# American Institute of Mathematical Sciences

January  2022, 9(1): 1-26. doi: 10.3934/jcd.2021019

## Model reduction for a power grid model

 Pacific Northwest National Laboratory, Richland, WA 99354, USA

* Corresponding author: Panos Stinis

Received  November 2020 Revised  November 2021 Published  January 2022 Early access  December 2021

Fund Project: The work presented here was supported by the U.S. Department of Energy (DOE) Office of Science, Office of Advanced Scientific Computing Research (ASCR) as part of the Multifaceted Mathematics for Rare, Extreme Events in Complex Energy and Environment Systems (MACSER) project. Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830

We examine the complexity of constructing reduced order models for subsets of the variables needed to represent the state of the power grid. In particular, we apply model reduction techniques to the DeMarco-Zheng power grid model. We show that due to the oscillating nature of the solutions and the absence of timescale separation between resolved and unresolved variables, the construction of accurate reduced models becomes highly non-trivial because one has to account for long memory effects. In addition, we show that a reduced model that includes even a short memory is drastically better than a memoryless model.

Citation: Jing Li, Panos Stinis. Model reduction for a power grid model. Journal of Computational Dynamics, 2022, 9 (1) : 1-26. doi: 10.3934/jcd.2021019
##### References:
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##### References:
 [1] M. Ceriotti, G. Bussi and M. Parrinello, Colored-noise thermostats à la carte, J. Chem. Theory Comput., 6 (2010), 1170-1180.  doi: 10.1021/ct900563s. [2] A. J. Chorin, O. H. Hald and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968-2973.  doi: 10.1073/pnas.97.7.2968. [3] A. J. Chorin, O. H. Hald and R. Kupferman, Optimal prediction with memory, Phys. D, 166 (2002), 239-257.  doi: 10.1016/S0167-2789(02)00446-3. [4] A. J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Commun. Appl. Math. Comput. Sci., 1 (2006), 1-27.  doi: 10.2140/camcos.2006.1.1. [5] J. H. Chow, Power System Coherency and Model Reduction, Power Electronics and Power Systems, 94, Springer, New York, 2013. doi: 10.1007/978-1-4614-1803-0. [6] I. Dobson, B. A. Carreras, V. E. Lynch and D. E. Newman, Newman, Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization, Chaos, 17 (2007), 1-27.  doi: 10.1063/1.2737822. [7] D. Givon, R. Kupferman and A. Stuart, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), R55-R127.  doi: 10.1088/0951-7715/17/6/R01. [8] O. H. Hald and P. Stinis, Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions, Proc. Natl. Acad. Sci. USA, 104 (2007), 6527-6532.  doi: 10.1073/pnas.0700084104. [9] H. S. Lee, S.-H. Ahn and E. F. Darve, The multi-dimensional generalized Langevin equation for conformational motion of proteins, J. Chem. Phys., 150 (2019).  doi: 10.1063/1.5055573. [10] J. Li and P. Stinis, A unified framework for mesh refinement in random and physical space, J. Comput. Phys., 323 (2016), 243-264.  doi: 10.1016/j.jcp.2016.07.027. [11] J. Li and P. Stinis, Mori-Zwanzig reduced models for uncertainty quantification, J. Comput. Dyn., 6 (2019), 39-68.  doi: 10.3934/jcd.2019002. [12] A. E. Motter, S. A. Myers, M. Anghel and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nature Physics, 9 (2013), 191-197.  doi: 10.1038/nphys2535. [13] D. Osipov and K. Sun, Adaptive nonlinear model reduction for fast power system simulation, IEEE Trans. Power Syst., 33 (2018), 6746-6754. [14] J. Qi, J. Wang, H. Liu and A. D. Dimitrovski, Nonlinear model reduction in power systems by balancing of empirical controllability and observability covariances, IEEE Trans. Power Syst., 32 (2017), 114-126.  doi: 10.1109/TPWRS.2016.2557760. [15] I. Simonsen, K. Buzna and S. Peters, Bornholdt and D. Helbing, Transient dynamics increasing network vulnerability to cascading failures, Phys. Rev. Lett., 100 (2008).  