With the increasing integration of distributed solar inverters into power grids, voltage violations have become a critical issue limiting solar energy absorption. Volt-var control, implemented in grid-following solar inverters, effectively mitigates overvoltage by adjusting reactive power based on voltage deviations. However, improper design can lead to oscillation instability in weak grids due to interactions between control loops. Existing stability studies often neglect the dynamic coupling between volt-var control and traditional grid-following controls, resulting in inaccurate stability boundaries. This paper addresses these limitations by developing a simplified fifth-order single-input single-output (SISO) model for solar inverters with volt-var control, analyzing oscillation mechanisms using the Nyquist stability criterion, and deriving a practical stability region. The analysis reveals that system stability improves with weaker and slower volt-var responses, shorter control delays, and stronger grids. Additionally, the impact of DC voltage control bandwidth on stability depends on the line resistance-inductance ratio: lower bandwidths enhance stability for high ratios, while higher bandwidths are beneficial for low ratios. The proposed stability boundary, validated through hardware-in-the-loop experiments, reduces computational errors to below 5% for open-loop response times exceeding 1.4 s, addressing the applicability issues of existing criteria.
The proliferation of distributed solar inverters in modern power systems introduces challenges such as voltage violations, harmonic pollution, and power oscillations. Volt-var control, mandated by standards like IEEE 1547-2018, enables solar inverters to regulate grid voltage by injecting or absorbing reactive power. However, in weak grids, the interaction between volt-var control and inverter dynamics can provoke instability. Previous studies have focused on static stability or single-time-scale dynamics, overlooking multi-scale interactions. This work bridges the gap by incorporating the dynamic effects of volt-var control, reactive power control, and DC voltage control, providing a comprehensive stability analysis for solar inverter systems.
Small-Signal Modeling of Solar Inverters with Volt-Var Control
The small-signal model of a grid-connected solar inverter with volt-var control captures the system’s dynamic behavior under perturbations. The volt-var control block adjusts the reactive power reference based on the voltage at the point of common coupling (PCC). The control law is linearized as:
$$ \Delta Q_{\text{ref}} = -K \cdot \frac{1 – T_d s/2}{1 + T_d s/2} \cdot \frac{1}{1 + T_r s} \cdot \Delta U_{\text{pcc}} $$
where \( K \) is the droop coefficient, \( T_d \) is the communication delay, and \( T_r \) is the time constant of the open-loop response. The grid-following control includes current control, DC voltage control, reactive power control, and a phase-locked loop (PLL). The current control dynamics are represented by:
$$ G_i(s) = \frac{k_{pc} s + k_{ic}}{L_{f1} s^2 + k_{pc} s + k_{ic}} $$
where \( k_{pc} \) and \( k_{ic} \) are the proportional and integral gains of the current controller, and \( L_{f1} \) is the filter inductance. The DC voltage control maintains power balance and is modeled as:
$$ G_{\text{dvc}}(s) = \frac{k_{pdc} s + k_{idc}}{C_{\text{dc}} s^2 + k_{pdc} s + k_{idc}} $$
where \( k_{pdc} \) and \( k_{idc} \) are the controller gains, and \( C_{\text{dc}} \) is the DC-link capacitance. Reactive power control tracks the reference from the volt-var block:
$$ G_{\text{rpc}}(s) = \frac{k_{pq} s + k_{iq}}{s + k_{pq} s + k_{iq}} $$
The PLL provides synchronization and is modeled as:
$$ G_{\text{pll}}(s) = \frac{k_{ppll} s + k_{ipll}}{s^2 + k_{ppll} s + k_{ipll}} $$
The network impedance, combining transformer and line parameters, is included as \( Z_g = R_g + L_g s \). The full-order small-signal model integrates these components, but its complexity obscures stability insights. Thus, a reduced-order model is developed.
