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Optimal Design of Volt/VAR Control Rules of Inverters using Deep Learning

Sarthak Gupta, Vassilis Kekatos, Spyros Chatzivasileiadis

TL;DR

This work reformulates ORD as a deep learning problem to design a DNN that emulates Volt/VAR dynamics and showcases the merit of DNN-based ORD by benchmarking it against its MINLP counterpart.

Abstract

Distribution grids are challenged by rapid voltage fluctuations induced by variable power injections from distributed energy resources (DERs). To regulate voltage, the IEEE Standard 1547 recommends each DER inject reactive power according to piecewise-affine Volt/VAR control rules. Although the standard suggests a default shape, the rule can be customized per bus. This task of optimal rule design (ORD) is challenging as Volt/VAR rules introduce nonlinear dynamics, and lurk trade-offs between stability and steady-state voltage profiles. ORD is formulated as a mixed-integer nonlinear program (MINLP), but scales unfavorably with the problem size. Towards a more efficient solution, we reformulate ORD as a deep learning problem. The idea is to design a DNN that emulates Volt/VAR dynamics. The DNN takes grid scenarios as inputs, rule parameters as weights, and outputs equilibrium voltages. Optimal rule parameters can be found by training the DNN so its output approaches unity for various scenarios. The DNN is only used to optimize rules and is never employed in the field. While dealing with ORD, we also review and expand on stability conditions and convergence rates for Volt/VAR dynamics on single- and multi-phase feeders. Tests showcase the merit of DNN-based ORD by benchmarking it against its MINLP counterpart.

Optimal Design of Volt/VAR Control Rules of Inverters using Deep Learning

TL;DR

This work reformulates ORD as a deep learning problem to design a DNN that emulates Volt/VAR dynamics and showcases the merit of DNN-based ORD by benchmarking it against its MINLP counterpart.

Abstract

Distribution grids are challenged by rapid voltage fluctuations induced by variable power injections from distributed energy resources (DERs). To regulate voltage, the IEEE Standard 1547 recommends each DER inject reactive power according to piecewise-affine Volt/VAR control rules. Although the standard suggests a default shape, the rule can be customized per bus. This task of optimal rule design (ORD) is challenging as Volt/VAR rules introduce nonlinear dynamics, and lurk trade-offs between stability and steady-state voltage profiles. ORD is formulated as a mixed-integer nonlinear program (MINLP), but scales unfavorably with the problem size. Towards a more efficient solution, we reformulate ORD as a deep learning problem. The idea is to design a DNN that emulates Volt/VAR dynamics. The DNN takes grid scenarios as inputs, rule parameters as weights, and outputs equilibrium voltages. Optimal rule parameters can be found by training the DNN so its output approaches unity for various scenarios. The DNN is only used to optimize rules and is never employed in the field. While dealing with ORD, we also review and expand on stability conditions and convergence rates for Volt/VAR dynamics on single- and multi-phase feeders. Tests showcase the merit of DNN-based ORD by benchmarking it against its MINLP counterpart.
Paper Structure (12 sections, 2 theorems, 32 equations, 13 figures, 5 tables)

This paper contains 12 sections, 2 theorems, 32 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Feeder Modeling Preliminaries
  3. Control Rules for Single-Phase Feeders
  4. ORD for $1\phi$ Feeders via Deep Learning
  5. Designing a Digital Twin of Volt/VAR Dynamics
  6. DNN Training
  7. ORD for $1\phi$ Feeders as an MINLP
  8. ORD for $3\phi$ Feeders via Deep Learning
  9. Numerical Tests
  10. Tests on Single-Phase Feeder
  11. Tests on a Multiphase Feeder
  12. Conclusions

Key Result

Proposition 1

Suppose $\epsilon$-stable Volt/VAR rules are described by $\mathbf{z}$. The depth $T$ of the DNN in Fig. fig:DNN required to ensure $\|\Phi\left(\tilde{\mathbf{v}};\mathbf{z}\right) -\mathbf{v}^*(\mathbf{z})\|_2\leq \epsilon_1$ for all grid conditions $\tilde{\mathbf{v}}$ is

Figures (13)

  • Figure 1: The piecewise linear Volt/VAR control rule $f(v)$ provisioned by the IEEE 1547 standard IEEE1547. The $x$-axis corresponds to the local voltage magnitude and the $y$-axis to the inverter setpoint for reactive power injection.
  • Figure 2: Volt/VAR rule $f(v)$ expressed as a sum of ReLUs.
  • Figure 3: Volt/VAR rule $f(v)$ model using a NN with 1 hidden layer.
  • Figure 4: DNN-based digital twin for the Volt/VAR dynamics of \ref{['eq:dynamics1']}. The DNN is structured so that $T$ time steps are arranged horizontally. The modules $\text{VC}_n$'s implementing the Volt/VAR curves for each one of the $N$ inverters are stacked vertically. Skip connections propagate the input vector (grid scenario) $\tilde{\mathbf{v}}$ to each time instant to implement $\mathbf{v}^{t+1}=\mathbf{X}\mathbf{q}^{t+1}+\tilde{\mathbf{v}}$.
  • Figure 5: Recurrent representation (RNN) of the digital twin of Fig. \ref{['fig:DNN']}.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2