The integration of renewable energy sources into modern power grids is predominantly facilitated by on grid inverters. These power electronic interfaces are critical for converting and conditioning the variable output from sources like photovoltaics and wind turbines. The dynamic behavior and, crucially, the stability of these on grid inverters are paramount for the secure and reliable operation of the entire power system. The stability of an on grid inverter is not a fixed attribute but is intrinsically linked to its instantaneous operating point, defined by its output active power (P) and reactive power (Q). Conventional stability analysis methods, such as impedance-based analysis or eigenvalue analysis at a nominal operating point, provide valuable insights but offer a limited view. They fail to capture the stability characteristics across the entire, continuously varying operational envelope of the on grid inverter, which is subject to fluctuations in renewable generation and random load switching.
This work addresses this gap by proposing a comprehensive stability analysis framework for on grid inverters employing PQ control strategy. Unlike simple current-controlled on grid inverters, PQ-controlled schemes introduce additional complexity due to multi-loop coupling (involving Phase-Locked Loop (PLL), power outer loops, and current inner loops), multiple time scales, and inherent nonlinearities. Our method moves beyond point-wise analysis to characterize the complete stable operating region (SOR) in the P-Q plane. This region visually maps all feasible operating points where the on grid inverter maintains small-signal stability.
The core of our approach is the development of a detailed small-signal model for the PQ-controlled on grid inverter that explicitly embeds the operating point variables (P and Q). This model meticulously accounts for the cross-coupling dynamics between the PLL, the active/reactive power control loops, and the current regulator, all under the influence of grid impedance. From this model, we derive the closed-loop transfer functions whose poles determine stability. By performing a numerical sweep across all topologically feasible P-Q points (considering power circuit constraints), we compute the system’s eigenvalues and demarcate the boundary between stable and unstable operation. This results in a precise graphical representation of the stable operating region for the on grid inverter.
Furthermore, we leverage this framework to systematically investigate the impact of key system parameters on the stability of the on grid inverter. We analyze how variations in grid impedance (strength), grid voltage amplitude, and internal control parameters like PLL bandwidth reshape the stable operating region. Additionally, we provide a comparative analysis, contrasting the stable operating regions of a PQ-controlled on grid inverter with those of a conventional current-controlled on grid inverter under identical conditions, highlighting the distinct stability implications of the control strategy. The theoretical findings and the efficacy of the stable operating region analysis method are rigorously validated through time-domain simulations and experimental tests on a prototype on grid inverter system.
System Modeling of the PQ-Controlled On Grid Inverter
To analyze the stability across the full operating range, we first establish a small-signal model of the three-phase on grid inverter with PQ control. The system structure, including the power circuit and the control loops, is considered. The control system comprises the SRF-PLL for grid synchronization, the outer active/reactive power (PQ) loops, and the inner current regulation loops in the synchronous reference frame (d-q). The power circuit includes the inverter-side filter and the grid impedance.
The dynamics of the PLL are crucial. Defining the actual grid voltage phase as $$\theta_r$$ and the PLL estimated phase as $$\theta_c$$, the small-signal relation is $$\Delta \theta_c = \Delta \theta_r + \Delta \theta$$, where $$\Delta \theta$$ is the phase error. This creates two coordinate systems: the actual system (dr-qr) and the control system (dc-qc). The transformation of a space vector $$\mathbf{x}$$ and its conjugate between these frames is given by:
$$
\begin{aligned}
\Delta \mathbf{x}^c &= \Delta \mathbf{x} – j \mathbf{X}_0 \Delta \theta \\
\Delta \mathbf{x}^{c*} &= \Delta \mathbf{x}^* + j \mathbf{X}_0^* \Delta \theta
\end{aligned}
$$
where $$\mathbf{X}_0$$ is the steady-state value of $$\mathbf{x}$$.
The power circuit equations in the synchronous frame are:
$$
\begin{aligned}
\mathbf{e}_o &= \mathbf{Z}_{\text{filt}} \mathbf{i}_{\text{pcc}} + \mathbf{u}_{\text{pcc}} \\
\mathbf{u}_{\text{pcc}} &= \mathbf{Z}_{\text{grid}} \mathbf{i}_{\text{pcc}} + \mathbf{U}_g
\end{aligned}
$$
where $$\mathbf{Z}_{\text{filt}} = R_{\text{filt}} + sL_{\text{filt}} + j\omega_0 L_{\text{filt}}$$ and $$\mathbf{Z}_{\text{grid}} = R_{\text{grid}} + sL_{\text{grid}} + j\omega_0 L_{\text{grid}}$$.
