In modern power systems with high penetration of renewable energy, the grid tied inverter serves as the crucial interface between distributed generation resources and the AC grid. The proliferation of these power electronic converters introduces new stability challenges, among which the frequency coupling phenomenon has garnered significant attention. This effect manifests when a harmonic disturbance at one frequency not only excites a current response at the same frequency but also induces a coupled response at a mirror-symmetrical frequency about the fundamental. This behavior transforms the system from a simple single-input single-output (SISO) characteristic into a complex multiple-input multiple-output (MIMO) system, dramatically complicating impedance-based stability analysis.
The core origin of frequency coupling in a three-phase grid tied inverter lies in the inherent asymmetry of its control structure within the synchronous rotating (dq) reference frame. Traditional controllers, such as the Phase-Locked Loop (PLL) and the DC-voltage control loop, often employ independent PI regulators on the d- and q-axes. This structural asymmetry breaks the symmetry between positive- and negative-sequence dynamics, leading to significant off-diagonal coupling terms in the inverter’s sequence-domain admittance matrix. Consequently, a positive-sequence voltage perturbation at frequency $f_p$ will generate both a positive-sequence current at $f_p$ and a negative-sequence current at $f_n = 2f_1 – f_p$, where $f_1$ is the fundamental grid frequency. This coupling undermines the accuracy of the simpler SISO impedance model and necessitates the use of more intricate MIMO stability criteria, such as the generalized Nyquist criterion, for reliable analysis.
The primary control objective for suppressing this frequency coupling is to achieve a “reverse change” between the diagonal and off-diagonal subsystems of the admittance matrix. Ideally, this means increasing (or at least maintaining) the magnitude of the direct-path diagonal terms (e.g., $Y_{pp}$, $Y_{nn}$) while simultaneously and substantially decreasing the magnitude of the cross-coupling off-diagonal terms (e.g., $Y_{pn}$, $Y_{np}$). When the coupling terms become negligible compared to the diagonal terms, the MIMO system effectively decouples into an SISO system, greatly simplifying modeling and stability assessment. However, due to the inherent strong coupling, conventional control modifications often lead to changes in the same direction for both diagonal and off-diagonal terms, failing to achieve the desired decoupling.
Sequence Admittance Modeling of the Grid Tied Inverter with Multiple Coupling Factors
To accurately analyze the frequency coupling phenomenon, a detailed sequence admittance model of the grid tied inverter is essential. This model establishes the relationship between harmonic voltage disturbances at the Point of Common Coupling (PCC) and the resulting harmonic current injections from the inverter. The system typically consists of a three-phase voltage-source converter with an L-type filter, controlled by a nested loop structure featuring a DC-voltage outer loop and a current inner loop, synchronized to the grid via a PLL. Applying the harmonic linearization method under the assumption of a small disturbance, the frequency-domain expressions for the PCC voltage and the grid current can be represented as sums of their fundamental and perturbed positive- and negative-sequence components.
The modeling process must account for the disturbance propagation through three key asymmetric control links: the DC-voltage loop, the current loop, and the PLL. The derived sequence admittance matrix $\mathbf{Y}_{inv}$ for the grid tied inverter takes the following MIMO form, relating the positive- and negative-sequence current perturbations ($I_p$, $I_n$) to the corresponding voltage perturbations ($V_p$, $V_n$):
$$
\begin{bmatrix}
I_p \\
I_n
\end{bmatrix}
=
\begin{bmatrix}
Y_{pp}(f) & Y_{pn}(f) \\
Y_{np}(f) & Y_{nn}(f)
\end{bmatrix}
\begin{bmatrix}
V_p \\
V_n
\end{bmatrix}
= \mathbf{Y}_{inv}(f) \begin{bmatrix}
V_p \\ V_n
\end{bmatrix}
$$
Each element $Y_{ij}(s)$ is a transfer function. The diagonal terms $Y_{pp}$ and $Y_{nn}$ represent the direct admittance paths, while the off-diagonal terms $Y_{pn}$ and $Y_{np}$ quantify the frequency coupling strength. A key metric for the severity of coupling is the relative magnitude ratio between the off-diagonal and diagonal terms. When $|Y_{pn}|, |Y_{np}| \ll |Y_{pp}|, |Y_{nn}|$, the coupling is weak, and the system can be treated as decoupled. Analysis of the Bode plots for this model reveals that in a conventional control setup, the magnitudes of $Y_{pn}$ and $Y_{np}$ are comparable to those of $Y_{pp}$ and $Y_{nn}$ across a wide frequency range, confirming strong frequency coupling. Furthermore, isolating the contribution of each control loop indicates that the asymmetry in the DC-voltage control loop is a predominant factor inducing this coupling effect.
