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Suppression and stability analysis of frequency coupling effect in grid-connected inverters

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Abstract

With the proposal of the “dual carbon target” in China, the rapid development of renewable energy, mainly photovoltaic and wind power, has been promoted. However, a large amount of renewable energy connected to a grid results in its proportion in the power system going from a low state to a high state. Under a high proportion, the asymmetry of the control structure or parameters in the three-phase grid-connected inverter controller lead to a strong coupling relationship between the sub/super synchronous frequency components, or frequency coupling effect (FCE). This phenomenon can deteriorate the power quality of the inverter system, amplify the harm of frequency oscillation, and even cause system disassembly. To solve the above problems, a unified impedance model considering the FCE induced by the phase-locked loop (PLL), the current loop (CL), and the power outer loop (POL) is established. Based on the established output impedance model, a parameter optimization theoretical analysis method is designed considering the critical stability of the PLL bandwidth and the CL asymmetry degree. Meanwhile, an improved control strategy for the PLL and the POL is proposed. Experimental results show that the proposed parameter optimization and structure improvement strategies can effectively suppress the influence of frequency coupling and enhance system stability. Finally, simulation and experimental results verify the correctness of the theoretical analysis and the effectiveness of the proposed strategy.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (NSFC) (Fund Number: 52077191, 62003297).

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Authors and Affiliations

  1. Department of Electrical Engineering, Yanshan University, Qinhuangdao, Hebei, China

    Xiaohuan Wang, Aihua Yang, Yang Liu & Xiaojun Zhao

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  1. Xiaohuan Wang
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  2. Aihua Yang
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Correspondence to Aihua Yang.

Appendix

Appendix

$$\begin{gathered} A_{11} = s_{p} L + K_{m} V_{dc} (H_{i} (s_{p} - s_{0} ) - jk_{d} ) \hfill \\ {\kern 1pt} A_{12} = K_{m} V_{dc} H_{i}^{\sim } (s_{p} - s_{0} ) \hfill \\ A_{21} = K_{m} V_{dc} H_{i}^{\sim } (s_{p} - s_{0} ){\kern 1pt} \hfill \\ {\kern 1pt} {\kern 1pt} A_{22} = s_{p2} L + K_{m} V_{dc} (H_{i} (s_{p} - s_{0} ) + jk_{d} ), \hfill \\ \end{gathered}$$
(A1)

where Eq. (A1) is the output impedance element that is partially affected by the CL only when the CL considers the FCE. In addition, A11 reflects the effect of the positive sequence voltage disturbance on the response of the positive sequence current; A12 reflects the effect of the positive sequence voltage disturbance on the response of the coupling current; A21 reflects the effect of the coupling voltage disturbance on the response of the positive sequence current; and A22 reflects the effect of the coupling voltage disturbance on the response of the coupling current, where Km indicates the inverter gain.

$$\begin{gathered} B_{11} = K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) - jk_{d} )I_{1} \\ {\kern 1pt} {\kern 1pt} - H_{i}^{\sim } (s_{p} - s_{0} )I_{1}^{*} + M_{1} ] \\ B_{12} = - K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) - jk_{d} )I_{1} \\ {\kern 1pt} - H_{i}^{\sim } (s_{p} - s_{0} )I_{1}^{*} + M_{1} ] \\ B_{21} = - K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) + jk_{d} )I_{1}^{*} \\ {\kern 1pt} - H_{i}^{\sim } (s_{p} - s_{0} )I_{1} + M_{1}^{*} ] \\ B_{22} = K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) + jk_{d} )I_{1}^{*} \\ {\kern 1pt} {\kern 1pt} - H_{i}^{\sim } (s_{p} - s_{0} )I_{1} + M_{1}^{*} ], \\ \end{gathered}$$
(A2)

where Eq. (A2) represents the corresponding output impedance matrix elements affected by the PLL when the CL and PLL are considered for the FCE. In addition, the variable elements B11, B12, B21, and B22 in the matrix represent the effects of the disturbance voltages Vp and Vp2 through the PLL on the inverter output voltage.

