The transition towards sustainable energy sources is a global imperative, with renewable generation forming the cornerstone of modern power systems. Unlike conventional synchronous generators, renewable sources like photovoltaics and wind turbines are interfaced with the grid through power electronic converters, primarily utility interactive inverters. While offering advantages in flexibility and control, the large-scale integration of these inverters introduces significant dynamic interaction challenges with the grid. A primary concern is the emergence of wideband oscillations, spanning from several hertz to hundreds of hertz, which can threaten system stability and power quality. These instabilities stem from the complex impedance coupling between multiple parallel-connected utility interactive inverters and the grid, especially under weak grid conditions characterized by high grid impedance.
Analyzing these interactions requires moving beyond traditional state-space models, which become intractable for numerous paralleled units. The impedance-based analysis method, viewing the system as a source-load interconnection, provides a powerful and scalable framework. This approach necessitates accurate wideband impedance models of the aggregate system of utility interactive inverters. This article systematically explores the modeling, impedance characterization, and stability assessment for parallel-operated utility interactive inverters, focusing on revealing wideband instability mechanisms.
Structure and Control of Parallel Inverter Systems
A typical grid-connected inverter system comprises a power stage and a control stage. The power stage includes the DC source (e.g., PV array, battery), a three-phase voltage-source converter (VSC) bridge, and an L or LCL output filter. The control system is typically implemented in the synchronous rotating dq-reference frame for independent control of active and reactive power. The core control loops for a utility interactive inverter include:
- Phase-Locked Loop (PLL): Synchronizes the inverter’s control frame with the grid voltage at the Point of Common Coupling (PCC).
- Current Control Loop: Regulates the inverter’s output current to follow reference commands derived from power/voltage outer loops. Both Proportional-Integral (PI) and more advanced controllers like Proportional-Resonant (PR) or active disturbance rejection control (ADRC) are employed.
- DC-Link Voltage or Power Control (Outer Loop): Provides the active current reference to maintain DC bus voltage or deliver a specific power setpoint.
In a parallel configuration, multiple such utility interactive inverter units are connected to the same PCC. Their collective behavior is not a simple summation due to potential interactions through shared grid impedance and control couplings. The basic electrical relationship for one phase of the inverter’s AC side, neglecting the filter capacitor for an L-filter, is given by Kirchhoff’s voltage law:
$$
v_{im,x} – v_{pcc,x} = L_f \frac{di_{x}}{dt}
$$
where $$ x \in \{a, b, c\} $$, $$ v_{im} $$ is the inverter bridge pole voltage, $$ v_{pcc} $$ is the PCC voltage, $$ i $$ is the output current, and $$ L_f $$ is the filter inductance. The pole voltage is determined by the DC-link voltage and the modulation signals generated by the controller.
Wideband Aggregated Impedance Modeling Methodology
The core of the analysis is to derive a linearized frequency-domain model for the output impedance of a utility interactive inverter as seen from the PCC. For parallel systems, understanding the “sequence impedance” is crucial due to the decoupled nature of positive- and negative-sequence components in balanced three-phase systems. The modeling employs the harmonic linearization technique, where small-signal positive- and negative-sequence voltage perturbations at a frequency $$ f_p $$ are superimposed on the fundamental grid voltage, and the corresponding current responses are analyzed.
1. Switching Cycle and Modulation Model
Assuming ideal switching and averaging over a switching period, the relationship between the modulation signal $$ m_{dq} $$ in the dq-frame and the generated average inverter voltage can be established. The key is to link the perturbation components in the modulation signals to the perturbation currents and voltages. The modulation signal is the output of the current controller, transformed by the PLL angle.
2. Control System and Sampling Delay Model
Digital control introduces inevitable delays that significantly impact high-frequency impedance. The total delay $$ G_d(s) $$ typically includes computational delay (often modeled as a 1.5-sample delay $$ e^{-1.5sT_s} $$) and the effect of PWM hold. A simplified representation for the sampling and signal conditioning path for voltage and current can be a first-order filter plus a delay:
$$
G_{sample}(s) = \frac{1}{1 + s / \omega_c} \cdot e^{-sT_d}
$$
where $$ \omega_c $$ is the sensor/anti-aliasing filter cutoff frequency and $$ T_d $$ is the total equivalent delay. This model is critical for accurately predicting phase shifts at higher frequencies.
