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Coordinating Multiple Sources for Service Restoration to Enhance Resilience of Distribution Systems

Ying Wang, Yin Xu, Jinghan He, Chen-Ching Liu, Kevin P. Schneider, Mingguo Hong, Dan T. Ton

arXiv:1810.06907v3math.OC

TL;DR

Extreme-event outages can isolate critical loads from utility sources, motivating coordinated use of microgrids and DGs. The paper proposes a two-stage restoration method combining topology selection with semidefinite optimization and an iterative treatment of load-status integers. Simulations on modified IEEE 13-node and 123-node systems validate efficient restoration-strategy determination in most scenarios.

  • Problem

    Extreme events can damage distribution facilities and leave interrupted loads unable to receive utility power, motivating restoration with local DGs, MGs, and other sources.

  • Method

    A two-stage method selects post-restoration topology, then formulates load and source decisions as a relaxed semidefinite program solved iteratively for integer load statuses.

  • Results

    The method determines optimal restoration strategies efficiently in most scenarios, with simulations validating its effectiveness on modified IEEE 13-node and 123-node test systems.

  • Takeaways & Limitations

    Coordinating multiple sources can better allocate generation capacity and restore more critical loads after extreme-event outages.

  • Takeaways & Limitations

    The snapshot restoration model should be extended to multiple time steps to represent fuel, storage state-of-charge, DG ramping, and mobile-source scheduling constraints.

Abstract

from arXiv · show

When a major outage occurs on a distribution system due to extreme events, microgrids, distributed generators, and other local resources can be used to restore critical loads and enhance resiliency. This paper proposes a decision-making method to determine the optimal restoration strategy coordinating multiple sources to serve critical loads after blackouts. The critical load restoration problem is solved by a two-stage method with the first stage deciding the post-restoration topology and the second stage determining the set of loads to be restored and the outputs of sources. In the second stage, the problem is formulated as a mixed-integer semidefinite program. The objective is maximizing the number of loads restored, weighted by their priority. The unbalanced three-phase power flow constraint and operational constraints are considered. An iterative algorithm is proposed to deal with integer variables and can attain the global optimum of the critical load restoration problem by solving a few semidefinite programs under two conditions. The effectiveness of the proposed method is validated by numerical simulation with the modified IEEE 13-node test feeder and the modified IEEE 123-node test feeder under plenty of scenarios. The results indicate that the optimal restoration strategy can be determined efficiently in most scenarios.

I. NOMENCLATURE

This section defines the sets, parameters, variables, and graph notation used in the restoration optimization models and iterative algorithm.

  • The notation includes bus, load-bus, DG-bus, line, phase, adjacency, impedance, and graph-related sets and parameters.
  • Additional notation specifies phase sets, directed lines, admittances, rated source power, and the root node of a tree.
  • The binary load-status variable γ_i equals 1 for a restored load and 0 otherwise.
  • The binary line-status variable a_ij indicates whether line (i,j) is energized, while b_ij indicates whether node i parents node j.
  • Load weights, load power, DG ratings, and phase-level power quantities parameterize restoration objectives and source constraints.
  • The notation distinguishes optimal values of the relaxed CLR-sdp objective and the primal CLR objective.

II. INTRODUCTION

The paper addresses restoration after extreme-event outages by coordinating local sources, then develops a two-stage optimization method with stated optimality conditions.

  • Extreme events can damage distribution infrastructure and prevent utility sources from reaching interrupted loads, motivating local-resource restoration.
  • Interconnecting microgrids, DGs, and other local sources forms aggregated islands that can allocate generation resources to restore more critical loads.
  • The proposed method first decides post-restoration topology, then selects restored loads and determines DG power outputs.
  • Relaxing load-status integer variables transforms the primal mixed-integer semidefinite program into an SDP without integer variables.
  • An iterative algorithm addresses non-integer load statuses, converges in few iterations, and attains the global optimum in most cases.
  • Sufficient global-optimality conditions require sufficiently large branch thermal limits and adequate VAR compensation on critical buses.
  • The paper focuses on pooling multiple small-capacity sources to restore as many critical loads as possible.

C. Mathematical Formulation

The critical load restoration model pools multiple sources, enforces electrical and operational constraints, and separates topology selection from load and source decisions.

  • C. Mathematical Formulation: The paper targets restoration of as many critical loads as possible by pooling the capacities of multiple power sources.
  • C. Mathematical Formulation: Target islands connect outage zones that are not completely isolated, and the resulting network is modeled as CLR-basic.
  • C. Mathematical Formulation: The objective maximizes restored loads weighted by priority, while constraints model unbalanced three-phase power flow, source injections, voltages, and line currents.
  • C. Mathematical Formulation: Radial topology constraints impose spanning-tree structure, including energized parent-child lines, one parent per non-root node, and no parent for the root.
  • C. Mathematical Formulation: Unbalanced power-flow equations and many integer variables make CLR-basic a nonconvex optimization problem.
  • A. A Two-Stage Solution Method: The two-stage method assigns topology-related integer variables to Stage 1 and determines restored loads and source outputs in Stage 2.
  • B. A Heuristic to Determine the Topology in Stage 1: Stage 1 uses a graph-theoretic heuristic that selects a minimum-diameter spanning tree, computable in O(mn + n^2logn) time.
  • B. A Heuristic to Determine the Topology in Stage 1: For the illustrated graph, topology G1 has smaller line (2,3) power loss and reduced voltage drops at nodes 3 and 4 than G2.

