In modern power systems, the increasing penetration of renewable energy sources has made the accurate analysis of fault characteristics of photovoltaic power plants a critical issue for relay protection and system stability. The behavior of a solar inverter during grid faults is primarily governed by the control parameters of its current loop. However, due to technical confidentiality, manufacturers rarely disclose the exact values of these parameters. Therefore, identifying the current loop control parameters of a solar inverter from measurable electrical quantities is an essential yet challenging task. This paper presents a novel identification method that leverages both the fault transient and the post-fault recovery process to separate and estimate the proportional and integral coefficients of the current loop PI controller. The proposed method tackles the well-known problem of low sensitivity of the integral coefficient when estimated simultaneously with the proportional coefficient, which often leads to poor identification accuracy. By exploiting the physical characteristics of the system during two distinct control stages, the parameters are fully decoupled and identified sequentially. The effectiveness of the proposed approach is validated through hardware-in-the-loop tests on a real-time digital simulator (RTDS) platform.
This work focuses on the grid-connected solar inverter with a typical voltage-oriented control structure. The current loop controller is modeled in the synchronous dq reference frame, and the differential equations describing the fault and recovery dynamics are derived analytically. The analysis reveals that during the fault period, the d-axis current response contains both proportional and integral coefficients in a coupled manner, making simultaneous identification prone to errors caused by sensitivity differences. In contrast, during the active power recovery stage with a ramp-type current reference, after the natural exponential transients decay, the steady-state tracking error becomes solely dependent on the integral coefficient and the known system resistance, thus providing an excellent opportunity to estimate the integral coefficient independently. Once the integral coefficient is fixed, the proportional coefficient can be identified from the fault transient data with high accuracy. This parameter separation principle constitutes the core contribution of this paper.
The proposed method also addresses practical issues such as measurement noise and low-pass filtering effects. A data compensation technique is introduced to correct the deviation caused by the low-pass filter used for signal conditioning. The identification procedure is formulated as a two-step optimization problem, where the recovery slope is first determined using least-squares fitting, followed by the integral coefficient using a genetic particle swarm optimization (GAPSO) algorithm, and finally the proportional coefficient is estimated using the same optimization tool. The complete workflow is presented and discussed in detail.
Extensive simulations and hardware-in-the-loop experiments are conducted to verify the correctness and robustness of the proposed method. The results demonstrate that the proportional and integral coefficients of the solar inverter current loop can be identified with errors below 4% even in the presence of moderate noise. Compared with traditional simultaneous identification approaches, the proposed stepwise method significantly improves the accuracy of the low-sensitivity integral coefficient, reducing the average error from over 39% to less than 3%. Furthermore, the method shows good adaptability to different voltage sag depths and recovery slopes, confirming its practical applicability for field data analysis.
1. Modeling of the Grid-Connected Solar Inverter Control System
This study is carried out on a typical three-phase grid-connected solar inverter system. The main circuit consists of a photovoltaic array, a DC-link capacitor, a three-phase inverter, an LCL filter, and the AC grid. The control system adopts a dual-loop structure with an outer voltage loop and an inner current loop. Since the d-axis and q-axis current controllers are symmetrical, the PI parameters for both axes are considered identical. The block diagram of the system is shown in Figure 1 (not referenced here). The inverter output voltage and current are transformed into the synchronous dq frame using the phase angle provided by a phase-locked loop (PLL). The current loop includes feedforward decoupling terms to eliminate the cross-coupling between the d and q axes and to compensate for the grid voltage disturbance.
Neglecting the capacitor branch of the LCL filter for fundamental frequency analysis, the circuit equation in the dq frame can be expressed as:
$$
\begin{cases}
u_d = -L\dfrac{di_d}{dt} – Ri_d + \omega L i_q + e_d \\
u_q = -L\dfrac{di_q}{dt} – Ri_q – \omega L i_d + e_q
\end{cases}
\tag{1}
$$
where \(R\) and \(L\) are the equivalent resistance and inductance between the inverter output and the point of common coupling (PCC), \(\omega\) is the angular frequency, \(e_d\) and \(e_q\) are the dq components of the PCC voltage, and \(i_d\), \(i_q\) are the dq components of the output current.
