# Admittance Remodeling Strategy of Grid-Connected Inverter Based on Improving GVF

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Admittance Model and Stability Analysis of Grid-Connected Inverter

#### 2.1. Output Admittance Model of Inverter System

_{1}, capacitor C

_{f}and inductor L

_{2}. i

_{1}is the current of L

_{1}, i

_{c}is the current of C

_{f}, i

_{g}is the grid-connected current, k

_{c}is the i

_{c}feedback coefficient, H

_{1}is the i

_{g}feedback coefficient, G

_{c}(s) is the proportional resonance controller, U

_{dc}is the ideal DC voltage output of the pre-stage photovoltaic power generation unit, u

_{pcc}is the common-point voltage, i

_{r}is the i

_{g}reference value. The phase is synchronized with the u

_{pcc}phase by CCF-PLL, and the amplitude is always the given value I

_{r}. G

_{c}(s) is the current controller. Z

_{g}is the impedance of the power grid. For the analysis of extreme cases, Z

_{g}is assumed to be pure inductive, denoting Z

_{g}= sL

_{g}.

_{f}(s) is the traditional grid voltage proportional feedforward, and the expression is 1/k

_{pwm}. G

_{PLL}(s) is the transfer function of SRF-PLL [8]. G

_{CCF}(s) is the transfer function of CCF [24]. Gc(s) is the current controller. G1(s) and G2(s) are the transfer functions after equivalent transformation in the current loop [32], and their expressions are as follows:

_{pwm}represents the gain of the inverter. While a bipolar sinusoidal pulse width modulation is used for the inverter, the transfer magnitude of the inverter bridge k

_{pwm}can be approximated by U

_{dc}/2 [33]. k

_{pp}is the proportional coefficient and k

_{pi}is the integral coefficient. k

_{CCF}is an adaptive adjustment coefficient, and, in order to avoid the interaction between CCF and SRF-PLL [14], k

_{CCF}is set to 0.2.

_{g}(s) can be derived from the superposition theorem as follows:

_{r}(s) and u

_{PCC}(s) is as follows:

_{p}(s) represents CCF-PLL admittance, Y

_{i}(s) represents inverter output admittance, and Y

_{v}(s) represents GVF admittance, whose expressions are as follows:

_{p}represents the proportional coefficient, k

_{r}represents the harmonic coefficient, ω

_{i}represents the bandwidth coefficient, ω

_{0}represents fundamental frequency angular frequency.

_{o}(s) of the inverter system is:

_{o1}(s), whose expression is Y

_{o1}(s) = Y

_{p}(s) + Y

_{i}(s).

#### 2.2. Stability Analysis

_{g}(s) can be rewritten as:

_{g}(s) is grid admittance, and the expression is 1/Z

_{g}(s).

_{o}(s) does not contain unstable poles under the strong grid. The stability of the system under a weak grid is determined by Y

_{o}(s)/Y

_{g}(s), and the stability condition is as follows: As $\left|{Y}_{\mathrm{o}}\left(s\right)\right|=\left|{Y}_{\mathrm{g}}\left(s\right)\right|$, the phase difference between Y

_{o}(s) and Y

_{g}(s) is less than 180°; that is, the admittance ratio of Y

_{o}(s)/Y

_{g}(s) phase margin (PM) is greater than 0°. Since the phase of Y

_{g}(s) is −90° in all frequency bands, the stability condition can be rewritten as: when, and only when $\left|{Y}_{\mathrm{o}}\left(s\right)\right|=\left|{Y}_{\mathrm{g}}\left(s\right)\right|$, the phase of Y

_{o}(s) is less than 90°.

_{o1}(s) and Y

_{o}(s) when SCR = 2. In this figure, the phase of output admittance of Y

_{o1}(s) in the frequency band of f

_{min}(133 Hz)~f

_{max}(2375 Hz) is from 19°~53°, the amplitude is about −25 dB, and the PM is 54.2°, indicating that the system is stable. After GVF is added, the phase of the total output admittance Y

_{o}(s) is advanced in the f

_{min}~f

_{max}frequency band, and the f

_{1}(178 Hz)~f

_{2}(644 Hz) frequency band is greater than 90°, resulting in a PM of −8.8° (f

_{c}= 439 Hz), and the system is unstable. It can be seen that the addition of GVF will affect the phase of the total output admittance of the system, and the influence frequency range is f

_{min}~f

_{max}, making the phase margin of the CCF-PLL inverter system in a weak grid negative, which seriously affects the stability of the system. Therefore, it is necessary to eliminate the negative effects of GVF.

