The pervasive integration of renewable energy sources, primarily through power electronic converters, has fundamentally altered the dynamics of modern power systems. As a researcher deeply involved in this field, I observe that large-scale renewable energy bases present inherent oscillation risks, where broadband instability events can occur, posing significant challenges to stable grid operation. In analyzing these challenges, impedance-based analysis has proven to be a highly effective method for studying the broadband characteristics of utility interactive inverters, offering clear insights into the mechanisms behind instability. This article details a structured methodological approach for modeling these systems, from the fundamental single-inverter model to the aggregated model of multiple units operating in parallel over unequal line lengths.
The core of our investigation begins with the fundamental building block: a single three-phase, two-level, constant-current controlled utility interactive inverter with an L-type filter. The primary circuit and control structure form the basis for our modeling effort. The controller employs a standard dq-frame current control scheme with PI regulators, and its synchronization to the grid is achieved through a Phase-Locked Loop (PLL). The inherent nonlinearity introduced by the PLL and the coordinate transformation processes necessitates a small-signal linearization approach to derive a tractable and accurate impedance model.
1. Impedance Modeling Foundation for a Single Utility Interactive Inverter
The first critical step is to linearize the system’s nonlinear components. We begin with the coordinate transformation operator. The modulation voltage transformation from the dq-frame to the stationary αβ-frame is given by \( \mathbf{m_{\alpha\beta}}(t) = e^{j\theta_p(t)} \mathbf{m_{dq}}(t) \). Introducing small perturbations around the steady-state operating point (denoted by subscript ‘0’) and neglecting second-order terms leads to the linearized relationship in the frequency domain:
$$ \Delta \mathbf{m’_{dq}}(s) = \Delta \mathbf{m_{dq}}(s) + j \mathbf{m_{dq0}} \Delta \theta_p(s) $$
where \( \Delta \mathbf{m’_{dq}}(s) = e^{-j\theta_0} \Delta \mathbf{m_{\alpha\beta}}(s) \). Similarly, for the output current transformation \( \mathbf{i_{odq}}(t) = e^{-j\theta_p(t)} \mathbf{i_{o\alpha\beta}}(t) \), the linearized form is:
$$ \Delta \mathbf{i_{odq}}(s) = \Delta \mathbf{i’_{odq}}(s) – j \mathbf{i_{odq0}} \Delta \theta_p(s) $$
where \( \Delta \mathbf{i’_{odq}}(s) = e^{-j\theta_0} \Delta \mathbf{i_{o\alpha\beta}}(s) \).
The PLL, being a critical source of frequency coupling, must also be linearized. The linearized model relates the perturbation in the PLL output angle to the perturbation in the input voltage in the dq-frame:
$$ \Delta \theta_p(s) = H(s) [ \Delta \mathbf{u’_{odq}}(s) – \Delta \mathbf{u’_{odq}}^*(s) ] $$
where the transfer function \( H(s) \) is defined as:
$$ H(s) = \frac{G_{PLL}(s)}{2j[s + G_{PLL}(s) u_{od0}]} \quad \text{and} \quad G_{PLL}(s) = K_{p,PLL} + \frac{K_{i,PLL}}{s} $$
The asterisk (*) denotes the complex conjugate, highlighting the coupling between positive- and negative-sequence components introduced by the PLL.
By integrating these linearized models with the plant dynamics (filter inductor \(L_f\), controller \(G_c(s)\), and computation delay \(G_d(s)\)), we construct the complete small-signal block diagram of the utility interactive inverter in the dq-frame. From this diagram, we derive the relationship between the perturbed output current and the perturbed terminal voltage in the dq-frame:
$$
\begin{bmatrix}
\Delta \mathbf{i’_{odq}}(s) \\[6pt]
\Delta \mathbf{i’_{odq}}^*(s)
\end{bmatrix}
=
\begin{bmatrix}
G_{1,dq}(s) & G_{2,dq}(s) \\[6pt]
G_{2,dq}^*(s) & G_{1,dq}^*(s)
\end{bmatrix}
\begin{bmatrix}
\Delta \mathbf{u’_{odq}}(s) \\[6pt]
\Delta \mathbf{u’_{odq}}^*(s)
\end{bmatrix}
$$
The elements \(G_{1,dq}(s)\) and \(G_{2,dq}(s)\) are complex transfer functions that incorporate all system parameters, including the PLL effects. This 2×2 matrix is the admittance matrix in the dq-frame. To obtain a model with clearer physical meaning at the point of common coupling (PCC), we shift the frequency reference back to the stationary αβ-frame. This yields the final admittance matrix model for a single utility interactive inverter:
$$
\begin{bmatrix}
\Delta \mathbf{i_{o\alpha\beta}}(s) \\[6pt]
\Delta \mathbf{i_{o\alpha\beta}}(s-j2\omega_0)
\end{bmatrix}
=
\begin{bmatrix}
Y_{11}(s) & Y_{12}(s) \\[6pt]
Y_{21}(s) & Y_{22}(s)
\end{bmatrix}
\begin{bmatrix}
\Delta \mathbf{u_{o\alpha\beta}}(s) \\[6pt]
\Delta \mathbf{u_{o\alpha\beta}}(s-j2\omega_0)
\end{bmatrix}
$$
where \( \omega_0 \) is the fundamental grid frequency. The off-diagonal terms \(Y_{12}(s)\) and \(Y_{21}(s)\) are non-zero precisely due to the PLL, demonstrating the frequency-coupling characteristics. The corresponding impedance matrix \( \mathbf{Z_{inv}} \) is the inverse of this admittance matrix.