Reduced-Order Small-Signal Model
Eigenvalue and participation factor analyses identify the dominant modes influencing stability. The fifth-order model retains the dynamics of volt-var control, reactive power control, and DC voltage control, while neglecting faster dynamics like current control and PLL. The state-space representation of the reduced model is:
$$ \begin{bmatrix} \Delta \dot{U}_{td} \\ \Delta \dot{U}_{tq} \\ \Delta \dot{I}_{fd} \\ \Delta \dot{I}_{fq} \\ \Delta \dot{U}_{dc} \end{bmatrix} = A_r \begin{bmatrix} \Delta U_{td} \\ \Delta U_{tq} \\ \Delta I_{fd} \\ \Delta I_{fq} \\ \Delta U_{dc} \end{bmatrix} + B_r \Delta U_{\text{pcc}} $$
where \( A_r \) and \( B_r \) are reduced system matrices. The model simplification is validated through time-domain simulations, showing close agreement with the full-order model. The key parameters for the reduced model are summarized in Table 1.
| Parameter | Description | Value |
|---|---|---|
| \( K \) | Droop coefficient | 20–30 |
| \( T_r \) | Open-loop response time constant | 1–90 s |
| \( T_d \) | Communication delay | 0.04–0.07 s |
| \( \alpha_{\text{dvc}} \) | DC voltage control bandwidth | 10–50 Hz |
| \( \alpha_{\text{rpc}} \) | Reactive power control bandwidth | 5–50 Hz |
The reduced model enables a clear analysis of the interactions between control loops. For instance, the transfer function from voltage perturbation to reactive power output is derived as:
$$ G_{\text{path}}(s) = G_{\text{vvc}}(s) G_{\text{rpc}}(s) + G_{\text{dvc}}(s) G_l(s) $$
where \( G_{\text{vvc}}(s) \) is the volt-var control transfer function, \( G_{\text{rpc}}(s) \) is the reactive power control transfer function, \( G_{\text{dvc}}(s) \) is the DC voltage control transfer function, and \( G_l(s) \) represents the line dynamics.
Stability Analysis Based on Nyquist Criterion
The stability of the solar inverter system is assessed using the Nyquist criterion. The open-loop transfer function \( G_{\text{path}}(s) Z_{gs}(s) \) is analyzed, where \( Z_{gs}(s) = R_g + L_g s – R_g C_u \) and \( C_u \) is a constant derived from steady-state operating points. The system is stable if the Nyquist plot of \( G_{\text{path}}(j\omega) Z_{gs}(j\omega) \) does not encircle the point (-1, 0). The phase and gain margins are evaluated to quantify stability.
The impact of various parameters on stability is summarized below:
- Grid Strength: A lower short-circuit ratio (SCR) reduces stability margins. For example, decreasing SCR from 3 to 2 shifts the dominant eigenvalues to the right-half plane.
- Volt-Var Parameters: Increasing the droop coefficient \( K \) or delay \( T_d \) degrades stability, while increasing the open-loop response time \( T_r \) enhances it.
- Reactive Power Control Bandwidth: The relationship between bandwidth \( \alpha_{\text{rpc}} \) and stability is nonlinear. Low bandwidths (e.g., 5 Hz) improve stability by slowing the response, while high bandwidths (e.g., 40 Hz) reduce phase lag.
- DC Voltage Control Bandwidth: The effect depends on the line resistance-inductance ratio \( R_g/L_g \). For high \( R_g/L_g \), lower bandwidths improve stability; for low \( R_g/L_g \), higher bandwidths are beneficial.
The frequency response of \( G_{\text{path}}(j\omega) \) is plotted for different parameters. For instance, Figure 1 shows Bode plots for varying SCR values, illustrating the reduction in gain margin as the grid weakens.