## 3. Inverter Admittance Remodeling Strategy Based on Improved GVF

#### 3.1. The Influence of Traditional GVF on System Stability

_{v}(s). It can be seen that its amplitude is about −25 dB in the f

_{min}~f

_{max}frequency band, and its phase is from 48.9°~227° (the phase at f

_{min}is 227°). The high phase of Y

_{v}(s)) should be the reason for Y

_{o}(s) phase advance.

_{o}(s) phase, a phasor diagram in circuit theory is applied to analyze it. By the foregoing, the inside f

_{min}~f

_{max}spectrum, Y

_{o1}(s) the amplitude of which basically remains the same, and Y

_{v}(s) have the same amplitude, so for convenience of analysis, it can be assumed Y

_{v}(s) and Y

_{o1}(s) values are equal; remember Y

_{v}(s) phase is φ

_{v}, Y

_{o1}(s) phase is φ

_{o1}, Y

_{o}(s) phase is φ

_{o}.φ

_{o1}= α(19° ≤ α ≤ 53°), to Y

_{o1}phasor for reference. Phasor diagram is as shown in Figure 6, including β = 60° − α.

_{v}= 180° − α, φ

_{o}= 90°, the critical stability; when φ

_{v}< 180° − α, φ

_{o}< 90°, satisfy the stability criterion based on impedance, system stability; but when φ

_{v}> 180° − α, φ

_{o}> 90°, the PM is less than zero, at this time when the system is not stable, the results and analysis results of the Bode diagram are basically identical.

#### 3.2. Improve the GVF Control Strategy

_{min}~f

_{max}spectrum φ

_{v}< 180° − α, but to meet the requirements of PM ≥ 30° (φ

_{o}≤ 60°), will be expected to make φ

_{v}≤ α + 2β. Therefore, an improved control strategy of GVF is proposed, and an all-pass filter is added to the G

_{f}(s) branch to correct the phase of Y

_{v}(s) and improve the phase margin of the inverter in the weak grid. Figure 7 shows the equivalent control block diagram of a grid-connected inverter after improved GVF, where G

_{AF}(s) is the all-pass filter transfer function, and the expression is:

_{v}(s) at f

_{min}is 227°, and its phase-frequency curve monotonically decreases in the frequency band from f

_{min}to f

_{max}. Therefore, as long as the phase at this point meets the requirements, the phase at other frequencies can also meet the requirements. Make f

_{min}with phase compensation point, then the lag compensation angle θ = (α + 2β) − 227°. A larger α corresponds to a smaller value of φ

_{v}that ensures the stability of the system; therefore, for analyzing the worst case, taking α = 53°, thus β = 7°, the system stability is obtained, and leaving at least 30° phase margin requirements for φ

_{v}≤ 67° or less. It should at least provide θ = 160° phase compensation. By substituting f

_{min}= 133 Hz and θ = −160° into Formula (12), a = 0.0068 can be obtained.

_{v_AF}(s). Compared with Figure 5, it can be seen that the addition of an all-pass filter only changes the phase of Y

_{v}(s) but does not change its amplitude. The phase of Y

_{v_AF}(s) at f

_{min}is 66.7°, less than 67°, and the phase in the whole band affected by the GVF phase is less than 67°, meeting the requirement of stability.

_{o_AF}(s) with different SCR values. By comparison with Y

_{o}(s) in Figure 4, it can be seen that the phase of total output admittance Y

_{o}(s) in the frequency band affected by GVF is reduced after the proposed control strategy is added. The phase margin of the system increases from −8.8°, −4.1°, 3.2° and 7.9° to 34.2°, 45.6°, 67.2° and 84° when the SCR is 2, 3, 6, and 10, respectively, which greatly improves the stability of the system.

## 4. Simulation and Experimental Results

#### 4.1. Simulated Analysis

_{g}= 12.8 mH). When no GVF is added before t = 0.12 s, no oscillation distortion occurs in the grid-connected current. When t = 0.12 s and GVF is added, the grid-connected current oscillates obviously. This indicates that the stability of the grid-connected inverter using CCF-PLL is better without GVF in weak power grids, but becomes unstable after introducing GVF, due to grid current oscillations.

_{g}(t) exhibits distortion and instability when SCR = 3.1 (L

_{g}= 8.28 mH). As the grid impedance increases, the distortion phenomenon of the current waveform becomes more severe when SCR = 2. The results show that traditional GVF can reduce the stability margin of the grid-connected inverter system using CCF-PLL in a weak grid, and aggravate the system’s instability with the increase in grid impedance.