| Matrix Element | Physical Meaning | Primary Influencing Factors |
|---|---|---|
| \(Y_{11}(s)\) | Direct admittance from voltage at frequency \(\omega\) to current at \(\omega\). | Current controller, filter, delay. Dominant at medium/high frequencies. |
| \(Y_{22}(s)\) | Direct admittance from voltage at frequency \(2\omega_0-\omega\) to current at \(2\omega_0-\omega\). | Same as \(Y_{11}\), but frequency-shifted. |
| \(Y_{12}(s)\), \(Y_{21}(s)\) | Coupling admittances linking positive- and negative-sequence frequencies. | PLL dynamics and steady-state operating point (\(m_{dq0}\), \(i_{dq0}\)). Dominant at low frequencies near PLL bandwidth. |
2. Broadband Aggregation Modeling for Multiple Parallel Utility Interactive Inverters
In practical renewable power plants, hundreds of utility interactive inverters operate in parallel. A critical challenge is to develop an accurate aggregated model that represents the collective broadband behavior of this cluster at the PCC. The aggregation is not a simple parallel addition of individual inverter impedances when line impedances are considered.
2.1. Aggregation Without Line Impedances
In the ideal case where all inverters are connected to the PCC with negligible cable impedance, the aggregation is straightforward. The total current injected at the PCC is the sum of the currents from all \(N\) inverters. Since the PCC voltage \( \mathbf{V} \) is common to all, the aggregated admittance matrix \( \mathbf{Y_{agg}} \) is simply the sum of the individual admittance matrices \( \mathbf{Y_{inv,k}} \):
$$ \mathbf{Y_{agg}} = \sum_{k=1}^{N} \mathbf{Y_{inv,k}} $$
This holds true for both the direct and coupling elements of the matrix.
2.2. Aggregation With Unequal Line Impedances
The realistic scenario involves each utility interactive inverter connected to the PCC via a cable with impedance \( \mathbf{Z_{line,k}} = sL_{k} \) (considering inductive impedance for simplicity). The terminal voltage \( \mathbf{V_{k}} \) at the k-th inverter’s output is no longer equal to the PCC voltage \( \mathbf{V_{pcc}} \). The inverter’s admittance matrix \( \mathbf{Y_{inv,k}} \) is initially defined with respect to its own terminal voltage \( \mathbf{V_{k}} \), which has a local phase reference defined by its own PLL measurement. To aggregate them at a common global reference (the PCC voltage phase), two steps are required.
First, the inverter’s admittance must be transformed to the global dq-reference frame defined by the PCC voltage phase angle. This involves a rotation transformation \( \mathbf{T_k} = \text{diag}(e^{-j\delta_k}, e^{j\delta_k}) \), where \( \delta_k \) is the steady-state phase angle difference between the inverter terminal voltage and the PCC voltage. The transformed admittance \( \mathbf{Y_{pcc,k}} \) is:
$$ \mathbf{Y_{pcc,k}} = \mathbf{T_k^{-1}} \mathbf{Y_{inv,k}} \mathbf{T_k} $$
Second, the total impedance of the k-th branch, including its line, is the sum of the inverter’s output impedance and the line impedance: \( \mathbf{Z_{branch,k}} = \mathbf{Y_{pcc,k}^{-1}} + \mathbf{Z_{line,k}} \). Finally, the aggregated admittance at the PCC of the entire multi-inverter system is the parallel combination of all branch impedances:
$$ \mathbf{Y_{agg}} = \left( \sum_{k=1}^{N} \mathbf{Z_{branch,k}^{-1}} \right) = \sum_{k=1}^{N} \left( \mathbf{Y_{pcc,k}^{-1}} + \mathbf{Z_{line,k}} \right)^{-1} $$
Neglecting the transformation step \( \mathbf{T_k} \) or the line impedance \( \mathbf{Z_{line,k}} \) can lead to significant errors in the aggregated model, particularly in the coupling terms, as will be demonstrated later.