3. Phase-Locked Loop (PLL) Frequency Coupling Model
The PLL is a primary source of impedance coupling between sequences. A small-signal positive-sequence voltage perturbation at frequency $$ f_p $$ can cause a perturbation in the estimated PLL angle $$ \Delta\theta $$. This angle perturbation, when used in the park transforms for current measurement and modulation, creates frequency coupling—mapping a positive-sequence perturbation at $$ f_p $$ in the stationary frame to components at $$ f_p – f_1 $$ in the dq-frame (where $$ f_1 $$ is the fundamental frequency), and vice-versa for negative-sequence. The linearized transfer function from PCC voltage perturbation to PLL angle perturbation is essential. For a standard SRF-PLL with a PI regulator $$ H_{PLL}(s) = k_{p,PLL} + k_{i,PLL}/s $$, the sensitivity can be derived.
4. Current Controller Model in the DQ-Frame
The current controller, whether PI or ADRC, processes the error between the reference and measured dq-currents. Under perturbation, the measured currents contain components induced by the voltage perturbations and modulated by the PLL’s angle perturbation. The controller’s output, the dq modulation signal $$ m_{dq} $$, thus contains components linked to both positive- and negative-sequence voltage perturbations. The transfer function of the controller itself (e.g., $$ G_{PI}(s) = k_p + k_i/s $$ or an ADRC transfer function) forms part of this relationship.
5. Composite Sequence Impedance Model
By integrating the models from steps 1-4 and solving for the ratio of the voltage perturbation to the current response in the stationary frame, the positive-sequence $$ Z_p(j\omega) $$ and negative-sequence $$ Z_n(j\omega) $$ output impedance of the utility interactive inverter are obtained. The final expressions clearly show different frequency couplings. For a system using dq-frame PI control and SRF-PLL, the impedances have distinct forms due to the PLL’s effect. The aggregated impedance of N parallel, identical utility interactive inverters is approximately $$ Z_{agg} = Z_{inv} / N $$, assuming balanced parameters and negligible inter-inverter cable impedances.
The following table summarizes typical parameters used for modeling a single utility interactive inverter unit.
| System Parameter | Symbol | Typical Value/Range |
|---|---|---|
| Rated Voltage (L-L, RMS) | $$ V_g $$ | 400 V |
| Rated Power | $$ S_n $$ | 10 kVA |
| DC-Link Voltage | $$ V_{dc} $$ | 700 V |
| Filter Inductance | $$ L_f $$ | 1.5 mH |
| Switching / Sampling Frequency | $$ f_s $$ | 10 kHz |
| Current Controller PI Gains | $$ k_{p,i}, k_{i,i} $$ | 0.5, 100 |
| PLL PI Gains | $$ k_{p,PLL}, k_{i,PLL} $$ | 50, 800 |
| Fundamental Frequency | $$ f_1 $$ | 50 Hz |
Impedance Characterization and Stability Analysis
Wideband Impedance Characteristics
Using the derived model, Bode plots of $$ Z_p(j\omega) $$ and $$ Z_n(j\omega) $$ can be generated to analyze the impact of various parameters. The operating point and controller parameters significantly shape the impedance profile.
Effect of Output Power: The output power level (directly related to the d-axis current $$ I_d $$) strongly influences the low-to-mid frequency range of the impedance. As power increases, the magnitude of the impedance typically decreases, which can reduce stability margins. The following table shows how key impedance features change with power for a given utility interactive inverter design.
| Output Power (pu) | Impedance Magnitude at 100 Hz (dBΩ) | Phase at 100 Hz (deg) | Resonant Frequency (Hz) |
|---|---|---|---|
| 0.3 | 35.2 | -15 | 850 |
| 0.6 | 32.1 | -18 | 840 |
| 1.0 | 29.5 | -22 | 830 |
Effect of Control Strategy: Replacing a PI current controller with a Quasi-Proportional Resonant (QPR) controller designed for stationary frame operation can alter the impedance shape, often removing the phase lag introduced by the dq-transformation at frequencies near the fundamental and its harmonics, potentially improving stability.