_{g}(t) before and after the improved GVF under SCR = 2. As shown in Figure 15, the THD before improvement is 19.59%; there are a large number of harmonics in a grid-connected current, and the largest component of harmonics is mainly concentrated in the frequency band f

_{1}(178 Hz)~f

_{2}(644 Hz) whose Y

_{o}(s) phase is greater than 90°; which is consistent with the theoretical analysis in Section 2.1. After the improvement, the THD is 1.13%, and the harmonic distortion rate of a grid-connected current is reduced by 94.2%, which greatly reduces the harmonic content in f

_{1}~f

_{2}bands and improves the stability of the system. The results show that the proposed strategy can ensure the stable operation of the grid-connected inverter when the grid impedance is large.

#### 4.2. Experiment Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wu, W.; Liu, Y.; He, Y.; Chung, H.S.H.; Liserre, M.; Blaabjerg, F. Damping methods for resonances caused by LCL-filter-based current controlled grid-tied power inverters: An overview. IEEE Trans. Ind. Electron.
**2017**, 64, 7402–7413. [Google Scholar] [CrossRef] - Yu, L.; Sun, H.; Zhao, B.; Xu, S.; Zhang, J.; Li, Z. ShiyunShort Circuit Ratio Index Analysis and Critical Short Circuit Ratio Calculation of Renewable Energy Grid-connected System. CSEE
**2022**, 42, 919–929. [Google Scholar] - Chen, X.; Zhang, Y.; Wang, S.; Chen, J.; Gong, C. Impedance-phased dynamic control method for grid-connected inverters in a weak grid. IEEE Trans. Power Electron.
**2016**, 32, 274–283. [Google Scholar] [CrossRef] - Li, M.; Zhang, X.; Guo, Z.; Wang, J.; Wang, Y.; Lib, F.; Zhao, W. The Control Strategy for the Grid-Connected Inverter through Impedance Reshaping in q-Axis and its Stability Analysis Under a Weak Grid. IEEE J. Emerg. Sel. Top. Power Electron.
**2020**, 9, 3229–3242. [Google Scholar] [CrossRef] - Huanhai, X.; Wei, D.; Xiaoming, Y.; Deqiang, G.; Kang, W.; Huan, X. Generalized Short Circuit Ratio for Multi Power Electronic Based Devices Infeed to Power Systems. CSEE
**2016**, 36, 6013–6027. [Google Scholar] - Fang, T.; Zhang, H.; Wu, H.; Zhang, Y. Robustness Enhancement of Coping with Dual Factors for Grid-Connected Inverter in Weak Grid Based on Synthesis-Admittance-Phasor Scheme. IEEE Trans. Ind. Electron.
**2022**, 14, 754–769. [Google Scholar] [CrossRef] - Du, Y.; Sun, Q.; Yang, X.; Cui, L.; Zhang, J.; Wang, F. Adaptive Virtual Impedance of Grid-Tied Inverters to Enhance the Stability in a Weak Grid. J. Electr. Eng. Technol.
**2019**, 14, 1235–1246. [Google Scholar] [CrossRef] - Xu, J.; Qian, Q.; Zhang, B.; Xie, S. Harmonics and stability analysis of single-phase grid-connected inverters in distributed power generation systems considering phase-locked loop impact. IEEE Trans. Sustain. Energy
**2019**, 10, 1470–1480. [Google Scholar] [CrossRef] - Ali, Z.; Christofides, N.; Hadjidemetriou, L.; Kyriakides, E. Multi-functional distributed generation control scheme for improving the grid power quality. IET Power Electron.
**2019**, 12, 30–43. [Google Scholar] [CrossRef] - Fang, J.; Li, X.; Li, H.; Tang, Y. Stability Improvement for Three-Phase Grid-Connected Converters Through Impedance Reshaping in Quadrature-Axis. IEEE Trans. Power Electron.
**2018**, 33, 8365–8375. [Google Scholar] [CrossRef] - Xu, J.; Qian, Q.; Xie, S.; Zhang, B. Grid-voltage feedforward based control for grid-connected LCL-filtered inverter with high robustness and low grid current distortion in weak grid. In Proceedings of the 2016 IEEE Applied Power Electronics Conference and Exposition, IEEE, Long Beach, CA, USA, 20–24 March 2016; pp. 1919–1925. [Google Scholar]
- Guo, X.; Guerrero, J.M. Abc-frame complex-coefficient filter and controller based current harmonic elimination strategy for three-phase grid connected inverter. Mod. Power Syst. Clean Energy
**2016**, 4, 87–93. [Google Scholar] [CrossRef] - Lee, K.J.; Lee, J.P.; Shin, D.; Yoo, D.W.; Kim, H.J. A novel grid synchronization PLL method based on adaptive low-pass notch filter for grid-connected PCS. IEEE Trans. Ind. Electron.
**2013**, 61, 292–301. [Google Scholar] [CrossRef] - Tu, C.; Gao, J.; Li, Q. Research on adaptability of grid-connected inverter with complex coefficient-filter structure phase locked loop to weak grid. Trans. Electrotech. Soc.
**2020**, 35, 2632–2642. [Google Scholar] - Lin, Z.; Chen, Z.; Yajuan, L.; Bin, L.; Jinhong, L.; Bao, X. Phase-reshaping strategy for enhancing grid-connected inverter robustness to grid impedance. IET Power Electron.
**2018**, 11, 1434–1443. [Google Scholar] [CrossRef] - Lin, Z.; Ruan, X.; Wu, L.; Zhang, H.; Li, W. Multi resonant Component-Based Grid-Voltage-Weighted Feedforward Scheme for Grid-Connected Inverter to Suppress the Injected Grid Current Harmonics under Weak Grid. IEEE Trans. Power Electron.
