Robust Stability Analysis of Grid-Connected Converters Based on Parameter-Dependent Lyapunov Functions
This paper deals with the problem of robust stability analysis of grid-connected converters with LCL filters controlled through a digital signal processor and subject to uncertain grid inductance. To model the uncertain continuous-time plant and the digital control gain, a discretization procedure,...
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| Published in | Journal of control, automation & electrical systems Vol. 28; no. 2; pp. 159 - 170 |
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| Main Authors | , , , , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.04.2017
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2195-3880 2195-3899 |
| DOI | 10.1007/s40313-017-0301-7 |
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| Abstract | This paper deals with the problem of robust stability analysis of grid-connected converters with LCL filters controlled through a digital signal processor and subject to uncertain grid inductance. To model the uncertain continuous-time plant and the digital control gain, a discretization procedure, described in terms of a Taylor series expansion, is employed to determine an accurate discrete-time model. Then, a linear matrix inequality-based condition is proposed to assess the robust stability of the polynomial discrete-time augmented system that includes the filter state variables, the states of resonant controllers and the delay from the digital control implementation. By means of a parameter-dependent Lyapunov function, the proposed strategy has as main advantage to provide theoretical certification of stability of the uncertain continuous-time closed-loop system, circumventing the main disadvantages of previous approaches that employ approximate discretized models, neglecting the errors. Numerical simulations illustrate the benefits of the discretization technique and experimental results validate the proposed approach. |
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| AbstractList | This paper deals with the problem of robust stability analysis of grid-connected converters with LCL filters controlled through a digital signal processor and subject to uncertain grid inductance. To model the uncertain continuous-time plant and the digital control gain, a discretization procedure, described in terms of a Taylor series expansion, is employed to determine an accurate discrete-time model. Then, a linear matrix inequality-based condition is proposed to assess the robust stability of the polynomial discrete-time augmented system that includes the filter state variables, the states of resonant controllers and the delay from the digital control implementation. By means of a parameter-dependent Lyapunov function, the proposed strategy has as main advantage to provide theoretical certification of stability of the uncertain continuous-time closed-loop system, circumventing the main disadvantages of previous approaches that employ approximate discretized models, neglecting the errors. Numerical simulations illustrate the benefits of the discretization technique and experimental results validate the proposed approach. |
| Author | Braga, Márcio F. Montagner, Vinicius F. Tognetti, Eduardo S. Peres, Pedro L. D. Maccari, Luiz A. Morais, Cecília F. Oliveira, Ricardo C. L. F. |
| Author_xml | – sequence: 1 givenname: Márcio F. surname: Braga fullname: Braga, Márcio F. organization: Department of Electrical Engineering, Institute of Exact and Applied Sciences, Federal University of Ouro Preto – UFOP – sequence: 2 givenname: Cecília F. surname: Morais fullname: Morais, Cecília F. organization: School of Electrical and Computer Engineering, University of Campinas – UNICAMP – sequence: 3 givenname: Luiz A. surname: Maccari fullname: Maccari, Luiz A. organization: Research Group of Control, Automation, Robotics and Systems, Federal University of Santa Catarina – UFSC – sequence: 4 givenname: Eduardo S. surname: Tognetti fullname: Tognetti, Eduardo S. email: estognetti@ene.unb.br organization: