In the rapidly evolving landscape of renewable energy integration, the accurate modeling and parameter identification of solar inverters have become paramount for ensuring grid stability and reliability. As a researcher deeply involved in power system dynamics, I have observed that traditional methods for identifying controller parameters in solar inverters often face challenges due to the cascading nature of inner and outer control loops. These methods, which involve creating short-circuits on the primary side to induce voltage dips or injecting step disturbances into controller reference values, frequently lead to non-unique identifiability of parameters. This issue arises because the coupled actions of the controllers make it difficult to decouple their effects during excitation. To address this, I propose a novel approach called the Virtual Measurement Excitation (VME) method, which applies disturbances directly to secondary-side measurement signals. The key advantage of VME lies in its ability to shield changes in other measurement signals while exciting a specific one, thereby decoupling the inner and outer loop controller actions and solving the identifiability problem. Based on this, I present a stepwise identification method for solar inverter controller parameters, which sequentially determines the parameters without relying heavily on optimization algorithms. This article delves into the mathematical foundations, implementation details, and validation through simulation case studies, aiming to provide a robust framework for enhancing the modeling accuracy of solar inverters in grid simulations.
The core of a grid-connected solar inverter system is its power electronic interface, which converts DC power from photovoltaic panels into AC power synchronized with the grid. Modern solar inverters typically employ a double-loop control structure: an outer voltage loop and an inner current loop, both often utilizing Proportional-Integral (PI) controllers. This structure ensures stable DC-link voltage regulation and precise current injection with low harmonic distortion. The dynamic behavior of a solar inverter is governed by a set of differential equations derived from its circuit topology and control logic. Understanding these equations is crucial for parameter identification. For a three-phase voltage-source inverter connected to the grid through an L-filter (representing the combined inductance of the output filter and transformer), the system can be modeled in the synchronous rotating dq-reference frame. The DC-side dynamics involve the capacitor voltage \( U_{DC} \), which is influenced by the photovoltaic current \( I_{pv} \) and the inverter input current \( I_{DC} \). The relationship is given by:
$$ I_{DC} – C \frac{dU_{DC}}{dt} = I_{pv} – I_{DC} $$
where \( C \) is the DC-link capacitance. On the AC side, the inverter output voltages \( u_d \) and \( u_q \) (in dq coordinates) are related to the grid voltages \( e_d \) and \( e_q \), the output currents \( i_d \) and \( i_q \), and the coupling inductance \( L \). The equations are:
$$ e_d = u_d – L \frac{di_d}{dt} + \omega L i_q $$
$$ e_q = u_q – L \frac{di_q}{dt} – \omega L i_d $$
where \( \omega \) is the grid angular frequency. These equations form the basis for deriving the controller dynamics. The double-loop control strategy aims to regulate the DC-link voltage to a reference value \( U_{DC\_ref} \) while injecting current with a desired power factor (typically unity). The outer voltage loop PI controller processes the error between \( U_{DC} \) and \( U_{DC\_ref} \) to generate the reference for the d-axis current \( i_{d\_ref} \). The q-axis current reference \( i_{q\_ref} \) is usually set to zero for unity power factor operation. The inner current loop PI controllers then adjust the inverter output voltages \( u_d \) and \( u_q \) to track these current references. Mathematically, the voltage controller can be expressed as:
$$ \frac{dx_1}{dt} = K_{iu} (U_{DC\_ref} – U_{DC}) $$
$$ i_{d\_ref} = K_{pu} (U_{DC\_ref} – U_{DC}) + x_1 $$
where \( x_1 \) is an internal state variable, \( K_{pu} \) is the proportional gain, and \( K_{iu} \) is the integral gain of the outer voltage loop. For the current controllers, the equations are:
$$ \frac{dx_2}{dt} = K_{ii} (i_{d\_ref} – i_d) $$
$$ u_d = e_d – \omega L i_q – K_{pi} (i_{d\_ref} – i_d) – K_{ii} x_2 $$
$$ \frac{dx_3}{dt} = K_{ii} (i_{q\_ref} – i_q) $$
$$ u_q = e_q + \omega L i_d – K_{pi} (i_{q\_ref} – i_q) – K_{ii} x_3 $$
where \( x_2 \) and \( x_3 \) are internal states, \( K_{pi} \) is the proportional gain, and \( K_{ii} \) is the integral gain of the inner current loop. The parameters to be identified are thus \( K_{pu} \), \( K_{iu} \), \( K_{pi} \), \( K_{ii} \), and \( L \). These parameters critically influence the stability and response of the solar inverter, making their accurate identification essential for reliable system simulation and analysis.
