With the rapid integration of renewable energy into power systems, photovoltaic (PV) systems have become a pivotal component. The grid-connected inverter, as the core interface between PV arrays and the grid, plays a crucial role in ensuring system stability and reliability. Accurate modeling of the inverter’s control parameters, especially during fault conditions like low-voltage ride-through (LVRT), is essential for simulating and analyzing grid behavior. However, manufacturers often keep control strategies and parameters confidential for commercial reasons, leading to discrepancies between simulation models and actual field measurements. This paper addresses this challenge by proposing a multi-stage parameter identification method for the LVRT model of grid-connected PV inverters, leveraging the Snow Ablation Optimizer (SAO) algorithm. The method accounts for parameter coupling and correlations, effectively mitigating plateau phenomena and improving identification accuracy.
The LVRT capability of a grid-connected inverter ensures that it remains connected and supports the grid during voltage dips, as mandated by standards such as GB/T32892-2016. Traditional identification methods often focus on steady-state parameters, neglecting the transient dynamics during faults and recovery phases. This can result in inaccurate models that fail to capture the inverter’s true behavior. Our approach explicitly incorporates the recovery transient process in the LVRT model, enabling a comprehensive representation of the inverter’s output characteristics. By analyzing the coupling between inner-loop PI control parameters and LVRT parameters, we develop a multi-stage identification strategy that decouples and sequentially identifies parameters based on their correlations. The SAO algorithm, inspired by the sublimation and ablation behaviors of snow, is employed for optimization due to its global search capabilities and fast convergence. Simulation results under various voltage dip scenarios demonstrate the effectiveness and robustness of our method.
The structure of this paper is as follows: First, we establish a mathematical model for the PV grid-connected system, including the LVRT control with recovery transients. Second, we analyze the coupling and correlations among parameters, leading to the multi-stage identification strategy. Third, we detail the SAO algorithm and its application. Finally, simulation case studies validate the method, comparing it with traditional approaches and testing its stability under different fault conditions.
Modeling of PV Grid-Connected System with LVRT
A typical PV grid-connected system consists of a PV array, a DC/AC inverter, and control loops. During normal operation, the inverter uses dual-loop control: an outer loop for power regulation and an inner loop for current control. When a grid fault occurs, the system switches to LVRT mode to provide reactive current support and limit active current. The LVRT process can be divided into three stages: steady-state, fault steady-state, and recovery transient. The recovery transient is critical as it prevents sudden current changes that could stress the DC bus and grid.
The inner-loop current control in the dq-frame is given by:
$$ u_d^* = k_{ip}(i_d^* – i_d) + k_{ii} \int (i_d^* – i_d) dt – \omega L i_q + e_d $$
$$ u_q^* = k_{ip}(i_q^* – i_q) + k_{ii} \int (i_q^* – i_q) dt + \omega L i_d + e_q $$
where \( k_{ip} \) and \( k_{ii} \) are the proportional and integral coefficients of the inner-loop PI controller, \( i_d^* \) and \( i_q^* \) are the reference currents, \( u_d^* \) and \( u_q^* \) are the reference voltages, \( \omega \) is the angular frequency, \( L \) is the inductance, and \( e_d \) and \( e_q \) are the grid voltages. During LVRT, the reference currents are determined by:
$$ i_q^* = K_{1\_Iq\_LV}(0.9 – U_{PCC}) + I_{qset\_LV} $$
$$ i_d^* = K_{1\_Ip\_LV} \times U_{PCC} + I_{pset\_LV} $$
where \( K_{1\_Iq\_LV} \) and \( K_{1\_Ip\_LV} \) are reactive and active current coefficients, \( I_{qset\_LV} \) and \( I_{pset\_LV} \) are current setpoints, and \( U_{PCC} \) is the per-unit voltage at the point of common coupling. After fault clearance, the active current recovers with a ramp rate:
$$ i_d^* = k_{lvrt\_p}(t – t_1) + i_{p0} $$
$$ i_q^* = i_{q0} $$
where \( k_{lvrt\_p} \) is the recovery rate, \( t_1 \) is the fault clearance time, and \( i_{p0} \) and \( i_{q0} \) are currents at \( t_1 \). The key parameters to be identified are: \( k_{ip} \), \( k_{ii} \), \( K_{1\_Iq\_LV} \), \( I_{qset\_LV} \), \( K_{1\_Ip\_LV} \), \( I_{pset\_LV} \), and \( k_{lvrt\_p} \). Outer-loop parameters are less sensitive and can be set to typical values.
