We present a method for identifying the control mode and control parameters of the electromechanical transient model for a solar inverter based on the Multi-Objective Differential Evolution (MODE) algorithm. Accurately modeling the electromechanical transient behavior of solar inverters is crucial for grid stability, especially during fault conditions. The low voltage ride through (LVRT) capability of a solar inverter directly impacts the security of the power system. However, the control parameters and modes of these inverters are often considered proprietary by manufacturers, making it difficult to obtain precise simulation models. To address this challenge, we propose a systematic identification approach that combines hardware-in-the-loop testing, key point extraction, and an advanced multi-objective optimization algorithm.
We first conduct a hardware-in-the-loop (HIL) test using the RT-LAB real-time simulation platform to generate the necessary LVRT operating condition data. This platform allows us to interface with a real solar inverter controller while maintaining the controllability of a digital simulation environment. We then extract the steady-state key points from the test data during the LVRT period to establish an identification dataset. We employ the MODE algorithm to identify the control parameters under both the specified power control mode and the specified current control mode. Within the MODE framework, we introduce an adaptive parameter tuning strategy based on population diversity and use a non-dominated sorting method to enhance algorithm performance. Finally, we compare the simulation results of different control modes against the measured data to determine the actual LVRT control mode of the solar inverter. Our results demonstrate that the proposed method can accurately and reliably identify the control mode and parameters of the solar inverter’s electromechanical transient model.
| Case Set | Type | $P_0$ | $u_t$ |
|---|---|---|---|
| Training | 3-Phase | 0.2 | 0.35 |
| 3-Phase | 1.0 | 0.75 | |
| 2-Phase | 0.2 | 0.35 | |
| 2-Phase | 1.0 | 0.50, 0.75 | |
| Testing | 3-Phase | 0.2 | 0.50 |
| 3-Phase | 1.0 | 0.50 | |
| 2-Phase | 0.2 | 0.20 | |
| 2-Phase | 1.0 | 0.20 |
1. Introduction
The large-scale integration of photovoltaic (PV) power generation, which heavily relies on the solar inverter, introduces significant uncertainty into power system operation. To accurately analyze the grid-integration characteristics of these systems, the development of precise simulation models is essential. The accuracy of these models is primarily determined by the control parameters of the solar inverter. However, due to proprietary restrictions, these parameters are often “black-boxed,” making them difficult to obtain directly through conventional means. Therefore, parameter identification has become a key technique for acquiring these unknown control parameters.
Existing research has explored various methods for identifying the parameters of a solar inverter. Some studies have proposed stepwise identification methods for control parameters by applying disturbances to secondary measurement signals. Other approaches have considered the transient characteristics of faults, establishing staged identification processes for PI regulator parameters and limiter parameters. While these studies have made progress, they primarily focus on the PI loop identification, which often fails to accurately describe the dynamic characteristics of the solar inverter during LVRT events. Given that the solar inverter must possess LVRT capability, accurately identifying the LVRT control parameters is of paramount importance for building high-fidelity simulation models.
To precisely identify the LVRT control parameters of a solar inverter, researchers have analyzed the impact of the AC current response loop on LVRT and proposed generic parameter identification strategies. Some studies have optimized control parameters based on common inverter control methods, considering power angle stability and transient overvoltage. Other works have suggested integrated identification strategies for control mode and parameters, but they have shown limitations in verifying parameter adaptability using measured data. Adaptive particle swarm optimization (APSO) algorithms have also been applied to identify LVRT control parameters from measured data, but the stability of these algorithms often requires further improvement. Artificial intelligence plays a vital role in modern power systems, including parameter identification. Some methods combine coarse parameter tuning with intelligent algorithms, while others use neural networks and optimization algorithms based on stepwise identification to enhance accuracy. However, these studies often focus on overall fitting effect, leading to potential overfitting in specific cases.
To overcome these limitations, we propose a multi-objective identification strategy to enhance the adaptability of the identified control parameters across different operating conditions. Among the various multi-objective optimization algorithms, the Multi-Objective Differential Evolution (MODE) algorithm stands out due to its simplicity, fast convergence, and effectiveness in solving complex optimization problems. This makes it highly suitable for the high-precision parameter identification required by solar inverter control models. In this work, we present a method based on the MODE algorithm to identify the LVRT control mode and parameters of the electromechanical transient model of a solar inverter. We first determine the optimization scope using typical values of the inverter’s control mode and parameters. Then, a multi-objective identification strategy is employed, combining non-dominated sorting with the MODE algorithm. We use an adaptive operator based on population diversity to guide the search. Finally, we determine the most likely control mode by verifying the simulation performance of the identified parameters under different control strategies.