$$\begin{gathered} E_{ip} = \frac{3}{4}H_{di} \left( {s_{p} } \right)G_{p} \left( {s_{p} - s_{0} } \right)\mathop {\mathop V\limits^{.} }\nolimits_{1}^{*} + \frac{3}{4}H_{qi} \left( {s_{p} } \right)G_{q} \left( {s_{p} - s_{0} } \right)\left( { - \mathop {\mathop V\limits^{.} }\nolimits_{1}^{*} } \right) \hfill \\ E_{ip2} = \frac{3}{4}H_{di} \left( {s_{p} } \right)G_{p} \left( {s_{p} - s_{0} } \right)\mathop {\mathop V\limits^{.} }\nolimits_{1}^{{}} + \frac{3}{4}H_{qi} \left( {s_{p} } \right)G_{q} \left( {s_{p} - s_{0} } \right)\mathop {\mathop V\limits^{.} }\nolimits_{1}^{{}} \hfill \\ F_{vp} = \frac{3}{4}H_{di} \left( {s_{p2} } \right)G_{p} \left( {s_{p2} - s_{0} } \right)\mathop {\mathop I\limits^{.} }\nolimits_{1}^{*} + \frac{3}{4}H_{qi} \left( {s_{p2} } \right)G_{q} \left( {s_{p2} - s_{0} } \right)\mathop {\mathop I\limits^{.} }\nolimits_{1}^{*} \hfill \\ F_{vp2} = \frac{3}{4}H_{di} \left( {s_{p2} } \right)G_{p} \left( {s_{p2} - s_{0} } \right)\mathop {\mathop I\limits^{.} }\nolimits_{1}^{{}} + \frac{3}{4}H_{qi} \left( {s_{p2} } \right)G_{q} \left( {s_{p2} - s_{0} } \right)\left( { - \mathop {\mathop I\limits^{.} }\nolimits_{1}^{{}} } \right), \hfill \\ \end{gathered}$$
(A3)

where Eq. (A3) represents the elements of the output matrix corresponding to being partially affected by the POL when the CL, PLL, and POL are considered for frequency coupling effects. The variables Eip, Eip2, Fvp, and Fvp2 represent the effects of the perturbation voltages Vp and Vp2 passing through the POL on the output current components.

$$Z_{i} = 1/Y_{i} = sL + K_{m} V_{dc} (H_{i} - jk_{d} )$$
(A4)
$$Y_{I} = {{(\frac{1}{2}K_{m} V_{dc} T_{PLL} H_{i}^{\sim } I_{1} )} \mathord{\left/ {\vphantom {{(\frac{1}{2}K_{m} V_{dc} T_{PLL} H_{i}^{\sim } I_{1} )} {[sL + K_{m} V_{dc} (H_{i} - jk_{d} )]}}} \right. \kern-0pt} {[sL + K_{m} V_{dc} (H_{i} - jk_{d} )]}}$$
(A5)
$$Y_{PLL} = \frac{{ - K_{m} V_{dc} T_{PLL} [(H_{i} - jk_{d} )I_{1} + M_{1} ]}}{{2[sL + K_{m} V_{dc} (H_{i} - jk_{d} )]}}$$
(A6)
$$\begin{gathered} B_{11}^{^{\prime}} = K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) - jk_{d} )I_{1} + M_{1} ] \hfill \\ B_{12}^{^{\prime}} = - K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) - jk_{d} )I_{1} + M_{1} ] \hfill \\ B_{21}^{^{\prime}} = - K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) + jk_{d} )I_{1}^{*} + M_{1}^{*} ] \hfill \\ B_{22}^{^{\prime}} = K_{m} V_{dc} T_{PLL} (s_{p} - s_{0} )[(H_{i} (s_{p} - s_{0} ) + jk_{d} )I_{1}^{*} + M_{1}^{*} ] \hfill \\ \end{gathered}$$
(A7)

Equation (A7) represents the impedance matrix elements affected by the PLL part of the system output only when the PLL considers the frequency coupling effect. The variable elements \({{B}^{\mathrm{^{\prime}}}}_{11}\), \({{B}^{\mathrm{^{\prime}}}}_{12}\), \({{B}^{\mathrm{^{\prime}}}}_{21}\) and \({{B}^{\mathrm{^{\prime}}}}_{22}\) have the same influence as in Eq. (A2).