The Virtual Measurement Excitation (VME) method is a groundbreaking technique that overcomes the limitations of traditional excitation methods. Instead of perturbing the physical system (e.g., causing voltage dips on the grid), VME introduces disturbances directly into the measurement signals that feed into the solar inverter’s controller. This is achieved through auxiliary circuits that can modify voltage or current measurements before they are processed by the control algorithms. For instance, to apply a VME to the DC-link voltage measurement \( U_{DC} \), a circuit can be used to superimpose a desired disturbance waveform (e.g., a step or sinusoid) onto the actual measured signal. Similarly, for current measurements, Park transformation can be applied to decompose currents into dq components, modify a specific component (like the q-axis current), and then reconstruct the three-phase signals for the controller. The primary advantage of VME is selective excitation: when applying a disturbance to one measurement signal, changes in other signals can be masked. For example, while exciting the DC-link voltage measurement, the grid voltage and current measurements can be held constant at their steady-state values. This decouples the actions of the inner and outer loops, allowing for independent identification of their parameters. The implementation of VME requires digital signal processors (DSPs) to generate precise disturbance waveforms and switch between real and virtual measurements. This approach not only enhances safety by avoiding physical disturbances but also provides flexibility in the type and frequency of excitations, which is beneficial for identifying parameters across different dynamic regimes.
Building on VME, I propose a stepwise identification method for solar inverter controller parameters. This method breaks down the identification process into sequential steps, each targeting a subset of parameters. The first step focuses on identifying the outer voltage loop parameters \( K_{pu} \) and \( K_{iu} \). To do this, the solar inverter is operated with a DC power supply replacing the photovoltaic panels, ensuring stable input. The grid-side voltage and current measurements are kept real, while a VME is applied to the DC-link voltage measurement \( U_{DC} \). Initially, \( U_{DC} \) is set to its reference value \( U_{DC\_ref} \), so the inverter outputs no active power (i.e., \( i_d = 0 \)). Then, a square wave VME with amplitude \( \Delta U_{DC1} \) and duration \( T \) is applied to \( U_{DC} \). The response of the d-axis current \( i_d \) is measured. After the disturbance ends, the steady-state value \( i_{d0} \) is related to the integral gain by:
$$ i_{d0} = K_{iu} \Delta U_{DC1} T $$
Since \( \Delta U_{DC1} \) and \( T \) are known, \( K_{iu} \) can be directly calculated. However, due to harmonic noise in practical systems, this step might not yield precise results for \( K_{pu} \). Therefore, a second VME is applied: a sinusoidal disturbance of amplitude \( \Delta U_{DC2} \) and frequency 2 Hz superimposed on \( U_{DC} \). The response \( i_d(t) \) is then analyzed. From the controller equations, the relationship can be derived as:
$$ i_d(t) = i_{d0} + K_{pu} \Delta U_{DC2} \sin(4\pi t) – \frac{K_{iu} \Delta U_{DC2}}{4\pi} (\cos(4\pi t) – 1) $$
By rearranging, we obtain:
$$ K_{pu} \sin(4\pi t) = \frac{i_d(t) – i_{d0}}{\Delta U_{DC2}} + \frac{K_{iu}}{4\pi} (\cos(4\pi t) – 1) $$
With \( i_{d0} \), \( \Delta U_{DC2} \), and \( K_{iu} \) known, the amplitude of the sinusoidal component in the left-hand side gives \( K_{pu} \). This can be extracted using Fourier analysis. Thus, both outer loop parameters are identified without optimization.
The second step targets the inner current loop parameters \( K_{pi} \), \( K_{ii} \), and the inductance \( L \). Here, the key is to decouple the inner loop from the outer loop by shielding certain measurements. Using VME circuits, the grid voltage measurement and the DC-link voltage measurement are masked, meaning they are held constant at their steady-state values (e.g., \( e_d = e_{d0} \), \( e_q = e_{q0} \), and \( U_{DC} = U_{DC\_ref} \)). This prevents the outer loop from reacting during the excitation. Then, a sinusoidal VME is applied to the q-axis component of the grid current measurement. Specifically, the measured q-axis current \( i_q \) is replaced with a virtual signal \( i_q^* = i_{q0} + \Delta i_q \), where \( i_{q0} = 0 \) (steady-state) and \( \Delta i_q = -A \sin(4\pi t) \) with amplitude \( A \). The d-axis current measurement remains real (\( i_d = i_{d0} \)). This VME excites the inner current loop. Substituting into the current controller equations and simplifying (assuming slow variations so that derivative terms are negligible), we get:
$$ i_{d0} – i_d = \frac{K_{pi} A}{L} \sin(4\pi t) – \frac{K_{ii} A}{4\pi L} (\cos(4\pi t) – 1) $$
where \( i_d \) is the actual d-axis current response. The left-hand side \( (i_{d0} – i_d) \) can be measured. By performing Fourier analysis on this signal, the DC component gives \( K_{ii} A / (4\pi L) \), and the amplitude of the 2 Hz component at \( t = 0.125 \) s gives \( K_{pi} A / L \). Since \( A \) is known, the ratios \( K_{ii}/L \) and \( K_{pi}/L \) are determined. Thus, the inner loop parameters are identified up to the inductance value.
