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Critical Load Restoration using Distributed Energy Resources for Resilient Power Distribution System

Shiva Poudel, Anamika Dubey

arXiv:1912.04535v1eess.SY

TL;DR

Extreme weather can cause prolonged distribution outages that interrupt critical services, while traditional restoration approaches may be inapplicable when the main grid is unavailable. The paper formulates an MILP-based DER restoration approach that selects robust radial restoration plans, and simulations show it can restore the maximum number of critical loads while accounting for feeder damage and DER availability.

  • Problem

    Extreme weather can cause prolonged outages and loss of critical services, while traditional restoration approaches may be inapplicable when the main grid is unavailable.

  • Method

    The paper formulates an MILP that allocates DERs, selects restoration paths, satisfies operational and connectivity constraints, and minimizes effective unavailability for robust radial restoration.

  • Results

    The approach restores a maximum number of critical loads while accounting for distribution-feeder damage, DER availability, and robustness to post-restoration failures.

  • Takeaways & Limitations

    The restored topology can incorporate network failure probability and DER availability to produce restoration plans robust to a second disaster strike.

Abstract

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Extreme weather events have a significant impact on the aging and outdated power distribution infrastructures. These high-impact low-probability (HILP) events often result in extended outages and loss of critical services, thus, severely affecting customers' safety. This calls for the need to ensure resilience in distribution networks by quickly restoring the critical services during a disaster. This paper presents an advanced feeder restoration method to restore critical loads using distributed energy resources (DERs). A resilient restoration approach is proposed that jointly maximizes the amount of restored critical loads and optimizes the restoration times by optimally allocating grid's available DER resources. The restoration problem is modeled as a mixed-integer linear program with the objective of maximizing the resilience to post-restoration failures while simultaneously satisfying grid's critical connectivity and operational constraints and ensuring a radial operation for a given open-loop feeder configuration. Simulations are performed to demonstrate the effectiveness of the proposed approach using IEEE 123-node feeder with 5 DERs supplying 11 critical loads and IEEE 906-bus feeder with 3 DERs supplying 17 critical loads. The impacts of DER availability and fuel reserve on restored networks are assessed and it is shown that the proposed approach is successfully able to restore a maximum number of critical loads using available DERs.

NOMENCLATURE · I. INTRODUCTION

The paper frames resilient distribution restoration around critical-load service during disasters, using DERs within a reliability-aware MILP formulation. Its nomenclature defines the graph, restoration, DER, network, and operational variables used throughout the approach.

  • NOMENCLATURE: The distribution system is represented as a graph G = (V, E), with critical-load nodes Cl, DER nodes M, and loop-related path sets.The nomenclature also defines parent nodes, children nodes, and path indices for restoration modeling.
  • NOMENCLATURE: Restoration decisions include node-DER and node-path assignments, critical-load pickup, path reliability, restored-subtree networks, and restoration unavailability.These variables support assigning DERs and paths while evaluating restored-network reliability.
  • NOMENCLATURE: DER and network operating quantities include availability, restoration time, reserve energy, line count, active and reactive injections, node voltage, and DER power capacities.The nomenclature specifies reserve energy in kWh and active/reactive capacities in kW and kVar.
  • I. INTRODUCTION: Extreme weather can cause prolonged electricity outages that eliminate critical services and threaten customer safety, motivating resilient distribution restoration.Approximately 78% of outages from 1992 to 2010 were attributed to extreme weather events, affecting around 178 million metered customers.
  • I. INTRODUCTION: Grid resilience requires withstanding and recovering from high-impact low-probability events, including restoring critical loads while the main grid is unavailable.Traditional distribution restoration approaches are inapplicable under this disaster condition, necessitating advanced methods.
  • I. INTRODUCTION: Prior research uses microgrids and DERs to isolate faults, serve local loads, and restore critical feeder loads, but often omits post-restoration distribution failures.Post-disaster feeder failures affect resilience alongside the capacity and duration of restored critical-load service.
  • I. INTRODUCTION: The proposed framework optimally allocates DERs to restore critical loads while maximizing post-restoration reliability and modeling tie-switches and open-loop feeder configurations.It defines a resilience metric incorporating post-restoration failures in distribution lines and DERs and formulates the framework as a MILP.