With the decoupling control strategy, the current loop control equations are written as:
$$
\begin{cases}
u_d = k_p(i_{dref}-i_d) + k_i\int(i_{dref}-i_d)dt – \omega L i_q + e_d \\
u_q = k_p(i_{qref}-i_q) + k_i\int(i_{qref}-i_q)dt + \omega L i_d + e_q
\end{cases}
\tag{2}
$$
where \(k_p\) and \(k_i\) represent the proportional and integral coefficients of the current loop PI controllers, and \(i_{dref}\), \(i_{qref}\) are the dq-axis current reference values.
The above model serves as the basis for deriving the analytical expressions of the solar inverter during fault and recovery transients. The control strategy during low voltage ride-through (LVRT) is implemented according to grid code requirements, where the d-axis current reference is determined by a maximum current limit and the q-axis current reference is set to provide reactive power support based on the PCC voltage sag depth.
2. Analytical Characterization of Fault and Recovery Transients
2.1 Fault Stage Analysis
When a three-phase short-circuit fault occurs, the PCC voltage drops below 90% of the nominal value, the controller disables the outer voltage loop and enters the LVRT mode. The d-axis current reference is set to:
$$
i_{dref} = \min(\sqrt{I_{max}^2 – i_{qref}^2}, I_N)
\tag{3}
$$
where \(I_{max}\) is the maximum allowable current during the fault (typically 1.1 p.u.), and \(I_N\) is the rated current. The q-axis current reference is given by:
$$
i_{qref} = \begin{cases}
0, & U_{PCC} > 0.9 \\
K(0.9 – U_{PCC})I_N, & 0.2 \le U_{PCC} \le 0.9 \\
1.05 I_N, & U_{PCC} < 0.2
\end{cases}
\tag{4}
$$
where \(K\) is the reactive current support coefficient (taken as 1.5 in this study) and \(U_{PCC}\) is the per-unit PCC voltage.
Assuming that the PCC voltage steps to a steady-state value immediately after the fault, the d-axis current response can be derived by combining (1) and (2) and applying the Laplace transform method. The resulting second-order differential equation is:
$$
\frac{d^2 i_d}{dt^2} + \left(\frac{R+k_p}{L}\right)\frac{di_d}{dt} + \frac{k_i}{L}i_d = \frac{k_i}{L}i_{dref} + \frac{k_p}{L}\frac{di_{dref}}{dt}
\tag{5}
$$
Since the current cannot change instantaneously due to the inductor, the initial conditions are:
$$
i_d(t_0^+) = i_d(t_0^-) = i_{d0}, \quad \left.\frac{di_d}{dt}\right|_{t_0^+} = \left.\frac{di_d}{dt}\right|_{t_0^-}
\tag{6}
$$
The complete solution for the d-axis current during the fault is:
$$
i_d(t) = C_1 e^{r_1(t-t_0)} + C_2 e^{r_2(t-t_0)} + i_{dref}(t_0^+), \quad t>t_0
\tag{7}
$$
where the constants \(C_1, C_2\) and the characteristic roots \(r_1, r_2\) are functions of \(k_p, k_i, R, L\), and the initial current values. The detailed expressions are omitted here for brevity.
From (7), it is evident that the fault transient current depends on both \(k_p\) and \(k_i\) in a coupled manner. The sensitivity of the response to \(k_i\) is generally much lower than that to \(k_p\). Therefore, estimating both parameters simultaneously from the fault transient data suffers from an ill-conditioned optimization problem, which often leads to unacceptable errors in the integral coefficient.
2.2 Recovery Stage Analysis
After the fault is cleared, the solar inverter is required to restore its active power output at a rate of at least 30% of rated power per second, according to grid codes. In this paper, a ramp-type recovery characteristic is adopted for the d-axis current reference:
$$
i_{dref,rec}(t) = k_d(t – t_1) + i_{dref0}, \quad t \ge t_1
\tag{8}
$$
where \(k_d\) is the recovery slope, \(t_1\) is the fault clearing time, and \(i_{dref0}\) is the steady-state d-axis current reference during the fault. The q-axis current reference is forced to zero after the fault is cleared.