Department of Electrical Engineering, University of Brasilia – UnB – sequence: 5 givenname: Vinicius F. surname: Montagner fullname: Montagner, Vinicius F. organization: Power Electronics and Control Research Group, Federal University of Santa Maria – UFSM – sequence: 6 givenname: Ricardo C. L. F. surname: Oliveira fullname: Oliveira, Ricardo C. L. F. organization: School of Electrical and Computer Engineering, University of Campinas – UNICAMP – sequence: 7 givenname: Pedro L. D. surname: Peres fullname: Peres, Pedro L. D. organization: School of Electrical and Computer Engineering, University of Campinas – UNICAMP |
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| Cites_doi | 10.1109/TIE.2008.2008774 10.1109/TIA.2010.2049974 10.1016/0005-1098(96)00063-5 10.1109/MIE.2010.935862 10.1007/s40313-013-0003-8 10.1007/978-1-4471-3365-0 10.1109/CACSD.2004.1393890 10.1137/1.9781611970777 10.1109/CDC.1996.572667 10.1109/CDC.2007.4434084 10.1109/TCST.2013.2292879 10.1109/TPEL.2013.2279015 10.1109/TPEL.2009.2016097 10.1109/TCSII.2006.883832 10.1109/31.7598 10.1109/TII.2013.2263506 10.1109/TAC.2007.900848 10.1109/TIA.2013.2266892 10.1109/TPEL.2010.2095429 10.1109/41.192662 10.1016/0005-1098(76)90006-6 10.1109/TCST.2013.2280717 10.1109/TPEL.2005.861185 10.1007/s40313-014-0166-y 10.1109/TII.2012.2222424 10.1049/iet-pel.2012.0032 10.1109/TPEL.2004.839821 10.1093/ietfec/e89-a.8.2166 10.1080/00207179.2010.536996 10.1109/TCST.2011.2114348 10.1016/j.conengprac.2011.03.004 10.1109/CDC.2013.6760901 |
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| Keywords | Linear matrix inequalities Grid-connected converters Sampled-data systems Parameter-dependent Lyapunov function Robust stability analysis |
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| References | Pena-AlzolaRLiserreMBlaabjergFSebastianRDannehlJFuchsFSystematic design of the lead-lag network method for active damping in LCL-filter based three phase convertersIEEE Transaction on Industrial Informatics2014101435210.1109/TII.2013.2263506 Su, H., Wang, J., & Chu, J. (1998). Robust memoryless H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal H\it }}_\infty $$\end{document} control for uncertain linear time-delay systems. In Proceedings of the 1998 American control conference, Philadelphia, PA, USA (pp. 3730–3731). Maccari, L., Massing, J. R., Schuch, L., Rech, C., Pinheiro, H., & Montagner, V. F., et al. (2012). Robust H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal H\it }_\infty $$\end{document} control for grid connected PWM inverters with LCL filters. InProceedings of the 10th IEEE / IAS international conference on industry applications (INDUSCON) (pp. 1–6). CE, Brazil: Fortaleza. WadaNSaitoKSaekiMModel predictive control for linear parameter varying systems using parameter dependent Lyapunov functionIEEE Transactions on Circuits and Systems II: Express Briefs200653121446145010.1109/TCSII.2006.883832 Hara, S., Yamamoto, Y., & Fujioka, H. (1996). Modern and classical analysis/synthesis methods in sampled-data control – A brief overview with numerical examples. In Proceedings of the 35th IEEE conference on decision and control, Kobe, Japan (pp. 1251–1256). Braga, M. F., Morais, C. F., Tognetti, E. S., Oliveira, R. C. L. F., & Peres, P. L. D. (2013). A new procedure for discretization and state feedback control of uncertain linear systems. In Proceedings of the 52nd IEEE conference on decision and control, Florence, Italy (pp. 6397–6402). YinJDuanSLiuBStability analysis of grid-connected inverter with LCL filter adopting a digital single-loop controller with inherent damping characteristicIEEE Transaction on Industrial Informatics2013921104111210.1109/TII.2012.2222424 Sturm, J. F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11(1–4), 625–653. http://sedumi.ie.lehigh.edu/. MaccariLMassingJRSchuchLRechCPinheiroHOliveiraRCLFLMI-based control for grid-connected converters with LCL filters under uncertain parametersIEEE Transactions on Power Electronics20142973776378510.1109/TPEL.2013.2279015 AgorretaJBorregaMLópezJMarroyoLModeling and control of N-paralleled