The increasing integration of renewable energy sources, primarily via grid-connected inverters, has fundamentally altered the dynamics of modern power systems. While enabling the transition to sustainable energy, this shift introduces significant challenges related to power quality and system stability. A prominent issue is the complex phenomenon of harmonic resonance, arising from dynamic interactions between the control systems of numerous grid-connected inverters and the network impedance. These resonances can lead to severe harmonic amplification, causing voltage distortion, equipment overheating, and in extreme cases, instability. Therefore, developing effective methods to assess the severity of potential resonances and pinpoint their critical locations within a multi-inverter system is paramount for ensuring secure and reliable operation.
Traditional analysis often focuses on identifying resonant frequencies by examining the minima of the system’s nodal admittance matrix eigenvalues. However, this approach primarily considers the magnitude of the modal impedance, which indicates the level of modal voltage excited by a unit modal current. It overlooks the damping characteristics inherent to the resonant mode, a crucial factor determining whether the resonance is well-damped or potentially unstable. A resonance with negative damping can lead to growing oscillations, posing a far greater threat than a passive, positively damped resonance. This work addresses this gap by proposing a comprehensive framework for resonance strength evaluation and key node identification in systems with multiple grid-connected inverters.
The cornerstone of the proposed method is Modal Analysis or Resonant Mode Analysis (RMA). For a system with \(n\) nodes, the harmonic node-voltage equation in the frequency domain is given by:
$$\Delta \mathbf{U}(\omega) = \Delta \mathbf{Y}(\omega)^{-1} \Delta \mathbf{I}(\omega)$$
where \(\Delta \mathbf{U}(\omega)\) and \(\Delta \mathbf{I}(\omega)\) are the node voltage and current injection vectors, respectively, and \(\Delta \mathbf{Y}(\omega)\) is the nodal admittance matrix. By performing an eigendecomposition, this coupled system can be transformed into decoupled modal circuits:
$$\Delta \mathbf{Y}(\omega) = \Delta \mathbf{L}(\omega) \Delta \mathbf{\Lambda}(\omega) \Delta \mathbf{T}(\omega)$$
where \(\Delta \mathbf{\Lambda}(\omega)\) is the diagonal eigenvalue matrix, and \(\Delta \mathbf{L}(\omega)\) and \(\Delta \mathbf{T}(\omega)\) are the left and right eigenvector matrices, with \(\Delta \mathbf{T}(\omega) = \Delta \mathbf{L}(\omega)^{-1}\). Defining modal voltages \(\Delta \mathbf{U}'(\omega) = \Delta \mathbf{T}(\omega) \Delta \mathbf{U}(\omega)\) and modal currents \(\Delta \mathbf{J}(\omega) = \Delta \mathbf{T}(\omega) \Delta \mathbf{I}(\omega)\), the system simplifies to:
$$
\begin{bmatrix}
\Delta U’_1(\omega) \\
\Delta U’_2(\omega) \\
\vdots \\
\Delta U’_n(\omega)
\end{bmatrix}
=
\begin{bmatrix}
\Delta z_1(\omega) & 0 & \cdots & 0 \\
0 & \Delta z_2(\omega) & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \Delta z_n(\omega)
\end{bmatrix}
\begin{bmatrix}
\Delta J_1(\omega) \\
\Delta J_2(\omega) \\
\vdots \\
\Delta J_n(\omega)
\end{bmatrix}
$$
Here, \(\Delta z_i(\omega) = \lambda_i^{-1}(\omega)\) is defined as the modal impedance. A parallel resonance mode \(i\) occurs at a frequency \(\omega_{r,i}\) where \(\lambda_i(\omega_{r,i}) \approx 0\), causing \(\Delta z_i(\omega_{r,i})\) to approach a very large magnitude. A key insight is that each modal impedance \(\Delta z_i(s)\) (with \(s = j\omega\)) can be conceptually represented by an equivalent RLC parallel circuit, as shown in the derivation below. This analogy allows for a more physical interpretation of resonance characteristics.