The output of the current controller in the control frame is:
$$\mathbf{e}_{\text{ref}}^c = F_c(s) ( \mathbf{i}_{\text{ref}}^c – \mathbf{i}_{\text{pcc}}^c )$$
where $$F_c(s) = K_{pc} + K_{ic}/s$$. The current reference $$\mathbf{i}_{\text{ref}}^c$$ is generated by the power controller:
$$\mathbf{i}_{\text{ref}}^c = F_p(s)(P_{\text{ref}} – P^c) + j F_p(s)(Q_{\text{ref}} – Q^c) = F_p(s)(\mathbf{S}_{\text{ref}}^* – \mathbf{S}^{c*})$$
where $$\mathbf{S} = P + jQ$$ is the complex power and $$F_p(s) = K_{ps} + K_{is}/s$$.
The power in the control frame is calculated as:
$$\mathbf{S}^c = P^c + jQ^c = \frac{3}{2} \mathbf{u}_{\text{pcc}}^{c} \mathbf{i}_{\text{pcc}}^{c*}$$
Its small-signal perturbation is:
$$\Delta \mathbf{S}^c = \frac{3}{2} ( \mathbf{U}_0 \Delta \mathbf{i}_{\text{pcc}}^{c*} + \mathbf{I}_0^* \Delta \mathbf{u}_{\text{pcc}}^{c} )$$
The PLL’s dynamics are linearized as:
$$\Delta \theta = \frac{F_{\text{PLL}}(s)}{U_{d0}} \cdot \frac{\Delta u_{\text{pcc},q}^r}{s} = f_{\text{PLL}}(s) \cdot \text{Im}(\Delta \mathbf{u}_{\text{pcc}})$$
where $$F_{\text{PLL}}(s) = K_{pp} + K_{ip}/s$$ and $$U_{d0}$$ is the d-axis steady-state PCC voltage.
By combining all transformations, controller equations, and circuit equations, and after eliminating intermediate variables, we derive the fundamental relationship between the perturbation in grid current and the perturbations in power references. This process involves substituting the PLL relation, coordinate transformations, and the power circuit model into the controller equations. The final expression can be arranged as:
$$
\Delta \mathbf{i}_{\text{pcc}} = \mathbf{G}_c(s) \Delta \mathbf{S}_{\text{ref}}^* + \mathbf{G}_v(s) \Delta \mathbf{u}_{\text{pcc}} + j \mathbf{E}_0 \Delta \theta
$$
After further manipulation to eliminate $$\Delta \mathbf{u}_{\text{pcc}}$$ and $$\Delta \theta$$, we obtain a closed-form expression. From this, the individual d- and q-axis current perturbations as functions of the active and reactive power reference perturbations are extracted:
$$
\begin{aligned}
\Delta i_{\text{pcc},d} &= G_{d1}(s) \Delta P_{\text{ref}} + G_{d2}(s) \Delta Q_{\text{ref}} \\
\Delta i_{\text{pcc},q} &= G_{q1}(s) \Delta P_{\text{ref}} + G_{q2}(s) \Delta Q_{\text{ref}}
\end{aligned}
$$
The four transfer functions $$G_{d1}(s)$$, $$G_{d2}(s)$$, $$G_{q1}(s)$$, and $$G_{q2}(s)$$ share a common denominator polynomial. The stability of the on grid inverter system at any given operating point is determined by the roots of this denominator polynomial. If all poles are in the left-half of the s-plane, the on grid inverter is stable at that P-Q point.
The key achievement of this modeling step is that the coefficients of these closed-loop transfer functions are not constant; they are functions of the steady-state operating point variables $$P_0$$ and $$Q_0$$ (embedded within the steady-state vectors $$\mathbf{U}_0$$, $$\mathbf{I}_0$$, and $$\mathbf{E}_0$$). This allows us to assess stability across the continuum of operating points for the on grid inverter.
