Accurate knowledge of internal control parameters is fundamental for analyzing the transient behavior and fault characteristics of grid-connected solar inverters. This analysis forms the essential basis for relay protection setting and stability assessment in modern power networks with high penetration of renewable energy. However, due to technical confidentiality, equipment manufacturers typically do not disclose the precise parameters of their proprietary control algorithms. Among these, the proportional and integral gains of the current control loop play a decisive role in shaping the solar inverter’s output current during grid disturbances, particularly during Low Voltage Ride-Through (LVRT) events. The inability to access these values creates significant challenges for power system engineers who need to perform accurate simulations and protection coordination studies.
The core difficulty in parameter identification often stems from an underdetermined system: the number of available equations derived from measurements is fewer than the number of unknown parameters to be identified, leading to non-convergence or inaccurate solutions. Existing approaches to tackle this can be broadly categorized into two groups. The first group, known as the rank-augmentation method, involves injecting additional probing signals (e.g., time-domain perturbations or frequency sweeps) into the system to create more independent equations. While effective, these methods require special injection devices or specific operational conditions, limiting their practicality for field applications. The second group, the order-reduction method, simplifies the model or separates the parameters to reduce the number of unknowns. A common challenge within this group, especially when using optimization algorithms like Particle Swarm Optimization (PSO), is the simultaneous identification of parameters with vastly different sensitivities. High-sensitivity parameters, such as the proportional gain, can dominate the optimization process, leading to poor identification accuracy for low-sensitivity parameters like the integral gain, even when they are mathematically separated in the model equations.
This article addresses this specific issue by proposing a novel, stepwise identification method for the current loop parameters of a grid-following solar inverter. The method leverages the distinct dynamic behaviors exhibited during a fault and the subsequent active power recovery period mandated by grid codes. Through a detailed analytical derivation, we demonstrate that the response during the steady-state phase of active power recovery is solely dependent on the integral coefficient, effectively decoupling it from the proportional coefficient. This inherent decoupling allows for a two-stage identification process: first, the integral coefficient is identified using data from the power recovery stage; second, with the integral coefficient fixed, the proportional coefficient is identified using data from the initial fault transient stage. This mechanistic separation fundamentally prevents the high-sensitivity parameter from impairing the identification accuracy of the low-sensitivity one. The proposed methodology is validated using a Hardware-in-the-Loop (HIL) test system built on a Real-Time Digital Simulator (RTDS) platform, confirming its effectiveness, stability, and noise tolerance.
System Modeling of a Grid-Connected Solar Inverter
The foundation of the parameter identification method lies in a standard model of a three-phase, two-level voltage source converter (VSC) used as a solar inverter. The system employs a grid-following control strategy with a cascaded outer voltage (or power) loop and an inner current loop in the synchronous reference frame (d-q frame). During severe voltage sags that trigger LVRT, the outer loop is typically disabled to prioritize fast current control, and the current references are set by grid code requirements (e.g., reactive current injection). The control structure is based on a standard PI-based current regulator with decoupling and feedforward terms.
The electrical dynamics from the inverter terminals to the Point of Common Coupling (PCC) can be described in the d-q frame as:
$$ L \frac{di_d}{dt} = u_d – R i_d + \omega L i_q – e_d $$
$$ L \frac{di_q}{dt} = u_q – R i_q – \omega L i_d – e_q $$
where \(i_d\) and \(i_q\) are the d- and q-axis currents, \(u_d\) and \(u_q\) are the corresponding inverter output voltages, \(R\) and \(L\) are the equivalent resistance and inductance of the filter and connecting transformer, \(\omega\) is the grid angular frequency, and \(e_d\), \(e_q\) are the grid voltages at the PCC.
The current control law for the inner loop is given by:
$$ u_d = k_p (i_{dref} – i_d) + k_i \int (i_{dref} – i_d) dt – \omega L i_q + e_d $$
$$ u_q = k_p (i_{qref} – i_q) + k_i \int (i_{qref} – i_q) dt + \omega L i_d + e_q $$
Here, \(k_p\) and \(k_i\) are the proportional and integral gains of the PI controller, which are the target parameters for identification. \(i_{dref}\) and \(i_{qref}\) are the current references. During an LVRT event, these references are typically defined as:
$$ i_{qref} = K \cdot (0.9 – U_{pcc}) \cdot I_N, \quad \text{for } 0.2 \leq U_{pcc} \leq 0.9 $$
$$ i_{dref} = \min(\sqrt{I_{max}^2 – i_{qref}^2}, I_{dpre-fault}) $$
where \(K\) is the reactive current support factor, \(U_{pcc}\) is the per-unit PCC voltage, \(I_N\) is the rated current, and \(I_{max}\) is the maximum allowed current during the fault.
