The rapid integration of renewable energy sources, primarily wind and solar, into modern power systems has introduced significant challenges related to wideband oscillatory instability. The grid-tied inverter, serving as the critical power electronic interface for these distributed generation systems and a key component in DC microgrids, plays a pivotal role in determining grid interconnection stability. Particularly under weak grid conditions characterized by high grid impedance, these inverters become susceptible to destabilizing interactions between their fast-switching controls and the network, leading to poorly-damped or growing oscillations in the subsynchronous and supersynchronous frequency ranges.
This article presents a circuit-equivalent methodology to demystify the instability mechanisms of a grid-tied inverter. By visualizing the inverter’s control loops—namely the current controller, phase-locked loop (PLL), and feedforward terms—as virtual circuit elements (resistors, inductors, capacitors), we construct an intuitive physical model. This model vividly reveals how the interaction between the current loop and the PLL inherently introduces a negative resistance in the inverter’s output impedance under weak grid conditions. This negative resistance is the fundamental root cause of oscillatory instability. Based on this circuit-based insight, we systematically derive design principles for active damping control strategies, compare the circuit characteristics and stabilization capabilities of different damping filter types, and validate the analysis through detailed simulation and hardware-in-the-loop (HIL) experiments.
1. Equivalent Circuit Modeling of the Grid-Tied Inverter
A typical three-phase two-level voltage source converter (VSC) with L-filter interfacing an AC grid is considered. Its standard control structure in the synchronous reference frame (dq-frame) includes a cascaded control loop: an inner current controller, an outer power or DC voltage controller (not explicitly analyzed for impedance modeling), a PLL for grid synchronization, and decoupling/feedforward terms. The dynamic relationship between the inverter’s terminal voltage and current, reflecting the internal controller dynamics, can be expressed as a multi-component equation:
$$ \begin{bmatrix} \Delta \mathbf{i}_{dq}^s \end{bmatrix} = \underbrace{-\mathbf{H}_I^{-1} \mathbf{G}_{dec} \Delta \mathbf{u}_{dq}^s}_{\Delta \mathbf{i}_{dqI}^s} \quad \underbrace{-\mathbf{H}_I^{-1} \mathbf{G}_{IPLL} \Delta \mathbf{u}_{dq}^s}_{\Delta \mathbf{i}_{dqPLL}^s} \quad \underbrace{-\mathbf{H}_I^{-1} \mathbf{G}_{dec} \Delta \mathbf{u}_{dq}^s}_{\Delta \mathbf{i}_{dqdec}^s} \quad \underbrace{-k_{vf} \mathbf{H}_I^{-1} \mathbf{G}_{VPLL} \Delta \mathbf{u}_{dq}^s}_{\Delta \mathbf{i}_{dqvf}^s} $$
Where $\Delta \mathbf{i}_{dq}^s$ and $\Delta \mathbf{u}_{dq}^s$ are the small-signal current and voltage vectors in the stationary frame mapped to the dq-frame via the PLL angle. The terms $\Delta \mathbf{i}_{dqI}^s$, $\Delta \mathbf{i}_{dqPLL}^s$, $\Delta \mathbf{i}_{dqdec}^s$, and $\Delta \mathbf{i}_{dqvf}^s$ represent the current components attributable to the inner current loop, PLL, decoupling term, and voltage feedforward, respectively. The matrices $\mathbf{H}_I$, $\mathbf{G}_{IPLL}$, $\mathbf{G}_{dec}$, and $\mathbf{G}_{VPLL}$ contain the transfer functions of the respective controllers and plant. The key to circuit visualization lies in interpreting each of these current components as being driven by the terminal voltage through an equivalent admittance.
1.1 Current Controller Equivalent Circuit
Considering only the inner current controller with PI regulation ($H_I(s) = K_{pi} + K_{ii}/s$) and the physical filter inductance $L_f$, the impedance in the dq-domain is diagonal if decoupling is considered:
$$ \mathbf{Z}_{I,dq}(s) = \begin{bmatrix} A_I(s) & 0 \\ 0 & A_I(s) \end{bmatrix}, \quad \text{where} \quad A_I(s) = K_{pi} + \frac{K_{ii}}{s} + sL_f $$
This diagonal element $A_I(s)$ can be directly interpreted as a series R-L-C branch. The proportional gain $K_{pi}$ acts as a virtual resistor $R_{vir}$, the integral gain $K_{ii}/s$ acts as a virtual capacitor $C_{vir}$ (since $1/(sC) \equiv K_{ii}/s$), and the filter inductor $sL_f$ is the physical inductance. Thus, per axis (d or q), the current controller can be modeled as a simple series circuit: a resistor $R_{vir}=K_{pi}$, a capacitor $C_{vir}=1/K_{ii}$, and an inductor $L_f$.
