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Harmonic Stability Analysis for Multi-Parallel Inverter-Based Grid-Connected Renewable Power System Using Global Admittance^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

#### 2.1. System Description

#### 2.2. Admittance Modeling of a Single-Inverter System

_{i}(s) is the current-loop controller, such as PR or VPI; D(s) is the current-loop disturbance gain; and P(s) is the controlled object, such as the L or LCL filter. According to Equation (1), the NORTON equivalent model consisting of a controllable current source and current-loop admittance can be obtained from the perspective of circuit port equivalence, as shown in Figure 5. Unlike the current-loop model in block diagram form, that in admittance form can be used to analyze the relationship between the output current and PCC voltage; then, the response expression of each part of the parallel system can be obtained to analyze the stability.

_{m}is the equivalent integral coefficient of each resonant frequency; ω

_{m}is the resonant angular frequency; θ

_{m}is the resonance frequency compensation angle; T

_{d}is the control system delay; and L’ and R’ are the inductance and resistance values set based on the zero-polar offset of the controller and the controlled object, respectively:

#### 2.3. Admittance Modeling of the VSIs System

_{p}(s) is the equivalent admittance of the passive device at the front side of the power grid; Y

_{g}(s) is the equivalent grid admittance; Y

_{n}(s) is the current-loop admittance of each module in the parallel system; I′

_{n}(s) is the equivalent current source of each module; and I

_{n}(s) is the actual output current of each module. To facilitate analysis, the voltage source in Figure 6a is converted to the NORTON equivalent form, as shown in Figure 6b. The grid voltage is composed of the equivalent current source, E(s)Y

_{g}(s), and the equivalent grid admittance, Y

_{g}(s).

## 3. Traditional Impedance-Based Stability Criterion

#### 3.1. Stability Criterion for a Single Grid-Connected Inverter System

_{g}, in series with an ideal voltage source, V

_{g}, and a single grid-connected inverter can be represented as a current source, I

_{s}(s), in parallel with an output admittance, Y

_{o}(s). An overall single-inverter grid-connected system can be represented as shown in Figure 7.

_{o}(s)/Y

_{g}(s)) satisfies the Nyquist criterion.

#### 3.2. Stability Criterion for a Multi-Parallel Grid-Connected Inverter System

_{p}(s) represents the passive components that are composed of power transformers or reactive compensation components. In such a system, the impedance-based approach for a single-inverter grid-connected system no longer applies.

_{cl,i}denotes the closed-loop gain of the current control loop, which can be designed to be stable in advance. Similar to the assumption above, all the inverters and grid are individually stable. Thus, the stability of the ith inverter can be predicted as a single-inverter grid-connected system, and a multi-parallel VSI grid-connected system will be stable if all the inverters are stable. It is clear from this approach that analyses must be implemented multiple times to investigate the stability of the entire system, thereby increasing the computational burden. In addition, all the MLG

_{i}values must be recalculated when the parameters of the grid and the number of inverters change.

#### 3.3. Unified Impedance-Based Stability Criterion

_{s,i}(s), and the output admittance is the summation of Y

_{o,i}(s) together with Y

_{p}(s). A global (GMLG) is defined as follows:

## 4. The Proposed Global Admittance-Based Stability Criterion

#### 4.1. Description of the Basic Principle

_{s}, is a contour that encircles the right half of the complex plane in a clockwise direction. According to the argument principle:

_{total}(s), is defined as the summation of all the admittances, including the grid admittance, other passive admittances, and output admittances of inverters. Thus, global admittance can be expressed as follows:

_{g,i}(s), of the ith inverter can be calculated as follows:

_{g,i}(s) depends on three different excitations, namely, the reference current of the ith inverter, the reference currents of other inverters, and the grid voltage.

_{g}(s) and Y

_{p}(s) are the transfer functions of passive components, they have no right half panel poles. Based on the condition that both the unloaded inverters and grid voltage are stable, Y

_{o,i}(s), I

_{s,i}(s), and E(s) have no right half plane poles either. Thus, the impact of the numerators in Equation (13) can be neglected when assessing the current stability, and the stability of I

_{g,i}(s) only depends on 1/Y

_{total}(s); specifically, it depends on the locations of Y

_{total}(s) zeros.

