# Analysis of an H∞ Robust Control for a Three-Phase Voltage Source Inverter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Control Scheme

## 3. State Space Model of the Augmented Plant

## 4. Formulation of the Standard H∞ Control Problem

^{®}.

## 5. Design Example

#### 5.1. Design of H∞ Current Controller

#### 5.2. Design of the H∞ Voltage Controller

#### 5.3. Design of PI and DB Predictive Control Schemes

## 6. Simulation Results

#### 6.1. Steady-State Performance in Stand-Alone Mode

#### 6.1.1. Steady-State Performance with a Resistive Load

#### 6.1.2. Steady-State Performance with Non-Linear Load

#### 6.1.3. Transient Response with Resistive Load

## 7. Conclusion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Liserre, M.; Fuchs, F.W.; Blaabjerg, F.; Dannehl, J.; Pena-Alzola, R.; Sebastian, R. Systematic Design of the Lead-Lag Network Method for Active Damping in LCL-Filter Based Three Phase Converters. IEEE Trans. Ind. Inform.
**2013**, 10, 43–52. [Google Scholar] [CrossRef] - Dou, C.X.; Jin, S.J.; Jiang, G.T.; Bo, Z.Q. Multi-agent based control framework for microgrids. In Proceedings of the 2009 Asia-Pacific Power and Energy Engineering Conference, Wuhan, China, 27–31 March 2009; pp. 1–4. [Google Scholar] [CrossRef]
- Sahoo, A.K.; Shahani, A.; Basu, K.; Mohan, N. LCL filter design for grid-connected inverters by analytical estimation of PWM ripple voltage. In Proceedings of the 2014 IEEE Applied Power Electronics Conference and Exposition- APEC 2014, Fort Worth, TX, USA, 16–20 March 2014; pp. 1281–1286. [Google Scholar] [CrossRef]
- Holmes, D.G.; Lipo, T.A.; McGrath, B.P.; Kong, W.Y. Optimized Design of Stationary Frame Three Phase AC Current Regulators. IEEE Trans. Power Electron.
**2009**, 24, 2417–2426. [Google Scholar] [CrossRef] - Sangwongwanich, A.; Abdelhakim, A.; Yang, Y.; Zhou, K. Control of Single-Phase and Three-Phase DC/AC Converters; Elsevier Inc.: Amsterdam, The Netherlands, 2018; ISBN 9780128052457. [Google Scholar]
- Hornik, T.; Zhong, Q.C. A current-control strategy for voltage-source inverters in microgrids based on H∞and Repetitive Control. IEEE Trans. Power Electron.
**2011**, 26, 943–952. [Google Scholar] [CrossRef] - Lee, T.; Chang, J. H∞ Loop-Shaping Controller Designs for the Single-Phase Inverters. IEEE Trans. Power Electron.
**2001**, 16, 473–481. [Google Scholar] - Chowdhury, M.A.; Kashem, S.B.A. H∞ loop-shaping controller design for a grid- connected single-phase photovoltaic system. Int. J. Sustain. Eng.
**2018**, 11, 196–204. [Google Scholar] [CrossRef] - Komurcugil, H. Rotating-sliding-line-based sliding-mode control for single-phase UPS inverters. IEEE Trans. Ind. Electron.
**2012**, 59, 3719–3726. [Google Scholar] [CrossRef] - Tahir, S.; Wang, J.; Baloch, M.; Kaloi, G. Digital Control Techniques Based on Voltage Source Inverters in Renewable Energy Applications: A Review. Electronics
**2018**, 7, 18. [Google Scholar] [CrossRef] - Quan, X.; Huang, A.Q.; Dou, X.; Wu, Z.; Hu, M. A novel adaptive control for three-phase inverter. In Proceedings of the 2018 IEEE Applied Power Electronics Conference and Exposition (APEC), San Antonio, TX, USA, 4–8 March 2018; pp. 1014–1018. [Google Scholar] [CrossRef]
- Espi, J.M.; Castello, J.; Garcia-Gil, R.; Garcera, G.; Figueres, E. An Adaptive Robust Predictive Current Control for Three-Phase Grid-Connected Inverters. IEEE Trans. Ind. Electron.