$$\begin{gathered} Z_{g} (s_{p} ) = Z_{eq} (s_{p} ) = 1/((Y_{p} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}})) \\ = 1/(\frac{{1 - \frac{1}{2}K_{m} V_{dc} \frac{{m_{0} k_{pp} /s(1 + m_{0} k_{pi} /sk_{pp} )}}{{1 + V_{1} m_{0} k_{pp} /s(1 + m_{0} k_{pi} s/k_{pp} )}}[(H_{i} - jk_{d} )I_{1} + M_{1} ]}}{{s_{p} L + K_{m} V_{dc} (H_{i} - jk_{d} )}} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}}) \\ \end{gathered}$$
(A8)
$$\begin{gathered} Arg\left[ {Z_{g} (s,m_{0} )Y_{eq} (s,m_{0} )} \right] \\ = \angle Z_{g} (s_{p} ) - \\ \angle (\frac{{1 - \frac{1}{2}K_{m} V_{dc} \frac{{m_{0} k_{pp} /s(1 + m_{0} k_{pi} /sk_{pp} )}}{{1 + V_{1} m_{0} k_{pp} /s(1 + m_{0} k_{pi} s/k_{pp} )}}[(H_{i} - jk_{d} )I_{1} + M_{1} ]}}{{s_{p} L + K_{m} V_{dc} (H_{i} - jk_{d} )}} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}}) \\ = 180^{ \circ } \\ \end{gathered}$$
(A9)
$$\begin{gathered} Z_{g} (s_{p} ) = Z_{eq} (s_{p} ) = 1/((Y_{p} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}})) \\ = 1/(\frac{{1 + \frac{1}{2}K_{m} V_{dc} \frac{{n_{0} k_{pp} /s(1 + n_{0} k_{pi} /sk_{pp} )}}{{1 + V_{1} n_{0} k_{pp} /s(1 + n_{0} k_{pi} s/k_{pp} )}}H_{i}^{\sim } I_{1} }}{{sL + K_{m} V_{dc} (H_{i} - jk_{d} )}} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}}) \\ \end{gathered}$$
(A10)
$$\begin{gathered} Arg\left[ {Z_{g} (s,n_{0} )Y_{eq} (s,n_{0} )} \right] = \angle Z_{g} (s_{p} ) - \angle (\frac{{1 + \frac{1}{2}K_{m} V_{dc} \frac{{n_{0} k_{pp} /s(1 + n_{0} k_{pi} /sk_{pp} )}}{{1 + V_{1} n_{0} k_{pp} /s(1 + n_{0} k_{pi} s/k_{pp} )}}H_{i}^{\sim } I_{1} }}{{sL + K_{m} V_{dc} (H_{i} - jk_{d} )}} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}}) = 180^{ \circ } \hfill \\ \angle (\frac{{1 + \frac{1}{2}K_{m} V_{dc} \frac{{n_{0} k_{pp} /s(1 + n_{0} k_{pi} /sk_{pp} )}}{{1 + V_{1} n_{0} k_{pp} /s(1 + n_{0} k_{pi} s/k_{pp} )}}H_{i}^{\sim } I_{1} }}{{sL + K_{m} V_{dc} (H_{i} - jk_{d} )}} - \frac{{{\text{Y}}_{12} (s_{p2} )Y_{21} (s_{p} )}}{{Y_{g} (s_{p2} ) + Y_{22} (s_{p2} )}}) = - 90^{ \circ } \hfill \\ \end{gathered}$$
(A11)

Equations (A8) and (A9) are the tuning formulas for the bandwidth boundary of the PLL considering the frequency coupling effect, and Eqs. (A10) and (A11) are the tuning formulas for the critical asymmetry degree of the current loop considering the frequency coupling effect. These four equations are quoted from Eq. (3) in [20].

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Wang, X., Yang, A., Liu, Y. et al. Suppression and stability analysis of frequency coupling effect in grid-connected inverters. J. Power Electron. 23, 945–958 (2023). https://doi.org/10.1007/s43236-023-00591-6

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