1.2 Phase-Locked Loop (PLL) Equivalent Circuit
The PLL’s dynamic effect couples the d and q axes. Its equivalent current contribution can be derived and mapped into circuit elements. The q-axis channel exhibits a significant circuit interpretation:
$$ \Delta i_{qPLL}^s = -G_{PLLq}(s) \Delta u_{sq}^s, \quad \text{where} \quad G_{PLLq}(s) = \frac{K_{pi} + \frac{K_{ii}}{s} + sL_f}{G_{PLL}(s) I_{d0}} $$
Here, $G_{PLL}(s)$ is the closed-loop transfer function of the PLL. This expression reveals that from the q-axis voltage to the q-axis current, the PLL effectively presents an admittance that is the original current controller admittance divided by $(G_{PLL}(s) I_{d0})$. This division by the PLL’s dynamics, particularly around its bandwidth, is the source of negative incremental resistance. The equivalent q-axis circuit can be drawn as a series connection of a negative resistor, a negative capacitor, and a frequency-dependent inductor, all scaled by the inverse of the PLL dynamics. The d-axis effect of the PLL appears as a current source dependent on the q-axis voltage.
1.3 Composite Model and Domain Transformation
Combining the equivalent circuits for all control blocks, a complete dq-domain circuit model of the grid-tied inverter is constructed. To analyze stability using the impedance-based Nyquist criterion, this model must be transformed into the sequence domain (positive and negative sequence). The transformation from the dq-domain impedance matrix $\mathbf{Z}_{dq}^{vsc}(s)$ to the sequence domain impedance $Z_P(s)$ and $Z_N(s)$ involves a frequency-shift operation ($s \rightarrow s \mp j\omega_0$) and a linear transformation matrix $\mathbf{A}$:
$$ \mathbf{Z}_{\pm}^{vsc}(s) = \mathbf{A}^{-1} \mathbf{Z}_{dq}^{vsc}(s) \mathbf{A} $$
Where $\mathbf{Z}_{\pm}^{vsc}(s)$ is the impedance in the so-called modified sequence domain. The final positive-sequence impedance $Z_P(s)$ seen by the grid is derived from the elements of $\mathbf{Z}_{\pm}^{vsc}(s)$. The equivalent circuit in the positive-sequence domain ultimately shows the inverter as a frequency-dependent R-L-C network in parallel with controlled sources, the parameters of which are complex functions of all controller gains and operating points.
| Control Block | dq-Domain Circuit Element (Per Axis) | Physical Interpretation |
|---|---|---|
| Inner Current Loop (PI) | $R_{vir}=K_{pi}$, $C_{vir}=1/K_{ii}$, $L_{phy}=L_f$ | Provides intended tracking and positive damping. |
| Phase-Locked Loop (PLL) | Negative $R$, Negative $C$, coupled sources | Introduces negative resistance near its bandwidth. |
| Voltage Feedforward | Modifies $R_{vir}$, adds coupling | Can cancel plant impedance but may reduce stability margin. |
| AC Grid | $Z_g(s) = sL_g + j\omega_0 L_g$ | Source impedance, inductive at low frequencies. |
2. Instability Mechanism: The Emergence of Negative Resistance
The core instability mechanism for a grid-tied inverter in a weak grid is the manifestation of a negative real part in its output impedance within the subsynchronous/supersynchronous frequency range. The circuit model provides a lucid explanation.
2.1 Origin of Negative Resistance: The analysis shows that under normal operation, only the PLL interaction path introduces components with a negative real part. Focusing on the q-axis channel, the equivalent impedance contributed by the PLL-current loop interaction is approximately:
$$ Z_{PLL,q}(s) \approx -\frac{1}{G_{PLL}(s) I_{d0}} \left( K_{pi} + \frac{K_{ii}}{s} \right) $$
The term $-\frac{1}{G_{PLL}(s) I_{d0}}$ is central. The PLL’s closed-loop transfer function $G_{PLL}(s)$ has a phase lag around its bandwidth. For frequencies below and around the PLL bandwidth, the real part of this term can become negative, effectively multiplying the current controller’s virtual impedance $(K_{pi} + K_{ii}/s)$ by a negative factor. This creates the net negative resistance. The current controller parameters $K_{pi}$ and $K_{ii}$ determine the magnitude of this negative impedance.
2.2 Influence of Controller Parameters: The sensitivity of the negative resistance to current controller gains varies with frequency:
- Integral Gain ($K_{ii}$): Dominates the negative resistance magnitude at lower frequencies (within ~0.5 pu of the current loop bandwidth). Increasing $K_{ii}$ increases the negative resistance significantly in this range.
