optimization (GAPSO) accelerates convergence. When the average fitness improvement falls below 0.1, Tent chaos mapping is applied to the global best for local refinement:
$$ o_{i+1} = \begin{cases}
\frac{o_i}{h}, & 0 < o_i \leq h \\
\frac{1 – o_i}{1 – h}, & h < o_i \leq 1
\end{cases} $$
$$ \beta = 0.1 \frac{(Gen_{\max} – Gen + 1)^2}{Gen_{\max}^2} $$
$$ o_{new,i+1} = \begin{cases}
o_1 + \beta o_{i+1}, & i \mod 2 = 0 \\
o_1 – \beta o_{i+1}, & i \mod 2 \neq 0
\end{cases} $$
$$ X_{new,i+1} = X_{\min} + (X_{\max} – X_{\min}) o_{new,i+1} $$
Here, \( h = 0.5 \), \( Gen \) is the current iteration, and \( Gen_{\max} \) is the maximum iterations. This enhances precision while avoiding local minima.
Identification proceeds in stages:
- Fault Stage: Identify \( K_q \) with \( K_{iP} \), \( K_{iI} \) fixed.
- Recovery Stage: Identify \( K_{dp} \) with \( K_{iP} \), \( K_{iI} \) fixed.
- Normal Stage: Identify \( K_{uP} \), \( K_{uI} \) with \( K_{iP} \), \( K_{iI} \) fixed.
- Refine \( K_{iP} \), \( K_{iI} \) using all stages.
Simulations based on a 1 MW solar power plant validate the method. The solar inverter model includes a 10.4 mF DC capacitor, 2.3 mH filter inductance, and 0.01 Ω resistance. A grid fault at 1 s triggers LVRT. Comparative results between synchronous and stepwise identification are shown below.
| Parameter | Range | Basis |
|---|---|---|
| \( K_{uP} \) | 1.5 – 3.5 | Current-voltage deviation ratio |
| \( K_{uI} \) | 1.3542 – 5.35 | Integral effect over time |
| \( K_{iP} \) (1st-order) | 14.45 – 144.5 | System bandwidth analysis |
| \( K_{iI} \) (1st-order) | 6.283 – 62.83 | System bandwidth analysis |
| \( K_{iP} \) (2nd-order) | 20.4323 – 204.323 | Damping and natural frequency |
| \( K_{iI} \) (2nd-order) | 4539.8 – 45398 | Damping and natural frequency |
| \( K_q \) | 0.355 – 3.554 | Fault current characteristics |
| \( K_{dp} \) | 0.231 – 2.312 | Recovery current slope |
The current loop order is determined by comparing fitness values with fixed voltage parameters. Lower fitness for second-order ranges confirms the system design, guiding parameter selection for solar inverters.
| \( K_{uP} \), \( K_{uI} \) | \( K_{iP} \), \( K_{iI} \) (1st-order) | Fitness (1st-order) | \( K_{iP} \), \( K_{iI} \) (2nd-order) | Fitness (2nd-order) |
|---|---|---|---|---|
| 2, 3 | 40.5, 50.3 | 0.5103 | 80, 25000 | 0.4294 |
| 3.23, 5.15 | 102.7, 34.3 | 0.1447 | 115.7, 26000 | 0.1295 |
| 1.72, 2.13 | 15.23, 7.11 | 0.6598 | 21.72, 5123.34 | 0.4713 |
Synchronous identification using standard PSO yields poor accuracy for low-sensitivity parameters, as high-sensitivity ones dominate the search direction.
| Parameter | Identified Value | Error (%) |
|---|---|---|
| \( K_q \) | 2.0036 | 0.180 |
| \( K_{dp} \) | 1.4121 | 5.860 |
| \( K_{uP} \) | 3.3025 | 10.080 |
| \( K_{uI} \) | 2.3881 | 19.405 |
| \( K_{iP} \) | 84.0459 | 40.080 |
| \( K_{iI} \) | 24112 | 20.560 |
In contrast, stepwise identification with CGAPSO achieves high precision across all parameters, demonstrating the method’s efficacy for solar inverter modeling.
| Parameter | Identified Value | Error (%) |
|---|---|---|
| \( K_q \) | 2.0001 | 0.005 |
| \( K_{dp} \) | 1.5050 | 0.330 |
| \( K_{uP} \) | 3.0316 | 1.050 |
| \( K_{uI} \) | 2.0166 | 0.830 |
| \( K_{iP} \) | 60.5160 | 0.860 |
| \( K_{iI} \) | 20183 | 0.915 |
The identified model’s output closely matches actual solar inverter data, with active and reactive currents aligning during all operational phases. This confirms the model’s accuracy in representing solar inverter behavior under normal and fault conditions. The stepwise CGAPSO approach effectively mitigates the influence of high-sensitivity parameters on low-sensitivity ones, ensuring comprehensive identification. Future work will expand parameter sets to include limiter values and other control elements, further enhancing solar inverter model fidelity for grid stability studies.
In conclusion, this study presents a robust framework for solar inverter parameter identification. By leveraging sensitivity analysis, operational data, and an advanced optimization algorithm, the method delivers precise control parameters for solar inverters. The resulting models faithfully replicate real-world dynamics, aiding in the reliable integration of solar energy into power systems. The proposed approach is scalable to various solar inverter configurations, underscoring its practical utility in renewable energy applications.