# Large-Signal Stabilization of Three-Phase VSR with Constant Power Load

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Obtaining a large-signal model of three-phase VSR with CPL based on mixed potential theory, whose stable boundary is derived when load power jumps.
- (2)
- Proposing a voltage control scheme to improve the large-signal stability based on ADRC, and whose control stability is proven.
- (3)
- Deriving the stable boundary of VSR with CPL based on ADRC, which proves that the proposed control scheme expands the load power jump range effectively.

## 2. Large-Signal Stability Analysis Based on Mixed Potential Theory under PI Control

#### 2.1. Introduction to the Mixed Potential Theory

_{1}, …, i

_{r}represents inductor currents, and v

_{r+}

_{1}, …, v

_{r+s}are capacitor voltages. P is defined as:

_{μ}can not be represented by i

_{μ}, the current potential can be written by voltage potential and Equation (2):

#### 2.2. Stability Problems of the Three-Phase VSR with CPL

_{CPL}to rise, in order to maintain its output constant, the input voltage v

_{CPL}of the CPL will decrease. However, this will lead to an increase in the voltage on the resistance Rs to make the current i

_{CPL}rise again. This response, similar to positive feedback, takes the circuit away from the operating point, and causes it to lose stability. The larger the disturbance, the more significant stability problems caused by CPL will be.

#### 2.3. Analysis of Power Jump Range of Three-Phase VSR with CPL

_{d,q}are the control voltage in the dq coordinates respectively, i

_{d,q}are the grid current in the dq coordinates respectively, i

_{o}is the output current of the equivalent current source on the DC side, and v

_{dc}is the output voltage of the VSR.

_{d,q}are the control voltage in the dq coordinates respectively, R is the equivalent series resistance of the grid side, i

_{L}is the load current of the VSR.

_{CPL}is the power of the CPL.

_{dcref}is the output voltage reference of the VSR, I

_{qref}is the current reference of the q axis, ω is the frequency of the grid, K

_{ip}and K

_{ii}are the proportion and integral coefficients of PI regulator of d axis current and q axis current respectively, K

_{vp}, K

_{vi}are the proportion and integral coefficients of PI regulator in voltage loop, respectively.

**A**(i) and

**B**(v) are solved as:

## 3. Control Scheme and Stability Analysis Based on ADRC

#### 3.1. Voltage Loop Control Scheme Based on the First Order ADRC

_{1}, x

_{2}is the state variables, and $h=\dot{f}\left(y,\dot{y},u\right)$.

_{1}and β

_{2}make z

_{1}and z

_{2}in ESO achieve good tracking effect on x

_{1}and x

_{2}. Ignore the estimation error of ESO, that is, ${z}_{1}\to {x}_{1}$ and ${z}_{2}\to {x}_{2}$. Meanwhile, if we let

#### 3.2. Parameters Simplification and Stability Proof

_{c}= 100. According to engineering experience, ω

_{0}is generally five to ten times ω

_{c}, and is set as ω

_{0}= 800 here.

_{1}and z

_{2}of ESO in the frequency domain are given by:

_{p}> 0, the three-phase VSR based on ADRC is stable.

#### 3.3. Large Sigal Stability Analysis of Three-Phase VSR with CPL Based on ADRC

_{0}= b, then

**A**(i) and

**B**(v) are solved as:

_{p}= K

_{vp}), the difference between Equations (15) and (36) is determined by item M in Equation (15). The instability leads by large-signal disturbance mainly refers to the shock and divergence of the bus voltage when the CPL power increases. Therefore, under the premise of restoring stability, the minimum bus voltage must appear at the first oscillatory trough after the disturbance, as shown in Figure 7. In the process, the bus voltage reduces from V

_{dcref}to the minimum value V

_{dcmin}, energy in the capacitor is continuously extracted to meet the power requirement of the CPL, so e

_{d}i

_{d}− P

_{CPL_PI}< 0, or M < 0. It can be concluded that when there is a large-signal disturbance, the power jump range of VSR based on ADRC is larger than PI control. In other words, the stability is better.

## 4. Experiments

_{CPL_o}is 300 V, and switching frequency is 20 kHz.

_{dc}and grid current i

_{a}, i

_{b}, and i

_{c}begin to oscillate until the protection was triggered and the system shut down. This indicates that this large-signal disturbance (load power suddenly increases by five times) caused the system to lose stability, and that the conventional PI control failed.