#### 4.2. The Proposed Stability Criterion

_{total}(s). The system will be stable only and if only all the zeros of Y

_{total}(s) are located in the left half panel. An improved Nyquist criterion is proposed to assess the distribution of Y

_{total}(s) poles. The Nyquist contour, Γ

_{s}, is still a contour that encircles the right half of the complex plane clockwise. Based on the argument principle, the difference between the zeros and poles of Y

_{total}(s) in the right panel can be obtained by the number of times that the clockwise Nyquist curve of Y

_{total}(s) encircles the origin (0, j0). As mentioned above, Y

_{g}(s), Y

_{p}(s), and Y

_{o,i}(s) all have no right half panel poles; thus, Y

_{total}(s) has no right half panel poles either. Consequently, the number of laps is equal to the number of right half panel zeros of Y

_{total}(s). The grid-connected system is stable if and only if the Nyquist curve of Y

_{total}(s) does not encircle the origin (0, j0).

_{total}(s) for an unstable grid-connected system. Figure 9a shows that the Nyquist plot of the unstable system encircles the origin, meeting the improved Nyquist stability criterion. There are multiple intersection points of the Nyquist plot and the real axis; these points are defined as resonance points that determine the system resonance state. The real part of a resonance point is defined as the resonance damping factor, R

_{d}, where:

_{d}is negative, the Nyquist plot encircles the origin (0, j0), reflecting an unstable system. By contrast, if the minimum R

_{d}is positive, this means no encirclement around the origin (0, j0), and the system is stable. The larger the minimum R

_{d}value is, the more stable the system. The authors of [30] proposed a passivity-based stabilization approach for a grid-connected system in which the output admittances of inverters are required to have positive real parts at full frequency. This approach sacrifices the bandwidth of the controller in the purist of stability, and the dynamics of the controller are weakened because of the reduced bandwidth. However, the global admittance-based criterion indicates that the grid-connected system is stable if and only if all the resonance points of Y

_{total}(s) are located at the right half panel instead of the output admittance of each inverter with a positive real part. Therefore, the bandwidth of the controllers can be designed in a wide frequency domain. Consequently, the dynamics of the inverters can be improved.

_{d}mapped from the Nyquist plot to the Bode plot reflects a step change from −180 to 180° in the phase frequency response curve. Thus, the grid-connected system stability can be predicted based on the Bode plot of Y

_{total}(s). Compared to the stability criteria proposed in [28,29], global admittance is given in summation form in this case instead of ratio form; this approach not only reduces the computational burden but also reveals the influence of each component on system stability.

#### 4.3. Discussion of the Case in which the Parameters are Different

_{total}at the resonance point, which is approximately 1600 Hz. The phase of each inverter output admittance is different, which means that the influence of each inverter on system instability is different. The real part of the output admittance of each inverter is shown in Table 3, where R

_{d,}

_{1}, R

_{d,}

_{2}, and R

_{d,p}are expressed in Equations (16) to (18) and the relationship between these variables and R

_{d}is expressed in Equation (19). The real part of the output admittance of inverter 1 at the resonance point is positive, and that of inverter 2 is negative, which implies that inverter 2 is responsible for the system instability.

## 5. Simulation and Experimental Results

#### 5.1. Simulation Verification

_{g}(s), the dashed line represents the total output admittance of the four inverters, 4*Y(s), and the solid line represents the global admittance, Y

_{total}(s). The interaction of Y

_{g}(s) and 4*Y(s) is the resonance point of the system, and the corresponding frequency is the resonance frequency. The resonance frequency of the four-parallel VSI system is approximately 2000 Hz. The magnitude of Y

_{total}(s) at the resonance frequency is the absolute value of R

_{d}and equals 0.02. The sign of R

_{d}can be obtained from the phase–frequency curve of Y

_{total}(s), as shown in (b). The step changes from −90 to 90° in the phase curve of Y

_{total}(s) indicates that the sign of R

_{d}is positive, which indicates that the four-parallel grid-connected system is stable. The fast Fourier transform (FFT) analysis of the PCC voltage is shown in (c). Resonance occurs at 2000 Hz, and the amplitude of the resonance component is only 6 V. The PCC voltage and total current from the inverters are stable, as shown in (d) and (e), respectively, and these findings are consistent with the anticipated results.

_{d}is 0.1 and the resonance frequency equals 2250 Hz. The step changes from −180 to 180° in the phase-frequency plot of Y

_{total}(s) in (b) imply that the sign of R

_{d}is negative, and R

_{d}equals −0.1. Based on the global admittance-based stability criterion, the six-parallel VSI grid-connected system is unstable. The FFT analysis of the PCC voltage is shown in (c). The amplitude of the resonance component is close to 400 V, which is even larger than that of the fundamental component. The simulation results of the PCC voltage and the total current from inverters, as shown in (d) and (e), respectively, indicate that the system starts losing stability at approximately 0.2 s.