At the moment of recovery initiation, both the current reference and the actual current are continuous:
$$
i_d(t_1^+) = i_d(t_1^-) = i_{dref,rec}(t_1^+) = i_{dref0}
\tag{9}
$$
Substituting (8) and (9) into the system equation and solving the differential equation yields the recovery-stage current expression:
$$
i_d(t) = C_3 e^{r_1(t-t_1)} + C_4 e^{r_2(t-t_1)} + k_d(t – t_1) + i_{dref0} + A, \quad t>t_1
\tag{10}
$$
where the constants \(C_3, C_4\) depend on the initial conditions, and \(A\) is given by:
$$
A = \frac{k_d R}{k_i}
\tag{11}
$$
Define the deviation between the actual current and the reference current as:
$$
e(t) = i_d(t) – i_{dref,rec}(t) = C_3 e^{r_1(t-t_1)} + C_4 e^{r_2(t-t_1)} + A
\tag{12}
$$
Equation (12) is crucial. The two exponential terms decay with time constants determined by the characteristic roots \(r_1\) and \(r_2\). After these transient terms become negligible, the deviation converges to a constant value \(A\), which is independent of \(k_p\) and depends only on the recovery slope \(k_d\), the integral coefficient \(k_i\), and the system resistance \(R\). Since \(k_d\) can be readily estimated from the recovery data and \(R\) is generally known from line parameters, the integral coefficient \(k_i\) can be solved directly from the steady-state deviation. This excellent property provides a strong identifiability for \(k_i\) during the recovery stage, completely decoupled from \(k_p\).
It is worth emphasizing that this behavior is a manifestation of the Type-1 system tracking a ramp input with a constant steady-state error. The solar inverter current loop, containing an integrator in the PI controller, exhibits a finite steady-state error when the reference is a ramp signal. The magnitude of this error is inversely proportional to the integral gain and directly proportional to the slope and the plant resistance. This physical insight forms the theoretical foundation of the proposed identification method.
3. Parameter Separation Strategy and Data Compensation
3.1 Separation of Proportional and Integral Coefficients
The fault and recovery processes are divided into two time intervals:
- Fault interval (\(t_0 \le t < t_1\)): the current response contains both \(k_p\) and \(k_i\) in a coupled manner.
- Recovery interval (\(t \ge t_1\)): after the exponential terms decay, the current response deviation depends only on \(k_i\), \(k_d\), and \(R\).
This temporal separation naturally allows us to estimate \(k_i\) first from the recovery stage, fix its value, and then estimate \(k_p\) from the fault stage. The proposed two-step procedure completely eliminates the impact of sensitivity differences between the two parameters.
The key steps of the parameter separation are summarized in Table 1.
| Stage | Available equation | Parameters involved | Identification goal |
|---|---|---|---|
| Fault stage | Equation (7) | \(k_p\), \(k_i\), \(R\), \(L\) | \(k_p\) (after \(k_i\) is fixed) |
| Recovery stage (after transient decay) | Equation (12) with \(e=A\) | \(k_i\), \(k_d\), \(R\) | \(k_i\) |
3.2 Compensation for Low-Pass Filter Effects
In real-world measurements, the current signals are often processed by a low-pass filter to attenuate switching harmonics. This filter introduces an additional lag and a steady-state deviation when the input signal is a ramp. Consider a first-order low-pass filter with time constant \(T\). The filtered ramp response is:
$$
y(t) = k_d t – k_d T + k_d T e^{-t/T}
\tag{13}
$$
After the exponential term decays, the filtered output lags the ideal ramp by \(k_d T\). This deviation, if not compensated, would be incorrectly attributed to the control parameter \(A\), leading to an overestimation of \(k_i\). Since the filter time constant \(T\) is usually known from the controller design, the compensation can be easily applied by subtracting \(k_d T\) from the measured deviation.