Characterizing the Stable Operating Region for the On Grid Inverter
Feasible Operating Range and Topological Constraints
Not every combination of (P, Q) is physically feasible for a given on grid inverter system due to the limits imposed by the power circuit parameters, grid voltage, and impedance. Before analyzing stability, we must identify the set of operating points that satisfy the steady-state power flow equations. This set is called the Topologically Constrained Region (TCR), denoted $$\sigma_{\text{tcr}}$$.
The steady-state relationships are:
$$ P_0 = 1.5 (u_{\text{pcc},d} i_{\text{pcc},d}) $$
$$ Q_0 = -1.5 (u_{\text{pcc},d} i_{\text{pcc},q}) $$
and
$$ u_{\text{pcc},d} = R_{\text{grid}} i_{\text{pcc},d} – \omega_0 L_{\text{grid}} i_{\text{pcc},q} + U_{g,d} $$
$$ u_{\text{pcc},q} = \omega_0 L_{\text{grid}} i_{\text{pcc},d} + R_{\text{grid}} i_{\text{pcc},q} + U_{g,q} $$
Assuming the grid voltage is aligned with the d-axis ($$U_{g,q}=0$$), we can combine these equations to eliminate the current components and obtain an equation in terms of $$u_{\text{pcc},d}$$:
$$ u_{\text{pcc},d}^4 + (2d_2 – U_g^2)u_{\text{pcc},d}^2 + (d_1^2 + d_2^2) = 0 $$
where
$$ d_1 = \omega_0 L_{\text{grid}} Q_0 – R_{\text{grid}} P_0 $$
$$ d_2 = -\omega_0 L_{\text{grid}} P_0 – R_{\text{grid}} Q_0 $$
A real positive solution for $$u_{\text{pcc},d}$$ exists only if the discriminant of this equation is non-negative. This condition defines the TCR for the on grid inverter.
Definition and Construction of the Stable Operating Region
The Stable Operating Region (SOR), denoted $$\sigma_{\text{sor}}$$, is defined as the subset of the Topologically Constrained Region $$\sigma_{\text{tcr}}$$ where the on grid inverter model exhibits small-signal stability. Conversely, the Unstable Operating Region (UOR), $$\sigma_{\text{uor}}$$, is the subset where at least one pole of the closed-loop system has a non-negative real part.
The step-by-step procedure to construct the SOR for the on grid inverter is as follows:
- Define the analysis range for active power ($$P_{\text{ref}}$$) and reactive power ($$Q_{\text{ref}}$$) and the resolution (step sizes $$\Delta P$$, $$\Delta Q$$).
- For each candidate operating point $$\Phi_{ij} = (P_i, Q_j)$$, check if it satisfies the topological constraint (lies within $$\sigma_{\text{tcr}}$$). If not, discard it.
- For each feasible operating point $$\Phi_{ij}$$, substitute the steady-state values $$P_0 = P_i$$ and $$Q_0 = Q_j$$ into the derived small-signal model to compute the coefficients of the closed-loop transfer functions.
- Calculate the poles of the system (roots of the common denominator polynomial).
- Classify the point $$\Phi_{ij}$$ as stable if all poles have negative real parts, and unstable otherwise.
- Collect all stable points to form $$\sigma_{\text{sor}}$$ and visualize the boundary between $$\sigma_{\text{sor}}$$ and $$\sigma_{\text{uor}}$$ in the P-Q plane.
This method provides a complete map of stability for the PQ-controlled on grid inverter across its entire capability curve.
Case Study: SOR under Rated Parameters
We demonstrate the method using a typical on grid inverter system. The key circuit and control parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | $$V_{dc}$$ | 800 V |
| Grid Voltage (RMS, line-to-line) | $$U_g$$ | 220 V |
| Filter Inductance | $$L_{\text{filt}}$$ | 2.5 mH |
| Grid Inductance | $$L_{\text{grid}}$$ | 7 mH |
| Grid Resistance | $$R_{\text{grid}}$$ | 0.01 Ω |
| Base Frequency | $$f_1$$ | 50 Hz |
| Control Loop | Parameter | Value |
|---|---|---|
| Phase-Locked Loop (PLL) | $$K_{pp}, K_{ip}$$ | 2.38, 869 |
| Power Loop (PI) | $$K_{ps}, K_{is}$$ | 0.0015, 1 |
| Current Loop (PI) | $$K_{pc}, K_{ic}$$ | 11, 4836 |
| Rated Power | $$P_N, Q_N$$ | 15 kW, 0 kvar |
With these parameters, the poles of the closed-loop system at the rated operating point (15 kW, 0 kvar) are all in the left-half plane, confirming nominal stability. We then sweep P from 0 to 22.5 kW and Q from 0 to 22.5 kvar. The resulting stable operating region is shown conceptually (the boundary is determined numerically). The SOR occupies a contiguous area in the P-Q plane near the origin. As power output increases beyond a certain boundary, the on grid inverter transitions into instability. The shape of this boundary is a key outcome of the analysis.