- Proportional Gain ($K_{pi}$): Becomes more influential at higher frequencies. Its effect on negative resistance is non-monotonic at very low frequencies.
The PLL bandwidth determines the frequency range over which this negative resistance effect is prominent. A wider PLL bandwidth spreads the negative resistance region over a broader frequency spectrum, increasing the risk of interaction with grid resonances.
2.3 Path to System Instability: The negative resistance $R_{neg}(f)$ appears in the inverter’s positive-sequence impedance $Z_P(j\omega)$. For stability, the total damping (real part) of the loop gain formed by the inverter impedance and the grid impedance $Z_g(j\omega)$ must be positive. The grid impedance is purely inductive ($L_g$), providing a small positive resistor $R_g = \text{Re}(Z_g) \approx 0$ at low frequencies. The stability condition can be simplified as:
$$ \text{Re}(Z_P(j\omega)) + \text{Re}(Z_g(j\omega)) > 0 \quad \forall \omega $$
In a weak grid, $L_g$ is large, making the grid impedance magnitude high. At certain frequencies where $\text{Re}(Z_P(j\omega))$ is negative and its magnitude satisfies $|\text{Re}(Z_P)| > R_g$, the net damping becomes negative. This leads to an undamped or negatively damped resonance with the grid inductance, causing oscillations to grow. The circuit model clearly shows this as a negative resistor in series with the grid inductor, forming an active R-L circuit prone to self-excitation.
3. Design of Active Damping Stabilization Control
To mitigate the instability, an active damping control is added. The principle is to inject a stabilizing signal that equivalently increases the positive damping (resistance) of the inverter in the critical frequency range, counteracting the PLL-induced negative resistance. Based on the circuit equivalence, the most effective point to add this damping is in the current control loop, often as a filtered feedback of capacitor voltage or a virtual impedance in the voltage feedforward path.
3.1 Design Principles from Circuit Equivalence:
From the circuit model analysis, an effective active damping compensator $G_{AD}(s)$ should:
- Exhibit Inductive Dominance: Its equivalent circuit should present an inductive behavior ($sL_{vir}$) in the problem frequency band. This equivalently reduces the sensitivity of the negative resistance to the integral gain $K_{ii}$, which is the dominant contributor.
- Satisfy Net Damping Criterion: After addition, the total equivalent resistance in the q-axis circuit $R_{q+} + R_{q-}$ must be greater than a stability threshold $R_{th} > 0$ across all frequencies.
- Maintain Baseband Performance: It must not interfere with the fundamental frequency control performance of the grid-tied inverter (e.g., power tracking, DC bus regulation).
- Adhere to Modulation Limits: The additional virtual impedance must not cause the modulator to saturate, respecting the condition $M_{index} \leq M_{th}$.
3.2 Comparative Analysis of Active Damping Types
Various filter-based active damping strategies can be represented as virtual impedances. Their circuit equivalence and effectiveness differ significantly.
| Active Damping Type | Transfer Function $G_{AD}(s)$ | Equivalent Circuit | Key Characteristic |
|---|---|---|---|
| Virtual Resistor | $R_v$ | Pure Resistor | Simple, but effect is non-monotonic and frequency-limited. |
| Resistor + High-Pass Filter (HPF) | $\frac{R_v s T}{1 + s T}$ | $R_v$ in series with $L_{vir}=R_v T$ | Inductive at low-freq. Good for sub/super-sync. damping. |
| Resistor + Low-Pass Filter (LPF) | $\frac{R_v}{1 + s T}$ | Frequency-dependent $R$ & $C$ | Capacitive tendency. Less effective for low-freq. instability. |
| Band-Pass Filter (BPF) | $\frac{R_v \cdot 2\xi\omega_n s}{s^2 + 2\xi\omega_n s + \omega_n^2}$ | Tuned R-L-C series | Targeted damping at center freq. $\omega_n$. Narrowband. |
| Band-Stop/Notch Filter (BRF) | $R_v \frac{s^2 + \omega_n^2}{s^2 + 2\xi\omega_n s + \omega_n^2}$ | Tuned parallel R-L-C | Blocks interaction at $\omega_n$. Can be sensitive to freq. shift. |
| Impedance + HPF (Proposed) | $\frac{R_v + s L_{v1}}{1 + s T}$ where $T=L_{v1}/R_v$ | Pure Inductor $L_{v1}$ at low freq. | Strong inductive characteristic. Provides robust, wideband damping against PLL-induced negative resistance. |
The effectiveness is evaluated by how much each method increases the real part of the inverter’s impedance $\text{Re}(Z_P(j\omega))$ in the critical frequency range (e.g., 10-100 Hz). Analysis shows that methods presenting an equivalent inductive impedance, like the Impedance-HPF type, are most effective because they directly counteract the negative resistance mechanism described in Section 2. The inductor $L_{v1}$ provides a positive impedance $j\omega L_{v1}$ whose influence grows with frequency, effectively neutralizing the negative resistance that also scales with frequency due to the $K_{ii}/s$ term.