**2011**, 58, 3537–3546. [Google Scholar] [CrossRef] [Green Version] - Colak, I.; Kabalci, E.; Bayindir, R. Review of multilevel voltage source inverter topologies and control schemes. Energy Convers. Manag.
**2011**, 52, 1114–1128. [Google Scholar] [CrossRef] - Trivedi, A.; Singh, M. Repetitive Controller for VSIs in Droop-Based AC-Microgrid. IEEE Trans. Power Electron.
**2017**, 32, 6595–6604. [Google Scholar] [CrossRef] - Mohamed, I.S.; Zaid, S.A.; Abu-Elyazeed, M.F.; Elsayed, H.M. Classical methods and model predictive control of three-phase inverter with output LC filter for UPS applications. In Proceedings of the 2013 International Conference on Control, Decision and Information Technologies (CoDIT), Hammamet, Tunisia, 6–8 May 2013; pp. 483–488. [Google Scholar] [CrossRef]
- Cortés, P.; Ortiz, G.; Yuz, J.I.; Rodríguez, J.; Vazquez, S.; Franquelo, L.G. Model predictive control of an inverter with output LC filter for UPS applications. IEEE Trans. Ind. Electron.
**2009**, 56, 1875–1883. [Google Scholar] [CrossRef] - Mattavelli, P. An improved deadbeat control for UPS using disturbance observers. IEEE Trans. Ind. Electron.
**2005**, 52, 206–212. [Google Scholar] [CrossRef] - Ibrahim Mohamed, Y.A.R.; El-Saadany, E.F. An improved deadbeat current control scheme with a novel adaptive self-tuning load model for a three-phase PWM voltage-source inverter. IEEE Trans. Ind. Electron.
**2007**, 54, 747–759. [Google Scholar] [CrossRef] - Hornik, T.; Zhong, Q.-C. H∞ repetitive voltage control of grid-connected inverters with a frequency adaptive mechanism. IET Power Electron.
**2010**, 3, 925. [Google Scholar] [CrossRef] - Laub, A.J.; Heath, M.T.; Paige, C.C.; Ward, R.C. Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms. IEEE Trans. Automat. Contr.
**1987**, 32, 115–122. [Google Scholar] [CrossRef] - Rasool, M.A.U.; Khan, M.M.; Faiz, M.T.; Zhang, W.; Tahir, S. An Optimized Disturbance Observer Based Digital Deadbeat Control Technique for Three-Phase Voltage Source Inverter. In Proceedings of the 2018 International Conference on Electronics and Electrical Engineering Technology, Tianjin, China, 19–21 September 2018; pp. 27–33. [Google Scholar] [CrossRef]
- Pichan, M.; Rastegar, H.; Monfared, M. Deadbeat Control of the Stand-Alone Four-Leg Inverter Considering the Effect of the Neutral Line Inductor. IEEE Trans. Ind. Electron.
**2017**, 64, 2592–2601. [Google Scholar] [CrossRef] - Chung, I.Y.; Liu, W.; Cartes, D.A.; Collins, E.G.; Moon, S. Il Control methods of inverter-interfaced distributed generators in a microgrid system. IEEE Trans. Ind. Appl.
**2010**, 46, 1078–1088. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Standard feedback robust control system, (

