The large-scale integration of renewable energy sources, primarily interfaced through power electronic converters, has fundamentally transformed modern power systems. Among these interfaces, the LCL-filtered grid-connected inverter (GCI) is prevalent due to its superior harmonic attenuation capabilities. However, the proliferation of such inverters creates complex multi-infeed systems where dynamic interactions between multiple GCIs, the network grid, and varying grid impedances can lead to destabilizing harmonic oscillations. These oscillations threaten power quality and overall system stability. Existing studies on mitigating these oscillations often involve adjusting control parameters across all inverters, which is impractical and lacks a targeted approach. This work addresses this gap by proposing a systematic method to first identify the key oscillation source within a multi-infeed system and then apply a targeted, asymmetric impedance reshaping strategy to that specific GCI, thereby enhancing the stability of the entire system with minimal intervention.
The stability of a grid connected inverter in a multi-infeed environment is influenced by a confluence of factors: its own control parameters (e.g., current/PLL controller gains), the network’s physical structure, line impedances, and the equivalent grid impedance at the point of common coupling (PCC). Traditional impedance-based or eigenvalue analyses often model the system as a whole, making it difficult to pinpoint which specific grid connected inverter is the primary contributor to an emerging instability. Our approach begins by constructing a novel current oscillation mode gain model from the perspective of node current feedback. This model elegantly incorporates the coupling characteristics of the network grid (through the bus impedance matrix) and the individual dynamics of each grid connected inverter (through its Norton equivalent admittance).
The derivation starts with the detailed modeling of a single grid connected inverter, accounting for the small-signal effects of critical components like the Phase-Locked Loop (PLL) and the DC-link dynamics. The PLL, essential for synchronization, introduces cross-coupling terms between the d- and q-axes. Its small-signal perturbation model is given by:
$$
\begin{cases}
\Delta x^c_{\text{d}} = \Delta x^g_{\text{d}} + \dfrac{G_{\text{PLL}}(s) x^c_{\text{q0}}}{s + G_{\text{PLL}}(s) U_{\text{pccd}}} \Delta u^g_{\text{pccq}} \\
\Delta x^c_{\text{q}} = \Delta x^g_{\text{q}} – \dfrac{G_{\text{PLL}}(s) x^c_{\text{d0}}}{s + G_{\text{PLL}}(s) U_{\text{pccd}}} \Delta u^g_{\text{pccd}}
\end{cases}
$$
where $G_{\text{PLL}}(s) = k_{p-\text{PLL}} + k_{i-\text{PLL}}/s$, $x$ represents state variables, superscripts $g$ and $c$ denote the grid and controller dq frames respectively, and $U_{\text{pccd}}$ is the steady-state d-axis PCC voltage. Combining this with the power balance from the DC side and the LCL filter plant model, we derive a multi-input multi-output (MIMO) output admittance model $\mathbf{Y}_{\text{eq}}(s)$ for a single inverter.
