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Optimal Voltage Control Using Online Exponential Barrier Method

Peng Zhang, Baosen Zhang

TL;DR

The paper addresses voltage regulation in distribution networks with high inverter based DER penetration and inaccurate network models. It introduces an online exponential barrier method that augments the objective with exponential barrier terms and updates the inverter setpoints using live voltage measurements, leveraging the linearized relation x = Rp + Xq to steer the system toward optimality while respecting safety. Key contributions include closed-form barrier parameter selection, safety guarantees at convergence under model errors, and an exponential convergence rate, with validation on a 56-bus radial feeder showing robustness to large estimation errors. The approach enables fast, safe voltage control with limited model information and is compatible with online system identification.

Abstract

This paper address the optimal voltage control problem of distribution systems with high penetration of inverter-based renewable energy resources, under inaccurate model information. We propose the online exponential barrier method that explicitly leverages the online feedback from grids to enhance the robustness to model inaccuracy and incorporates the voltage constraints to maintain the safety requirements. We provide analytical results on the optimal barrier parameter selection and sufficient conditions for the safety guarantee of converged voltages. We also establish theoretical results on the exponential convergence rate with proper step-size. The effectiveness of the proposed framework is validated on a 56-bus radial network, where we significantly improve the robustness against model inaccuracy compared to existing methods.

Optimal Voltage Control Using Online Exponential Barrier Method

TL;DR

The paper addresses voltage regulation in distribution networks with high inverter based DER penetration and inaccurate network models. It introduces an online exponential barrier method that augments the objective with exponential barrier terms and updates the inverter setpoints using live voltage measurements, leveraging the linearized relation x = Rp + Xq to steer the system toward optimality while respecting safety. Key contributions include closed-form barrier parameter selection, safety guarantees at convergence under model errors, and an exponential convergence rate, with validation on a 56-bus radial feeder showing robustness to large estimation errors. The approach enables fast, safe voltage control with limited model information and is compatible with online system identification.

Abstract

This paper address the optimal voltage control problem of distribution systems with high penetration of inverter-based renewable energy resources, under inaccurate model information. We propose the online exponential barrier method that explicitly leverages the online feedback from grids to enhance the robustness to model inaccuracy and incorporates the voltage constraints to maintain the safety requirements. We provide analytical results on the optimal barrier parameter selection and sufficient conditions for the safety guarantee of converged voltages. We also establish theoretical results on the exponential convergence rate with proper step-size. The effectiveness of the proposed framework is validated on a 56-bus radial network, where we significantly improve the robustness against model inaccuracy compared to existing methods.

Paper Structure

This paper contains 12 sections, 6 theorems, 29 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. MODELING AND PROBLEM FORMULATION
  3. Modeling of distribution systems
  4. Optimal voltage control
  5. Algorithm
  6. Optimality analysis and safety guarantee
  7. Simulation results
  8. Environment Setup
  9. Results on optimality
  10. Results on safety
  11. Conclusion
  12. Acknowledgment

Key Result

Theorem 1

(Optimality condition) Under Assumption 1 and model $\mathbf{B}$, there exists the unique barrier parameter $\mathbf{\alpha} \in \mathbb{R}_+^n$ with fixed $\mathbf{\beta} \in \mathbb{R}_+^{n}$, such that the augmented cost function $\tilde{c}(\mathbf{u} | \alpha, \mathbf{B})$ is strongly convex and and $\alpha_j = 0$, $j = N\setminus {A}$, with ${A}$ as the set of active constraints at $\mathbf{u

Figures (2)

  • Figure 1: Comparison on the regulated voltages under online barrier method (red diamonds) and directly solving the LCQP (green hexagons) with inaccurate model $\mathbf{\hat{B}}_2$. The proposed method yields a solution close to that of the fully known model (orange squares), while no control (blue circles) exhibits significant deviations. The safety limit is denoted by the gray dashed line.
  • Figure 2: Comparison on the maximum voltage during intermediate steps under (a) online exponential barrier method and (b) online primal-dual method, with models $\mathbf{B}$, $\mathbf{\hat{B}}_1$ and $\mathbf{\hat{B}}_2$. Although both are robust to the model inaccuracies, (b) exhibits significant violations in the process.

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 2 more