Parametric Analysis of the On Grid Inverter’s Stable Operating Region
The SOR framework is a powerful tool for sensitivity analysis. We investigate how the stable operating region of the on grid inverter changes with variations in external grid conditions and internal control parameters.
Impact of Grid Impedance
Grid strength, characterized by the Short-Circuit Ratio (SCR) or equivalently the grid inductance $$L_{\text{grid}}$$, significantly affects the stability of an on grid inverter. We analyze the SOR for different values of $$L_{\text{grid}}$$ and $$R_{\text{grid}}$$.
- Grid Inductance ($$L_{\text{grid}}$$): As $$L_{\text{grid}}$$ increases (weaker grid), the stable operating region $$\sigma_{\text{sor}}$$ shrinks considerably. The stability boundary moves towards lower power levels. This confirms that weak grid conditions exacerbate stability challenges for the PQ-controlled on grid inverter.
- Grid Resistance ($$R_{\text{grid}}$$): In contrast, an increase in $$R_{\text{grid}}$$ generally has a stabilizing effect or a less detrimental impact compared to inductance. The stable operating region may slightly expand or its boundary may shift to higher power levels as $$R_{\text{grid}}$$ increases. This is because resistance introduces damping, while inductance mainly contributes to phase lag that can undermine stability.
The three-dimensional plot of the SOR boundary as a function of P, Q, and $$L_{\text{grid}}$$ (or $$R_{\text{grid}}$$) visually demonstrates this relationship.
Impact of Grid Voltage
Grid voltage sags and swells are common disturbances. We analyze the SOR for grid voltages $$U_g$$ ranging from 0.85 pu to 1.15 pu. The results show:
- Voltage sag (e.g., from 1.0 pu to 0.9 pu) reduces the size of the stable operating region. Operating points that were stable at nominal voltage may become unstable during a sag.
- Voltage swell (e.g., from 1.0 pu to 1.1 pu) can enlarge the stable operating region, potentially stabilizing points that were marginally unstable at nominal voltage.
- The sensitivity of the SOR boundary to voltage changes appears relatively uniform across the analyzed range for the on grid inverter.
Impact of PLL Bandwidth
The PLL is a critical component whose bandwidth ($$f_{\text{bw}}$$) is typically tuned for a nominal condition. Its performance degrades at off-nominal operating points. We integrate the PLL bandwidth into our model by relating its PI gains to $$f_{\text{bw}}$$ and the damping ratio $$\zeta$$ (set to 0.707):
$$ \omega_n = U_{d0} \sqrt{K_{ip}} $$
$$ \zeta = \frac{K_{pp} U_{d0}}{2 \sqrt{K_{ip}}} $$
$$ f_{\text{bw}} = \frac{\omega_n}{2\pi} \sqrt{1 + 2\zeta^2 + \sqrt{1 + (2\zeta^2)^2}} $$
By varying $$f_{\text{bw}}$$ from 10 Hz to 80 Hz, we observe:
- The stable operating region $$\sigma_{\text{sor}}$$ contracts as the PLL bandwidth increases. A faster PLL can introduce negative damping interactions with other loops in the on grid inverter, especially under weak grid conditions.
- The sensitivity is more pronounced at higher output power levels. For example, at unity power factor, the stability limit might drop from 19 kW to 9 kW as bandwidth increases from 30 Hz to 70 Hz.
This analysis provides guidance for selecting a robust PLL bandwidth that ensures an adequate stable operating region for the expected power range of the on grid inverter.
Comparative Analysis: PQ vs. Current Controlled On Grid Inverter
It is instructive to compare the stability characteristics of a PQ-controlled on grid inverter with a simpler grid-following on grid inverter that uses direct current control (CC). The CC inverter has a similar structure but lacks the outer power loops; its current references are directly setpoints.