_{d}equals −0.6, the resonance frequency is 2250 Hz, the amplitude of the resonance component of the PCC voltage is close to 10,000 V, and the instability occurs at 0.02 s. By comparing the above data with those in case I and case II, the following conclusions can be drawn. R

_{d}decreases as the number of inverters increases. The system will be unstable if R

_{d}is less than zero. The resonance frequency and the amplitude of the resonance component of the PCC voltage increase as R

_{d}decreases. The smaller R

_{d}is, the shorter the time that the instability occurs. The conclusions of the three cases are shown in Table 4.

#### 5.2. Experimental Verification

_{1}~R

_{3}) of the grid impedance, the real value of the resonance point is changed, which will result in a stable parallel system (when R

_{d}> 0).

_{d}> 0, and the parallel system becomes stable, which verifies the effectiveness of the proposed analysis method.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Blaabjerg, F.; Chen, Z.; Kjaer, S.B. Power electronics as efficient interface in dispersed power generation systems. IEEE Trans. Power Electron.
**2004**, 19, 1184–1194. [Google Scholar] [CrossRef] - Rocabert, J.; Luna, A.; Blaabjerg, F.; Rodríguez, P. Control of Power Converters in AC Microgrids. IEEE Trans. Power Electron.
**2012**, 27, 4734–4749. [Google Scholar] [CrossRef] - Liserre, M.; Teodorescu, R.; Blaabjerg, F. Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values. IEEE Trans. Power Electron.
**2006**, 21, 263–272. [Google Scholar] [CrossRef] - Turner, R.; Walton, S.; Duke, R. Stability and Bandwidth Implications of Digitally Controlled Grid-Connected Parallel Inverters. IEEE Trans. Ind. Electron.
**2010**, 57, 3685–3694. [Google Scholar] [CrossRef] - Wang, F.; Duarte, J.L.; Hendrix, M.A.M.; Ribeiro, P.F. Modeling and Analysis of Grid Harmonic Distortion Impact of Aggregated DG Inverters. IEEE Trans. Power Electron.
**2011**, 26, 786–797. [Google Scholar] [CrossRef] - Wang, X.; Blaabjerg, F.; Chen, Z. Autonomous Control of Inverter-Interfaced Distributed Generation Units for Harmonic Current Filtering and Resonance Damping in an Islanded Microgrid. IEEE Trans. Ind. Appl.
**2014**, 50, 452–461. [Google Scholar] [CrossRef] - Zhang, H.; Wang, X.; Harnefors, L.; Gong, H.; Hasler, J.; Nee, H. SISO Transfer Functions for Stability Analysis of Grid-Connected Voltage-Source Converters. IEEE Trans. Ind. Appl.
**2019**, 55, 2931–2941. [Google Scholar] [CrossRef] - Wang, X.; Blaabjerg, F.; Wu, W. Modeling and Analysis of Harmonic Stability in an AC Power-Electronics-Based Power System. IEEE Trans. Power Electron.
**2014**, 29, 6421–6432. [Google Scholar] [CrossRef] - Turner, R.; Walton, S.; Duke, R. A case study on the application of the Nyquist stability criterion as applied to interconnected loads and sources on grids. IEEE Trans. Ind. Electron.
**2013**, 60, 2740–2749. [Google Scholar] [CrossRef] - Cao, W.; Fan, D.; Liu, K.; Zhao, J.; Ruan, L.; Wu, X. Harmonic Stability Assessment based on Global Admittance for Multi-Paralleled Grid-Connected VSIs using Modified Nyquist Criterion. In Proceedings of the International Power Electronics Conference (IPEC-Niigata 2018 -ECCE Asia), Niigata, Japan, 20–24 May 2018. [Google Scholar]
- Wang, X.; Blaabjerg, F. Harmonic Stability in Power Electronic-Based Power Systems: Concept, Modeling, and Analysis. IEEE Trans. Smart Grid.
