Stable Online Control of Linear Time-Varying Systems

This repository contains source code necessary to reproduce the results presented in the following paper: Stable Online Control of Linear Time-Varying Systems (http://proceedings.mlr.press/v144/qu21a/qu21a.pdf).

In this work, we propose COvariance Constrained Online Linear Quadratic (COCO-LQ) control, that guarantees input-to-state stability for a large class of LTV systems while also minimizing the control cost. The proposed method incorporates a state covariance constraint into the semi-definite programming (SDP) formulation of the LQ optimal controller. We empirically demonstrate the performance of COCO-LQ in both synthetic experiments and a power system frequency control example.

Authors: Guannan Qu, Yuanyuan Shi, Sahin Lale, Steven Low, Anima Anandkumar and Adam Wierman

Accepted and Presented at the 3rd Annual Conference on Learning for Dynamics and Control (L4DC).

Experiments

Synthetic Time-Varying Systems

1. Switching Linear Systems

2. Time-variant Systems

Frequency Control with Renewable Generation

We consider a power system frequency control problem with time-varying renewable generation and thus, system inertia. The state space model of power system frequency dynamics follow,

$$ \begin{bmatrix} \dot{\theta}\\ \dot{\omega} \end{bmatrix} = \begin{bmatrix} 0 & I \\ -M_t^{-1} L & -M_t^{-1} D \end{bmatrix} \begin{bmatrix} \theta \\ \omega \end{bmatrix} + \begin{bmatrix} 0 \\ M_t^{-1}\end{bmatrix} P_{in} $$

where the state variable is defined as the stacked vector of the voltage angle $\theta$ and frequency $\omega$. $M_t = diag(m_{t,i})$ is the inertia matrix, where $m_{t,i}$ represents the equivalent rotational inertia at bus i and time t. $M_t$ is time-varying and depends on the mix of online generators, since only thermal generators provide rotational inertia and renewable generation does not. $D = diag(d_i)$ is the damping matrix, where $d_i$ is the generator damping coefficient. $L$ is the network susceptance matrix. The control variable pin corresponds to the electric power generation.

We test on the IEEE WECC 3-machine 9-bus system, where the system is changing between two states: a high renewable generation scenario where $m_{t,i} = 2$ (i.e., 80 percent renewable with zero inertia and 20 percent of thermal generation with 10s inertia), and a low renewable generation scenario where $m_{t,i} = 8$ (i.e., 20 percent renewable and 80 percent thermal generation), with additional random fluctuations between $[0, 0.2]$.

Citation

@inproceedings{qu2021stable,
  title={Stable online control of linear time-varying systems},
  author={Qu, Guannan and Shi, Yuanyuan and Lale, Sahin and Anandkumar, Anima and Wierman, Adam},
  booktitle={Learning for Dynamics and Control},
  pages={742--753},
  year={2021},
  organization={PMLR}
}