grid-connected inverters with LCL filter coupled due to grid impedance in PV plantsIEEE Transactions on Power Electronics201126377078510.1109/TPEL.2010.2095429 Althoff, M., Stursberg, O., & Buss, M. (2007). Reachability analysis of linear systems with uncertain parameters and inputs. InProceedings of the 46th IEEE conference on decision and control, New Orleans, LA (pp. 726–732). Löfberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the 2004 IEEE international symposium on computer aided control systems design, Taipei, Taiwan (pp. 284–289). FrancisBAWonhamWMThe internal model principle of control theoryAutomatica197612545746542925710.1016/0005-1098(76)90006-60344.93028 LiserreMTeodorescuRBlaabjergFStability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance valuesIEEE Transactions on Power Electronics200621126327210.1109/TPEL.2005.861185 MezaCBielDJeltsemaDScherpenJLyapunov-based control scheme for single-phase grid-connected PV central invertersIEEE Transactions on Control Systems Technology201220252052910.1109/TCST.2011.2114348 ChenJYangFHanQLModel-free predictive H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\infty }$$\end{document} control for grid-connected solar power generation systemsIEEE Transactions on Control Systems Technology20142252039204710.1109/TCST.2013.2292879 MukherjeeNDeDAnalysis and improvement of performance in LCL filter-based PWM rectifier/inverter application using hybrid damping approachIET Power Electronics20136230932510.1049/iet-pel.2012.0032 ÅströmKJWittenmarkBComputer controlled systems: Theory and design1984Englewood Cliffs, NJPrentice Hall Inc MattavelliPSpiazziGTentiPPredictive digital control of power factor preregulators with input voltage estimation using disturbance observersIEEE Transactions on Power Electronics200520114014710.1109/TPEL.2004.839821 BoydSEl GhaouiLFeronEBalakrishnanVLinear matrix inequalities in system and control theory1994Philadelphia, PASIAM Studies in Applied Mathematics10.1137/1.97816119707770816.93004 DharSDashPKA finite time fast terminal sliding mode I-V control of grid-connected PV arrayJournal of Control, Automation and Electrical Systems201526331433510.1007/s40313-014-0166-y ParkerSMcGrathBHolmesDRegions of active damping control for LCL filtersIEEE Transactions on Industry Applications201450142443210.1109/TIA.2013.2266892 DannehlJWesselsCFuchsFLimitations of voltage-oriented PI current control of grid-connected PWM rectifiers with filtersIEEE Transactions on Industrial Electronics200956238038810.1109/TIE.2008.2008774 KothareMVBalakrishnanVMorariMRobust constrained model predictive control using linear matrix inequalitiesAutomatica1996321013611379142003810.1016/0005-1098(96)00063-50897.93023 AckermannJRobust control: Systems with uncertain physical parameters1993LondonSpringer10.1007/978-1-4471-3365-00853.93002 OppenheimerEPMichelANApplication of interval analysis techniques to linear systems: Part ii-The interval matrix exponential functionIEEE Transactions on Circuits and Systems198835101230124296077510.1109/31.7598 LeeSWonSModel Predictive Control for linear parameter varying systems using a new parameter dependent terminal weighting matrixIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E200689A82166217210.1093/ietfec/e89-a.8.2166 Olalla, C., Queinnec, I., Leyva, R., & Aroudi, A. E. (2011). Robust optimal control of bilinear DC-DC converters. Control Engineering Practice, 19(7), 688–699. GuerreroJBlaabjergFZhelevTHemmesKMonmassonEJemeiSDistributed generation: Toward a new energy paradigmIEEE Industrial Electronics Magazine201041526410.1109/MIE.2010.935862 KawamuraAChuarayapratipRHaneyoshiTDeadbeat control of PWM inverter with modified pulse patterns for uninterruptible power supplyIEEE Transactions on Industrial Electronics198835229530010.1109/41.192662 DannehlJFuchsFHansenSThøgersenPInvestigation of active damping approaches for PI-based current control of grid-connected pulse width modulation converters with LCL filtersIEEE Transactions on Industry Applications20104641509151710.1109/TIA.2010.2049974 BragaMFMoraisCFTognettiESOliveiraRCLFPeresPLDDiscretization