Multi-Stage Identification Strategy Considering Parameter Correlations
Direct simultaneous identification of all parameters leads to plateau phenomena due to coupling and correlations. For instance, the inner-loop parameters \( k_{ip} \) and \( k_{ii} \) are coupled with LVRT parameters in the control equations. To address this, we first analyze parameter correlations using Hessian matrix-based method. The sensitivity matrix \( \mathbf{Q}(\theta) \) is constructed from output trajectories, and the Hessian matrix is approximated as \( \mathbf{H}(\theta) = \mathbf{Q}(\theta)^T \mathbf{Q}(\theta) \). The correlation factor \( \lambda_j(\mathbf{H}) \) is computed, and parameters with \( \lambda_j(\mathbf{H}) > 10^5 \) are considered correlated.
Based on this analysis, parameters are classified into three categories:
- Independent in both active and reactive currents (IAR): \( k_{ip} \), \( k_{ii} \).
- Active-current-related (ACR): \( K_{1\_Iq\_LV} \), \( I_{qset\_LV} \).
- Reactive-current-related (RCR): \( K_{1\_Ip\_LV} \), \( I_{pset\_LV} \), \( k_{lvrt\_p} \).
The multi-stage identification strategy proceeds as follows:
- Stage 1: Identify IAR parameters by minimizing the objective function considering both active and reactive currents:
$$ J = \frac{1}{n} \sum_{t=1}^{n} (i_d – i_{d\_ref})^2 + \frac{1}{n} \sum_{t=1}^{n} (i_q – i_{q\_ref})^2 $$
All parameters are searched within theoretical bounds. - Stage 2: Estimate ACR parameters using only reactive current error:
$$ J = \frac{1}{n} \sum_{t=1}^{n} (i_q – i_{q\_ref})^2 $$
IAR parameters are fixed from Stage 1, others are searched broadly. - Stage 3: Estimate RCR parameters using only active current error:
$$ J = \frac{1}{n} \sum_{t=1}^{n} (i_d – i_{d\_ref})^2 $$
IAR parameters fixed, others searched broadly. - Stage 4: Correct ACR and RCR parameters by minimizing the full objective function with narrowed search ranges around estimates from Stages 2 and 3.
This staged approach decouples parameters, reduces overfitting, and enhances accuracy.
Snow Ablation Optimizer Algorithm
The Snow Ablation Optimizer (SAO) is a metaheuristic algorithm inspired by the sublimation and ablation of snow. It features four phases: initialization, exploration, exploitation, and a dual-population mechanism. The algorithm balances global and local search, making it suitable for parameter identification.
Initialization: The population is randomly generated within bounds:
$$ \mathbf{Z} = \mathbf{R} + \alpha \times (\mathbf{U} – \mathbf{R}) $$
where \( \mathbf{Z} \) is the population matrix, \( \mathbf{U} \) and \( \mathbf{R} \) are upper and lower bounds, and \( \alpha \) is a random vector in [0,1].
Exploration: Simulates Brownian motion of water vapor, encouraging global search:
$$ \mathbf{Z}_h(m+1) = \mathbf{Z}_h(m) + \beta_h(m) \otimes \left[ \alpha (\mathbf{A}_{elite}(m) – \mathbf{Z}_h(m)) + (1-\alpha)(\bar{\mathbf{Z}}(m) – \mathbf{Z}_h(m)) \right] $$
where \( \beta_h(m) \) is a Brownian step vector, \( \mathbf{A}_{elite}(m) \) is a random elite individual, and \( \bar{\mathbf{Z}}(m) \) is the centroid of the population.
