Qualitative stability and synchronicity analysis of power network models in port-Hamiltonian form

Authors

Volker Mehrmann, Riccardo Morandin, Simona Olmi, Eckehard Schöll

Abstract

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies, the analysis and control of the stability and the synchronization of power networks is an important topic that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model with inertia. The usual formulation in the form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model with inertia as a polynomial port-Hamiltonian system of differential-algebraic equations, with a quadratic Hamiltonian function including a generalized order parameter. This leads to a robust representation of the system with respect to disturbances: it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations, and it explicitly displays all possible constraints and allows for robust simulation methods. The model is immersed into a system of model hierarchies that will be helpful for applying adaptive simulations in future works. We illustrate the advantages of the modified modeling approach with analytics and numerical results.

Citation

• Journal: Chaos: An Interdisciplinary Journal of Nonlinear Science
• Year: 2018
• Volume: 28
• Issue: 10
• Pages:
• Publisher: AIP Publishing
• DOI: 10.1063/1.5054850

BibTeX

@article{Mehrmann_2018,
  title={{Qualitative stability and synchronicity analysis of power network models in port-Hamiltonian form}},
  volume={28},
  ISSN={1089-7682},
  DOI={10.1063/1.5054850},
  number={10},
  journal={Chaos: An Interdisciplinary Journal of Nonlinear Science},
  publisher={AIP Publishing},
  author={Mehrmann, Volker and Morandin, Riccardo and Olmi, Simona and Schöll, Eckehard},
  year={2018}
}

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References

• Filatrella, G., Nielsen, A. H. & Pedersen, N. F. Analysis of a power grid using a Kuramoto-like model. The European Physical Journal B vol. 61 485–491 (2008) – 10.1140/epjb/e2008-00098-8
• Power System Stability and Control (1994)
• Olmi, S., Navas, A., Boccaletti, S. & Torcini, A. Hysteretic transitions in the Kuramoto model with inertia. Physical Review E vol. 90 (2014) – 10.1103/physreve.90.042905
• Salam, F., Marsden, J. & Varaiya, P. Arnold diffusion in the swing equations of a power system. IEEE Transactions on Circuits and Systems vol. 31 673–688 (1984) – 10.1109/tcs.1984.1085570
• Nishikawa, T. & Motter, A. E. Comparative analysis of existing models for power-grid synchronization. New Journal of Physics vol. 17 015012 (2015) – 10.1088/1367-2630/17/1/015012
• Rohden, M., Sorge, A., Timme, M. & Witthaut, D. Self-Organized Synchronization in Decentralized Power Grids. Physical Review Letters vol. 109 (2012) – 10.1103/physrevlett.109.064101
• Dörfler, F., Chertkov, M. & Bullo, F. Synchronization in complex oscillator networks and smart grids. Proceedings of the National Academy of Sciences vol. 110 2005–2010 (2013) – 10.1073/pnas.1212134110
• Monshizadeh, N., De Persis, C., van der Schaft, A. J. & Scherpen, J. M. A. A Novel Reduced Model for Electrical Networks With Constant Power Loads. IEEE Transactions on Automatic Control vol. 63 1288–1299 (2018) – 10.1109/tac.2017.2747763
• van der Schaft, A. & Stegink, T. Perspectives in modeling for control of power networks. Annual Reviews in Control vol. 41 119–132 (2016) – 10.1016/j.arcontrol.2016.04.017
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Advanced Dynamics and Control of Structures and Machines (2004)
• Modeling and Simulation of Dynamic Systems Using Bond Graphs (2008)
• van der Schaft, A. J. Port-Hamiltonian Differential-Algebraic Systems. Surveys in Differential-Algebraic Equations I 173–226 (2013) doi:10.1007/978-3-642-34928-7_5 – 10.1007/978-3-642-34928-7_5
• Byrnes, C. I., Isidori, A. & Willems, J. C. Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control vol. 36 1228–1240 (1991) – 10.1109/9.100932
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• Fiaz, S., Zonetti, D., Ortega, R., Scherpen, J. M. A. & van der Schaft, A. J. A port-Hamiltonian approach to power network modeling and analysis. European Journal of Control vol. 19 477–485 (2013) – 10.1016/j.ejcon.2013.09.002
• Schröder, M., Timme, M. & Witthaut, D. A universal order parameter for synchrony in networks of limit cycle oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science vol. 27 (2017) – 10.1063/1.4995963
• Line Integral Methods for Conservative Problems (2015)
• Differential-Algebraic Equations: Analysis and Numerical Solution (2006)
• FORTUNA, L., FRASCA, M. & SARRA FIORE, A. A NETWORK OF OSCILLATORS EMULATING THE ITALIAN HIGH-VOLTAGE POWER GRID. International Journal of Modern Physics B vol. 26 1246011 (2012) – 10.1142/s0217979212460113