The impedance of a parallel RLC circuit is:
$$Z_{RLC}(s) = \frac{1}{\frac{1}{R} + \frac{1}{sL} + sC} = \frac{s/C}{s^2 + s\frac{1}{RC} + \frac{1}{LC}}$$
By equating this form to the modal impedance \(\Delta z_i(s)\), we can express it in a standard second-order form:
$$\Delta z_i(s) = \frac{s \Delta L(s)}{s^2 \Delta L(s) \Delta C(s) + s \frac{\Delta L(s)}{\Delta R(s)} + 1} = \frac{ (s / \Delta C(s))}{s^2 + s (1 / \Delta R(s) \Delta C(s)) + (1 / \Delta L(s) \Delta C(s))}$$
Defining \(\omega_{r,i}^2 = 1/(\Delta L(s)\Delta C(s))\) and \(2\alpha_i = 1/(\Delta R(s)\Delta C(s))\), we get:
$$\Delta z_i(s) = \frac{s / \Delta C(s)}{s^2 + 2\alpha_i s + \omega_{r,i}^2}$$
At resonance (\(s = j\omega_{r,i}\)), the relationship simplifies. The quality factor \(Q_i\) for the \(i\)-th mode, characterizing its damping and bandwidth, is defined as:
$$Q_i = \frac{\omega_{r,i}}{2\alpha_i} = \omega_{r,i} \Delta R(s) \Delta C(s)$$
The sign and magnitude of \(Q_i\) are critical. A positive \(Q_i\) indicates positive damping (\(\Delta R > 0\)), while a negative \(Q_i\) signifies negative damping (\(\Delta R < 0\)), which can lead to unstable, growing harmonic oscillations. The magnitude \(|Q_i|\) indicates the sharpness of the resonance peak; a larger \(|Q_i|\) implies a narrower bandwidth and a more severe harmonic amplification for injections near \(\omega_{r,i}\). This quality factor can be extracted directly from the Nyquist plot of \(\Delta z_i(\omega)\). For any point on the curve, the angle \(\gamma\) satisfies:
$$\tan \gamma = \frac{\text{Im}(\Delta z_i(\omega))}{\text{Re}(\Delta z_i(\omega))} = \frac{\omega_{r,i}^2 – \omega^2}{2\alpha_i \omega}$$
Knowing \(\omega_{r,i}\) from the impedance magnitude peak, and measuring \(\gamma\) at a known frequency \(\omega\), one can solve for \(\alpha_i\) and subsequently compute \(Q_i\). This provides a direct metric to assess the strength and stability of each resonant mode: modes with high \(|Q_i|\) require attention, and modes with \(Q_i < 0\) are particularly problematic.
While the quality factor evaluates the inherent severity of a resonant mode, it does not indicate where in the physical network this mode is most excitable or observable. A resonant mode might have a high \(Q_i\) but participate only weakly at a particular node, meaning harmonic injections at that node would not significantly excite it. To identify the key nodes associated with a specific resonant mode, we analyze the participation of modal impedances in the physical node impedances.
The physical impedance between node \(p\) and \(q\), \(\Delta Z_{pq}(\omega)\), relates the current injection at node \(q\) to the voltage at node \(p\): \(\Delta U_p(\omega) = \Delta Z_{pq}(\omega) \Delta I_q(\omega)\). Using the eigen-decomposition, it can be shown that:
$$\Delta Z_{pq}(\omega) = \sum_{i=1}^{n} \Delta L_{pi}(\omega) \Delta T_{iq}(\omega) \Delta z_i(\omega) = \sum_{i=1}^{n} \text{PF}_{pq}^i(\omega) \cdot \Delta z_i(\omega)$$
Here, \(\text{PF}_{pq}^i = \Delta L_{pi} \Delta T_{iq}\) is defined as the Participation Factor of mode \(i\) in the node impedance \(\Delta Z_{pq}\). For self-impedance (\(p = q\)), \(\text{PF}_{pp}^i\) indicates how excitable/observable mode \(i\) is at node \(p\) itself. For mutual impedance (\(p \ne q\)), \(\text{PF}_{pq}^i\) indicates how effectively a current injection at node \(q\) can excite mode \(i\) and have it observed as a voltage at node \(p\). By comparing \(\text{PF}_{pq}^i\) for a given resonant mode \(i\) across all relevant node pairs \((p, q)\), we can identify the critical nodes or node pairs where this resonance is most likely to be triggered and observed. This guides targeted monitoring and mitigation efforts.