Theoretical Analysis of Fault and Recovery Characteristics
The proposed identification method is built upon a detailed analytical solution of the solar inverter’s current response during two distinct phases: the fault inception and the subsequent active power recovery after fault clearance.
Fault Stage Dynamics
Assuming a symmetrical fault occurs at time \(t_0\), causing a step change in the current references, the dynamics can be derived by combining the electrical and control equations. The resulting second-order differential equation for the d-axis current (with a similar form for the q-axis) is:
$$ \frac{d^2 i_d}{dt^2} + \frac{R+k_p}{L} \frac{d i_d}{dt} + \frac{k_i}{L} i_d = \frac{k_p}{L} \frac{d i_{dref}}{dt} + \frac{k_i}{L} i_{dref} $$
Solving this equation with initial conditions reflecting the continuity of inductor current yields the time-domain response:
$$ i_d(t) = C_1 e^{r_1 (t-t_0)} + C_2 e^{r_2 (t-t_0)} + i_{dref}(t_0^+), \quad t > t_0 $$
where \(r_{1,2}\) are the roots of the characteristic equation \(s^2 + \frac{R+k_p}{L}s + \frac{k_i}{L}=0\), and constants \(C_1, C_2\) depend on \(k_p\), \(k_i\), \(R\), \(L\), and the pre- and post-fault reference values. Crucially, this transient response is a function of both \(k_p\) and \(k_i\). If one attempts to identify them simultaneously from this stage, the low sensitivity of \(k_i\) can lead to significant errors.
Active Power Recovery Stage Dynamics
After grid voltage is restored, grid codes require the solar inverter to ramp its active power back to the pre-fault level at a specified rate. A common implementation uses a linear ramp for the d-axis current reference:
$$ i_{dref}^{rec}(t) = k_d (t – t_1) + i_{dref}^0, \quad t \geq t_1 $$
where \(t_1\) is the recovery start time, \(k_d\) is the ramp slope, and \(i_{dref}^0\) is the d-axis reference just before recovery starts. The q-axis reference is typically set to zero. Solving the system dynamics for this ramp input yields a response consisting of a forced component tracking the ramp and the natural response from the fault stage decaying further:
$$ i_d(t) = C_3 e^{r_1 (t-t_1)} + C_4 e^{r_2 (t-t_1)} + k_d (t – t_1) + i_{dref}^0 + A, \quad t > t_1 $$
The key insight lies in the constant term \(A\), which is given by:
$$ A = -\frac{R k_d}{k_i} $$
After the natural exponential terms \(C_3 e^{r_1 (t-t_1)} + C_4 e^{r_2 (t-t_1)}\) have decayed sufficiently, the difference between the actual current \(i_d(t)\) and its reference \(i_{dref}^{rec}(t)\) becomes constant:
$$ \Delta i_d = i_d(t) – i_{dref}^{rec}(t) \approx A = -\frac{R k_d}{k_i} $$
This steady-state tracking error is a fundamental property of a Type-1 system (one integrator) when following a ramp input. Remarkably, this error \(\Delta i_d\) depends only on the integral gain \(k_i\), the ramp slope \(k_d\), and the resistance \(R\). It is independent of the proportional gain \(k_p\). This provides the theoretical foundation for cleanly separating the identification of the two parameters.
| Stage | Current Reference | Steady-State Current Error (\(\Delta i\)) | Dependence on \(k_p\) | Dependence on \(k_i\) |
|---|---|---|---|---|
| Fault (Step Response) | Constant | Zero | Strong (via transient shape) | Moderate/Weak (via transient shape) |
| Recovery (Ramp Response) | Linear Ramp | Constant: \(-\frac{R k_d}{k_i}\) | None | Strong (inverse relationship) |
Proposed Stepwise Parameter Identification Method
Based on the analytical findings, we propose a sequential identification procedure that leverages data from both the fault recovery stage and the initial fault stage.