C. Solution Method for Stage 2

Stage 2 reformulates critical load restoration as a mixed-integer semidefinite program and relaxes it to an SDP to address nonconvex power flow constraints and integer load statuses.

  • Stage 2 formulation: After topology selection, Stage 2 determines restored loads and source power outputs under power-flow and operational constraints.The formulation shares variables and constraints with optimal power flow but adds load-status and topology decisions.
  • Semidefinite reformulation: The original model is transformed into a mixed-integer semidefinite program using slack-variable transformations for voltage and current products.The resulting CLR model uses semidefinite matrix variables associated with buses and lines.
  • Semidefinite relaxation: The convex relaxation replaces binary load statuses with variables in [0,1] and removes the rank constraint.The relaxed CLR-sdp is optimized over load statuses and semidefinite variables subject to constraints (10)–(16).
  • Exactness and challenge: CLR-sdp is exact only when its solution satisfies the rank constraint and contains no non-integer load-status values.Non-integer values represent partially restored loads, which are infeasible when loads can only be restored whole.
  • Exactness and challenge: Non-integer load statuses arise when generation capacity or voltage and current limits prevent further load restoration.These conditions make recovering a globally optimal feasible solution for the primal CLR challenging.

VI. AN ITERATIVE ALGORITHM

The proposed iterative algorithm repeatedly solves the relaxed SDP, fixes or excludes fractional load statuses, and uses priority weights and network limits to obtain an integer restoration plan.

  • Priority weighting: Loads are assigned priority levels with decreasing weights, and sufficiently separated weights prioritize higher-level loads.The weighting conditions ensure that higher-priority restoration dominates lower-priority restoration in the objective.
  • Algorithm procedure: Each iteration adds constraints that fix at least one fractional load status, while the ADDCONSTRAINTS flowchart summarizes the decision procedure.The iteration continues by resolving CLR-sdp with the new constraints.
  • Algorithm procedure: The algorithm solves CLR-sdp iteratively until all load-status variables are integer, then conducts an OPF to determine source outputs.The final OPF computes s, V, S, and I for the load statuses selected by the iterative process.
  • Constraint addition: ADDCONSTRAINTS first fixes fully restored loads, then sets certain fractional statuses to zero when priority or operating-limit tests rule them out.The procedure evaluates non-integer loads by weight, voltage, and line-current bounds.
  • Constraint addition: When fractional loads share a priority level and insufficient DG capacity is decisive, the algorithm computes the maximum number restorable and abandons loads with smaller fractional values.This is handled through the load-count calculation and subsets ℒc3 and ℒc4.

C. Convergence Analysis

The convergence analysis shows that the iterative process terminates because each iteration fixes at least one previously fractional load-status variable to an integer.

  • Convergence: Each iteration fixes at least one load-status variable to 0 or 1.ADDCONSTRAINTS ensures this progress whenever fractional statuses remain.
  • Convergence: Because the number of load-status variables is finite, all their values are determined after finitely many iterations.Therefore, convergence of the proposed algorithm is guaranteed.

D. Global Optimum Verification

The paper provides an optimality criterion under which the iterative SDP solutions certify the global optimum of the primal critical load restoration problem.

  • Optimality criterion: If the stated inequality holds and no constraint is added in Step 4 during the specified iterations, the global optimum of CLR is attained.The criterion is evaluated using the solutions obtained at each iteration.
  • Optimality criterion: The iteration indexing defines the initial solve as iteration 0 and the final iteration count as m.A proof of the criterion is provided in Appendix B.

E. Sufficient Conditions to Attain Global Optimum

The paper identifies two sufficient conditions under which its iterative semidefinite-program approach attains the global optimum, then demonstrates the procedure on a modified IEEE 13-node feeder.