To demonstrate the application of this method, we consider a multi-inverter grid-connected system representative of a wind farm collection network. The system comprises multiple chain-connected grid-connected inverters along several radial feeder lines, all connected to a common Point of Common Coupling (PCC). Each grid-connected inverter uses an LCL filter and standard vector control in the synchronous reference frame. The aggregated system model is an impedance network where each inverter is represented by its Norton equivalent: an ideal current source in parallel with its output impedance \(\Delta Z_s(\omega)\). The network includes line impedances \(\Delta Z_L(\omega)\) and grid impedance \(\Delta Z_g(\omega)\). System parameters for the grid-connected inverters and network are summarized in the table below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Current PI, \(k_{pi}\) | 0.15 | Grid Frequency, \(f_1\) | 50 Hz |
| Current PI, \(k_{ii}\) | 10 | DC-link Voltage, \(u_{dc}\) | 1.12 kV |
| Filter Inductor, \(L_c\) | 0.5 mH | DC-link Capacitor, \(C_{dc}\) | 0.09 F |
| Filter Capacitor, \(C_f\) | 200 µF | PLL PI, \(k_{pPLL}\) | 70 |
| Line Resistance, \(R_1\) | 0.05 Ω | PLL PI, \(k_{iPLL}\) | 1800 |
| Line Inductance, \(L_1\) | 0.2 mH | Voltage Feedforward, \(K_f\) | 0.42 |
| Grid Resistance, \(R_g\) | 0.02 Ω | Decoupling Gain, \(K_{dq}\) | 0.157 |
| Grid Inductance, \(L_g\) | 0.1 mH | Switching Freq., \(f_s\) | 20 kHz |
Applying modal analysis to this system reveals several resonant modes within a practical frequency range. The calculated quality factors for these modes provide immediate insight. For instance, consider two dominant modes: a low-frequency mode (Mode 1, \(f_{r1} \approx 467.3\) Hz) and a high-frequency mode (Mode 2, \(f_{r2} \approx 1241.5\) Hz) under baseline parameters. Their Nyquist plots yield positive quality factors (\(Q_1=5.1\), \(Q_2=3.2\)), indicating positive but potentially sharp resonances. Sensitivity analysis shows that increasing the current controller integral gain \(k_{ii}\) or decreasing the PLL proportional gain \(k_{pPLL}\) increases the magnitude of \(Q\), making the resonances sharper. More critically, a specific combination of parameters (\(k_{ii}=10, k_{pPLL}=30\)) results in \(Q_2\) becoming negative (\(Q_2 = -7.2\)), signaling that Mode 2 has transitioned to a negatively damped, unstable condition.
Time-domain simulations confirm the assessment. Under conditions where \(Q\) values are positive, the harmonic voltage distortion at the resonant frequencies correlates with the magnitude of \(Q\). When \(Q_2\) becomes negative, the harmonic content at \(f_{r2}\) increases dramatically, validating that a negative quality factor signifies a much more severe resonance threat due to the underlying negative damping.
The next step is to identify where in the network these concerning resonances are most relevant. Using the participation factor analysis, we examine the contribution of different resonant modes to specific node impedances, such as the self-impedance at Node 2, \(\Delta Z_{22}\), and the mutual impedance between the PCC and Node 2, \(\Delta Z_{PCC,2}\). A subset of participation factors at the resonant frequencies \(f_{r1}\) and \(f_{r2}\) is illustrative.
| Resonant Mode | PF at \(f_{r1}\) for \(\Delta Z_{22}\) | PF at \(f_{r2}\) for \(\Delta Z_{PCC,2}\) | ||
|---|---|---|---|---|
| Mode 1 | 0.39 | (High) | 0.02 | (Low) |
| Mode 2 | 0.15 | (Medium) | 0.15 | (Medium) |
| Mode 5 | 0.38 | (High) | 0.07 | (Low) |
The table shows that Mode 1 has a high participation factor in the self-impedance of Node 2 but a very low one in the mutual impedance from the PCC to Node 2. This means Mode 1 is highly excitable and observable at Node 2 itself, but a current injection at the PCC will barely excite it at Node 2. Conversely, Mode 2 has a significant participation factor in the mutual impedance \(\Delta Z_{PCC,2}\), indicating it can be effectively excited by injections at the PCC and observed at Node 2. This analysis successfully identifies Node 2 as a key location for Mode 1, and the PCC-Node 2 path as a key path for Mode 2. Injecting test currents at these critical locations confirmed that the resulting harmonic voltage amplification was consistent with the participation factor analysis.
In conclusion, the integration of a large number of grid-connected inverters necessitates advanced tools for harmonic resonance management. The method proposed in this work, combining modal analysis with a quality factor derived from an equivalent RLC circuit interpretation and a participation factor for node impedance, provides a powerful two-stage assessment framework. First, the quality factor \(Q\) evaluates the inherent strength and stability (damping) of each resonant mode, clearly flagging modes with high positive \(Q\) (severe passive resonance) and, most importantly, those with negative \(Q\) (unstable resonance). Second, the participation factor \(\text{PF}_{pq}^i\) pinpoints the specific nodes and injection-observation pairs where a concerning resonant mode is most excitable and observable. This targeted identification is crucial for implementing effective and efficient mitigation strategies, such as installing passive or active damping at the key nodes, or adjusting the control parameters of the grid-connected inverters that contribute most to the problematic mode. This approach offers system planners and operators a refined methodology to proactively assess and address harmonic resonance risks in the evolving landscape of inverter-dominated power systems.