For a system with $n$ inverters, the network’s nodal equations are described by the bus admittance matrix $\mathbf{Y}_{\text{bus}}(s)$. By focusing on the nodes where GCIs are connected, we extract the mutual coupling impedance matrix $\mathbf{Z}_m(s)$. The interaction between the inverter cluster and the grid is then captured by the current relationship:
$$
\mathbf{I}_s(s) – \mathbf{I}_o(s) = \mathbf{Z}_m(s) \mathbf{Y}_{\text{eq}}(s) \mathbf{I}_o(s)
$$
where $\mathbf{I}_s(s)$ is the vector of inverter equivalent current sources and $\mathbf{I}_o(s)$ is the vector of output currents. Rearranging this yields the current oscillation mode gain model, a closed-loop transfer matrix from source currents to output currents:
$$
\mathbf{I}_o(s) = [\mathbf{I} + \mathbf{Y}_{\text{eq}}(s)\mathbf{Z}_m(s)]^{-1} \mathbf{I}_s(s) \triangleq \mathbf{G}_m(s) \mathbf{I}_s(s)
$$
This model, $\mathbf{G}_m(s)$, is the cornerstone of our analysis. Its eigenvalues, $\lambda_i(s)$, directly indicate the system’s modal response. A mode with a gain $|\lambda_i(j\omega)| \gg 1$ at a particular frequency $\omega$ signifies potential harmonic oscillation. To identify the source, we perform eigenvalue decomposition on $\mathbf{G}_m(s)$ and compute the participation factor (PF) for each inverter. The participation factor matrix $\mathbf{PF}$ is derived from the right eigenvector matrix $\mathbf{Q}(s)$:
$$
\mathbf{G}_m(s) = \mathbf{Q}(s) \boldsymbol{\Lambda}(s) \mathbf{Q}^{-1}(s)
$$
$$
\begin{bmatrix}
I’_{o1} \\
\vdots \\
I’_{on}
\end{bmatrix}
=
\begin{bmatrix}
\lambda_1 & & \\
& \ddots & \\
& & \lambda_n
\end{bmatrix}
\begin{bmatrix}
I’_{s1} \\
\vdots \\
I’_{sn}
\end{bmatrix}
$$
The participation factor of the $k$-th inverter in the $i$-th mode is calculated as $PF_{ik} = Q_{ik}Q_{ik}^{-1}$. The inverter with the largest PF corresponding to the critical mode (the $\lambda_i$ with the largest peak magnitude) is identified as the dominant harmonic oscillation source. This provides a clear, physics-informed target for remedial action.
| Parameter | G1 Value | G2 Value | G3 Value |
|---|---|---|---|
| Power Rating (P) | 31.8 kW | 31.8 kW | 31.8 kW |
| Inverter-side Inductor (L₁) | 3 mH | 3 mH | 3 mH |
| Grid-side Inductor (L₂) | 1 mH | 1.2 mH | 1.5 mH |
| Current Ctrl. Integral Gain (kᵢ) | 500 | 1250 | 1250 |
| Damping Resistor (R_d) | 8 Ω | 8 Ω | 8 Ω |
| Line Impedance (Z_g) | 0.324 mH | 0.533 mH | 0.126 mH |
Applying this method to a case study with three GCIs (parameters in Table above) reveals a critical oscillatory mode. The participation factor analysis conclusively identifies G1 as the primary oscillation source, responsible for the largest contribution to the unstable mode. This identification remains robust even with ±20% uncertainty in line impedance parameters, demonstrating the method’s practical utility.
Having identified the oscillation source G1, we then analyze the oscillation mechanism from its perspective. The rest of the system (other GCIs, lines, grid) is aggregated into an equivalent load impedance $\mathbf{Z}_{go}(s)$. The stability of this equivalent source-load system can be assessed using the impedance-based Nyquist criterion generalized for MIMO systems, or more intuitively, by examining the minor loop gain in the individual d-d and q-q channels. For the studied case, the analysis reveals that while the d-d channel interaction has positive phase margin, the q-q channel interaction exhibits a negative phase margin at the crossover frequency, pinpointing the q-axis dynamics as the root cause of instability. This insight is crucial: the instability is primarily driven by the q-axis impedance characteristics of the oscillation source grid connected inverter.
Based on this understanding, we propose a two-step, asymmetric impedance reshaping control strategy applied only to the identified oscillation source G1. The first step counteracts the destabilizing effect of the PLL. It has been established that the PLL introduces a positive feedback loop in the q-axis current reference path, reducing damping. This effect can be canceled by adding a compensating signal. The compensation terms $G_{\text{PLL1,qq}}(s)$ and $G_{\text{PLL2,qq}}(s)$ are derived as:
$$
G_{\text{PLL1,qq}}(s) = -\frac{G_{\text{PLL}}(s) I^c_{\text{2d0}}}{s + G_{\text{PLL}}(s) U_{\text{pccd}}}, \quad G_{\text{PLL2,qq}}(s) = -\frac{G_{\text{PLL}}(s) U^c_{\text{Md0}}}{s + G_{\text{PLL}}(s) U_{\text{pccd}}}
$$
where $I^c_{\text{2d0}}$ and $U^c_{\text{Md0}}$ are steady-state operating points. Adding these to the q-axis control loop effectively eliminates the PLL-induced positive feedback.