Exploitation: Uses a degree-day model for local search around the best solution:
$$ \mathbf{Z}_h(m+1) = M \times \mathbf{G}(m) + \beta \otimes \left[ \beta (\mathbf{G}(m) – \mathbf{Z}_h(m)) + (1-\beta)(\bar{\mathbf{Z}}(m) – \mathbf{Z}_h(m)) \right] $$
$$ M = \left( 0.35 + 0.25 \frac{e^{m/m_{max}} – 1}{e – 1} \right) \times T(m) $$
where \( \mathbf{G}(m) \) is the best solution, \( M \) is the ablation rate, and \( T(m) \) is a temperature factor.
Dual-Population Mechanism: The population is split into two subpopulations for exploration and exploitation, with sizes dynamically adjusted over iterations to maintain balance.
The SAO algorithm demonstrates superior convergence speed and accuracy compared to traditional algorithms like Particle Swarm Optimization (PSO), as shown in benchmark tests. Its ability to escape local optima makes it ideal for identifying parameters of grid-connected inverters.
Simulation Case Studies
We validate the proposed method using a PV grid-connected system model based on actual field data. The system is simulated in RT-LAB hardware-in-the-loop platform, with faults applied to cause voltage dips of 20%, 40%, 60%, and 80% of nominal voltage. The SAO algorithm is configured with a population size of 50 and 50 iterations.
Multi-Stage Identification Process
For an 80% voltage dip, parameter correlations are analyzed. The Hessian matrix shows that \( K_{1\_Iq\_LV} \) and \( I_{qset\_LV} \) are correlated with active current, while \( K_{1\_Ip\_LV} \), \( I_{pset\_LV} \), and \( k_{lvrt\_p} \) are correlated with reactive current. Thus, classification aligns with the theory. The multi-stage identification results are summarized in Table 1.
| Parameter | Stage 1 | Stage 2 | Stage 3 | Stage 4 | True Value |
|---|---|---|---|---|---|
| \( k_{ip} \) | 6.11 | 6.11 | 6.11 | 6.11 | 6 |
| \( k_{ii} \) | 1.03 | 1.03 | 1.03 | 1.03 | 1 |
| \( K_{1\_Iq\_LV} \) | 1.89 | 1.55 | 1.75 | 1.70 | 1.75 |
| \( I_{qset\_LV} \) | -0.84 | -1.90 | -3.60 | -2.15 | -2.10 |
| \( K_{1\_Ip\_LV} \) | 2.32 | 0.56 | 1.70 | 1.93 | 1.89 |
| \( I_{pset\_LV} \) | 40.13 | 61.35 | 40.90 | 40.79 | 40.18 |
| \( k_{lvrt\_p} \) | 1.18 | 1.56 | 1.00 | 1.03 | 1.05 |
Stage 1 identifies IAR parameters accurately. Stages 2 and 3 estimate ACR and RCR but show overfitting when using single current errors. Stage 4 corrects these by considering both currents, yielding close matches to true values. The simulated outputs align well with measured data, confirming the strategy’s effectiveness.
Comparison with Traditional Sensitivity-Based Identification
Traditional methods fix less sensitive parameters and identify others simultaneously, often leading to overfitting. For the same 80% dip, traditional identification results in larger errors, as shown in Table 2. The multi-stage method reduces overall fitting error from 1.65 to 0.074, demonstrating superior accuracy.
| Parameter | Multi-Stage Value | Error (%) | Traditional Value | Error (%) |
|---|---|---|---|---|
| \( k_{ip} \) | 6.11 | 1.84 | 6.15 | 2.5 |
| \( k_{ii} \) | 1.02 | 2.0 | 0.97 | 3.1 |
| \( K_{1\_Iq\_LV} \) | 1.70 | 2.86 | 1.86 | 6.43 |
| \( I_{qset\_LV} \) | -2.15 | 2.33 | -2.01 | 4.48 |
| \( K_{1\_Ip\_LV} \) | 1.93 | 2.11 | 1.99 | 5.42 |
| \( I_{pset\_LV} \) | 40.79 | 1.52 | 38.52 | 4.12 |
| \( k_{lvrt\_p} \) | 1.03 | 1.90 | 1.10 | 4.76 |
Effectiveness of SAO Algorithm
Replacing SAO with PSO in the multi-stage strategy results in higher errors and slower convergence. SAO achieves an average fitness value of 0.0074 over 50 runs, while PSO yields 0.021. SAO converges in 29 iterations versus 35 for PSO, highlighting its efficiency. The performance of SAO in parameter identification for grid-connected inverters is robust, making it a preferred choice for such optimization problems.