and control of polytopic systems with uncertain sampling rates and network-induced delaysInternational Journal of Control201487112398241132577721308.93131 JungersMOliveiraRCLFPeresPLDMPC for LPV systems with bounded parameter variationsInternational Journal of Control2011842436277387510.1080/00207179.2010.5369961222.93136 KatayamaHAokiHStraight-line trajectory tracking control for sampled-data underactuated shipsIEEE Transactions on Control Systems Technology20142241638164510.1109/TCST.2013.2280717 GabeIMontagnerVFPinheiroHDesign and implementation of a robust current controller for VSI connected to the grid through an LCL filterIEEE Transactions on Power Electronics20092461444145210.1109/TPEL.2009.2016097 OliveiraRCLFPeresPLDParameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxationsIEEE Transactions on Automatic Control200752713341340233276210.1109/TAC.2007.900848 MoraesVMCastelanEBMorenoUFFull-order dynamic output-feedback compensator for time-stamped networked control systemsJournal of Control, Automation and Electrical Systems2013241223210.1007/s40313-013-0003-8 301_CR15 J Yin (301_CR37) 2013; 9 A Kawamura (301_CR18) 1988; 35 VM Moraes (301_CR27) 2013; 24 301_CR35 J Agorreta (301_CR3) 2011; 26 BA Francis (301_CR12) 1976; 12 MV Kothare (301_CR19) 1996; 32 301_CR34 N Wada (301_CR36) 2006; 53 J Ackermann (301_CR2) 1993 J Dannehl (301_CR10) 2009; 56 I Gabe (301_CR13) 2009; 24 M Jungers (301_CR16) 2011; 84 RCLF Oliveira (301_CR30) 2007; 52 H Katayama (301_CR17) 2014; 22 C Meza (301_CR26) 2012; 20 MF Braga (301_CR7) 2014; 87 L Maccari (301_CR24) 2014; 29 S Parker (301_CR32) 2014; 50 S Dhar (301_CR11) 2015; 26 301_CR29 EP Oppenheimer (301_CR31) 1988; 35 S Lee (301_CR20) 2006; 89A 301_CR22 N Mukherjee (301_CR28) 2013; 6 301_CR23 S Boyd (301_CR5) 1994 P Mattavelli (301_CR25) 2005; 20 301_CR1 J Guerrero (301_CR14) 2010; 4 M Liserre (301_CR21) 2006; 21 J Dannehl (301_CR9) 2010; 46 KJ Åström (301_CR4) 1984 301_CR6 J Chen (301_CR8) 2014; 22 R Pena-Alzola (301_CR33) 2014; 10 |
| References_xml | – reference: Maccari, L., Massing, J. R., Schuch, L., Rech, C., Pinheiro, H., & Montagner, V. F., et al. (2012). Robust H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal H\it }_\infty $$\end{document} control for grid connected PWM inverters with LCL filters. InProceedings of the 10th IEEE / IAS international conference on industry applications (INDUSCON) (pp. 1–6). CE, Brazil: Fortaleza. – reference: OppenheimerEPMichelANApplication of interval analysis techniques to linear systems: Part ii-The interval matrix exponential functionIEEE Transactions on Circuits and Systems198835101230124296077510.1109/31.7598 – reference: Sturm, J. F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11(1–4), 625–653. http://sedumi.ie.lehigh.edu/. – reference: BoydSEl GhaouiLFeronEBalakrishnanVLinear matrix inequalities in system and control theory1994Philadelphia, PASIAM Studies in Applied Mathematics10.1137/1.97816119707770816.93004 – reference: MattavelliPSpiazziGTentiPPredictive digital control of power factor preregulators with input voltage estimation using disturbance observersIEEE Transactions on Power Electronics200520114014710.1109/TPEL.2004.839821 – reference: GabeIMontagnerVFPinheiroHDesign and implementation of a robust current controller for VSI connected to the grid through an LCL filterIEEE Transactions on Power Electronics20092461444145210.1109/TPEL.2009.2016097 – reference: DannehlJWesselsCFuchsFLimitations of voltage-oriented PI current control of grid-connected PWM rectifiers with filtersIEEE Transactions on Industrial Electronics200956238038810.1109/TIE.2008.2008774 – reference: Löfberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the 2004 IEEE international symposium on computer aided control systems design, Taipei, Taiwan (pp. 284–289). – reference: YinJDuanSLiuBStability analysis of grid-connected inverter with LCL filter adopting a digital single-loop controller with inherent damping characteristicIEEE Transaction on Industrial Informatics2013921104111210.1109/TII.2012.2222424 – reference: MukherjeeNDeDAnalysis and