The compensated steady-state error is:
$$
e_{comp} = \frac{k_d R}{k_i}
\tag{14}
$$
Therefore, the integral coefficient is estimated as:
$$
\hat{k}_i = \frac{\hat{k}_d R}{e_{comp}}
\tag{15}
$$
where \(\hat{k}_d\) is the estimated recovery slope from the filtered data. Since the filter does not change the slope of the ramp, \(\hat{k}_d\) can be obtained accurately using least-squares fitting on the filtered recovery data.
4. Implementation of the Identification Procedure
4.1 Overall Workflow
The proposed identification method is implemented according to the flowchart shown in Figure 2 (not referenced here). The main steps are as follows:
- Collect three-phase voltage and current data during the fault and recovery process from the PCC of the solar inverter.
- Transform the three-phase currents into the dq frame using the PLL angle. The PLL parameters can be obtained using known techniques, but they are assumed to be known in this work.
- Extract the recovery-stage data. Apply low-pass filtering (if required) and estimate the recovery slope \(k_d\) using a sliding-window least-squares linear regression. Continue the window until the slope estimate converges, i.e., the difference between consecutive estimates is below a threshold.
- Compensate for the low-pass filter deviation using the known time constant \(T\).
- Compute the steady-state error \(e_{comp}\) from the compensated recovery data after the exponential transients have died out.
- Solve for \(k_i\) using Equation (15).
- Fix \(k_i\) to its estimated value and substitute it into the fault-stage expression (Equation (7)). Use a genetic particle swarm optimization (GAPSO) algorithm to search for \(k_p\) that minimizes the sum of squared errors between the measured and calculated d-axis currents during the fault transient.
- Output the final estimates \(\hat{k}_p\) and \(\hat{k}_i\).
4.2 Optimization Algorithm
Although the integral coefficient can be solved analytically from the steady-state error using Equation (15), an optimization-based approach is also viable for the sake of robustness and automation. In this paper, the GAPSO algorithm is employed for the final identification of \(k_i\) as well as \(k_p\). The fitness function is defined as:
$$
Q = \sum_{i=1}^{n} [i_{d,real}(i) – i_{d,calc}(i)]^2
\tag{16}
$$
where \(i_{d,real}\) is the measured d-axis current and \(i_{d,calc}\) is the calculated value using the analytical expression.
To improve the convergence performance, an adaptive inertia weight is used:
$$
\omega = \begin{cases}
\omega_{min} + \frac{(\omega_{max}-\omega_{min})(Q – Q_{min})}{Q_{avg} – Q_{min}}, & Q \le Q_{avg} \\
\omega_{max}, & Q > Q_{avg}
\end{cases}
\tag{17}
$$
where \(\omega_{max}\) and \(\omega_{min}\) are the maximum and minimum inertia weights, respectively, and \(Q_{min}\), \(Q_{avg}\) are the minimum and average fitness values among all particles.
The GAPSO algorithm combines the global search ability of genetic algorithms (crossover and mutation) with the local exploitation of particle swarm optimization. This hybrid approach helps escape local optima and provides accurate and stable estimation results.
5. Simulation Setup
To validate the proposed identification method, a hardware-in-the-loop (HIL) test system is built on the RTDS real-time digital simulator. The primary solar inverter model is implemented in RTDS, while the actual controller hardware communicates with the simulator via high-speed optical fibers. The controller samples the electrical quantities, executes the LVRT and recovery control algorithms, and sends the switching signals back to the simulator. The system configuration is shown in Figure 3 (not referenced here). The parameters of the solar inverter system are listed in Table A1 in the Appendix (not shown here but summarized in Table 2 below).
| Parameter | Symbol | Value |
|---|---|---|
| Rated power | \(S_N\) | 500 kVA |
| Grid voltage (line-to-line) | \(U_N\) | 380 V |
| Grid frequency | \(f_N\) | 50 Hz |
| DC-link voltage | \(U_{dc}\) | 800 V |
| Filter resistance | \(R\) | 0.01 p.u. |
| Filter inductance | \(L\) | 0.15 p.u. |
| Current loop proportional gain | \(k_p\) | 0.285 |
| Current loop integral gain | \(k_i\) | 6.0 |
| Recovery slope | \(k_d\) | 0.6–1.0 |
The HIL system has been verified against field disturbance recording data from a real wind-solar hybrid plant. The comparison of dq-axis currents showed excellent agreement, confirming that the controller hardware accurately mimics the behavior of practical solar inverters. Since the field controller parameters are confidential, the formal identification tests are performed on the HIL system with known parameters, allowing direct verification of the estimated values.