Stability Under Different Voltage Dips
The multi-stage strategy is tested under voltage dips of 60%, 40%, and 20%. Identification results remain accurate, with errors below 3% for all parameters, as summarized in Tables 3-5. The simulated currents match measured data closely in each case, proving the method’s stability and adaptability to various fault conditions.
| Parameter | Identified Value | Error (%) | Search Range |
|---|---|---|---|
| \( k_{ip} \) | 6.08 | 1.4 | [0, 12] |
| \( k_{ii} \) | 1.02 | 2.0 | [0, 2] |
| \( K_{1\_Iq\_LV} \) | 1.81 | 3.16 | [0, 4] |
| \( I_{qset\_LV} \) | -2.05 | 2.4 | [-4, 0] |
| \( K_{1\_Ip\_LV} \) | 1.84 | 2.41 | [0, 4] |
| \( I_{pset\_LV} \) | 39.41 | 1.92 | [0, 80] |
| \( k_{lvrt\_p} \) | 1.06 | 1.0 | [0, 2] |
| Parameter | Identified Value | Error (%) | Search Range |
|---|---|---|---|
| \( k_{ip} \) | 6.07 | 1.2 | [0, 12] |
| \( k_{ii} \) | 0.99 | 1.0 | [0, 2] |
| \( K_{1\_Iq\_LV} \) | 1.79 | 2.53 | [0, 4] |
| \( I_{qset\_LV} \) | -2.06 | 2.1 | [-4, 0] |
| \( K_{1\_Ip\_LV} \) | 1.91 | 1.55 | [0, 4] |
| \( I_{pset\_LV} \) | 39.70 | 1.2 | [0, 80] |
| \( k_{lvrt\_p} \) | 1.06 | 1.0 | [0, 2] |
| Parameter | Identified Value | Error (%) | Search Range |
|---|---|---|---|
| \( k_{ip} \) | 5.93 | 1.3 | [0, 12] |
| \( k_{ii} \) | 1.01 | 1.0 | [0, 2] |
| \( K_{1\_Iq\_LV} \) | 1.70 | 2.6 | [0, 4] |
| \( I_{qset\_LV} \) | -2.14 | 2.0 | [-4, 0] |
| \( K_{1\_Ip\_LV} \) | 1.92 | 1.5 | [0, 4] |
| \( I_{pset\_LV} \) | 39.78 | 1.0 | [0, 80] |
| \( k_{lvrt\_p} \) | 1.07 | 2.0 | [0, 2] |
These results underscore the robustness of our multi-stage identification strategy for grid-connected inverters across diverse operating conditions. The on grid inverter parameters are accurately captured, ensuring reliable model performance.
Conclusion
This paper presents a multi-stage parameter identification method for the low-voltage ride-through model of grid-connected PV inverters, utilizing the Snow Ablation Optimizer algorithm. By incorporating the recovery transient process, we establish a comprehensive LVRT model that reflects actual inverter dynamics. The coupling between inner-loop PI control parameters and LVRT parameters is analyzed, leading to a classification based on correlations. The multi-stage strategy sequentially identifies parameters, mitigating plateau phenomena and overfitting. Simulation studies under various voltage dips demonstrate that our method achieves higher accuracy compared to traditional sensitivity-based approaches, with errors below 3% for all key parameters. The SAO algorithm proves effective in optimization, offering fast convergence and precise results. This work contributes to improved modeling of grid-connected inverters, enhancing the reliability and stability of PV-integrated power systems. Future research may extend the method to other renewable energy inverters and complex grid scenarios.