improvement of performance in LCL filter-based PWM rectifier/inverter application using hybrid damping approachIET Power Electronics20136230932510.1049/iet-pel.2012.0032 – reference: LeeSWonSModel Predictive Control for linear parameter varying systems using a new parameter dependent terminal weighting matrixIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E200689A82166217210.1093/ietfec/e89-a.8.2166 – reference: Althoff, M., Stursberg, O., & Buss, M. (2007). Reachability analysis of linear systems with uncertain parameters and inputs. InProceedings of the 46th IEEE conference on decision and control, New Orleans, LA (pp. 726–732). – reference: OliveiraRCLFPeresPLDParameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxationsIEEE Transactions on Automatic Control200752713341340233276210.1109/TAC.2007.900848 – reference: DharSDashPKA finite time fast terminal sliding mode I-V control of grid-connected PV arrayJournal of Control, Automation and Electrical Systems201526331433510.1007/s40313-014-0166-y – reference: KothareMVBalakrishnanVMorariMRobust constrained model predictive control using linear matrix inequalitiesAutomatica1996321013611379142003810.1016/0005-1098(96)00063-50897.93023 – reference: ÅströmKJWittenmarkBComputer controlled systems: Theory and design1984Englewood Cliffs, NJPrentice Hall Inc – reference: MezaCBielDJeltsemaDScherpenJLyapunov-based control scheme for single-phase grid-connected PV central invertersIEEE Transactions on Control Systems Technology201220252052910.1109/TCST.2011.2114348 – reference: JungersMOliveiraRCLFPeresPLDMPC for LPV systems with bounded parameter variationsInternational Journal of Control2011842436277387510.1080/00207179.2010.5369961222.93136 – reference: MoraesVMCastelanEBMorenoUFFull-order dynamic output-feedback compensator for time-stamped networked control systemsJournal of Control, Automation and Electrical Systems2013241223210.1007/s40313-013-0003-8 – reference: ChenJYangFHanQLModel-free predictive H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\infty }$$\end{document} control for grid-connected solar power generation systemsIEEE Transactions on Control Systems Technology20142252039204710.1109/TCST.2013.2292879 – reference: LiserreMTeodorescuRBlaabjergFStability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance valuesIEEE Transactions on Power Electronics200621126327210.1109/TPEL.2005.861185 – reference: WadaNSaitoKSaekiMModel predictive control for linear parameter varying systems using parameter dependent Lyapunov functionIEEE Transactions on Circuits and Systems II: Express Briefs200653121446145010.1109/TCSII.2006.883832 – reference: KatayamaHAokiHStraight-line trajectory tracking control for sampled-data underactuated shipsIEEE Transactions on Control Systems Technology20142241638164510.1109/TCST.2013.2280717 – reference: Olalla, C., Queinnec, I., Leyva, R., & Aroudi, A. E. (2011). Robust optimal control of bilinear DC-DC converters. Control Engineering Practice, 19(7), 688–699. – reference: Pena-AlzolaRLiserreMBlaabjergFSebastianRDannehlJFuchsFSystematic design of the lead-lag network method for active damping in LCL-filter based three phase convertersIEEE Transaction on Industrial Informatics2014101435210.1109/TII.2013.2263506 – reference: Braga, M. F., Morais, C. F., Tognetti, E. S., Oliveira, R. C. L. F., & Peres, P. L. D. (2013). A new procedure for discretization and state feedback control of uncertain linear systems. 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| SubjectTerms | Closed loops Continuous time systems Control Control and Systems Theory Digital signal processors Discrete time systems Discretization Electrical Engineering Engineering Feedback control Inductance Liapunov functions Linear matrix inequalities Mechatronics Microprocessors Numerical models Parameters Polynomials Robotics Robotics and Automation Robustness Series expansion Stability analysis Taylor series |
| Title | Robust Stability Analysis of Grid-Connected Converters Based on Parameter-Dependent Lyapunov Functions |
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