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Load Restoration in Islanded Microgrids: Formulation and Solution Strategies

Shourya Bose, Yu Zhang

TL;DR

This work addresses load restoration in islanded microgrids subject to extreme events, formulating a non-convex NLP that embeds DistFlow physics, internal frequency regulation via droop control, and ESS complementarity. It compares two real-time solution paradigms: (i) MPC with convex relaxations of DistFlow and ESS constraints to yield tractable subproblems, and (ii) constrained policy optimization (CPO), a tailored RL approach that learns a policy to satisfy operational constraints while optimizing restoration. The authors introduce new convex relaxations for ESS complementarity and inverter voltage dependence on reactive power, and develop a CPO framework with a QCLP surrogate and gradient-efficient updates suitable for load restoration. Simulation results on 36-bus and 141-bus islanded MGs show CPO offering robustness to RES forecast errors and tighter adherence to nonlinear constraints than MPC, albeit with longer training times; MPC remains strong under perfect forecasts. The findings suggest CPO as a practical, constraint-respecting alternative to MPC for resilient, real-time MG load restoration, with opportunities to extend to topology switching and multi-phase systems.

Abstract

Adverse circumstances such as extreme weather events can cause significant disruptions to normal operation of electric distribution systems (DS), which includes isolating parts of the DS due to damaged transmission equipment. In this paper, we consider the problem of load restoration in a microgrid (MG) that is islanded from the upstream DS. The MG contains sources of distributed generation such as microturbines and renewable energy sources, as well as energy storage systems (ESS). We formulate the load restoration task as a non-convex optimization problem. This problem embodies the physics of the MG by leveraging a branch flow model, while incorporating salient phenomenon in islanded MGs such as the need for internal frequency regulation, and complementarity requirements arising in ESS operations. Since the formulated optimization problem is non-convex, we introduce a convex relaxation which can be solved through model predictive control as a baseline method. However, in order to solve the problem considering its full non-convexity, we leverage a policy-learning method called constrained policy optimization, a tailored version of which is used as our proposed algorithm. The aforementioned approaches, along with an additional deep learning method are compared through extensive simulations.

Load Restoration in Islanded Microgrids: Formulation and Solution Strategies

TL;DR

This work addresses load restoration in islanded microgrids subject to extreme events, formulating a non-convex NLP that embeds DistFlow physics, internal frequency regulation via droop control, and ESS complementarity. It compares two real-time solution paradigms: (i) MPC with convex relaxations of DistFlow and ESS constraints to yield tractable subproblems, and (ii) constrained policy optimization (CPO), a tailored RL approach that learns a policy to satisfy operational constraints while optimizing restoration. The authors introduce new convex relaxations for ESS complementarity and inverter voltage dependence on reactive power, and develop a CPO framework with a QCLP surrogate and gradient-efficient updates suitable for load restoration. Simulation results on 36-bus and 141-bus islanded MGs show CPO offering robustness to RES forecast errors and tighter adherence to nonlinear constraints than MPC, albeit with longer training times; MPC remains strong under perfect forecasts. The findings suggest CPO as a practical, constraint-respecting alternative to MPC for resilient, real-time MG load restoration, with opportunities to extend to topology switching and multi-phase systems.

Abstract

Adverse circumstances such as extreme weather events can cause significant disruptions to normal operation of electric distribution systems (DS), which includes isolating parts of the DS due to damaged transmission equipment. In this paper, we consider the problem of load restoration in a microgrid (MG) that is islanded from the upstream DS. The MG contains sources of distributed generation such as microturbines and renewable energy sources, as well as energy storage systems (ESS). We formulate the load restoration task as a non-convex optimization problem. This problem embodies the physics of the MG by leveraging a branch flow model, while incorporating salient phenomenon in islanded MGs such as the need for internal frequency regulation, and complementarity requirements arising in ESS operations. Since the formulated optimization problem is non-convex, we introduce a convex relaxation which can be solved through model predictive control as a baseline method. However, in order to solve the problem considering its full non-convexity, we leverage a policy-learning method called constrained policy optimization, a tailored version of which is used as our proposed algorithm. The aforementioned approaches, along with an additional deep learning method are compared through extensive simulations.

Paper Structure

This paper contains 10 sections, 4 theorems, 37 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Load Restoration Problem Formulation
  3. Convex Relaxation and Solution using Model Predictive Control
  4. Solution using Constrained Policy Optimization
  5. Simulation Results
  6. Conclusion
  7. Proof of Lemma \ref{['lemma:ConvexHull']}
  8. Proof of Lemma \ref{['lemma:DroopBus']}
  9. Proof of Theorem \ref{['th:OptApprox']}
  10. Proof of Lemma \ref{['lemma:partialDerivative']}

Key Result

Lemma 1

The constraint $\sqrt{v_{i,t}} = \sqrt{v_i^*} - k_Q(\Im(s_{i,t})-Q_i^*)$ can be relaxed to a second-order cone given as

Figures (5)

  • Figure 1: Schematic of load restoration solution with MPC, with look-ahead window $H=3$.
  • Figure 2: Schematic of RL framework to solve load restoration.
  • Figure 3: A 36-bus MG that is adapted from the IEEE 37-bus distribution feeder.
  • Figure 4: Simulation results for the 36-bus MG.
  • Figure 5: Simulation results for the 141-bus MG.

Theorems & Definitions (10)

  • Lemma 1: Convex relaxation of \ref{['eq:droop_reactive']}
  • proof
  • Lemma 2: Convex hull of nonconvex ESS CC feasible set
  • proof
  • Proposition 1: Parameters in problem \ref{['eq:PCPOrelaxedprob']}
  • proof
  • Remark 1: Implementation of Proposition \ref{['th:OptApprox']}
  • Lemma 3
  • proof
  • Remark 2: Evaluating partials from total derivatives