Accurate state-estimation is a vital prerequisite for fast feedback control methods such as Nonlinear Model Predictive Control (NMPC). For efficient process control, it is of great importance that the estimation process is carried out as fast as possible to provide the feedback mechanism with fresh information and enable fast reactions in case of any disturbances. We discuss how Multi-Level Iterations (MLI), known from NMPC, can be applied to the Moving Horizon Estimation (MHE) method for estimating the states and parameters of a system described by a Differential Algebraic Equation model. A challenging field of application for the proposed MLI-MHE method are electric microgrids. These push current control approaches to their limits due to the rising penetration of volatile renewable energy sources and the fast electrical system dynamics. We investigate the closed-loop control performance of the proposed MLI-MHE algorithm in combination with an NMPC controller for a realistic sized microgrid as a numerical example.
Eine genaue Zustandsschätzung stellt eine wichtige Voraussetzung für die Anwendung von Feedback-Steuerungsmethoden wie Nichtlinearer Modellprädikativer Regelung (NMPC) dar. Um eine effiziente Prozesssteuerung zu gewährleisten, ist es notwendig dem Feedback-Mechanismus eine aktuelle Schätzung des Systemzustandes zur Verfügung zu stellen, um eine schnelle Reaktion auf unvorhergesehene Störungen zu ermöglichen. Wir diskutieren, wie die sogenannten Multi-Level Iterationen (MLI), welche schon im Kontext von NMPC bekannt sind, auf das Problem der Zustands- und Parameterschätzung auf bewegten Horizonten (Moving Horizon Estimation, MHE) für DAE Systeme angewandt werden können. Ein herausforderndes Anwendungsgebiet für die vorgeschlagene MLI-MHE-Methode sind elektrische Microgrids. Herkömmliche Regelungsansätze stoßen hierbei aufgrund der schnellen elektrischen Systemdynamik und des steigenden Anteils volatiler erneuerbarer Eenergiequellen im Stomnetz schnell an ihre Grenzen. Wir untersuchen die Closed-Loop Performance des vorgeschlagenen MLI-MHE Algorithmus in Kombination mit einem NMPC Controller anhand eines Microgrids von realistischer Größe als numerisches Beispiel.
Funding source: Bundesministerium für Bildung und Forschung
Award Identifier / Grant number: 05M18VHA
Funding statement: This research was funded by the German Federal Ministry of Education and Research (BMBF) in the research project MOReNet (Grant No 05M18VHA).
About the authors
Jürgen Gutekunst studied mathematics at the University of Tübingen, Germany and received his diploma degree in 2011. In 2019, he obtained a Ph.D. in Applied Mathematics from the University of Heidelberg, Germany. He is currently working as a postdoctoral researcher at the Institute for Applied Mathematics at Heidelberg University. His research interests include Economic Nonlinear Model Predictive Control and online state and parameter estimation.
Robert Scholz received the B.Sc. in computational mathematics at Otto von Guericke University Magdeburg in 2013 and his M.Sc. in mathematics at Heidelberg University in 2016. Currently he is a Ph.D. student at the Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University. His research focus is on Nonlinear Model Predictive Control and parameter estimation of electrical energy networks.
Armin Nurkanović received the B.Sc. degree from the Faculty of Electrical Engineering, Tuzla, Bosnia and Herzegovina, in 2015, and the M.Sc. degree from the Department of Electrical and Computer Engineering, Technical University of Munich, Munich, Germany, in 2018. He is currently working toward a Ph.D. degree at the Systems Control and Optimization Laboratory, Department of Microsystems Engineering, University of Freiburg, Germany, and at Siemens Corporate Technology, Munich, Germany. His research interests include numerical methods for Model Predictive Control, nonlinear optimization and nonsmooth dynamic systems.
Amer Mešanović received his M.Sc. degree in electrical engineering from the Technical University Munich, Germany. Currently he is pursuing a Ph.D. at the Otto von Guericke University in Magdeburg and working as a research scientist at Siemens AG, Munich. His research interests include control, modelling, and parameter estimation in power systems.