  • Sufficient conditions: Global optimality requires sufficiently large branch thermal limits and sufficient VAR compensation on critical buses.Under this condition, current and voltage constraints are not activated.
  • Sufficient conditions: Global optimality also requires sufficiently different kW demands among loads assigned to the same priority level.This prevents an additional constraint from being added in the fifth step of ADDCONSTRAINTS.
  • Test system: The modified IEEE 13-node feeder contains two microgrids, three distributed generators, an energy storage unit, added switches, and two tie lines.Loads are divided into three levels with weighting factors 100, 10, and 0.2.
  • Iterative procedure: The illustrated stage-two restoration strategy is obtained in two iterations, with results reported in Table II.The ratio of the two largest eigenvalue magnitudes measures how close a solution matrix is to rank one; smaller ratios indicate closer rank-one structure.
  • Iterative procedure: The restoration strategy first fixes topology and then iteratively adds constraints for integer load-restoration decisions in the semidefinite program.The illustrated case starts with a non-integer value at bus 634 and adds constraints for buses 675, 645, and 646 before handling bus 634.
  • Results: The global optimal solution restores loads at buses 675, 645, 646, and 632, with DG outputs of 552.54 kW, 200.00 kW, 360.00 kW, and 231.76 kW, respectively.The reported power loss is 1.30 kW, and bus 633 is the voltage reference at 1 p.u. and 0 degrees.

B. Case II: The Modified 123-Node Test System

The modified IEEE 123-node system evaluates coordinated restoration across three microgrids under multiple faults, showing efficient and globally optimal recovery in tested scenarios while exposing snapshot and dynamic limitations.

  • System and scenario: Three microgrids operate grid-connected normally and islanded after extreme events; available sources include diesel generators, energy storage, and photovoltaic generators.The test system contains five specified line faults, and photovoltaic active power is assumed constant.
  • Restoration results: 880.3: The coordinated strategy restores eight first-level and eight second-level loads after three algorithm iterations.All solution matrices have numerical rank one, and the final restoration scheme is reported in Fig. 6.
  • Restoration results: 99.43%: Coordinating multiple sources increases the objective over operating each microgrid independently.The uncoordinated comparison assumes each islanded microgrid supplies only its own loads.
  • Restoration results: Coordination fully utilizes the three microgrids' generation capacities, restoring critical loads that isolated operation cannot serve.Loads 39 and 72 are restored through interconnected microgrids, while MG2 also supports some non-critical feeder loads.
  • Efficiency and comparison: 100 scenarios: The proposed algorithm and MOSEK produce identical solutions for the tested scenarios, while the linear model can select extra loads optimistically.The linear approximation ignores power losses; one or two additional loads are selected in 78 scenarios.
  • Limitations: The snapshot model omits outage-duration constraints, switching transients, and renewable uncertainty, requiring further research for broader restoration settings.Future extensions include multi-time-step resource, storage, ramping, transient, synchronization, and uncertainty constraints.

A. Formulations of CLR-misocp and CLR-milp

The alternative CLR formulations express restoration as mixed-integer second-order-cone or linear-approximation models, alongside the semidefinite formulation, with objectives centered on weighted load restoration.

  • CLR-misocp: CLR-misocp maximizes weighted restored-load priority while penalizing system power losses through a balancing factor.Its decision variables include binary load statuses, source injections, voltages, line powers, and squared currents.
  • CLR-misocp: The CLR-misocp power-flow relaxation imposes a second-order-conic constraint in place of the exact power-definition equation.The relaxation is exact for radial networks with a convex, non-decreasing objective under the stated conditions.
  • CLR-milp: CLR-milp maximizes the same weighted load-restoration objective using a linearized power-flow model under two assumptions.The assumptions are negligible line losses and nearly balanced voltages.
  • CLR-milp: The CLR-milp constraints simplify the semidefinite formulation, and its outputs are load statuses and power-flow information satisfying those constraints.The formulation includes binary restored-load variables and continuous network variables.

B. Proof of the Optimality Criterion

The proof establishes that the iterative treatment of integer load-status variables preserves global optimality under the stated weighting and restoration conditions. It analyzes successive iterations by showing which load statuses must be fixed and why the resulting solution remains optimal.

  • Optimality criterion: The optimality proof requires showing that loads fixed to γ_i=1 remain 1, loads fixed to γ_i=0 remain 0, and the step-5 treatment preserves global optimality.These are the three conditions identified as sufficient for the criterion.
  • Integer-status case: The relaxed solution is globally optimal after integer relaxation, and integer load statuses directly yield the global optimum of the primal CLR.The proof distinguishes this case from solutions containing non-integer load statuses.
  • Priority weighting: The weighting scheme prioritizes higher-level loads because each level's weight exceeds the total possible contribution from all lower-priority levels.This ensures that higher-priority loads are restored before lower-priority loads.
  • First iteration: For the first iteration, non-integer statuses arise from load-related constraints rather than insufficient generation capacity, forcing at least one load status among the relevant higher-level loads to zero.The proof uses this structure to establish the required status relationships for higher-priority levels.
  • Objective bound: The difference between the relaxed objective and the corresponding integer restoration objective bounds the possible objective improvement used in proving the second condition.This bound supports fixing statuses for loads that cannot belong to a globally optimal restoration.
  • Capacity-limited case: When the current priority level is capacity-limited, n_re is the estimated maximum number of loads at that level that can be fully restored.The proof then shows that the step-5 treatment of these loads does not compromise the primal problem's global optimality.
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