In modern power systems, the integration of distributed energy resources has led to a significant increase in power electronic interfaces, particularly three phase inverters. These devices play a crucial role in converting DC power from sources like solar and wind into AC power suitable for grid connection. However, the interaction between three phase inverters and the grid can lead to stability issues, especially in weak grid conditions. Impedance-based stability analysis has emerged as a powerful tool to assess these interactions, with harmonic linearization being a widely adopted method due to its clear physical interpretation. This paper presents a simplified harmonic linearization approach for modeling the sequence impedance of three phase LCL-type inverters, accounting for frequency coupling effects introduced by phase-locked loops (PLLs). The proposed method streamlines the modeling process by reducing complex coordinate transformations and convolution operations, making it more accessible for practical applications.
The control strategy of a three phase LCL inverter typically involves a cascaded structure with outer power loops and inner current loops. The LCL filter is used to attenuate switching harmonics, but it introduces resonance peaks that must be damped. Active damping methods, such as capacitor current feedback, are often employed to suppress these resonances without incurring significant losses. The PLL is critical for synchronizing the inverter with the grid voltage, but its dynamics can lead to frequency coupling, where a single-frequency disturbance produces responses at multiple frequencies. This coupling complicates the impedance model, necessitating a matrix representation that captures interactions between positive and negative sequences.
Traditional harmonic linearization methods involve extensive computations due to repeated coordinate transformations between abc and dq frames. For instance, when a positive-sequence voltage disturbance at frequency \(f_p\) is applied, the PLL generates phase angle perturbations that result in coupled components at frequencies like \(f_p – 2f_1\), where \(f_1\) is the fundamental frequency. The impedance matrix \(\mathbf{Z}\) is defined as:
$$\mathbf{Z} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix}$$
where \(Z_{11}\) represents the positive-sequence impedance, \(Z_{22}\) the negative-sequence impedance, and \(Z_{12}\), \(Z_{21}\) account for cross-coupling between sequences. In conventional modeling, deriving these elements requires solving convoluted equations in the frequency domain, which becomes cumbersome for complex control strategies.
The proposed simplified method bypasses these complexities by leveraging intermediate variables derived directly from the control strategy. Consider the d-axis and q-axis components of the inverter bridge voltage, \(e_d\) and \(e_q\). These can be decomposed into steady-state and perturbation components. For example, \(e_d\) is expressed as:
$$e_d = e_{d1} + e_{d2}$$
where \(e_{d1}\) contains frequencies related to perturbations, and \(e_{d2}\) represents steady-state terms. The key insight is that \(e_{d1}\) and \(e_{q1}\) can be obtained directly from the current control law in the abc frame, avoiding dq transformations. For a positive-sequence disturbance at \(f_p\), the bridge voltage in the abc frame at \(f_p\) is:
$$\hat{e}_a(f_p) = H_i(s – j\omega_0)(-\hat{I}_p) \pm jK_{dq}\hat{I}_p – K_C \hat{I}_{Cp}$$
where \(H_i(s)\) is the current controller transfer function, \(K_{dq}\) is the decoupling gain, \(K_C\) is the capacitor feedback gain, and \(\hat{I}_p\), \(\hat{I}_{Cp}\) are the positive-sequence grid and capacitor currents, respectively. Similarly, the coupled component at \(f_p – 2f_1\) is:
$$\hat{e}_a(f_p – 2f_1) = H_i(s – j\omega_0)(-\hat{I}_{po}) \mp jK_{dq}\hat{I}_{po} – K_C \hat{I}_{C0}$$
Here, \(\hat{I}_{po}\) is the coupled negative-sequence current, and \(\hat{I}_{C0}\) is the capacitor current at the fundamental frequency. The steady-state terms \(e_{d2}\) and \(e_{q2}\) are computed using the operating point values, which are straightforward to determine from power flow equations. This approach eliminates the need for convoluted operations involving Park transforms and simplifies the derivation of the impedance matrix elements.
The impedance elements are then derived by substituting these expressions into the LCL filter equations. For instance, \(Z_{11}\) is obtained as:
$$Z_{11} = \frac{-\hat{V}_p}{\hat{I}_p} = \frac{1 + L_1 C s^2 + (sL_1 + sL_2 + L_1 L_2 C s^3) Y_{11}}{Y_{11}}$$
where \(Y_{11}\) is the admittance corresponding to \(Z_{11}\). The full expressions for all impedance elements are provided in Table 1, which summarizes the simplified harmonic linearization model for a three phase inverter.