_{dc}was quickly regulated back to the reference 650 V. In this case, the cascade system composed of VSR and CPL ran well, and the steady performance was good. The THD of grid currents was 3.5%, 3.9%, and 4% respectively. Compared with Figure 9a, the power jump range of the VSR with CPL had been effectively expanded and stability had been improved.

_{CPL_o}quickly back to 300 V, and the oscillation is very small. The CPL output current i

_{CPL_o}and output power P

_{CPL}also quickly reached the target value, which satisfies the CPL characteristic. This shows that the results of Figure 9 is meaningful and sufficient.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mixed Potential Theory Stability Theorem 3

_{1}is the minimum eigenvalue of matrix ${\mathit{L}}^{-1/2}{\mathit{A}}_{\mathit{i}\mathit{i}}\left(i\right){\mathit{L}}^{-1/2}$, μ

_{2}is the minimum eigenvalue of matrix ${\mathit{C}}^{-1/2}{\mathit{B}}_{\mathit{v}\mathit{v}}\left(v\right){\mathit{C}}^{-1/2}$. If all the i and v in the circuits meet

## References

- Ramirez-Hernandez, J.; Araujo-Vargas, I.; Rivera, M.; Ramirez-Hernandez, J.; Araujo-Vargas, I.; Rivera, M. A Modular AC-DC Power Converter with Zero Voltage Transition for Electric Vehicles. Energies
**2017**, 10, 1386. [Google Scholar] [CrossRef] - Nami, A.; Liang, J.; Dijkhuizen, F.; Demetriades, G.D. Modular Multilevel Converters for HVDC Applications: Review on Converter Cells and Functionalities. IEEE Trans. Power Electron.
**2015**, 30, 18–36. [Google Scholar] [CrossRef] - Mehrasa, M.; Pouresmaeil, E.; Zabihi, S.; Vechiu, I.; Catalao, J.P.S. A multi-loop control technique for the stable operation of modular multilevel converters in HVDC transmission systems. Int. J. Electr. Power Energy Syst.
**2018**, 96, 194–207. [Google Scholar] [CrossRef] - Zhang, X.; Zhong, Q.C.; Ming, W.L. Stabilization of a Cascaded DC Converter System via Adding a Virtual Adaptive Parallel Impedance to the Input of the Load Converter. IEEE Trans. Power Electron.
**2015**, 31, 1826–1832. [Google Scholar] [CrossRef] - Du, W.; Zhang, J.; Zhang, Y.; Qian, Z. Stability Criterion for Cascaded System with Constant Power Load. IEEE Trans. Power Electr.
**2013**, 28, 1843–1851. [Google Scholar] [CrossRef] - Mishima, T.; Akamatsu, K.; Nakaoka, M. A High Frequency-Link Secondary-Side Phase-Shifted Full-Range Soft-Switching PWM DC–DC Converter with ZCS Active Rectifier for EV Battery Chargers. IEEE Trans. Power Electr.
**2013**, 28, 5758–5773. [Google Scholar] [CrossRef] - Shin, S.C.; Lee, H.J.; Kim, Y.H.; Lee, J.H.; Won, C.Y. Transient Response Improvement at Startup of a Three-Phase AC/DC Converter for a DC Distribution System in Commercial Facilities. IEEE Trans. Power Electron.
**2014**, 29, 6742–6753. [Google Scholar] [CrossRef] - Cao, W.; Mecrow, B.C.; Atkinson, G.J.; Bennett, J.W.; Atkinson, D.J. Overview of Electric Motor Technologies Used for More Electric Aircraft (MEA). IEEE Trans. Ind. Electron.
**2012**, 59, 3523–3531. [Google Scholar] [CrossRef] - Verma, A.K.; Jain, C.; Singh, B.; Shahani, D.T. Adaptive noise cancellation based harmonic elimination in grid integrated photovoltaic system. IET Renew. Power Gener.
**2016**, 10, 1096–1104. [Google Scholar] [CrossRef] - Huang, M.; Peng, Y.; Chi, K.T.; Liu, Y.; Sun, J.; Zha, X. Bifurcation and large-signal stability analysis of three-phase voltage source converter under grid voltage dips. IEEE Trans. Power Electron.
**2017**, 32, 8868–8879. [Google Scholar] [CrossRef] - Mehrasa, M.; Pouresmaeil, E.; Zabihi, S.; Catalao, J.P.S. Dynamic model, control and stability analysis of MMC in HVDC transmission systems. IEEE Trans. Power Deliv.
**2016**, 32, 1471–1482. [Google Scholar] [CrossRef] - Mehrasa, M.; Pouresmaeil, E.; Zabihi, S.; Rodrigues, E.M.G.; Catalão, J.P.S. A control strategy for the stable operation of shunt active power filters in power grids. Energy
**2016**, 96, 325–334. [Google Scholar] [CrossRef] - Pouresmaeil, E.; Shaker, H.R.; Mehrasa, M.; Shokridehaki, M.A.; Rodrigues, E.M.G. Stability analysis for operation of DG units in smart grids. In Proceedings of the IEEE International Conference on Power Engineering, Energy and Electrical Drives, Riga, Latvia, 11–13 May 2015; pp. 447–452. [Google Scholar]
- Pouresmaeil, E.; Bo, N.; Mehrasa, M.; Erdinc, O.; Catalao, J.P.S. A control algorithm for the stable operation of interfaced converters in microgrid systems. In Proceedings of the Innovative Smart Grid Technologies Conference Europe, Istanbul, Turkey, 12–15 October 2014; pp. 1–6. [Google Scholar]