**2019**, 10, 2858–2870. [Google Scholar] [CrossRef] - Hu, T. A Nonlinear-System Approach to Analysis and Design of Power-Electronic Converters with Saturation and Bilinear Terms. IEEE Trans. Power Electron.
**2011**, 26, 399–410. [Google Scholar] [CrossRef] - Wu, H.; Pickert, V.; Giaouris, D.; Ji, B. Nonlinear Analysis and Control of Interleaved Boost Converter Using Real-Time Cycle to Cycle Variable Slope Compensation. IEEE Trans. Power Electron.
**2017**, 32, 7256–7270. [Google Scholar] [CrossRef] - Amin, M.; Molinas, M. Small-Signal Stability Assessment of Power Electronics Based Power Systems: A Discussion of Impedance- and Eigenvalue-Based Methods. IEEE Trans. Ind. Appl.
**2017**, 53, 5014–5030. [Google Scholar] [CrossRef] - Kalcon, G.O.; Adam, G.P.; Anaya-Lara, O.; Lo, S.; Uhlen, K. Small-Signal Stability Analysis of Multi-Terminal VSC-Based DC Transmission Systems. IEEE Trans. Power Syst.
**2012**, 27, 1818–1830. [Google Scholar] [CrossRef] - Pinares, G.; Bongiorno, M. Modeling and Analysis of VSC-Based HVDC Systems for DC Network Stability Studies. IEEE Trans. Power Deliv.
**2016**, 31, 848–856. [Google Scholar] [CrossRef] - Beerten, J.; D’Arco, S.; Suul, J.A. Identification and Small-Signal Analysis of Interaction Modes in VSC MTDC Systems. IEEE Trans. Power Deliv.
**2016**, 31, 888–897. [Google Scholar] [CrossRef] - Kundur, P. Power System Stability and Control; McGraw-Hill: New York, NY, USA, 1994. [Google Scholar]
- Middlebrook, R.D.D. Input filter consideration in design and application of switching regulators. In Proceedings of the IEEE Industry Applications Society Annual Meeting, Chicago, IL, USA, 11–14 October 1976. [Google Scholar]
- Cao, W.; Ma, Y.; Yang, L.; Wang, F.; Tolbert, L.M. D-Q Impedance Based Stability Analysis and Parameter Design of Three-Phase Inverter-Based AC Power Systems. IEEE Trans. Ind. Electron.
**2017**, 64, 6017–6028. [Google Scholar] [CrossRef] - Cao, W.; Ma, Y.; Wang, F. Sequence-Impedance-Based Harmonic Stability Analysis and Controller Parameter Design of Three-Phase Inverter-Based Multibus AC Power Systems. IEEE Trans. Power Electron.
**2017**, 32, 7674–7693. [Google Scholar] [CrossRef] - Cho, Y.; Hur, K.; Kang, Y.C.; Muljadi, E. Impedance-Based Stability Analysis in Grid Interconnection Impact Study Owing to the Increased Adoption of Converter-Interfaced Generators. Energies
**2017**, 10, 1355. [Google Scholar] [CrossRef] - Liu, Z.; Liu, J.; Bao, W.; Zhao, Y. Infinity-norm of impedance-based stability criterion for three-phase AC distributed power systems with constant power loads. IEEE Trans. Power Electron.
**2015**, 30, 3030–3043. [Google Scholar] [CrossRef] - Vesti, S.; Suntio, T.; Oliver, J.A.; Prieto, R.; Cobos, J.A. Impedance-based stability and transient-performance assessment applying maximum peak criteria. IEEE Trans. Power Electron.
**2013**, 28, 2099–2104. [Google Scholar] [CrossRef] - Suntio, T.; Messo, T.; Berg, M.; Alenius, H.; Reinikka, T.; Luhtala, R.; Zenger, K. Impedance-Based Interactions in Grid-Tied Three-Phase Inverters in Renewable Energy Applications. Energies
**2019**, 12, 464. [Google Scholar] [CrossRef] - Sun, J. Impedance-based stability criterion for grid-connected inverters. IEEE Trans. Power Electron.
**2011**, 26, 3075–3078. [Google Scholar] [CrossRef] - Liu, F.; Liu, J.; Zhang, H.; Xue, D. Stability issues of Z + Z type cascade system in hybrid energy storage system (HESS). IEEE Trans. Power Electron.
**2014**, 29, 5846–5859. [Google Scholar] [CrossRef] - Wang, X.; Blaabjerg, F.; Liserre, M.; Chen, Z.; He, J.; Li, Y. An active damper for stabilizing power-electronics-based AC systems. IEEE Trans. Power Electron.
**2014**, 29, 3318–3329. [Google Scholar] [CrossRef] - Ye, Q.; Mo, R.; Shi, Y.; Li, H. A unified Impedance-based Stability Criterion (UIBSC) for paralleled grid-tied inverters using global minor loop gain (GMLG). In Proceedings of the IEEE Energy Conversion Congress and Exposition (ECCE), Montreal, QC, Canada, 20–24 September 2015. [Google Scholar]
- Harnefors, L.; Wang, X.; Yepes, A.G.; Blaabjerg, F. Passivity-Based Stability Assessment of Grid-Connected VSCs-An Overview. IEEE J. Emerg. Sel. Top. Power Electron.
**2016**, 4, 116–125. [Google Scholar] [CrossRef]

