# American Institute of Mathematical Sciences

doi: 10.3934/jcd.2021019
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## 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 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, doi: 10.3934/jcd.2021019
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##### References:
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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