2. Grid-Connected Solar Inverter System
2.1 System Structure and Control Strategy
The PV generation system shown in the standard literature consists primarily of a PV array, a solar inverter, an LCL filter, and the grid connection system. Under normal operating conditions, the solar inverter adopts a double-closed-loop control strategy based on voltage vector orientation. This consists of an outer voltage loop and an inner current loop. Maximum Power Point Tracking (MPPT) is employed to maximize the conversion of solar energy into electrical energy by prioritizing active power output. In this process, the three-phase grid voltages $u_{ga}, u_{gb}, u_{gc}$ and currents $i_{ga}, i_{gb}, i_{gc}$ are transformed into their dq-axis components $u_{gd}, u_{gq}$ and $i_{gd}, i_{gq}$ via a Park transformation. To achieve synchronous control of the grid current with the grid voltage, a Phase-Locked Loop (PLL) is required to provide accurate feedback of the grid voltage phase $\theta_{PLL}$. The tracking performance of $\theta_{PLL}$ directly influences the grid-connection characteristics of the solar inverter.
The DC voltage $u_{dc}$ is regulated by the voltage outer loop, producing the d-axis current reference $i_{gd,ref}$ as shown in equation (1). Under normal conditions, the q-axis current reference $i_{gq,ref}$ is generally set to zero.
$$i_{gd,ref} = k_{P1} (u_{dc,ref} – u_{dc}) + k_{I1} \int (u_{dc,ref} – u_{dc}) dt$$
Here, $k_{P1}$ and $k_{I1}$ are the proportional and integral coefficients of the voltage outer loop, respectively, and $u_{dc,ref}$ is the DC voltage reference. The current inner loop ensures that the actual currents $i_{gd}$ and $i_{gq}$ track their references quickly and accurately. It outputs the dq-axis components $u_{d,ref}$ and $u_{q,ref}$ of the inverter’s AC side voltage reference.
$$ \begin{cases} u_{d,ref} = u_{gd} – \omega L i_{gq} + k_{P2} (i_{gd,ref} – i_{gd}) + k_{I2} \int (i_{gd,ref} – i_{gd}) dt \\ u_{q,ref} = u_{gq} + \omega L i_{gd} + k_{P3} (i_{gq,ref} – i_{gq}) + k_{I3} \int (i_{gq,ref} – i_{gq}) dt \end{cases} $$
Where $\omega$ is the angular frequency of the grid voltage, and $L$ is the filter inductance. $k_{P2}, k_{I2}$ are the proportional and integral coefficients for the d-axis current, and $k_{P3}, k_{I3}$ are for the q-axis current. During a short-circuit fault, the point of common coupling (PCC) voltage deviates from its normal range. This leads to a power imbalance between the inverter’s output power and the MPPT power, potentially causing an overvoltage on the DC side.
$$P_{PV} – P = d \left( \frac{C u_{dc}^2}{2} \right) / dt$$
Here, $P_{PV}$ is the output power of the PV array, and $C$ is the DC-link capacitance. To maintain grid-connection capability, the solar inverter must switch to an LVRT control strategy. The inverter is required to prioritize reactive current injection $i_Q$ to support the PCC voltage.
$$i_Q = K (u_{lin} – u) i_N$$
Where $K$ is the ratio of the reactive current variation to the voltage variation, $u_{lin}$ is the low voltage threshold, $u$ is the per-unit value of the inverter’s AC side voltage, and $i_N$ is the rated current of the solar inverter.
2.2 LVRT Control Mode of the Electromechanical Transient Model
During the voltage sag period, the solar inverter must prioritize the injection of reactive current $i_Q$ to support the PCC voltage, while simultaneously managing the active current $i_P$ output within the total current limit. The actual commanded values are constrained by limiting circuits.
$$i_{Q,cmd} = \min(i_{Q, LVRT}, i_{Q, max})$$
$$i_{P,cmd} = \min(i_{P, LVRT}, \sqrt{i_{max}^2 – i_{Q}^2}, i_{P, max})$$
During LVRT, the primary control modes for the electromechanical transient model of the solar inverter include the specified power control and the specified current control. The specified current control mode defines the active and reactive current commands as a function of the voltage drop $u_t$ and the pre-fault currents $i_{P0}$ and $i_{Q0}$.