| Scenario | Assumption | Aggregated Admittance Formula | Accuracy |
|---|---|---|---|
| Ideal (No Lines) | Zero line impedance, common voltage reference. | \( \mathbf{Y_{agg}} = \sum \mathbf{Y_{inv,k}} \) | Accurate only for co-located inverters. |
| Realistic (With Lines) | Non-zero, unequal line impedances \(L_k\). | \( \mathbf{Y_{agg}} = \sum \left( (\mathbf{T_k^{-1}} \mathbf{Y_{inv,k}} \mathbf{T_k})^{-1} + \mathbf{Z_{line,k}} \right)^{-1} \) | Accurate representation of practical plant layout. |
| Incorrect Simplified | Includes lines but ignores reference frame transformation (\( \mathbf{T_k} \)). | \( \mathbf{Y_{agg}} = \sum \left( \mathbf{Y_{inv,k}^{-1}} + \mathbf{Z_{line,k}} \right)^{-1} \) | Inaccurate, especially for coupling admittance at low/medium frequencies. |
3. Admittance Measurement Method Considering Frequency Coupling
To validate the accuracy of the derived analytical models, a precise measurement technique is essential. Standard single-frequency injection methods fail to capture the frequency-coupling effects inherent in a utility interactive inverter. Therefore, a dual-frequency injection scheme is employed. The procedure involves two sequential perturbation tests at each frequency point of interest, \( \omega_p \).
Test 1: Inject a balanced three-phase voltage perturbation \( \mathbf{u_{s1}} \) at frequency \( \omega_p \) at the PCC. Measure the steady-state PCC voltage and current. Using FFT, extract the spectral components at the injection frequency \( \omega_p \) and its coupled frequency \( 2\omega_0 – \omega_p \). This yields the vectors \( \mathbf{u_{s1}}(\omega_p) \), \( \mathbf{i_{s1}}(\omega_p) \), \( \mathbf{u_{s1}}(2\omega_0-\omega_p) \), and \( \mathbf{i_{s1}}(2\omega_0-\omega_p) \).
Test 2: Inject a balanced three-phase voltage perturbation \( \mathbf{u_{s2}} \) at the coupled frequency \( 2\omega_0 – \omega_p \) (with the same magnitude and phase as in Test 1). Again, measure and extract the spectral components at both \( \omega_p \) and \( 2\omega_0 – \omega_p \), obtaining \( \mathbf{u_{s2}}(\omega_p) \), \( \mathbf{i_{s2}}(\omega_p) \), \( \mathbf{u_{s2}}(2\omega_0-\omega_p) \), and \( \mathbf{i_{s2}}(2\omega_0-\omega_p) \).
From these two tests, we have a system of equations at each frequency:
$$
\begin{bmatrix}
\mathbf{i}(\omega_p)_1 & \mathbf{i}(\omega_p)_2 \\[6pt]
\mathbf{i}^*(2\omega_0-\omega_p)_1 & \mathbf{i}^*(2\omega_0-\omega_p)_2
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\[6pt]
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
\mathbf{u}(\omega_p)_1 & \mathbf{u}(\omega_p)_2 \\[6pt]
\mathbf{u}^*(2\omega_0-\omega_p)_1 & \mathbf{u}^*(2\omega_0-\omega_p)_2
\end{bmatrix}
$$
The subscripts 1 and 2 denote measurements from Test 1 and Test 2, respectively. The complete 2×2 admittance matrix \( \mathbf{Y} \) is then solved by inverting the matrix of measured voltage vectors:
$$
\mathbf{Y} =
\begin{bmatrix}
\mathbf{i}(\omega_p)_1 & \mathbf{i}(\omega_p)_2 \\[6pt]
\mathbf{i}^*(2\omega_0-\omega_p)_1 & \mathbf{i}^*(2\omega_0-\omega_p)_2
\end{bmatrix}
\begin{bmatrix}
\mathbf{u}(\omega_p)_1 & \mathbf{u}(\omega_p)_2 \\[6pt]
\mathbf{u}^*(2\omega_0-\omega_p)_1 & \mathbf{u}^*(2\omega_0-\omega_p)_2
\end{bmatrix}^{-1}
$$
This method accurately captures the frequency-coupling characteristics and serves as the “ground truth” for model validation.