Stability Region Characterization
The stability boundary is derived by solving for the frequency \( \omega_g \) where the phase of \( G_{\text{path}}(j\omega) \) is -180°:
$$ \text{Im}\left[ G_{\text{path}}(j\omega_g) \right] = 0 $$
The stability condition is then:
$$ |G_{\text{path}}(j\omega_g)| < \frac{1}{|Z_{gs}(j\omega_g)|} $$
This inequality defines the stable region in the parameter space. The boundary is computed for typical operating conditions and compared with the full-order model. The fifth-order model reduces the maximum error to 0.82%, compared to 18.3% for a second-order model that neglects inverter dynamics. The stability region for key parameters is summarized in Table 2.
| Parameter | Stable Range | Condition |
|---|---|---|
| Droop Coefficient \( K \) | < 26 (for SCR=2) | \( T_r = 1 \) s, \( T_d = 0.05 \) s |
| Open-Loop Response Time \( T_r \) | > 1.4 s | \( K = 20 \), SCR=2 |
| Delay \( T_d \) | < 0.06 s | \( K = 20 \), SCR=2 |
| Reactive Power Bandwidth \( \alpha_{\text{rpc}} \) | 5–10 Hz or >40 Hz | Nonlinear stability region |
| DC Voltage Bandwidth \( \alpha_{\text{dvc}} \) | 10–50 Hz (case-dependent) | Depends on \( R_g/L_g \) |
The stability boundary can be visualized as a surface in the parameter space. For practical design, the droop coefficient \( K \) should be selected based on the grid strength and control bandwidths to avoid instability.
Validation Through Simulation and Experiment
The analysis is validated using time-domain simulations and hardware-in-the-loop (HIL) experiments. The simulation model, built in MATLAB/Simulink, includes a three-phase solar inverter with LCL filter. The parameters are listed in Table 3.
| Parameter | Value |
|---|---|
| Rated Power | 30 kVA |
| Rated Voltage | 380 V |
| DC-Link Voltage | 700 V |
| Filter Inductance \( L_{f1} \), \( L_{f2} \) | 700 μH, 150 μH |
| Filter Capacitance \( C_f \) | 10 μF |
| DC-Link Capacitance \( C_{\text{dc}} \) | 1 mF |
| Switching Frequency | 10 kHz |
Simulation results demonstrate that increasing \( K \) or \( T_d \) leads to oscillatory instability, while increasing \( T_r \) stabilizes the system. For example, with SCR=2, the system becomes unstable when \( K \) exceeds 24, exhibiting 6.8 Hz oscillations. The HIL experiments, conducted on a Typhoon HIL 604 platform with a TMS320F28379D DSP controller, confirm the stability boundary. The experimental setup injects disturbances and measures the dynamic response of reactive power and voltage.
The error between the theoretical stability boundary and experimental results is analyzed for various open-loop response times. As shown in Table 4, the error decreases below 5% for \( T_r > 1.4 \) s, validating the proposed boundary for practical applications.
| Open-Loop Response Time \( T_r \) (s) | Theoretical \( K \) Boundary | Experimental \( K \) Boundary | Error (%) |
|---|---|---|---|
| 1.0 | 26 | 24 | 7.7 |
| 1.2 | 34 | 32 | 5.8 |
| 1.4 | 42 | 40 | 4.8 |
| 1.6 | 50 | 48 | 4.0 |
Conclusion
This paper presents a comprehensive analysis of oscillation mechanisms and stability region characterization for grid-connected solar inverters with volt-var control. The key findings are:
- Volt-var control can induce low-frequency oscillations in weak grids, primarily influenced by interactions with reactive power control and DC voltage control. Current control dynamics have negligible impact.
- Stability is enhanced by weaker and slower volt-var responses, shorter delays, and stronger grids. The reactive power control bandwidth has a nonlinear effect on stability, with optimal ranges at low and high bandwidths.
- The DC voltage control bandwidth’s impact depends on the line resistance-inductance ratio: lower bandwidths stabilize systems with high ratios, while higher bandwidths benefit those with low ratios.
- The derived stability boundary, based on a fifth-order SISO model and Nyquist criterion, achieves errors below 5% for open-loop response times exceeding 1.4 s, providing a reliable tool for parameter design in solar inverter systems.
Future work will extend the analysis to multi-inverter systems and explore adaptive control strategies to enhance stability under varying grid conditions. The insights from this study contribute to the safe and efficient integration of distributed solar inverters into modern power grids.