The final step is to determine the inductance \( L \). This can be done either by direct measurement of the physical inductor or through an additional identification step using VME. For identification, a three-phase voltage dip VME can be applied to the grid voltage measurements while monitoring the solar inverter’s active and reactive power outputs. By comparing the simulated response with the measured one, \( L \) can be estimated using an optimization algorithm like particle swarm optimization. Since only one parameter is optimized, the process is efficient. Once \( L \) is known, \( K_{pi} \) and \( K_{ii} \) are computed from the ratios obtained earlier. The overall stepwise procedure ensures that all five parameters of the solar inverter controller are accurately identified.
To validate the proposed method, I conducted simulation studies using a detailed solar inverter model in a power system simulation environment. The solar inverter parameters were set to typical values, and the VME-based identification steps were implemented. The results demonstrated high accuracy, with parameter errors within acceptable limits. For instance, in a case study, the identified parameters closely matched the true values, confirming the feasibility of the approach. The use of VME allowed for clean excitation without interference from grid disturbances, and the stepwise decoupling eliminated cross-talk between loops. This validation underscores the practical utility of the method for real-world solar inverter parameter identification.
In conclusion, the stepwise identification method leveraging Virtual Measurement Excitation offers a robust solution to the parameter identifiability problem in solar inverter controllers. By decoupling the inner and outer loops through selective measurement disturbances, it enables precise and sequential identification of PI gains and inductance. This method enhances the modeling accuracy of solar inverters, which is crucial for grid integration studies and stability analysis. Future work will focus on hardware implementation of VME circuits and testing on actual solar inverters to further refine the technique. As renewable penetration grows, such advanced identification methods will play a vital role in ensuring grid reliability and performance.
The mathematical models and identification steps can be summarized in the following tables for clarity. Table 1 lists the solar inverter controller parameters and their descriptions, while Table 2 outlines the stepwise identification procedure.
| Parameter | Description | Typical Range |
|---|---|---|
| \( K_{pu} \) | Proportional gain of outer voltage loop | 0.1 – 10 |
| \( K_{iu} \) | Integral gain of outer voltage loop | 10 – 1000 |
| \( K_{pi} \) | Proportional gain of inner current loop | 0.01 – 1 |
| \( K_{ii} \) | Integral gain of inner current loop | 1 – 100 |
| \( L \) | Coupling inductance (filter + transformer) | 0.01 – 0.5 H |
| Step | Objective | VME Applied | Measurements Shielded | Key Equations |
|---|---|---|---|---|
| 1 | Identify \( K_{iu} \) | Square wave on \( U_{DC} \) | None | \( i_{d0} = K_{iu} \Delta U_{DC1} T \) |
| 2 | Identify \( K_{pu} \) | Sinusoid on \( U_{DC} \) | None | \( K_{pu} \sin(4\pi t) = \frac{i_d – i_{d0}}{\Delta U_{DC2}} + \frac{K_{iu}}{4\pi} (\cos(4\pi t) – 1) \) |
| 3 | Determine \( K_{ii}/L \) and \( K_{pi}/L \) | Sinusoid on \( i_q \) | Grid voltage and \( U_{DC} \) | \( i_{d0} – i_d = \frac{K_{pi} A}{L} \sin(4\pi t) – \frac{K_{ii} A}{4\pi L} (\cos(4\pi t) – 1) \) |
| 4 | Identify \( L \) | Voltage dip on grid voltage | None | Optimization using power response |
| 5 | Compute \( K_{pi} \) and \( K_{ii} \) | N/A | N/A | \( K_{pi} = (K_{pi}/L) \times L \), \( K_{ii} = (K_{ii}/L) \times L \) |
The effectiveness of the method relies on the precise generation of VME signals. In practice, this requires careful design of auxiliary circuits. For voltage measurements, a DSP can sample the real voltage, compute its phasor, and then synthesize a disturbed signal using a digital-to-analog converter. For current measurements, the Park transformation must be implemented in real-time to modify dq components. The shielding of measurements is achieved by continuously outputting the steady-state values during the excitation period. These technical aspects ensure that the VME is seamlessly integrated into the solar inverter’s control system without affecting its normal operation.
Moreover, the proposed method has significant implications for the deployment and maintenance of solar inverters in power grids. Accurate parameter identification enables better tuning of controllers, leading to improved dynamic performance and stability margins. For grid operators, having reliable models of solar inverters facilitates more accurate simulations for planning and fault analysis. The stepwise approach reduces computational burden by avoiding complex multi-parameter optimization, making it suitable for field applications. As solar energy continues to expand, such identification techniques will become standard tools for ensuring the seamless integration of inverter-based resources.
In summary, this article presents a comprehensive stepwise identification method for solar inverter controller parameters, centered on the innovative Virtual Measurement Excitation. By decoupling control loops and using targeted disturbances, it solves the identifiability problem and provides a practical pathway for accurate parameter estimation. The method is validated through simulations, and its implementation details are discussed. Future efforts will focus on real-world testing and adaptation to various solar inverter topologies, further solidifying its role in the advancement of renewable energy systems.