The second step introduces a phase compensator $G_p(s)$ in series with the original q-axis current controller $G_{dq}(s)$ to provide additional positive phase shift in the low-frequency range where the instability occurs. The compensator is a lead-lag type filter:
$$
G_p(s) = k_{ps} \frac{k_1 s + 1}{k_2 s + 1}
$$
The parameters ($k_{ps}, k_1, k_2$) are designed to achieve a desired phase margin (PM), typically between 30° and 60°, at the q-q impedance crossover frequency. The design constraint is formulated by ensuring the compensated inverter’s q-q output impedance $\mathbf{Z}_{\text{eq,qq}}(s)$ and the equivalent grid impedance $\mathbf{Z}_{\text{go,qq}}(s)$ satisfy the phase condition at their magnitude crossover frequency $\omega_c$:
$$
\text{PM} = 180^\circ – \left[ \arg(\mathbf{Z}_{\text{go,qq}}(j\omega_c)) – \arg(\mathbf{Z}_{\text{eq,qq}}(j\omega_c)) \right] \in [30^\circ, 60^\circ]
$$
Given $\mathbf{Z}_{\text{eq,qq}}(s) \approx Z_0(s) = \frac{1}{G_{dq}(s)G_p(s)K_{PWM}}$ at low frequency, the required phase from $G_p(s)$ is:
$$
\arg(G_p(j\omega_c)) = \arg(\mathbf{Z}_{\text{go,qq}}(j\omega_c)) – \arg(Z_{\text{cf}}(j\omega_c)) – \arg(G_{dq}(j\omega_c)K_{PWM}) – \text{PM} + 180^\circ
$$
where $Z_{\text{cf}}(s)$ is the equivalent capacitor branch impedance. An iterative tuning process yields optimal parameters (e.g., $k_{ps}=0.25, k_1=0.01, k_2=0.0012$).
The efficacy of the proposed method is validated through both impedance analysis and real-time hardware-in-the-loop (RT-Lab) experiments on a platform with three 10.6 kW GCIs. The system parameters are those listed in the table. Under traditional control, with a grid impedance $L_g = 2.2$ mH, the system becomes unstable, exhibiting severe harmonic oscillation in the currents. The measured Total Harmonic Distortion (THD) for G1’s current is 64.22%, significantly higher than for G2 (16.81%) and G3 (17.42%), experimentally confirming G1’s role as the oscillation source as predicted by the model.
When the proposed asymmetric control—PLL compensation plus q-axis phase advance—is applied only to G1, the system stability is dramatically improved. The impedance Bode plots show the phase margin in the critical q-q channel increases from -0.18° (unstable) to over 80°, providing ample stability. Experimentally, the system remains stable with $L_g$ up to 5.8 mH, a substantial increase from the original 2.2 mH limit. The system loses stability only when $L_g$ reaches 6.0 mH. This demonstrates that targeted stabilization of the identified source grid connected inverter effectively enhances the robustness of the entire multi-infeed system against grid impedance variations.
In conclusion, this work presents a comprehensive framework for managing harmonic stability in multi-grid connected inverter systems. The novel current oscillation mode gain model provides a clear and effective method for identifying the key harmonic oscillation source within a complex cluster. The subsequent asymmetric impedance reshaping strategy, combining PLL positive-feedback cancellation and targeted q-axis phase compensation, offers a precise and efficient solution to suppress oscillations and significantly widen the stable operating region. This approach moves beyond the blanket adjustment of all converters, providing a more practical and effective paradigm for ensuring the reliable integration of numerous grid connected inverter units into weak and variable grids.