6. Experimental Results and Analysis
6.1 Verification of the Analytical Recovery Model
To validate the theoretical derivation of the recovery stage, a symmetric fault was applied with the PCC voltage dropping to 20% of the nominal voltage. The recovery slope was set to 0.8. The measured d-axis current, the current reference, and the calculated current from Equation (10) are plotted in Figure 4 (not referenced here). The calculated response matches the simulated response very closely. During the initial part of the recovery, the difference between the actual current and the reference current changes dynamically due to the exponential terms. After a short transient, the difference becomes constant, as predicted by Equation (12).
Furthermore, by varying \(k_p\) from its true value by 0%, 20%, and 50% while keeping \(k_i\) at its true value, the steady-state error after the exponential decay remained unchanged, confirming that the recovery-stage deviation is independent of \(k_p\). This experimental evidence strongly supports the parameter separation principle.
6.2 Identification Results Under Different Recovery Slopes
Fault tests with voltage drop to 20% of nominal voltage were performed for three different recovery slopes: \(k_d = 0.6, 0.8, 1.0\). The proposed identification method was applied to the measured data. Each test was repeated multiple times, and the average estimated values are presented in Table 3.
| Parameter | Recovery slope | True value | Estimated value | Relative error (%) |
|---|---|---|---|---|
| \(k_d\) | 0.6 | 0.6 | 0.598 | 0.33 |
| 0.8 | 0.8 | 0.802 | 0.25 | |
| 1.0 | 1.0 | 1.001 | 0.10 | |
| \(k_i\) | 0.6 | 6.0 | 6.118 | 1.97 |
| 0.8 | 6.0 | 6.135 | 2.26 | |
| 1.0 | 6.0 | 6.154 | 2.57 | |
| \(k_p\) | 0.6 | 0.285 | 0.280 | 1.75 |
| 0.8 | 0.285 | 0.283 | 0.70 | |
| 1.0 | 0.285 | 0.282 | 1.05 |
The results show that the recovery slope \(k_d\) is estimated with very high accuracy (errors below 0.33% for all cases). The integral coefficient \(k_i\) exhibits errors between 1.97% and 2.57%, and the proportional coefficient \(k_p\) has errors between 0.70% and 1.75%. The slight increase in \(k_i\) error for larger recovery slopes is attributed to the shorter recovery time, which causes the exponential terms to decay less completely before the data window ends. Nevertheless, all errors remain below 3%, demonstrating the effectiveness of the proposed method under different operating conditions.
6.3 Stability and Repeatability
To assess the stability of the proposed identification method, tests were performed for three different voltage sag depths: 20%, 30%, and 40% of the nominal voltage. For each sag depth, 20 independent identification runs were executed. The mean error, error variance, and maximum error for each parameter are summarized in Table 4.
| Voltage drop | Parameter | Mean error (%) | Error variance (%) | Maximum error (%) |
|---|---|---|---|---|
| 20% \(U_N\) | \(k_d\) | 0.33 | 0.42 | 0.68 |
| \(k_p\) | 1.16 | 0.53 | 1.75 | |
| \(k_i\) | 2.26 | 0.30 | 2.57 | |
| 30% \(U_N\) | \(k_d\) | 0.72 | 0.56 | 1.05 |
| \(k_p\) | 1.86 | 0.78 | 2.02 | |
| \(k_i\) | 2.52 | 0.46 | 2.78 | |
| 40% \(U_N\) | \(k_d\) | 1.27 | 0.73 | 1.42 |
| \(k_p\) | 2.04 | 0.92 | 2.42 | |
| \(k_i\) | 3.13 | 0.34 | 3.45 |
All parameters maintain mean errors below 3.2% and maximum errors below 3.5% across the considered voltage sag range. The low variance values confirm that the identification process is stable and repeatable. The error generally increases with shallower voltage sags because the fault current is closer to the normal operating range, making the fault transient less informative. However, the proposed method still provides acceptable accuracy for all tested cases.