| Element | Expression |
|---|---|
| \(Z_{11}\) | \(\frac{1 + L_1 C s^2 + (sL_1 + sL_2 + L_1 L_2 C s^3) Y_{11}}{Y_{11}}\) |
| \(Z_{12}\) | \(\frac{(sL_1 + sL_2 + L_1 L_2 C s^3) Y_{12}}{Y_{12}}\) |
| \(Z_{21}\) | \(\frac{(sL_1 + sL_2 + L_1 L_2 C s^3) Y_{21}}{Y_{21}}\) |
| \(Z_{22}\) | \(\frac{1 + L_1 C s^2 + (sL_1 + sL_2 + L_1 L_2 C s^3) Y_{22}}{Y_{22}}\) |
Stability analysis must account for the interaction between the inverter impedance and the grid impedance. When the grid impedance \(Z_g\) is considered, the frequency coupling leads to additional loops where disturbance voltages at one sequence induce currents in the other. The effective positive-sequence impedance \(Z_{pp}\) and negative-sequence impedance \(Z_{nn}\) are derived as:
$$Z_{pp} = \frac{1}{Y_{11} + \frac{-Y_{12}Y_{21}(s – 2j\omega_0)}{Y_g(s – 2j\omega_0) + Y_{22}(s – 2j\omega_0)}}$$
$$Z_{nn} = \frac{1}{Y_{22} + \frac{-Y_{12}(s + 2j\omega_0)Y_{21}}{Y_g(s + 2j\omega_0) + Y_{11}(s + 2j\omega_0)}}$$
These expressions show that the stability of the system depends on both the direct impedances and the coupling terms. The Nyquist criterion is applied to the ratios \(Z_g / Z_{pp}\) and \(Z_g / Z_{nn}\) to assess stability. If either ratio encircles the critical point (-1, j0), the system is unstable. This approach provides a more accurate stability assessment than methods that ignore frequency coupling.
To validate the model, simulations were conducted using parameters typical for a three phase inverter system. The inverter rating was 10 kVA, with a DC link voltage of 700 V and grid voltage of 380 V (line-to-line). The LCL filter parameters were: \(L_1 = 2\) mH, \(L_2 = 1\) mH, and \(C = 10\) μF. The current controller was a PI type with gains \(K_p = 0.5\) and \(K_i = 100\), and the PLL had gains \(K_{pPLL} = 50\) and \(K_{iPLL} = 1000\). The capacitor current feedback gain \(K_C\) was set to 0.1. impedance measurements were performed by injecting small-signal disturbances at various frequencies and analyzing the response using FFT.
The results showed close agreement between the simulated and theoretical impedance curves. For example, \(Z_{11}\) exhibited inductive behavior at high frequencies, dominated by the grid-side inductor \(L_2\), while at low frequencies, the phase approached -90°, indicating potential stability issues. The coupling impedances \(Z_{12}\) and \(Z_{21}\) were significant at low frequencies but diminished above 3 kHz, highlighting the importance of frequency coupling in stability analysis. Table 2 compares the impedance magnitudes at key frequencies, demonstrating the accuracy of the simplified model.
| Frequency (Hz) | \(|Z_{11}|\) (Theory) (Ω) | \(|Z_{11}|\) (Simulation) (Ω) | \(|Z_{12}|\) (Theory) (Ω) | \(|Z_{12}|\) (Simulation) (Ω) |
|---|---|---|---|---|
| 50 | 15.2 | 15.1 | 14.8 | 14.7 |
| 100 | 18.5 | 18.3 | 17.9 | 17.8 |
| 500 | 25.3 | 25.1 | 20.1 | 19.9 |
| 1000 | 30.7 | 30.5 | 15.4 | 15.2 |
Stability analysis was further investigated under weak grid conditions, where the short-circuit ratio (SCR) is low. For a grid impedance of 22 mH (SCR ≈ 2.1), the Nyquist plots of \(Z_g / Z_{pp}\) and \(Z_g / Z_{nn}\) encircled the critical point, indicating instability. In contrast, methods neglecting frequency coupling showed no encirclement, leading to false stability predictions. Time-domain simulations confirmed the instability, with current waveforms exhibiting growing oscillations at frequencies around 40 Hz and 60 Hz, consistent with the impedance model predictions. This underscores the necessity of including coupling effects in stability assessments for three phase inverters.
The simplified modeling approach offers several advantages for practical applications. It reduces computational effort, making it suitable for real-time stability monitoring and controller design. Moreover, it can be extended to other inverter topologies and control strategies, such as those with virtual impedance or advanced damping techniques. Future work could focus on applying this method to multi-inverter systems, where interactions between multiple three phase inverters can lead to complex resonance phenomena. Additionally, the impact of grid background harmonics and unbalanced conditions on the impedance model warrants further investigation.
In conclusion, the proposed harmonic linearization method provides an efficient and accurate way to model the sequence impedance of three phase LCL inverters. By simplifying the derivation process and explicitly accounting for frequency coupling, it enhances the reliability of stability analysis in power electronic-dominated grids. The validation through simulations confirms its effectiveness, making it a valuable tool for engineers working on the integration of distributed energy resources. As the penetration of three phase inverters continues to grow, such modeling techniques will be essential for ensuring grid stability and facilitating the transition to renewable energy systems.