$$ \begin{cases} i_{P,LVRT} = k_{1,iP} u_t + k_{2,iP} i_{P0} + i_{P,set} \\ i_{Q,LVRT} = k_{1,iQ} (u_{lin} – u_t) + k_{2,iQ} i_{Q0} + i_{Q,set} \end{cases} $$
Here, $k_{1,iP}$ and $k_{2,iP}$ are the LVRT active current coefficients, and $i_{P,set}$ is the active current setpoint. Similarly, $k_{1,iQ}$ and $k_{2,iQ}$ are the LVRT reactive current coefficients, and $i_{Q,set}$ is the reactive current setpoint. The specified power control mode defines the LVRT power commands based on pre-fault power values and setpoints.
$$ \begin{cases} i_{P,LVRT} = P_{LVRT} / u_t = (k_P P_0 + P_{set}) / u_t \\ i_{Q,LVRT} = Q_{LVRT} / u_t = (k_Q Q_0 + Q_{set}) / u_t \end{cases} $$
Where $k_P$ and $k_Q$ are the LVRT power coefficients, and $P_{set}$ and $Q_{set}$ are the power setpoints. $P_0$ and $Q_0$ are the pre-fault active and reactive power values.
3. Identification Method
3.1 Extraction of LVRT Key Points
To accurately identify the control parameters of the solar inverter, we extract key points from the LVRT operating condition response. The LVRT period is divided into three stages: pre-disturbance (A), during-disturbance (B), and post-disturbance (C). Stage B is further subdivided into a transient interval (B1) and a steady-state interval (B2). The steady-state key points for active power $P_t$, reactive power $Q_t$, active current $i_{Pt}$, and reactive current $i_{Qt}$ are calculated by averaging the measured values over the steady-state interval B2. For example, $i_{Pt}$ is calculated as follows:
$$i_{Pt} = \frac{\int_{b_1}^{b_2} i_P(t)}{b_2 – b_1} dt$$
We use these key points from the measured cases to build the identification dataset, filtering out cases that are in the current limiting state. This dataset is then used to identify the power coefficients/setpoints for the specified power mode or the current coefficients/setpoints for the specified current mode.
3.2 Control Parameter Identification using MODE
The Differential Evolution (DE) algorithm is a population-based optimization method that mimics natural evolution. It uses mutation, crossover, and selection operations. We enhance the standard DE algorithm by incorporating a multi-objective strategy and an adaptive parameter tuning mechanism. The optimization process begins with population initialization.
$$ \begin{cases} T = [T_1, T_2, \dots, T_G]^T \\ T_i = [x_{i1}, x_{i2}, \dots, x_{in}] \\ x_{ij} = \min x_j + r (\max x_j – \min x_j) \end{cases} $$
Here, $T$ is the initial population of size $G$, $x_{ij}$ is the $j$-th parameter of the $i$-th individual, and $r \in (0,1)$ is a random number. The mutation operation generates a donor vector.
$$T_a^{(1)} = T_a + p_m (T_b – T_c)$$
Where $p_m$ is the mutation operator. The crossover operation creates a trial vector $T_a^{(1)}$ with probability $p_c$. To improve search speed, we use an adaptive parameter tuning strategy based on population diversity, measured by information entropy $H(T)$.
$$H(T) = -\sum_{j=1}^{n} \sum_{h=h_0}^{h_1} [p(x_j = h) \log_2 p(x_j = h)]$$
$$ \begin{cases} p_m = p_{m, min} + \frac{H(T)}{H_0(T)} (p_{m, max} – p_{m, min}) \\ p_c = p_{c, max} – \frac{H(T)}{H_0(T)} (p_{c, max} – p_{c, min}) \end{cases} $$
We introduce a multi-objective strategy using non-dominated sorting. The objective functions are defined to minimize both the sum of absolute errors and the maximum absolute error across all cases.
$$ \begin{cases} J_1 = \min(\sum_{k=1}^{m} |e_g|) \\ J_2 = \min(\max |e_g|) \end{cases} $$
In the non-dominated sorting process, individuals are ranked into layers (Pareto fronts). The individual’s performance is evaluated based on its objective functions. The population is updated by selecting the top $G$ individuals based on their rank and crowding distance $L_i$, ensuring both convergence and diversity.
$$L_i = L_{i,1} + L_{i,2}$$
3.3 Identification of Control Mode
To accurately determine the LVRT control mode of the solar inverter, we use an evaluation metric that considers the average deviation $\beta_X$ and the maximum deviation $\beta_{X, max}$ between the simulated and measured values of power and current.