4. Simulation Verification of Aggregated Models
To substantiate the proposed aggregation methodology, time-domain simulations of two parallel utility interactive inverters were conducted. The system parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link Voltage | \(V_{dc}\) | 750 V |
| Grid Line Voltage (RMS) | \(U_g\) | 415 V |
| Filter Inductor | \(L_f\) | 10 mH |
| Switching/Sampling Frequency | \(f_s\) | 10 kHz |
| Grid Inductance | \(L_g\) | 0.03 mH |
| Current PI Controller (\(K_p, K_i\)) | \(G_c(s)\) | 10, 1000 |
| PLL PI Controller (\(K_{p,PLL}, K_{i,PLL}\)) | \(G_{PLL}(s)\) | 1.08, 99.75 |
| Reference Current (\(i_d, i_q\)) | \(i_{dref}, i_{qref}\) | 10 A, 0 A |
4.1. Case 1: Two Identical Inverters Without Line Impedances
In this baseline case, both inverters are connected directly to the PCC. The aggregated analytical model is \( \mathbf{Y_{agg}} = \mathbf{Y_{inv1}} + \mathbf{Y_{inv2}} = 2 \cdot \mathbf{Y_{inv}} \). The simulated admittance measurement via the dual-frequency sweep method shows excellent agreement with the analytical model across all four elements of the admittance matrix, validating the accuracy of the single-inverter model and the simple summation rule for this ideal case.
4.2. Case 2: Two Identical Inverters With Unequal Line Impedances
This is the central case demonstrating the need for the full aggregation method. Inverter 1 is connected via a 10 mH line inductor, and Inverter 2 via a 5 mH line inductor. The aggregated model is now calculated using the correct formula:
$$ \mathbf{Y_{agg}} = \left( (\mathbf{T_1^{-1}} \mathbf{Y_{inv}} \mathbf{T_1})^{-1} + sL_1 \right)^{-1} + \left( (\mathbf{T_2^{-1}} \mathbf{Y_{inv}} \mathbf{T_2})^{-1} + sL_2 \right)^{-1} $$
The simulation results confirm a perfect match between this analytical model and the measured admittance from the simulated sweep test. This validates the proposed transformation and aggregation procedure for the realistic scenario of multiple utility interactive inverters connected via unequal paths.
4.3. Case 3: Highlighting the Error from an Incomplete Model
To emphasize the importance of the reference frame transformation \( \mathbf{T_k} \), an incomplete aggregated model is calculated by simply adding the line impedances to the inverter impedances without the transformation:
$$ \mathbf{Y_{agg,incorrect}} = \left( \mathbf{Y_{inv}^{-1}} + sL_1 \right)^{-1} + \left( \mathbf{Y_{inv}^{-1}} + sL_2 \right)^{-1} $$
Comparing this model with the simulation measurements reveals significant discrepancies, particularly in the phase of the coupling admittance terms (\(Y_{12}, Y_{21}\)) at medium frequencies. This error could lead to mistaken conclusions about system stability if used in an impedance-based analysis. The results underscore that accurate broadband aggregation modeling for parallel utility interactive inverters must account for both the line impedance and the proper alignment of each inverter’s model to a common reference frame.
5. Conclusion
In this work, I have presented a comprehensive methodology for broadband aggregation modeling of parallel-operated utility interactive inverters. The process begins with deriving a precise frequency-coupled admittance model for a single inverter, explicitly incorporating the effects of the current controller and the PLL. This model accurately captures the negative-resistance behavior at low frequencies and the coupling between positive- and negative-sequence harmonics.
The core contribution lies in extending this to the multi-inverter case. For the practical scenario where inverters are connected via unequal cable impedances, I have demonstrated that the aggregated admittance at the PCC is not a simple parallel sum of individual inverter admittances. A critical two-step process is required: first, transforming each inverter’s admittance to a common global reference frame defined by the PCC voltage; second, summing the branch admittances that include both the transformed inverter output impedance and the series line impedance.
The proposed dual-frequency injection measurement technique provides a reliable means to validate the derived models. Simulation studies confirm the high accuracy of the full aggregation method and clearly illustrate the errors introduced by neglecting the necessary coordinate transformations. The developed modeling framework provides a solid foundation for impedance-based stability analysis of large-scale renewable power plants, enabling the effective assessment and mitigation of broadband oscillation risks associated with clusters of utility interactive inverters.