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Online learning for robust voltage control under uncertain grid topology

Christopher Yeh, Jing Yu, Yuanyuan Shi, Adam Wierman

TL;DR

This work combines a nested convex body chasing algorithm with a robust predictive controller to achieve provably finite-time convergence to safe voltage limits in the online setting where there is uncertainty in both the network topology as well as load and generation variations.

Abstract

Voltage control generally requires accurate information about the grid's topology in order to guarantee network stability. However, accurate topology identification is challenging for existing methods, especially as the grid is subject to increasingly frequent reconfiguration due to the adoption of renewable energy. In this work, we combine a nested convex body chasing algorithm with a robust predictive controller to achieve provably finite-time convergence to safe voltage limits in the online setting where there is uncertainty in both the network topology as well as load and generation variations. In an online fashion, our algorithm narrows down the set of possible grid models that are consistent with observations and adjusts reactive power generation accordingly to keep voltages within desired safety limits. Our approach can also incorporate existing partial knowledge of the network to improve voltage control performance. We demonstrate the effectiveness of our approach in a case study on a Southern California Edison 56-bus distribution system. Our experiments show that in practical settings, the controller is indeed able to narrow the set of consistent topologies quickly enough to make control decisions that ensure stability in both linearized and realistic non-linear models of the distribution grid.

Online learning for robust voltage control under uncertain grid topology

TL;DR

This work combines a nested convex body chasing algorithm with a robust predictive controller to achieve provably finite-time convergence to safe voltage limits in the online setting where there is uncertainty in both the network topology as well as load and generation variations.

Abstract

Voltage control generally requires accurate information about the grid's topology in order to guarantee network stability. However, accurate topology identification is challenging for existing methods, especially as the grid is subject to increasingly frequent reconfiguration due to the adoption of renewable energy. In this work, we combine a nested convex body chasing algorithm with a robust predictive controller to achieve provably finite-time convergence to safe voltage limits in the online setting where there is uncertainty in both the network topology as well as load and generation variations. In an online fashion, our algorithm narrows down the set of possible grid models that are consistent with observations and adjusts reactive power generation accordingly to keep voltages within desired safety limits. Our approach can also incorporate existing partial knowledge of the network to improve voltage control performance. We demonstrate the effectiveness of our approach in a case study on a Southern California Edison 56-bus distribution system. Our experiments show that in practical settings, the controller is indeed able to narrow the set of consistent topologies quickly enough to make control decisions that ensure stability in both linearized and realistic non-linear models of the distribution grid.
Paper Structure (17 sections, 6 theorems, 40 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 6 theorems, 40 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Model
  4. Robust Online Voltage Control
  5. Algorithm
  6. Assumptions
  7. Main result
  8. Proofs
  9. Case Study
  10. Experimental Setup
  11. Experimental Results
  12. Linearized power flow with full control
  13. Nonlinear power flow with full control
  14. Nonlinear power flow with partial observation and partial control
  15. Varying $\delta$
  16. ...and 2 more sections

Key Result

Theorem 1

Under as:bounded_was:bounded_Xas:bounded_vpar, alg:robust_online_volt_control ensures that the voltage limits will be violated at most $\frac{2 \gamma(m)}{\rho} \mathop{\mathrm{diam}}\nolimits(\mathcal{X} \times [0,\overline\eta]) + 1$ times, where $\rho = \frac{\delta\epsilon}{1+\delta\|\overline{q

Figures (7)

  • Figure 1: Online robust voltage control framework
  • Figure 2: Voltage profile of 7 buses without control, simulated with (a) linear dynamics \ref{['eq:simplified-distflow']} and (b) nonlinear balanced AC dynamics \ref{['eq:nonlinear-distflow']}.
  • Figure 3: (a)-(d) Voltage profiles of 7 different buses simulated under linear system dynamics \ref{['eq:simplified-distflow']}. Dotted black lines indicate voltage limits $[\underline{v}, \overline{v}]$. (a)$\Pi$+SEL initialized with random $\hat{X} \in \mathcal{X}_\alpha$. (b) like (a) but the topology for buses 1-14 is known. (c) like (a) but the topology and line parameters for buses 1-14 are known. (d) like (a) but $\hat{X}=X^\star$ is fixed and known so only $\hat{\eta}$ is learned (e) Convergence of $\hat{X}_t$ towards true $X^\star$ (solid lines, left axis) and estimated $\hat{\eta}$ (dotted lines, right axis). Notice that even when $\|\hat{X}_t - X^\star\|_\triangle$ does not reach 0, the controller still performs quite well.
  • Figure 4: Parallels \ref{['fig:linear']}. Voltage profiles of 7 different buses simulated under balanced nonlinear AC power flow \ref{['eq:nonlinear-distflow']}.
  • Figure 5: Effect of varying $\delta$ on consistent model chasing. As in \ref{['fig:linear_error']}, convergence of $\hat{X}_t$ towards $X^\star$ is plotted in solid lines (left axis), and estimated $\hat{\eta}$ is plotted in dotted lines (right axis). In blue are results where we fix $\hat{\eta} = \eta^\star = 8.65$ and $\delta$ has no effect. (a) linear dynamics (b) nonlinear dynamics.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: $\left\lVert\cdot\right\rVert_\triangle$ and $\left\lVert\cdot\right\rVert_{\triangle,\delta}$
  • Theorem 1: Main Result
  • Definition 2: Consistent Parameter
  • Theorem 2
  • Lemma 1: SEL is consistent
  • Lemma 2: SEL is competitive
  • Theorem 3: $\Pi$ is $\rho$-robust
  • Lemma 3