Hans Georg Bock is a Director and the former Executive Director of Heidelberg University’s Interdisciplinary Center for Scientific Computing (IWR). He is a member both of the Heidelberg and the Russian Academy of Science. Before moving to Heidelberg in 1991, his scientific career led him to the Universities of Cologne, Bonn, Heidelberg and Augsburg, and to the German Aerospace Center (DLR) in Oberpfaffenhofen. Well-known, e. g., since his early work on the direct multiple shooting method and its many variants, Hans Georg Bock authored and co-authored over 250 scientific publications on innovative numerical algorithms of optimization and optimal control and their applications in science, engineering and in industry. Recent areas of research include Nonlinear Model Predictive Control, Mixed-Integer and Inverse Optimal Control and Optimum Experimental Design. Among other recognitions he was awarded an ERC Advanced Investigator Grant and two honorary doctorates from the Russian Academy of Science and the Vietnam Academy of Science and Technology.
Ekaterina Kostina is a Professor in Numerical Analysis at the Heidelberg University. After obtaining a Ph.D. in Mathematics from the Institute of Mathematics, National Academy of Sciences of Belarus, she was a senior scientist at the Institute of Mathematics. In 1997 her scientific career led her to Germany where she was an assistant professor at the IWR, Heidelberg University. From 2006 and 2015 she held a professorship in Numerical Optimization at Marburg Unversity, where she also was a co-initiator and a principal investigator of the Hessian research center on “Synthetic Microbiology”. She has published over 80 research publications, mostly in numerical optimization and process control, and is a member of the editorial board of “Optimization and Engineering”. She is also one of the founding members of the national “Committee for Mathematical Modeling, Simulation and Optimization (KOMSO)”. Her recent areas of research include parameter estimation, optimum experimental design, and modeling and analysis of processes under uncertainties and their numerical optimization.
1. A. Alessandri and M. Gaggero. Fast moving horizon state estimation for discrete-time systems using single and multi iteration descent methods. IEEE Transactions on Automatic Control, 62(9):4499–4511, 2017.10.1109/TAC.2017.2660438Search in Google Scholar
2. I. Bauer, H.G. Bock, and J.P. Schlöder. DAESOL – A BDF-code for the numerical solution of differential algebraic equations. Internal report, IWR, SFB 359, Universität Heidelberg, 1999.Search in Google Scholar
3. K. Baumgartner, A. Zanelli and M. Diehl. Zero-order moving horizon estimation. In 2019 IEEE 58th Conference on Decision and Control (CDC). IEEE, 2019.10.1109/CDC40024.2019.9029525Search in Google Scholar
4. V.M. Becerra, P.D. Roberts and G.W. Griffiths. Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations. Control Engineering Practice, 9(3):267–281, 2001.10.1016/S0967-0661(00)00110-6Search in Google Scholar
5. H.G. Bock. Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen, volume 183 of Bonner Mathematische Schriften. Universität Bonn, Bonn, 1987.Search in Google Scholar
6. H.G. Bock. Randwertproblemmethoden zur parameteridentifizierung in systemen nichtlinearer differentialgleichungen. Number 183. Der Math.-Naturwiss. Fakultät der Universität Bonn, 1987.Search in Google Scholar
7. H.G. Bock, M. Diehl, E.Kostina and J.P. Schlöder. Constrained optimal feedback control of systems governed by large differential algebraic equations. In Real-Time PDE-Constrained Optimization, pages 3–24. Society for Industrial and Applied Mathematics, 2007.10.1137/1.9780898718935.ch1Search in Google Scholar
8. H.G. Bock and K.J. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. IFAC Proceedings Volumes, 17(2):1603–1608, 1984.10.1016/S1474-6670(17)61205-9Search in Google Scholar