- Rahimi, A.M.; Emadi, A. Active Damping in DC/DC Power Electronic Converters: A Novel Method to Overcome the Problems of Constant Power Loads. IEEE Trans. Ind. Electron.
**2009**, 56, 1428–1439. [Google Scholar] [CrossRef] - Liu, X.; Zhou, Y.; Zhang, W.; Ma, S. Stability Criteria for Constant Power Loads with Multistage LC Filters. IEEE Trans. Veh. Technol.
**2011**, 60, 2042–2049. [Google Scholar] [CrossRef] - Zhang, M.; Li, Y.; Liu, F.; Luo, L.; Cao, Y.; Shahidehpour, M. Voltage Stability Analysis and Sliding Mode Control Method for Rectifier in DC Systems with Constant Power Loads. IEEE J. Emerg. Sel. Top. Power Electr.
**2017**, 5, 1621–1630. [Google Scholar] [CrossRef] - Mitchell, D.M. Damped EMI Filters for Switching Regulators. IEEE Trans. Electromagn. Compat.
**1978**, EMC-20, 457–463. [Google Scholar] [CrossRef] - Cespedes, M.; Xing, L.; Sun, J. Constant-Power Load System Stabilization by Passive Damping. IEEE Trans. Power Electr.
**2011**, 26, 1832–1836. [Google Scholar] [CrossRef] - Liu, B.; Ben, H.; Zhang, X.; Meng, T.; Wang, X. Stabilization of a cascaded AC/DC system based on small signal analysis, in Book Stabilization of a cascaded AC/DC system based on small signal analysis. In Proceedings of the International Conference on Electrical Machines and Systems, Sydney, Australia, 11–14 August 2017; pp. 1–6. [Google Scholar]
- Hatua, K.; Jain, A.K.; Banerjee, D.; Ranganathan, V.T. Active Damping of Output LC Filter Resonance for Vector-Controlled VSI-Fed AC Motor Drives. IEEE Trans. Ind. Electron.
**2012**, 59, 334–342. [Google Scholar] [CrossRef] - Dannehl, J.; Liserre, M.; Fuchs, F.W. Filter-Based Active Damping of Voltage Source Converters with LCL Filter. IEEE Trans. Ind. Electron.
**2011**, 58, 3623–3633. [Google Scholar] [CrossRef] - Liu, S.; Zhou, L.; Lu, W. Simple analytical approach to predict large-signal stability region of a closed-loop boost DC–DC converter. IET Power Electron.
**2013**, 6, 488–494. [Google Scholar] [CrossRef] - Weaver, W.W.; Iii, R.D.R.; Wilson, D.G.; Matthews, R.C. Metastability of Pulse Power Loads Using the Hamiltonian Surface Shaping Method. IEEE Trans. Energy Convers.
**2017**, 32, 820–828. [Google Scholar] [CrossRef] - Zhang, X.N.; Vilathgamuwa, D.M.; Foo, G.; Tseng, K.J.; Kandasamy, K.; Gupta, A.K.; Chandana, G. Cascaded sliding mode control for global stability of three phase AC/DC PWM rectifier with rapidly varying power electronic loads. In Proceedings of the 39th Annual Conference of the IEEE Industrial Electronics Society, Vienna, Austria, 10–14 November 2013; pp. 4580–4587. [Google Scholar]
- Magne, P.; Nahid-Mobarakeh, B.; Pierfederici, S. Dynamic Consideration of DC Microgrids with Constant Power Loads and Active Damping System—A Design Method for Fault-Tolerant Stabilizing System. IEEE J. Emerg. Sel. Top. Power Electron.
**2014**, 2, 562–570. [Google Scholar] [CrossRef] - Zhang, W.; Hou, Y.; Liu, X.; Zhou, Y. Switched Control of Three-Phase Voltage Source PWM Rectifier Under a Wide-Range Rapidly Varying Active Load. IEEE Trans. Power Electr.
**2012**, 27, 881–890. [Google Scholar] [CrossRef] - Zhang, P.H.; Yang, G.J.; Tie-Cai, L.I. Direct Voltage Control of Three-phase PWM Rectifier Based on Feedback Linearization. Proc. CSEE
**2010**, 30, 39–46. [Google Scholar] [CrossRef] - Han, J. From PID to Active Disturbance Rejection Control. IEEE Trans. Ind. Electron.
**2009**, 56, 900–906. [Google Scholar] [CrossRef] - Tian, J.; Zhang, S.; Zhang, Y.; Li, T. Active disturbance rejection control based robust output feedback autopilot design for airbreathing hypersonic vehicles. ISA Trans.
**2018**, 74, 45–59. [Google Scholar] [CrossRef] [PubMed] - Xiao, Y.; Hong, Y.; Chen, X.; Huo, W. Switching control of wind turbine sub-controllers based on an active disturbance rejection technique. Energies
**2016**, 9, 793. [Google Scholar] [CrossRef] - Brayton, R.K.; Moser, J.K. A theory of nonlinear networks. I. Q. Appl. Math.
**1964**, 2, 1–33. [Google Scholar] [CrossRef] - Jeltsema, D.; Scherpen, J.M.A. On Brayton and Moser’s missing stability theorem. IEEE Trans. Circuits Syst. II Express Briefs
**2005**, 52, 550–552. [Google Scholar] [CrossRef]