$$ \begin{cases} \beta_X = \left| \frac{\sum_{t=K_{start}}^{K_{end}} (X_{LVRT}(t) – X_{cmd}(t))}{K_{end} – K_{start} + 1} \right| \\ \beta_{X, max} = \max \left| X_{LVRT}(t) – X_{cmd}(t) \right| \end{cases} $$
If the evaluation indices for the specified current mode satisfy the thresholds $\beta_P < 0.1$, $\beta_{iP} < 0.1$, $\beta_{P, max} < 0.15$, and $\beta_{iP, max} < 0.15$, the mode is classified as specified current control. If $i_{Pt} \approx i_{P0}$, it is classified as pre-fault current control. Otherwise, it is classified as no additional control. A similar logic is applied to determine the reactive power control mode.
4. Case Study and Analysis
4.1 Hardware-in-the-Loop Test
To validate the proposed method for identifying the control mode and parameters of the solar inverter’s electromechanical transient model, we performed HIL tests on a real solar inverter controller using the RT-LAB real-time simulation platform. We tested various LVRT scenarios, including three-phase and two-phase short-circuit faults under different power conditions. The specific cases used for training and testing are listed in the first table.
4.2 Identification Results
We extracted key points from the LVRT cases and built the identification dataset. The MODE algorithm was used to identify the parameters for both the specified power and specified current control modes. The convergence of the Pareto front for the reactive current $i_Q$ identification is shown in the text. We used a fuzzy comprehensive evaluation method to select the best compromise solution from the Pareto front. The identified parameters were then plugged into the PSASP PV electromechanical transient model. The simulation waveforms for case 1 (symmetrical, $P_0=0.2$ pu, $u_t=0.50$ pu) and case 2 (asymmetrical, $P_0=0.2$ pu, $u_t=0.50$ pu) are presented in the following error analysis tables.
| Case | $\beta_P$ | $\beta_{P, max}$ | $\beta_{iP}$ | $\beta_{iP, max}$ | $\beta_Q$ | $\beta_{Q, max}$ | $\beta_{iQ}$ | $\beta_{iQ, max}$ |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.002 | 0.009 | 0.005 | 0.019 | 0.101 | 0.102 | 0.184 | 0.185 |
| 2 | 0.027 | 0.027 | 0.039 | 0.039 | 0.022 | 0.023 | 0.030 | 0.033 |
| 3 | 0.069 | 0.079 | 0.186 | 0.211 | 0.121 | 0.124 | 0.292 | 0.299 |
| 4 | 0.034 | 0.065 | 0.074 | 0.131 | 0.109 | 0.112 | 0.199 | 0.203 |
| 5 | 0.008 | 0.015 | 0.014 | 0.028 | 0.108 | 0.109 | 0.214 | 0.217 |
| 6 | 0.053 | 0.074 | 0.100 | 0.141 | 0.117 | 0.121 | 0.232 | 0.240 |
| Case | $\beta_P$ | $\beta_{P, max}$ | $\beta_{iP}$ | $\beta_{iP, max}$ | $\beta_Q$ | $\beta_{Q, max}$ | $\beta_{iQ}$ | $\beta_{iQ, max}$ |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.008 | 0.009 | 0.015 | 0.017 | 0.001 | 0.002 | 0.002 | 0.005 |
| 2 | 0.012 | 0.022 | 0.018 | 0.032 | 0.001 | 0.001 | 0.001 | 0.002 |
| 3 | 0.012 | 0.015 | 0.029 | 0.036 | 0.001 | 0.006 | 0.003 | 0.015 |
| 4 | 0.014 | 0.017 | 0.025 | 0.032 | 0.001 | 0.006 | 0.002 | 0.011 |
| 5 | 0.009 | 0.010 | 0.020 | 0.023 | 0.001 | 0.002 | 0.010 | 0.013 |
| 6 | 0.002 | 0.022 | 0.011 | 0.030 | 0.001 | 0.005 | 0.011 | 0.019 |
For Case 1, the specified power control mode yielded $\beta_P = 0.002$ and $\beta_{iP} = 0.005$, which were lower than the specified current mode. However, the reactive current errors $\beta_Q$ and $\beta_{iQ}$ were significantly lower in the specified current mode. This indicates that Case 1 uses a specified power control for active power and a specified current control for reactive power. For Cases 2 through 6, the specified current control mode provided the best fit for both active and reactive power, indicating that the solar inverter’s controller predominantly uses this mode under the tested conditions.