Theoretical Foundation: Achieving Reverse Change via Opposite Signs in Frequency-Domain Convolution
The central challenge is to devise a control strategy that systematically increases the diagonal admittance terms while decreasing the off-diagonal terms. The proposed solution is founded on a novel principle that leverages the intrinsic sign differences present in the frequency-domain representation of signals and operations within the dq-frame. The realization of this “reverse change” is decomposed into two sequential steps, each exploiting a specific sign reversal mechanism.
Step 1: Reverse Change within the Same Row of the Admittance Matrix. To alter the two elements in a single row of the coupling matrix in opposite directions, the introduced control variable must exhibit opposite signs for its positive- and negative-sequence perturbation components. Analyzing the dq-frame components under harmonic perturbation reveals this property exists for the q-axis variables. The q-axis components of PCC voltage ($v_q$) and grid current ($i_q$) can be expressed as:
$$
\begin{aligned}
V_{q}[ \pm(f_p – f_1)] &= \mp j V_p \pm j V_n \\
I_{q}[ \pm(f_p – f_1)] &= \mp j I_p \pm j I_n
\end{aligned}
$$
Clearly, the coefficients for $V_p$ and $V_n$ (and similarly for $I_p$ and $I_n$) have opposite signs ($\mp j$ vs. $\pm j$). In contrast, the d-axis components have coefficients with the same sign. Therefore, using $v_q$ and $i_q$ as feedback variables provides the necessary opposite-sign property to affect elements in the same row inversely.
Step 2: Reverse Change within the Same Column via q-axis Convolution. To subsequently achieve a reverse change between the two rows (i.e., between diagonal and off-diagonal terms), the control effect must be injected into the q-axis channel of the modulation signal. The disturbance in the q-axis modulation signal $M_q[\pm(f_p-f_1)]$ propagates to the inverter’s output voltage via convolution with the PLL’s sine term, $\sin\theta_{PLL}[\pm f_1]$. The critical observation is that $\sin\theta_{PLL}[+f_1] = -0.5j$ and $\sin\theta_{PLL}[-f_1] = +0.5j$, which are opposite in sign. This means the same q-axis modulation disturbance $M_q[+(f_p-f_1)] contributes to the first-row voltage via multiplication by $-0.5j$ and to the second-row voltage via multiplication by $+0.5j$. This sign reversal in the convolution process, inherent to the q-axis injection path, enables the reverse change between rows. Injecting the signal into the d-axis would involve convolution with $\cos\theta_{PLL}$, whose values at $\pm f_1$ are both $+0.5$, lacking this crucial sign reversal property.
By combining these two steps—using q-axis variables ($v_q$, $i_q$) to achieve row-wise reverse change and injecting them via the q-axis path to achieve column-wise reverse change through convolutional sign difference—the overall control strategy can effectively realize the “reverse change” between the diagonal and off-diagonal subsystems of the grid tied inverter’s admittance matrix.
Proposed Improved Voltage Control Method for Frequency Coupling Suppression
Based on the theoretical principle outlined above, a specific improved voltage control method is designed. The core of the method is to apply a symmetrical compensation to the DC-voltage control loop by introducing feedback from both d- and q-axis voltage and current components into the q-axis reference channel. This structure is depicted in the following control block diagram augmentation. The standard DC-voltage controller output, which primarily sets the d-axis current reference $I_{dr}$, is supplemented by a q-axis compensation signal.