**2020**, 9, 9784–9793. [Google Scholar] [CrossRef] - Yan, Q.; Wu, X.; Yuan, X.; Geng, Y. An improved grid-Voltage feedforward strategy for high-power three-phase grid-connected inverters based on the simplified repetitive predictor. IEEE Trans. Power Electron.
**2016**, 31, 3880–3897. [Google Scholar] [CrossRef] - Xu, J.; Xie, S.; Zhang, B.; Qian, Q. Robust grid current control with impedance-phase shaping for LCL-filtered inverters in weak and distorted grid. IEEE Trans. Power Electron.
**2018**, 33, 10240–10250. [Google Scholar] [CrossRef] - Wang, X.; Qin, K.; Ruan, X.; Pan, D.; He, Y.; Liu, F. A robust grid-voltage feedforward scheme to improve adaptability of grid-connected inverter to weak grid condition. IEEE Trans. Power Electron.
**2020**, 36, 2384–2395. [Google Scholar] [CrossRef] - Yang, S.; Tong, X.; Yin, J.; Wang, H.; Deng, Y.; Liu, L. BPF-based grid voltage feedforward control of grid-connected converters for improving robust stability. J. Power Electron.
**2017**, 17, 432–441. [Google Scholar] [CrossRef] - Xu, J.; Xie, S.; Qian, Q.; Zhang, B. Adaptive feedforward algorithm without grid impedance estimation for inverters to suppress grid current instabilities and harmonics due to grid impedance and grid voltage distortion. IEEE Trans. Ind. Electron.
**2017**, 64, 7574–7586. [Google Scholar] [CrossRef] - Khajeh, K.G.; Farajizadeh, F.; Solatialkaran, D. A full-feedforward technique to mitigate the grid distortion effect on parallel grid-tied inverters. IEEE Trans. Power Electron.
**2022**, 37, 8404–8419. [Google Scholar] [CrossRef] - Xu, J.; Xie, S.; Tang, T. Improved control strategy with grid-voltage feedforward for LCL-filter-based inverter connected to weak grid. IET Power Electron.
**2014**, 7, 2660–2671. [Google Scholar] [CrossRef] - Gao, J.; Tu, C.; Guo, Q.; Xiao, F.; Jiang, F.L.; Lu, B. Impedance Reshaping Control Method to Improve Weak Grid Stability of Grid-Connected Inverters. In Proceedings of the IECON 2020 the 46th Annual Conference of the IEEE Industrial Electronics Society, Singapore, 19–21 October 2020; pp. 1342–1346. [Google Scholar]
- Li, X.; Fang, J.; Tang, Y.; Wu, X. Robust design of LCL filters for single-current-loop-controlled grid-connected power converters with unit PCC voltage feedforward. IEEE J. Emerg. Sel. Top. Power Electron.
**2017**, 6, 54–72. [Google Scholar] [CrossRef] - Chen, B.; Zeng, C.B.; Miao, H.; Hong, C. Improved voltage feedforward method for improving robust stability of grid-connected inverters in weak grids. J. Electr. Power Sci. Technol.
**2021**, 36, 118–124. [Google Scholar] - Zeng, C.; Wang, H.; Li, S.; Miao, H. Grid-voltage-feedback active damping with lead compensation for LCL-type inverter connected to weak grid. IEEE Access
**2021**, 9, 106813–106823. [Google Scholar] [CrossRef] - Wang, H.; Zeng, C.; Miao, H. A phase compensation algorithm of a grid-connected inverter based on a feedforward multi-resonant grid voltage. Power Syst. Prot. Control
**2021**, 49, 81–89. [Google Scholar] - Wang, G.; Du, X.; Shi, Y.; Yang, Y.; Sun, P.; Li, G. Effects on oscillation mechanism and design of grid-voltage feedforward in grid-tied converter under weak grid. IET Power Electron.
**2019**, 12, 1094–1101. [Google Scholar] [CrossRef] - Xie, Z.; Chen, Y.; Wu, W.; Gong, W.; Guerrero, J.M. Stability enhancing voltage feed-forward inverter control method to reduce the effects of phase-locked loop and grid impedance. IEEE J. Emerg. Sel. Top. Power Electron.
**2020**, 9, 3000–3009. [Google Scholar] [CrossRef] - Pang, B.; Li, F.; Dai, H.; Nian, H. High Frequency Resonance Damping method for voltage source converter based on voltage feedforward control. Energies
**2020**, 13, 1591. [Google Scholar] [CrossRef] - Yang, D.; Ruan, X.; Wu, H. Impedance shaping of the grid-connected inverter with LCL filter to improve its adaptability to the weak grid condition. IEEE Trans. Power Electron.
**2014**, 29, 5795–5805. [Google Scholar] [CrossRef] - Xia, W.; Kang, J. Stability of LCL-filtered grid-connected inverters with capacitor current feedback active damping considering controller time delays. J. Mod. Power Syst. Clean Energy
**2017**, 5, 584–598. [Google Scholar] [CrossRef] - Zhu, K.; Sun, P.; Zhou, L.; Du, X.; Luo, Q. Frequency-Division Virtual Impedance Shaping Control Method for Grid-Connected Inverters in a Weak and Distorted Grid. IEEE Trans. Power Electron.
**2020**, 35, 8116–8129. [Google Scholar] [CrossRef] - Wang, H.; Chen, Y.; Wu, W.; Liao, S.; Wang, Z.; Li, G.; Guo, J. Impedance Reshaping Control Strategy for Improving Resonance Suppression Performance of a Series-Compensated Grid-Connected System. Energies
**2021**, 14, 2844. [Google Scholar] [CrossRef] - Sun, J. Impedance-based stability criterion for grid-connected inverters. IEEE Trans. Power Electron.
**2011**, 26, 3075–3078. [Google Scholar] [CrossRef] - Xue, T.; Sun, P.; Xu, Z.; Luo, Q. Feedforward phase compensation method of LCL grid-connected inverter based on all-pass filter in weak grid. IET Power Electron.
**2020**, 13, 4407–4416. [Google Scholar] [CrossRef]