**Figure 9.**Main operating waveforms of the VSR when load power is changed from 2 kW to 10 kW. (

**a**) with conventional PI control; (

**b**) with the proposed ADRC.

**Figure 10.**Main operating waveforms of the CPL when load power is changed from 2 kW to 10 kW. (

**a**) with conventional PI control; (

**b**) with the proposed ADRC.

Symbol | Quantity | Value |
---|---|---|

E_{abc} | Grid phase voltage | 220 V |

f_{g} | Grid frequency | 50 Hz |

f_{s} | PWM frequency | 16 kHz |

L | Input inductance | 3.2 mH |

V_{dcref} | Bus voltage | 650 V |

R | Equivalent resistance | 0.2 Ω |

C | Bus capacitance | 100 μF |

K_{ip} | Proportional gain of the current PI regulator | 5 |

K_{ii} | Integral gain of the current PI regulator | 100 |

K_{vp} | Proportional gain of the voltage PI regulator | 0.2 |

K_{vi} | Integral gain of the voltage PI regulator | 80 |

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**MDPI and ACS Style**

Liu, B.; Ben, H.; Zhang, X.
Large-Signal Stabilization of Three-Phase VSR with Constant Power Load. *Energies* **2018**, *11*, 1706.
https://doi.org/10.3390/en11071706

**AMA Style**

Liu B, Ben H, Zhang X.
Large-Signal Stabilization of Three-Phase VSR with Constant Power Load. *Energies*. 2018; 11(7):1706.
https://doi.org/10.3390/en11071706

**Chicago/Turabian Style**

Liu, Bo, Hongqi Ben, and Xiaobing Zhang.
2018. "Large-Signal Stabilization of Three-Phase VSR with Constant Power Load" *Energies* 11, no. 7: 1706.
https://doi.org/10.3390/en11071706