4.3 Validation of Multi-Objective Strategy
To verify the effectiveness of our multi-objective strategy, we compared the MODE algorithm against a standard single-objective DE algorithm. The average error $I_1$ and maximum error $I_2$ were used to evaluate performance on the testing set.
The results showed that while the DE algorithm might achieve a slightly lower $I_{1,iP}$ in some cases, the MODE algorithm consistently achieved lower $I_2$ values and better overall performance for the reactive power metrics $I_{1,Q}$, $I_{2,Q}$, $I_{1,iQ}$, and $I_{2,iQ}$. This demonstrates that the multi-objective approach enhances the adaptability and robustness of the identified parameters for the solar inverter across different operating conditions, effectively mitigating the risk of overfitting to a single criterion.
4.4 Algorithm Comparison
We further validated the accuracy of the MODE algorithm by comparing it with other multi-objective algorithms: APSO, NSGA-II, and MOCS. We used the average error percentage at the LVRT key points as the metric for comparison. The results are summarized in the following table.
| Case | APSO | NSGA-II | MOCS | MODE (Proposed) | ||||
|---|---|---|---|---|---|---|---|---|
| $i_{Pt}$ | $i_{Qt}$ | $i_{Pt}$ | $i_{Qt}$ | $i_{Pt}$ | $i_{Qt}$ | $i_{Pt}$ | $i_{Qt}$ | |
| 1 | 10.33 | 0.26 | 0.26 | 0.18 | 1.69 | 0.31 | 2.69 | 0.20 |
| 2 | 12.39 | 0.56 | 23.18 | 0.89 | 25.70 | 12.55 | 8.65 | 0.23 |
| 3 | 2.96 | 0.78 | 5.95 | 0.69 | 1.47 | 5.38 | 3.18 | 0.56 |
| 4 | 2.26 | 0.25 | 21.39 | 1.45 | 2.51 | 0.48 | 2.84 | 0.03 |
| 5 | 10.07 | 1.89 | 31.33 | 2.20 | 6.12 | 3.44 | 9.59 | 1.92 |
| 6 | 1.46 | 2.49 | 19.03 | 0.87 | 1.29 | 2.88 | 1.55 | 2.28 |
| 7 | 4.84 | 3.02 | 12.11 | 11.78 | 5.31 | 1.98 | 0.87 | 2.59 |
| 8 | 1.03 | 3.28 | 2.53 | 2.78 | 10.65 | 2.81 | 0.91 | 1.50 |
For the active current $i_{Pt}$, the average errors for APSO, NSGA-II, MOCS, and the proposed MODE algorithm were approximately 5.66%, 14.47%, 6.84%, and 3.78%, respectively. For the reactive current $i_{Qt}$, the average errors were 1.56%, 2.60%, 3.72%, and 1.16%. These results clearly demonstrate that the proposed MODE algorithm achieves significantly higher identification accuracy and robustness compared to the other state-of-the-art algorithms. The MODE algorithm’s ability to maintain population diversity while converging to the Pareto front allows it to find more globally optimal solutions for the solar inverter’s control parameters.
5. Conclusion
We proposed and validated a novel method for identifying the LVRT control mode and parameters of the electromechanical transient model for a solar inverter based on the Multi-Objective Differential Evolution (MODE) algorithm. Our work was motivated by the need for accurate simulation models to analyze the grid-integration characteristics of photovoltaic systems, especially given the proprietary “black-box” nature of inverter controllers.
The key conclusions from our research are as follows. First, compared to the standard single-objective DE algorithm, the proposed MODE-based approach significantly improves parameter identification accuracy by effectively simulating the LVRT characteristics across multiple operating conditions. Second, the use of non-dominated sorting within the MODE framework optimizes the population update process, which enhances search efficiency by maintaining diversity in the parameter space. This reduces the risk of the identification process becoming trapped in local optima. Third, in a comparative analysis against other advanced multi-objective algorithms such as APSO, NSGA-II, and MOCS, the proposed MODE algorithm demonstrated superior performance. The average identification errors for the LVRT active current $i_{P, LVRT}$ and reactive current $i_{Q, LVRT}$ were consistently below 3.8% and 1.2%, respectively, across various test cases. This confirms that our method provides high efficiency, accuracy, and adaptability for identifying the control parameters of a solar inverter. Our method is a reliable tool for power system engineers and researchers who need to build accurate models of solar inverters for stability studies and grid code compliance verification.