Using an equivalent modeling approach for the CC on grid inverter, we can construct its stable operating region under the same grid and circuit parameters. A comparative analysis reveals the following trends:
- Trend Consistency: Both types of on grid inverters exhibit similar qualitative trends: their SORs shrink with increasing grid inductance, shrink with decreasing grid voltage, and shrink with increasing PLL bandwidth.
- Quantitative Differences: The size and shape of the SOR differ.
- Under varying grid strength (SCR), the SOR boundaries for PQ and CC control are often close, with the PQ-controlled on grid inverter sometimes having a slightly larger region (e.g., +0.07 pu in stability limit at unit power factor).
- At higher grid voltages (e.g., 1.15 pu), the PQ-controlled on grid inverter can have a significantly larger SOR (e.g., stability limit of 1.47 pu) compared to the CC inverter (e.g., 1.20 pu).
- At lower PLL bandwidths (e.g., 25 Hz), the current-controlled on grid inverter may exhibit a larger SOR (e.g., 1.67 pu limit) than the PQ-controlled one (e.g., 1.47 pu). At higher bandwidths, their regions converge.
This comparison underscores that the addition of the power control outer loops modifies the dynamic interactions within the on grid inverter, altering its stability margins in a non-trivial way that depends on the specific operating condition and parameter set.
Validation and Discussion
The proposed stable operating region analysis method for the on grid inverter was validated through comprehensive time-domain simulations in MATLAB/Simulink and experimental tests on a down-scaled laboratory prototype. The validation confirmed the accuracy of the predicted stability boundary.
Simulation Results: Multiple test points were chosen from the stable, boundary, and unstable regions predicted by the SOR analysis for the on grid inverter. Simulations confirmed that:
- Points inside $$\sigma_{\text{sor}}$$ (e.g., 14 kW, 5 kvar) resulted in stable, low-THD current waveforms.
- Points on the predicted boundary showed increased oscillatory content but maintained stability.
- Points inside $$\sigma_{\text{uor}}$$ (e.g., 19 kW, 6 kvar) led to growing oscillations in current and power, confirming instability.
Further simulations validated the parametric studies. For instance, increasing $$L_{\text{grid}}$$ moved a previously stable point into instability, while increasing $$U_g$$ or decreasing $$f_{\text{bw}}$$ could stabilize a previously unstable operating point for the on grid inverter.
Experimental Results: A hardware prototype of the PQ-controlled on grid inverter was built. Experiments involving step changes in power reference across the predicted SOR boundary successfully reproduced the stability/instability transitions. The experimentally observed stability limits aligned well with those predicted by the theoretical SOR analysis, thereby validating the practical relevance of the method for real on grid inverter systems.
Conclusion
This work presents a novel stability analysis framework for PQ-controlled on grid inverters based on the concept of a Stable Operating Region (SOR). The core contributions are:
- Comprehensive Modeling: We developed a detailed small-signal model for the on grid inverter that embeds the operating point (P, Q) variables and accounts for the multi-loop, multi-timescale, and coupled dynamics of the PLL, power control, and current regulation. This model enables stability evaluation at any feasible operating condition without repetitive point-wise modeling.
- Stable Operating Region Characterization: By performing a numerical sweep over the P-Q plane and computing closed-loop poles at each point, we introduced a method to graphically demarcate the stable operating region of the on grid inverter. This provides an intuitive and complete picture of system stability across its entire operational range.
- Parametric Sensitivity and Comparison: Using the SOR as a tool, we systematically analyzed the impact of grid impedance, grid voltage, and PLL bandwidth on the stability of the on grid inverter. We further compared the SOR of a PQ-controlled on grid inverter with that of a current-controlled one, highlighting control-strategy-dependent stability characteristics.
The key findings are that the stable operating region for an on grid inverter shrinks with a weaker grid (higher inductance), expands with higher grid resistance, shrinks during voltage sags, and shrinks with increasing PLL bandwidth. The sensitivity to grid voltage is relatively uniform, while sensitivity to PLL bandwidth is more acute at higher power outputs.
The proposed SOR-based method offers a significant advancement over traditional point-analysis techniques for assessing the stability of on grid inverters in modern, variable power systems. It provides system designers and operators with a powerful visual tool to ensure robust operation of the on grid inverter across all anticipated working conditions, thereby enhancing the reliability of renewable energy integration.