**Figure 6.**Equivalent circuit of a parallel system. (

**a**) Voltage-source form and (

**b**) current-source form.

**Figure 9.**Nyquist and Bode plots of global admittance Y

_{total}(s) for an unstable grid-connected system. (

**a**) Nyquist curve; (

**b**) Bode plots.

**Figure 10.**Stability analysis using Bode plots when VSIs have different parameters (an unstable case). (

**a**) Bode plots; (

**b**) zoom of dotted box in (

**a**).

**Figure 11.**Frequency analysis and simulation results of four (case I), six (case II), and eight (case III) parallel VSI systems. (

**a**) Amplitude–frequency plot; (

**b**) phase–frequency plot; (

**c**) FFT analysis of the PCC voltage; (

**d**) PCC voltage; (

**e**) total output current.

**Figure 12.**Experiment platform. (

**a**) Multi-parallel grid-connected VSIs; (

**b**) VSI unit; (

**c**) ScopeCorder-DL850E.

**Figure 14.**Experimental waveform when the grid impedance changes (C = 600 µF, R = 0.2 Ω, Resonance frequency: 11th). (

**a**) The waveform when the harmonic instability occurs; (

**b**) zoom of dotted box in (

**a**).

**Figure 15.**Experimental waveform when the grid impedance changes (C = 600 µF, R = 0.6 Ω). (

**a**) The waveform when the system becomes stable; (

**b**) zoom of dotted box in (

**a**).

Method | Calculation Burden | Reflect the Instability-Causing Element | Ease of Stability-Oriented Redesign |
---|---|---|---|

MLG | massive | positive | direct |

GMLG | small | negative | complicated |

Parameters | Values | ||
---|---|---|---|

Inverter 1 | LCL-filter | L_{f,}_{1} | 1.3 mH |

R_{Lf,}_{1} | 0.1 Ω | ||

C_{f,}_{1} | 3 μF | ||

L_{g,}_{1} | 1.5 mH | ||

R_{Lg,}_{1} | 0.2 Ω | ||

PR current controller | K_{p,}_{1} | 10.5 | |

K_{i,}_{1} | 2000 | ||

K_{i,}_{5} | 500 | ||

K_{i,}_{7} | 500 | ||

K_{i,}_{11} | 500 | ||

Inverter 2 | LCL-filter | L_{f,}_{2} | 1.5 mH |

R_{Lf,}_{2} | 0.1 Ω | ||

C_{f,}_{2} | 4.7 μF | ||

L_{g,}_{2} | 1.8 mH | ||

R_{Lg,}_{2} | 0.2 Ω | ||

PR current controller | K_{p}_{2} | 18 | |

K_{i,}_{2} | 3000 | ||

K_{i,}_{5} | 500 | ||

K_{i,}_{7} | 500 | ||

K_{i,}_{11} | 500 | ||

DC-link voltage | V_{dc} | 700 V | |

Grid | L_{s} | 0.6 mH | |

R_{s} | 0.2 Ω | ||

PFC device capacity | Y_{PFC} | 30 μF |

ω_{f} (Hz) | R_{d} | R_{d,}_{1} | R_{d,}_{2} | R_{d, p} |
---|---|---|---|---|

1600 | −0.0041 | 0.0038 | −0.0133 | 0.0054 |

Case | Number of Inverters | Resonance Frequency | R_{d} | System Stability |
---|---|---|---|---|

I | 4 | 2000 Hz | 0.02 | stable |

II | 6 | 2250 Hz | −0.1 | unstable |

III | 8 | 2250 Hz | −0.6 | unstable |

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## Share and Cite

**MDPI and ACS Style**

Cao, W.; Liu, K.; Wang, S.; Kang, H.; Fan, D.; Zhao, J.
Harmonic Stability Analysis for Multi-Parallel Inverter-Based Grid-Connected Renewable Power System Using Global Admittance. *Energies* **2019**, *12*, 2687.
https://doi.org/10.3390/en12142687

**AMA Style**

Cao W, Liu K, Wang S, Kang H, Fan D, Zhao J.
Harmonic Stability Analysis for Multi-Parallel Inverter-Based Grid-Connected Renewable Power System Using Global Admittance. *Energies*. 2019; 12(14):2687.
https://doi.org/10.3390/en12142687

**Chicago/Turabian Style**

Cao, Wu, Kangli Liu, Shunyu Wang, Haotian Kang, Dongchen Fan, and Jianfeng Zhao.
2019. "Harmonic Stability Analysis for Multi-Parallel Inverter-Based Grid-Connected Renewable Power System Using Global Admittance" *Energies* 12, no. 14: 2687.
https://doi.org/10.3390/en12142687