Step 1: Identification of the Recovery Slope (\(k_d\))
The ramp slope \(k_d\) is not a control parameter but a setpoint defining the recovery rate. It can be directly and accurately estimated from the measured d-axis current or power data during the recovery phase after low-pass filtering. A moving data window with a least-squares linear fit is applied. The slope is considered converged when subsequent estimates differ by less than a threshold \(\epsilon\):
$$ |k_d^{(m)} – k_d^{(m-1)}| < \epsilon $$
Step 2: Identification of the Integral Coefficient (\(k_i\))
Once \(k_d\) is known and the exponential transients from the recovery start have decayed (determined by checking the slope convergence), the constant current error \(\Delta i_d\) is measured. Using the fundamental relationship:
$$ \Delta i_d = -\frac{R k_d}{k_i} $$
The integral gain is obtained directly by:
$$ k_i = -\frac{R k_d}{\Delta i_d} $$
This provides a very straightforward and accurate estimate of \(k_i\) that is completely independent of \(k_p\). In practice, to account for measurement noise and minor model inaccuracies, this calculation can be performed over a selected data window in the steady-recovery region, and the average value can be taken.
Step 3: Identification of the Proportional Coefficient (\(k_p\))
With \(k_i\) now accurately known and fixed, the model for the initial fault transient stage (the step response) contains only \(k_p\) as an unknown. The measured current data from the fault inception period is used. An optimization algorithm (e.g., Genetic Algorithm-Particle Swarm Optimization hybrid) is employed to find the value of \(k_p\) that minimizes the error between the measured current and the current predicted by the analytical solution \(i_d(t, k_p, k_i^{known})\). Since \(k_i\) is fixed, the underdetermined problem is resolved, and \(k_p\) can be identified with high accuracy.
Compensation for Low-Pass Filtering Effects
Measured signals are often filtered. A first-order low-pass filter with time constant \(T\) applied to a ramp signal \(k_d t\) introduces a steady-state offset of \(k_d T\). This offset adds directly to the observed \(\Delta i_d\), potentially biasing the \(k_i\) estimate. Therefore, after identifying \(k_d\), the measured current error must be compensated:
$$ \Delta i_d^{corrected} = \Delta i_d^{measured} – k_d T $$
The corrected value is then used in the formula for \(k_i\).
| Step | Data Source | Parameter Identified | Governing Equation/Principle | Key Advantage |
|---|---|---|---|---|
| 1 | Active Power Recovery Stage | Recovery Slope \(k_d\) | Linear least-squares fit to \(i_d(t)\) or \(P(t)\) | Direct calculation, independent of \(k_p, k_i\) |
| 2 | Steady-State Portion of Recovery Stage | Integral Gain \(k_i\) | $$ k_i = -\frac{R k_d}{\Delta i_d^{corrected}} $$ | Complete decoupling from \(k_p\); direct, non-iterative calculation. |
| 3 | Initial Fault Transient Stage | Proportional Gain \(k_p\) | Optimization matching analytical fault response \(i_d(t, k_p, k_i^{known})\) to measured data. | Problem reduced to single-variable optimization, improving accuracy and convergence. |
Experimental Validation and Results Analysis
The proposed method was validated using a Hardware-in-the-Loop (HIL) test bench. A detailed model of the power circuit (grid, filter, solar inverter) was simulated in Real-Time Digital Simulator (RTDS), while the actual solar inverter controller hardware (with known, preset parameters \(k_p = 0.285\), \(k_i = 6.0\)) was connected via a fiber-optic interface. This setup provided realistic, closed-loop test data. Three-phase symmetrical faults were applied at the PCC, causing voltage dips to approximately 20%, 30%, and 40% of nominal, followed by clearance and active power recovery with slopes \(k_d\) set between 0.6 and 1.0 pu/s.
| Component | Parameter | Value |
|---|---|---|
| Grid | Voltage (L-L) | 10 kV |
| Transformer | Rating, Connection | 2 MVA, Dy |
| Filter & Line | Resistance \(R\) | 0.05 Ω |
| Filter & Line | Inductance \(L\) | 2.0 mH |
| Solar Inverter | Rated Power | 1.5 MW |
| Solar Inverter (Actual) | Current Loop \(k_p\) | 0.285 |
| Solar Inverter (Actual) | Current Loop \(k_i\) | 6.0 |
| Control | Recovery Slope \(k_d\) | 0.6 – 1.0 pu/s |
Identification Accuracy and Stability
For a severe voltage dip to 20% with different recovery slopes, the method demonstrated high accuracy. The results, averaged over multiple runs, are shown below. The integral coefficient \(k_i\), often difficult to identify, was estimated with an error of less than 3%. The proportional gain \(k_p\) was identified with an error generally below 2%.