Stability Assessment Using the Generalized Nyquist Criterion (GNC)
For a system comprising a source impedance $$ Z_s(j\omega) $$ (the grid) and a load impedance $$ Z_L(j\omega) $$ (the aggregated utility interactive inverter), the minor-loop gain is defined as $$ L(j\omega) = Z_s(j\omega) / Z_L(j\omega) $$. According to the impedance-based stability criterion, the system is stable if the Nyquist plot of $$ L(j\omega) $$ does not encircle the critical point (-1, j0). The distance to this point indicates the stability margin.
Analysis involves plotting $$ L(j\omega) $$ for different scenarios. Key factors affecting stability include:
- Grid Strength (Short-Circuit Ratio – SCR): A weaker grid (higher $$ Z_s $$, lower SCR) makes the $$ L(j\omega) $$ contour larger, increasing the risk of encirclement.
- Number of Parallel Inverters: As N increases, $$ Z_L $$ decreases ($$ Z_L = Z_{inv}/N $$), which increases $$ L(j\omega) $$, potentially degrading stability.
- Control Parameters: Increasing PLL bandwidth or current controller gains can reshape the inverter impedance phase in a detrimental way, potentially causing a negative damping effect at specific frequencies.
The stability can be quantified by parameters like Phase Margin (PM) and Gain Margin (GM) derived from the minor-loop gain $$ L(j\omega) $$. The following formula finds the crossover frequency $$ \omega_c $$ where magnitude is 1 (0 dB):
$$
|Z_s(j\omega_c)| = |Z_L(j\omega_c)|
$$
The Phase Margin is then: $$ PM = 180^\circ + \angle(Z_s(j\omega_c)/Z_L(j\omega_c)) $$.
A comparative stability analysis for different control strategies under weak grid conditions might yield results summarized as follows:
| Control Configuration | Grid SCR | Phase Margin (PM) | Gain Margin (GM) | Stable? |
|---|---|---|---|---|
| Standard PI + SRF-PLL | 5 | 15° | 3 dB | Marginally |
| Standard PI + SRF-PLL | 3 | -5° | -2 dB | No |
| QPR + SRF-PLL | 3 | 25° | 5 dB | Yes |
| PI with PLL bandwidth reduced by 50% | 3 | 10° | 2 dB | Barely |
Validation via Time-Domain Simulation
The accuracy of the impedance model and the stability prediction must be validated. This is done by building a detailed switching model of the parallel utility interactive inverter system in a platform like MATLAB/Simulink or PLECS. A scenario predicted to be unstable by the Nyquist criterion (e.g., low SCR with high PLL gain) is simulated. The appearance of growing oscillations in the grid current or PCC voltage at the predicted frequency confirms the model’s validity. Conversely, for a stable design point, the system should exhibit well-damped dynamics following a disturbance.
Conclusion
The widespread integration of renewable energy sources via utility interactive inverters necessitates advanced analytical tools to ensure system stability. The wideband aggregated impedance modeling method, based on harmonic linearization and sequence impedance theory, provides a powerful and scalable framework for analyzing parallel inverter systems. This approach successfully captures the critical frequency-coupling effects introduced by controls like the PLL and reveals how parameters such as output power, grid strength, and controller design dictate the system’s impedance characteristics and, consequently, its stability margins. The model enables pre-assessment of stability risks and guides the design of controllers and system parameters for robust operation of large-scale plants comprising multiple parallel utility interactive inverter units. Future work may focus on extending these models to account for more complex grid conditions, unbalanced operation, and the interaction with other grid-forming power electronic devices.