The compensation signals are derived as follows: to handle the “cosine” and “constant” terms related to the power factor, the q-axis components $v_q$ and $i_q$ are processed through gains mirroring their coefficients in the admittance model. To handle the “sine” terms that become significant at non-unity power factor, the d-axis components $v_d$ and $i_d$ are processed through corresponding gains. The gains are designed based on the steady-state operating point parameters (e.g., fundamental current magnitude $I_1$, power factor angle $\phi_{i1}$, fundamental voltage $V_1$, filter inductance $L_f$). The mathematical formulation of the added compensation signal $\Delta I_{qr}^{comp}$ can be summarized as:
$$
\Delta I_{qr}^{comp} = G_{dc}(s) \left[ K_1 \cdot v_q + K_2 \cdot i_q + K_3 \cdot v_d + K_4 \cdot i_d \right]
$$
Where $G_{dc}(s)$ is the DC-voltage PI controller, and $K_1$ to $K_4$ are gain coefficients designed based on the steady-state operating point and the inverter parameters to precisely counteract the asymmetric terms originating from the standard DC-voltage loop.
The impact of this control method on the DC-voltage loop-related sub-matrix within the full admittance model is profound. If we denote the original coupling terms contributed by the DC-voltage loop as $F_pC_{vn}$ and $F_nC_{vp}$, the proposed method adds compensatory terms $-X$ and $-Y$ respectively. Since the signs of $X$ and $Y$ are opposite to those of the original terms, the net effect is a reduction in the magnitude of the off-diagonal coupling terms. Simultaneously, for the diagonal terms $F_pC_{vp}$ and $F_nC_{vn}$, the added compensation has the same sign as the original terms, leading to a slight increase or maintenance of their magnitude. This quantitatively confirms the achievement of the “reverse change.”
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| DC-link Voltage | $V_{dc0}$ | 700 | V |
| DC-link Capacitance | $C_{dc}$ | 7 | mF |
| Grid Filter Inductance | $L_f$ | 7 | mH |
| Fundamental Frequency | $f_1$ | 50 | Hz |
| Rated Grid Voltage (phase) | $V_1$ | 311 | V |
| DC Current | $I_{pv}$ | 75 | A |
| PLL Proportional Gain | $k_{p,PLL}$ | 0.0158 | – |
| DC-Voltage PI: $k_p$ | $k_{p,dc}$ | 2.23 | – |
| DC-Voltage PI: $k_i$ | $k_{i,dc}$ | 280.29 | – |
Analysis of Frequency Coupling Suppression and System Stability
Applying the proposed control method fundamentally alters the admittance characteristics of the grid tied inverter. Bode plot analysis for various power factors ($\lambda$) demonstrates the method’s effectiveness. For the diagonal terms $Y_{pp}$ and $Y_{nn}$, the magnitude shows a slight increase in the low-frequency region (below ~30 Hz) and remains virtually unchanged at higher frequencies compared to the conventional control. Crucially, for the off-diagonal coupling terms $Y_{pn}$ and $Y_{np}$, the magnitude is dramatically reduced across the entire frequency spectrum for all tested power factors. This significant attenuation of the coupling terms relative to the diagonal terms confirms successful decoupling. The grid tied inverter’s model can now be accurately approximated as an SISO system, where $Y_{pn} \approx 0$ and $Y_{np} \approx 0$.