3.3 Parameter Design Guidelines
For the proposed Impedance-HPF active damping ($G_{AD}(s) = (R_v + sL_{v1}) / (1 + sT)$):
- Virtual Inductor ($L_{v1}$): This is the key parameter. It should be chosen to provide sufficient positive reactance. A starting point is $L_{v1} \approx (0.1 \text{ to } 0.3) \cdot L_f$, where $L_f$ is the physical filter inductance.
- Time Constant ($T$): Set $T = L_{v1} / R_v$. This ensures the HPF corner frequency is consistent with the inductive behavior.
- Virtual Resistor ($R_v$): Provides baseline damping. It is chosen based on the stability threshold and modulation limit constraint: $R_{v,min} \leq R_v \leq R_{v,max}$. $R_{v,max}$ is derived from the modulator voltage limit condition.
- Implementation: The damping signal $v_{damp,dq} = G_{AD}(s) \cdot i_{dq}$ is subtracted from the current controller’s voltage reference or added as a feedback term.
4. Simulation and Experimental Validation
A detailed simulation model of a 4 MW grid-tied inverter was built in PSCAD/EMTDC with parameters: $L_f = 100 \mu H$, $K_{pi}=1$, $K_{ii}=100$, PLL bandwidth variable (10-50 Hz). The grid short-circuit ratio (SCR) was varied from 5 to 1.
4.1 Oscillation Suppression Comparison: With SCR=3 and PLL bandwidth increased to 50 Hz, the system exhibited growing oscillations at 35/65 Hz (sub/supersynchronous). Different active damping methods were activated at t=2.1s with comparable parameter effort ($R_v=0.1 \Omega$, target frequency ~15 Hz).
- Resistor-BPF/BRF: Oscillation amplitude was contained but not fully suppressed.
- Impedance-LPF: Marginally stabilized the system, but damping was weak.
- Impedance-HPF (Proposed): Rapidly suppressed the oscillations within 200 ms, demonstrating superior performance.
4.2 Dynamic Performance in Weak Grids: The proposed Impedance-HPF damping was tested under power step changes in increasingly weak grids.
- Without damping: Power steps caused sustained oscillations at SCR=3 and instability at SCR=1.
- With Impedance-HPF damping: The system remained stable with well-damped transients even at an extreme SCR=1, proving the controller’s robustness.
4.3 Controller Hardware-in-the-Loop (CHIL) Verification: Experiments on a StarSim real-time platform with a physical digital controller (MT1070RCP) confirmed the simulation findings. The growing oscillations under weak grid conditions were reliably reproduced. The proposed active damping control successfully stabilized the system within 200 ms, while other methods showed inadequate damping. Power step responses further validated the dynamic stability enhancement provided by the circuit-based design.
5. Conclusion
This article has presented a comprehensive circuit-equivalence approach to analyze and mitigate instability in grid-tied inverters. The key contributions and findings are:
- Intuitive Circuit Model: The control loops of a grid-tied inverter are successfully mapped to virtual R-L-C elements and dependent sources. This model provides an intuitive physical understanding beyond traditional state-space or impedance models.
- Clarity on Instability Mechanism: The analysis explicitly shows that the interaction between the current controller’s integral action and the phase-locked loop is the fundamental source of negative resistance in the inverter’s output impedance. The PLL dictates the frequency range of this effect, while the integral gain $K_{ii}$ primarily governs its magnitude.
- Systematic Stabilization Design: Based on the circuit insight, design principles for active damping are established, emphasizing the need for an equivalent inductive characteristic to counteract the negative resistance effectively.
- Superior Damping Strategy: Comparative analysis reveals that an Impedance-High Pass Filter based active damping control offers the most robust and effective mitigation against subsynchronous/supersynchronous oscillations caused by PLL interactions in weak grids. Its inherent inductive equivalence directly tackles the root cause of instability.
- Validation: Simulation and hardware-in-the-loop experiments confirm the theoretical analysis. The proposed damping method ensures stable operation of the grid-tied inverter even under very weak grid conditions (SCR down to 1), with excellent dynamic performance during power transients.
This circuit-based framework not only advances the fundamental understanding of inverter-grid interactions but also provides power electronics engineers with a practical and powerful tool for the design of stable and robust grid-tied inverter systems for renewable energy integration.