**b**) Block diagram of the H∞ robust voltage control scheme.

**Figure 6.**Simulation results with the H∞ robust controller in steady-state condition. (

**a**) Output voltage, (

**b**) Load current, (

**c**) α-frame voltage waveform, (

**d**) Voltage tracking Error, (

**e**) Voltage THD, (

**f**) Current THD.

**Figure 7.**Simulation results with the DB predictive controller in steady-state condition. (

**a**) Output voltage (

**b**) Load current, (

**c**) α-frame voltage waveform, (

**d**) Voltage tracking error, (

**e**) Voltage THD, (

**f**) Current THD.

**Figure 8.**Simulation results with the PI controller in steady-state condition with linear loads. (

**a**) Output voltage, (

**b**) Load current, (

**c**) d-frame voltage waveform, (

**d**) d-frame voltage tracking error, (

**e**) Voltage THD, (

**f**) Current THD.

**Figure 9.**Simulation results with the H∞ robust controller in steady-state condition with non-linear loads. (

**a**) Output voltage, (

**b**) Load current, (

**c**) THD value of voltage, (

**d**) THD value of current.

**Figure 10.**Simulation results with the DB predictive controller in steady-state condition. (

**a**) Output voltage, (

**b**) Load current, (

**c**) THD value of voltage, (

**d**) THD values of current.

**Figure 11.**Simulation results with the PI controller in steady-state condition. (

**a**) Output voltage, (

**b**) Load current, (

**c**) THD value of voltage, (

**d**) THD value of current.

**Figure 12.**Simulation results with the H∞ robust controller in transient condition. (

**a**) Output voltage, (

**b**) Load current, (

**c**) α-frame voltage waveform, (

**d**) Voltage tracking error.

**Figure 13.**Simulation results with the DB predictive controller in transient condition. (

**a**) Output voltage, (

**b**) Load current, (

**c**) α-frame voltage waveform, (

**d**) Voltage tracking error.

**Figure 14.**Simulation results with the PI controller in transient condition. (

**a**) Output voltage, (

**b**) Load current, (

**c**) d-frame voltage waveform, (

**d**) Voltage tracking error.

Description | Variable | Value |
---|---|---|

DC-Link Voltage | V_{DC} | 440 V |

Rated Power Output | Po | 9 kW |

Filter Capacitance | C_{f} | 15 μF |

Filter Inductance | L_{f} | 2.7 mH |

Filter Resistance | R_{f} | 0.1 Ω |

Damping Resistance | R_{d} | 1 Ω |

Frequency _{PWM} | f_{S} | 12.8 kHz |

Controller | Gain | Value |
---|---|---|

Current Loop Proportional Gain | k_{pc} | 3.0 |

Current Loop Integral Gain | k_{ic} | 5.21 |

Voltage Loop Proportional Gain | k_{pv} | 0.1756 |

Voltage Loop Integral Gain | k_{pi} | 0.25449 |

Luenburger Gain | L_{M} | ${\left[\begin{array}{cc}5.3519& 1.0460\end{array}\right]}^{T}$ |

Observer Gain | η | 0.1 |

Controller | H∞ Robust | DB Predictive | Proportional Integral |
---|---|---|---|

Linear Loads (Voltage) | 0.30% | 0.60% | 0.40% |

Linear Loads (Current) | 1.60% | 3.38% | 2.83% |

Controller | H∞ Robust | DB Predictive | Proportional Integral |
---|---|---|---|

Non-Linear Loads (Voltage) | 3.06% | 4.54% | 4.70% |

Non-Linear Loads (Current) | 85.96% | 93.34% | 98.98% |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rasool, M.A.U.; Khan, M.M.; Ahmed, Z.; Saeed, M.A.
Analysis of an H∞ Robust Control for a Three-Phase Voltage Source Inverter. *Inventions* **2019**, *4*, 18.
https://doi.org/10.3390/inventions4010018

**AMA Style**

Rasool MAU, Khan MM, Ahmed Z, Saeed MA.
Analysis of an H∞ Robust Control for a Three-Phase Voltage Source Inverter. *Inventions*. 2019; 4(1):18.
https://doi.org/10.3390/inventions4010018

**Chicago/Turabian Style**

Rasool, Muhammad Ahmad Usman, Muhammad Mansoor Khan, Zahoor Ahmed, and Muhammad Abid Saeed.
2019. "Analysis of an H∞ Robust Control for a Three-Phase Voltage Source Inverter" *Inventions* 4, no. 1: 18.
https://doi.org/10.3390/inventions4010018