6.4 Comparison with Conventional Simultaneous Identification
To benchmark the proposed stepwise identification method, a comparative study was conducted using three popular metaheuristic algorithms: particle swarm optimization (PSO), genetic particle swarm optimization (GAPSO), and differential evolution (DE). Two approaches were evaluated: the conventional simultaneous identification of \(k_p\) and \(k_i\) from the fault transient data, and the proposed stepwise identification using both fault and recovery data. The voltage drop was set to 20% \(U_N\), and each algorithm was run 20 times. The mean estimates are listed in Table 5.
| Algorithm | Method | Parameter | True value | Estimated value | Relative error (%) |
|---|---|---|---|---|---|
| PSO | Simultaneous | \(k_p\) | 0.285 | 0.277 | 2.80 |
| \(k_i\) | 6.0 | 9.06 | 51.03 | ||
| GAPSO | Simultaneous | \(k_p\) | 0.285 | 0.289 | 1.40 |
| \(k_i\) | 6.0 | 8.34 | 39.00 | ||
| DE | Simultaneous | \(k_p\) | 0.285 | 0.276 | 3.15 |
| \(k_i\) | 6.0 | 8.62 | 43.67 | ||
| PSO | Stepwise (proposed) | \(k_p\) | 0.285 | 0.279 | 2.10 |
| \(k_i\) | 6.0 | 6.129 | 2.17 | ||
| GAPSO | Stepwise (proposed) | \(k_p\) | 0.285 | 0.280 | 1.75 |
| \(k_i\) | 6.0 | 6.118 | 1.97 | ||
| DE | Stepwise (proposed) | \(k_p | 0.285 | 0.288 | 1.05 |
| \(k_i\) | 6.0 | 6.154 | 2.57 |
When using the conventional simultaneous identification approach, \(k_p\) converges to a value relatively close to the true value, but \(k_i\) deviates severely, with errors ranging from 39% to 51%. This confirms the well-known sensitivity issue: the low-sensitivity integral coefficient is overfitted or driven to spurious values when optimized together with the high-sensitivity proportional coefficient. In contrast, the proposed stepwise identification method yields \(k_i\) errors below 2.6% and \(k_p\) errors below 2.2%, regardless of the underlying optimization algorithm. These results clearly demonstrate that the accuracy improvement comes from the parameter separation strategy, not from the specific search algorithm. The proposed method is therefore algorithm-agnostic and can be combined with any suitable optimizer.
6.5 Noise Robustness Analysis
Practical field measurements are always contaminated by noise. To evaluate the robustness of the proposed method, Gaussian white noise with signal-to-noise ratios (SNR) of 40 dB, 30 dB, and 20 dB was added to the simulated current data. The voltage drop was set to 20% \(U_N\), and the recovery slope was 0.8. The identification procedure was repeated 20 times for each noise level. The average relative errors are listed in Table 6.
| Parameter | SNR = 40 dB | SNR = 30 dB | SNR = 20 dB |
|---|---|---|---|
| \(k_d\) | 0.30% | 1.32% | 5.78% |
| \(k_i\) | 2.43% | 3.67% | 12.51% |
| \(k_p\) | 1.19% | 1.65% | 15.27% |
For SNR levels of 40 dB and 30 dB, the identification errors remain within acceptable bounds: \(k_i\) errors are 2.43% and 3.67%, respectively, while \(k_p\) errors are 1.19% and 1.65%. The recovery slope estimation is also robust, with errors below 1.4% for these SNR levels. However, when the SNR drops to 20 dB, all parameter errors increase substantially, exceeding 5.78% for \(k_d\), 12.51% for \(k_i\), and 15.27% for \(k_p\). This indicates that the proposed method can tolerate noise down to an SNR of approximately 30 dB, which is typical for high-quality disturbance recorders in substations. For highly noisy environments, additional denoising techniques or a longer data window may be required to maintain accuracy.