doi: 10.1103/PhysRevLett.100.218701. [16] F. Sloothaak, S. C. Borst and B. Zwart, Robustness of power-law behavior in cascading line failure models, Stoch. Models, 34 (2018), 45-72.  doi: 10.1080/15326349.2017.1383165. [17] P. Stinis, A phase transition approach to detecting singularities of partial differential equations, Commun. Appl. Math. Comput. Sci., 4 (2009), 217-239.  doi: 10.2140/camcos.2009.4.217. [18] P. Stinis, Renormalized Mori-Zwanzig-reduced models for systems without scale separation, Proc. A., (2015), 13pp.  doi: 10.1098/rspa.2014.0446. [19] S. Tamrakar, M. Conrath and S. Kettemann, Propagation of disturbances in AC electricity grids, Scientific Reports, 8 (2018).  doi: 10.1038/s41598-018-24685-5. [20] D. Witthaut and M. Timme, Braess's paradox in oscillator networks, desynchronization and power outage, New J. Phys., 14 (2012).  doi: 10.1088/1367-2630/14/8/083036. [21] D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139.  doi: 10.1137/040615201. [22] H. Zheng and C. L. DeMarco, A bi-stable branch model for energy-based cascading failure analysis in power systems, North American Power Symposium 2010, Arlington, TX, 2010. doi: 10.1109/NAPS. 2010.5618968. [23] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001.
A 3-bus system.
Full system - Evolution of resolved variables. a) $\omega_1$ for generator 1, b) $\omega_2$ for generator 2 and c) $\alpha_2$ for generator 2.
Full system - Evolution of unresolved variables. a) $\alpha_3$ for load bus 3, b) $V_3$ for load bus 3.
Reduced system - Evolution of the memory integral. (a) For $\omega_1,$ (b) For $\omega_2.$.
Reduced model for 3-bus system. Comparison of reduced models with infinite memory and without memory. (a) Evolution of the resolved variable $\alpha_2,$ (b) Evolution of the resolved variable $\omega_1$, and (c) Evolution of the resolved variable $\omega_2.$
Reduced model for 3-bus system. Comparison of reduced models with variable memory length (including without memory). (a) Evolution of the resolved variable $\alpha_2$ and (b) Evolution of the resolved variable $\omega_1.$
Reduced model for 3-bus system. Comparison of reduced models with variable order of the finite-rank projection and timestep $\Delta t = 10^{-4}.$ (a) Evolution of the resolved variable $\alpha_2,$ (b) Evolution of the resolved variable $\omega_1.$
Reduced model for the 3-bus system. Evolution of the memory kernels $b_1^{\mu}(s)$ of the resolved variable $\omega_1$ on linear functions of the resolved variables. (a) Projection on the 1 degree Hermite polynomial $h^{(1,0,0)},$ (b) Projection on the 1 degree Hermite polynomial $h^{(0,1,0)},$ and (c) Projection on the 1 degree Hermite polynomial $h^{(0,0,1)}.$ We have also included in the insets the evolution near the time origin (see text for details).
Reduced model for 3-bus system. Evolution of the projections $f_1^{\mu}(s) = (Le^{sL}F_1(u_0,0),h^{\mu}(\hat{u}_0))$ of $Le^{sL}F_1(u_0,0)$ on linear functions of the resolved variables. (a) Projection on the 1 degree Hermite polynomial $h^{(1,0,0)},$ (b) Projection on the 1 degree Hermite polynomial $h^{(0,1,0)},$ and (c) Projection on the 1 degree Hermite polynomial $h^{(0,0,1)}.$ We have also included in the insets the evolution near the time origin (see text for details).
Reduced model for particle coupled to heat bath - Evolution of the position $x(t)$ of the particle (a) For $t_{memory} = 1,$ (b) For $t_{memory} = 2,$ and (c) For $t_{memory} = 3.$
Reduced model for particle coupled to heat bath - Evolution of the momentum $p(t)$ of the particle (a) For $t_{memory} = 1,$ (b) For $t_{memory} = 2,$ and (c) For $t_{memory} = 3.$
Parameters for the 3-bus system
 $M_1$ $M_2$ $b_1$ $b_2$ $b_3$ $D_1$ $D_2$ $0.052$ $0.0531$ $10$ $10$ $10$ $0.05$ $0.05$ $D_3$ $P_2$ $P_3$ $Q_3$ $\epsilon$ $V_1$ $V_2$ $0.005$ $-2.0$ $3.0$ $0.1$ $5$ $0.9$ $0.9$
 $M_1$ $M_2$ $b_1$ $b_2$ $b_3$ $D_1$ $D_2$ $0.052$ $0.0531$ $10$ $10$ $10$ $0.05$ $0.05$ $D_3$ $P_2$ $P_3$ $Q_3$ $\epsilon$ $V_1$ $V_2$ $0.005$ $-2.0$ $3.0$ $0.1$ $5$ $0.9$ $0.9$
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