11. M. Diehl. Real-Time Optimization for Large Scale Nonlinear Processes. PhD thesis, University of Heidelberg, 2001.Search in Google Scholar
12. M. Diehl, R. Findeisen, H.G. Bock, F. Allgöwer and J.P. Schlöder. Nominal stability of real-time iteration scheme for nonlinear model predictive control. IEE Proceedings - Control Theory and Applications, 152(3):296–308, 2005.10.1049/ip-cta:20040008Search in Google Scholar
13. M. Diehl, P. Kühl, H.G Bock and J.P. Schlöder. Schnelle Algorithmen für die Zustands- und Parameterschätzung auf bewegten Horizonten. Automatisierungstechnik, 54(12):602–613, 2006.10.1524/auto.2006.54.12.602Search in Google Scholar
14. H.J. Ferreau, C. Kirches, A. Potschka, H.G. Bock and M.Diehl. qpoases: A parametric active-set algorithm for quadratic programming. Mathematical Programming Computation, 6(4):327–363, 2014.10.1007/s12532-014-0071-1Search in Google Scholar
15. J.V. Frasch, L. Wirsching, S. Sager and H.G. Bock. Mixed-level iteration schemes for nonlinear model predictive control. IFAC Proceedings Volumes, 45(17):138–144, 2012.10.3182/20120823-5-NL-3013.00085Search in Google Scholar
16. A. Gelb. Applied Optimal Estimation. MIT Press, 1974.Search in Google Scholar
17. N.J. Gordon, D.J. Salmond and A.F.M. Smith. Novel approach to nonlinear/non-gaussian bayesian state estimation. IEE Proceedings F Radar and Signal Processing, 140(2):107, 1993.10.1049/ip-f-2.1993.0015Search in Google Scholar
18. E.L. Haseltine and J.B. Rawlings. Critical evaluation of extended Kalman filtering and moving-horizon estimation. Industrial & Engineering Chemistry Research, 44(8):2451–2460, 2005.10.1021/ie034308lSearch in Google Scholar
19. M. Ilić, R. Jaddivada and X. Miao. Modeling and analysis methods for assessing stability of microgrids. IFAC-PapersOnLine, 50(1):5448–5455, 2017.10.1016/j.ifacol.2017.08.1081Search in Google Scholar
20. T. Kraus, P. Kühl, L. Wirsching, H.G. Bock and M. Diehl. A moving horizon state estimation algorithm applied to the Tennessee Eastman benchmark process. In 2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems. IEEE, 2006.10.1109/MFI.2006.265620Search in Google Scholar
21. P. Kühl, M. Diehl, T. Kraus, J.P. Schlöder and H.G. Bock. A real-time algorithm for moving horizon state and parameter estimation. Computers & Chemical Engineering, 35(1):71–83, 2011.10.1016/j.compchemeng.2010.07.012Search in Google Scholar
22. D.B. Leineweber, I. Bauer, H.G. Bock and J.P. Schlöder. An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects. Computers & Chemical Engineering, 27(2):157–166, 2003.10.1016/S0098-1354(02)00158-8Search in Google Scholar
23. A. Nurkanović, A. Mešanović, A. Zanelli, J. Frey, G. Frison, S. Albrecht and M. Diehl. Real-time nonlinear model predictive control for microgrid operation. In Proceedings of the American Control Conference (ACC), Denver, USA, 2020.10.23919/ACC45564.2020.9147816Search in Google Scholar
25. C.V. Rao, J.B. Rawlings and D.Q. Mayne. Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations. IEEE Transactions on Automatic Control, 48(2):246–258, 2003.10.1109/TAC.2002.808470Search in Google Scholar
26. R. Scholz, A. Nurkanović, A. Mešanović, J. Gutekunst, H.G. Bock A. Potschka and E. Kostina. Multi-level iterations for microgrid control with automatic level choice. In Scientific Computing in Electrical Engineering. Springer International Publishing, 2020, submitted.10.1007/978-3-030-84238-3_29Search in Google Scholar
27. V.H. Schulz, H.G. Bock and M.C. Steinbach. Exploiting invariants in the numerical solution of multipoint boundary value problems for DAE. SIAM Journal on Scientific Computing, 19(2):440–467, 1998.10.1137/S1064827594261917Search in Google Scholar
28. L. Wirsching. Multi-Level Iteration Schemes with Adaptive Level Choice for Nonlinear Model Predictive Control. PhD thesis, 2018.Search in Google Scholar
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