7. Discussion
The proposed identification method leverages two distinct physical phenomena of a solar inverter:
- Fault transient response: driven by a step change in current reference, containing rich information about both \(k_p\) and \(k_i\).
- Recovery ramp tracking error: a constant steady-state error when tracking a ramp reference, governed solely by \(k_i\), \(k_d\), and \(R\).
This separation is possible because the solar inverter current loop is a Type-1 control system. The integral action produces infinite DC gain for step inputs but a finite tracking error for ramp inputs. By exploiting this characteristic, the low-sensitivity integral coefficient can be identified with high precision without being distorted by the proportional coefficient.
One important practical aspect is the accurate knowledge of the system resistance \(R\). In transmission and distribution networks, the resistance between the inverter output and the PCC can be estimated from cable lengths and conductor data. In cases where \(R\) is subject to uncertainty, the proposed method can be extended to estimate \(R\) and \(k_i\) jointly from the recovery stage, although this would introduce a slight coupling. However, for most practical applications, \(R\) is sufficiently well known.
Another consideration is the type of fault. The present work focuses on symmetrical three-phase faults, which produce balanced dq voltages and currents. For asymmetrical faults, the negative-sequence components and the PLL dynamics become more complex. The proposed method may be extended by using the positive-sequence dq frame and separating the sequence quantities, but this remains a topic for future research.
The method assumes a ramp-shaped active current reference during recovery, as required by some grid codes. If a different recovery profile is used, such as an exponential or power-rate-limited ramp, the analytical expressions in this paper can be adapted accordingly. The core principle—separating parameters based on the steady-state tracking error of a particular reference waveform—remains valid.
The computational burden of the proposed method is low. The recovery slope is obtained by linear least squares, and the integral coefficient is solved analytically via Equation (15). Only the proportional coefficient requires a metaheuristic search, using the fault-transient data. In this study, the GAPSO algorithm converged within 100 generations and 50 particles, which corresponds to a few seconds of computation time on a standard personal computer. This makes the method suitable for offline analysis of disturbance recordings.
8. Conclusion
This paper presented a systematic and practical approach for identifying the current loop control parameters of a grid-connected solar inverter. The key contributions are summarized as follows:
- Parameter separation based on physical characteristics: By analyzing the fault transient and the recovery ramp-tracking behavior, the proportional coefficient \(k_p\) and the integral coefficient \(k_i\) are fully decoupled. The integral coefficient is identified from the steady-state tracking error during the recovery stage, while the proportional coefficient is identified from the fault transient using the now-fixed \(k_i\). This eliminates the adverse effect of sensitivity differences that plagues conventional simultaneous identification.
- High identification accuracy: The RTDS-based hardware-in-the-loop experiments show that the relative errors of \(k_p\) and \(k_i\) are both below 3% for voltage sags down to 20% of nominal voltage and recovery slopes between 0.6 and 1.0. The method outperforms simultaneous identification by reducing the \(k_i\) error from over 39% to below 2.6%.
- Robustness and stability: Statistical analysis over 20 repeated runs confirms low variance and small maximum errors. The method tolerates measurement noise down to a signal-to-noise ratio of 30 dB, which is typical for modern digital recording equipment.
- Algorithm independence: The proposed stepwise procedure yields accurate results regardless of the optimization algorithm used in the final stage. PSO, GAPSO, and DE all produce comparable results when combined with the parameter separation strategy, demonstrating good portability.
- Practical applicability: The method only requires data from a natural fault event and the subsequent recovery period. No additional signal injection devices are needed, making it directly applicable to field disturbance records.
Future work will extend the method to asymmetrical faults and investigate the joint estimation of system resistance and control parameters when line parameters are uncertain. In addition, the impact of PLL dynamics on the recovery stage will be further examined to enhance the robustness of the method under weak grid conditions.
Overall, the proposed identification method provides a reliable and accurate tool for obtaining the current loop control parameters of a solar inverter from disturbance recording data. The physical insights underlying the parameter separation not only improve identification accuracy but also enhance the interpretability of the estimation process. This work contributes to the broader goal of accurate fault analysis and protection coordination in power systems with high penetration of inverter-based resources.