This decoupling directly simplifies stability analysis. For an SISO system, the impedance-based stability can be assessed using the simple ratio of the grid impedance $Z_g(s)$ to the inverter output impedance $Z_{eq}(s) \approx 1/Y_{pp}(s)$, evaluated via Nyquist or Bode plots of $Z_g / Z_{eq}$. When the system is strongly coupled (MIMO), the equivalent output impedance $Z_{eq, MIMO}(s)$ is a more complex function involving all four admittance terms:
$$
Z_{eq, MIMO}(s) = \frac{1 + Y_{nn}(s) Z_g(s)}{Y_{pp}(s) + [Y_{pp}(s)Y_{nn}(s) – Y_{pn}(s)Y_{np}(s)] Z_g(s)}
$$
With the proposed method active, since $Y_{pn}(s) \approx 0$ and $Y_{np}(s) \approx 0$, the expression collapses to $Z_{eq, SISO}(s) = 1 / Y_{pp}(s)$. Comparative Bode plots of $Z_{eq, MIMO}$ and $Z_{eq, SISO}$ under the proposed control show perfect overlap, validating the SISO approximation. Stability analysis for a weak grid condition (Short-Circuit Ratio, SCR = 1.47) reveals that while the conventional control leads to instability (unmatched encirclements in the Nyquist criterion), the system with the proposed improved voltage control remains stable, demonstrating its stabilizing effect.
Experimental Validation and Robustness Assessment
The performance of the proposed frequency coupling suppression method was rigorously validated using a Hardware-in-the-Loop (HIL) experimental platform based on RT-LAB OP5707 and a DSP F28379D controller, implementing the three-phase grid tied inverter system with parameters as listed in Table 1.
Experiment 1: Frequency Coupling under Conventional Control. With positive-sequence voltage disturbances injected at 80 Hz and 90 Hz (10% of fundamental amplitude) under a stiff grid, the grid current spectrum showed significant coupled components at the mirror frequencies 20 Hz and 10 Hz, with magnitudes (3.73% and 3.25%) comparable to the disturbance frequencies themselves (4.16% and 3.69%), confirming strong coupling. Switching to a weak grid (SCR=1.47) induced instability.
Experiment 2: Comparison with Symmetrical PLL (SPLL) Method. Under the same conditions, applying a state-of-the-art Symmetrical PLL control method increased the disturbance currents (80Hz: 4.87%, 90Hz: 4.01%) and reduced the coupled currents (20Hz: 0.82%, 10Hz: 1.01%), showing its partial decoupling capability. However, the system remained unstable when switched to the weak grid, indicating limited stability enhancement.
Experiment 3: Performance of the Proposed Method. With the proposed improved voltage control activated, the disturbance currents saw a modest increase (80Hz: 5.25%, 90Hz: 4.55%), while the coupled currents were suppressed to nearly negligible levels (20Hz: 0.02%, 10Hz: 0.03%). This dramatic reduction in coupling harmonics, far superior to the SPLL method, validates the core “reverse change” principle. Furthermore, the system remained perfectly stable when transitioned to the SCR=1.47 weak grid, proving its dual benefit of decoupling and stability enhancement.
Experiment 4: Robustness to Parameter Variation. To test robustness, key parameters (grid voltage, DC voltage, power factor) were changed significantly. Without re-tuning the proposed controller’s compensation gains, the method maintained excellent performance: coupled harmonics remained below 0.03% in both stiff and weak grid conditions, and stability was preserved. This demonstrates the method’s strong robustness and its applicability across different operating points for the grid tied inverter.
Conclusion
This paper has addressed the critical issue of frequency coupling in grid tied inverters, which arises primarily from asymmetric control structures like the DC-voltage loop. A novel control design principle was established, based on strategically exploiting the opposite signs inherent in the frequency-domain representation of q-axis variables and the convolution operation in the q-axis modulation path. This principle enables the systematic realization of a “reverse change” between the diagonal and off-diagonal terms of the inverter’s sequence admittance matrix.
Guided by this principle, an improved voltage control method was developed, featuring a symmetrical q-axis compensation within the DC-voltage loop. The method successfully decouples the positive- and negative-sequence dynamics of the grid tied inverter by significantly attenuating the cross-coupling admittance terms while slightly increasing or maintaining the direct-path terms. This effectively reduces the system order from a complex MIMO model to a simpler, SISO model, greatly facilitating stability analysis. Comprehensive theoretical analysis and HIL experimental results confirm the method’s superior effectiveness in suppressing frequency coupling, its ability to enhance stability in weak grids, and its robustness against parameter variations. The proposed approach provides a valuable and practical solution for ensuring the stable and high-quality integration of modern grid tied inverters into power systems with high renewable penetration.