| Recovery Slope \(k_d\) (pu/s) | Identified \(k_d\) (pu/s) | Error (%) | Identified \(k_i\) | Error (%) | Identified \(k_p\) | Error (%) |
|---|---|---|---|---|---|---|
| 0.6 | 0.598 | 0.33 | 6.118 | 1.97 | 0.280 | 1.75 |
| 0.8 | 0.802 | 0.25 | 6.135 | 2.26 | 0.283 | 0.70 |
| 1.0 | 1.001 | 0.10 | 6.154 | 2.57 | 0.282 | 1.05 |
The stability of the method was tested by performing 20 identifications for each fault depth. The table below shows the mean error, variance, and maximum error, confirming the method’s robustness under different operating conditions.
| Voltage Dip | Parameter | Mean Error (%) | Error Variance (%) | Max Error (%) |
|---|---|---|---|---|
| 20% | \(k_d\) | 0.33 | 0.42 | 0.68 |
| \(k_p\) | 1.16 | 0.53 | 1.75 | |
| \(k_i\) | 2.26 | 0.30 | 2.57 | |
| 30% | \(k_d\) | 0.72 | 0.56 | 1.05 |
| \(k_p\) | 1.86 | 0.78 | 2.02 | |
| \(k_i\) | 2.52 | 0.46 | 2.78 | |
| 40% | \(k_d\) | 1.27 | 0.73 | 1.42 |
| \(k_p\) | 2.04 | 0.92 | 2.42 | |
| \(k_i\) | 3.13 | 0.34 | 3.45 |
Comparison with Traditional Simultaneous Identification
To highlight the advantage of the proposed decoupling, we compared it with a traditional approach where both \(k_p\) and \(k_i\) are identified simultaneously from the fault transient data using the same optimization algorithm (GA-PSO). The results are starkly different.
| Method & Algorithm | Parameter | True Value | Identified Value | Error (%) |
|---|---|---|---|---|
| Simultaneous (GA-PSO) | \(k_p\) | 0.285 | 0.289 | 1.40 |
| \(k_i\) | 6.0 | 8.34 | 39.0 | |
| Proposed Stepwise (GA-PSO) | \(k_p\) | 0.285 | 0.280 | 1.75 |
| \(k_i\) | 6.0 | 6.118 | 1.97 |
The traditional method fails to accurately identify the low-sensitivity integral coefficient \(k_i\), yielding errors around 40%, while the proposed stepwise method maintains high accuracy for both parameters. This clearly demonstrates that the mechanistic decoupling is the crucial factor, not the choice of optimization algorithm.
Noise Tolerance
Practical measurements contain noise. To evaluate robustness, Gaussian white noise was added to the simulation data at different Signal-to-Noise Ratio (SNR) levels. The mean identification errors are summarized below.
| Parameter | Mean Error @ 40dB SNR (%) | Mean Error @ 30dB SNR (%) | Mean Error @ 20dB SNR (%) |
|---|---|---|---|
| \(k_d\) | 0.30 | 1.32 | 5.78 |
| \(k_i\) | 2.43 | 3.67 | 12.51 |
| \(k_p\) | 1.19 | 1.65 | 15.27 |
The method shows good tolerance to noise levels down to 30dB SNR, maintaining errors below 4% for the control parameters. Performance degrades at the higher noise level of 20dB SNR, which is expected as the fundamental signal features become obscured.
Conclusion
This article has presented a novel, stepwise methodology for identifying the current loop control parameters of a grid-connected solar inverter. The core innovation lies in exploiting the distinct dynamic signatures of the inverter during a grid fault and the subsequent active power recovery phase mandated by grid codes. Through rigorous analytical derivation, we proved that the steady-state tracking error during a linear power ramp recovery is solely a function of the integral gain \(k_i\), independent of the proportional gain \(k_p\). This inherent physical decoupling is harnessed to design a two-stage identification process: first, a direct calculation of \(k_i\) from recovery-stage data; second, an optimization-based identification of \(k_p\) using fault-stage data with \(k_i\) fixed as a known constant.
The proposed method effectively solves the long-standing problem of low accuracy in identifying low-sensitivity parameters when they are estimated simultaneously with high-sensitivity ones. Validation via a RTDS-based Hardware-in-the-Loop test system, using actual solar inverter controller hardware, confirmed the method’s effectiveness. The integral and proportional gains were identified with average errors consistently below 3% and 2%, respectively, under various fault depths and recovery rates. The method also demonstrated good stability across multiple identification runs and tolerable robustness against measurement noise down to 30dB SNR.
This parameter identification strategy provides power system researchers and engineers with a practical, accurate, and non-intrusive tool to characterize commercial solar inverters. The accurately identified parameters can significantly enhance the fidelity of simulation models used for protection coordination, stability analysis, and integration studies of power systems with high renewable penetration. Future work may explore the extension of this principle to identify parameters under asymmetric faults and in solar inverters employing grid-forming control strategies.