**Figure 2.**Equivalent control block diagram of simple-phase LCL grid-connected inverter. (

**a**) Without GVF; (

**b**) With GVF.

**Figure 15.**THD analysis before and after improving GVF. (

**a**) After improving GVF. (

**b**) Before improving GVF.

**Figure 16.**Experimental waveform of i

_{g}(t) before adding improvement GVF. (

**a**) SCR = 3.1. (

**b**) SCR = 2.

**Figure 17.**Experimental waveform of i

_{g}(t) after adding improvement GVF. (

**a**) SCR = 3.1. (

**b**) SCR = 2.

Parameter | Value | Parameter | Value |
---|---|---|---|

v_{g}/V | 220 | k_{p} | 0.55 |

v_{in}/V | 400 | k_{r} | 75 |

L_{1}/mH | 2 | ω_{i} | π |

L_{2}/mH | 0.5 | k_{pp} | 0.833 |

C_{f}/μF | 8 | k_{pi} | 107.88 |

k_{c} | 0.1 | H_{1} | 0.15 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, S.; Huang, S.; He, W.; Zhang, D.
Admittance Remodeling Strategy of Grid-Connected Inverter Based on Improving GVF. *Electronics* **2023**, *12*, 2122.
https://doi.org/10.3390/electronics12092122

**AMA Style**

Li S, Huang S, He W, Zhang D.
Admittance Remodeling Strategy of Grid-Connected Inverter Based on Improving GVF. *Electronics*. 2023; 12(9):2122.
https://doi.org/10.3390/electronics12092122

**Chicago/Turabian Style**

Li, Shengqing, Simin Huang, Weihua He, and Dong Zhang.
2023. "Admittance Remodeling Strategy of Grid-Connected Inverter Based on Improving GVF" *Electronics* 12, no. 9: 2122.
https://doi.org/10.3390/electronics12092122