A. Contributions · B. Assumptions • · II. RESTORATION PROBLEM - DEFINITIONS

The paper formulates restoration to maximize resilience against post-restoration failures while restoring as many critical loads as DER capacity and available paths permit. It incorporates tie-switches and defines radial, DER-supplied restored networks while optimizing restoration duration under explicit operational assumptions.

  • A. Contributions: The restoration objective maximizes resilience by decreasing the likelihood of post-restoration failures.The objective is explicitly defined as a function of post-restoration failure.
  • A. Contributions: The approach picks up a maximum number of critical loads depending on DER capacity and available restoration paths.This contribution applies under arbitrary disaster conditions.
  • A. Contributions: The formulation models tie-switches and potential alternate restoration paths within an open-loop distribution configuration.Distribution systems are radially operated but may include tie-switches.
  • B. Assumptions •: The distribution circuit is assumed to have enough remote-controlled switches for operating the proposed restoration plan.Remote terminal units and additional tie switches support advanced automation capabilities.
  • B. Assumptions •: Non-critical loads are disconnected before critical-load restoration using remotely controllable smart-meter infrastructure.Advanced Metering Infrastructure enables distribution operators to remotely disconnect customer supply.
  • B. Assumptions •: Each restoration path maintains radial topology, and DERs are not networked because existing grids typically lack advanced islanded-network control.Each restored network is supplied by one DER.
  • II. RESTORATION PROBLEM - DEFINITIONS: The framework maximizes restored critical-load resilience using post-restoration network reliability and DER availability indices while optimizing restoration duration.The graph-theoretic framework represents feeders, restored networks, and the restoration objective.

A. Graphical Representation … 3) Node-Path Assignment Variable:

The restoration model represents the distribution network as a probabilistic graph and decomposes it into DER-supplied restored subtree networks. Binary assignment variables specify DER ownership and select radial supply paths for nodes with multiple restoration options.

  • A. Graphical Representation: The distribution network is modeled as a probabilistic graph G = (V, E), with nodes representing buses and edges representing distribution lines.Each edge has a probability-of-failure index qe for disaster conditions.
  • A. Graphical Representation: The problem formulation explicitly defines the variables used by the critical load restoration model.
  • 1) Restored Subtree Network (RSN):: The approach decomposes G = (V, E) into n(M) restored subtree networks, each supplied by only one available DER.M is the set of nodes with DERs available for restoration, and Sk denotes the subtree supplied by DER k.
  • 2) Node-DER Assignment Variable:: Each critical load is assigned to only one non-networked DER through binary node-DER assignment variables vk_i.For a system with n nodes and m DERs, the formulation contains m × n node-DER assignment variables.
  • 2) Node-DER Assignment Variable:: The binary assignment vk_i indicates whether node i is restored using DER k and belongs to Sk.The value vk_i = 0 means node i does not belong to Sk.
  • 3) Node-Path Assignment Variable:: Because open-loop tie-switch configurations can create multiple supply paths, radial restoration requires selecting one path for eligible nodes.The binary variable yk_i,Pα = 1 indicates that node i is supplied by DER k along path α and requires energizing nodes in Pα.

4) Critical Load Pickup Variable: … 3) Critical Load Restoration Time:

The formulation represents physically feasible critical-load pickup, while jointly considering restored demand, restoration duration, and post-restoration reliability. DER availability and reserve energy determine restoration duration, which is allocated equitably across restored critical loads.

  • 4) Critical Load Pickup Variable:: A binary variable s_i is associated with each critical load to represent cases where physical damage prevents restoration through an available DER-connected path.This accommodates the possibility that not all critical loads can be restored after multiple distribution-line faults.
  • B. Restoration Objective: The restoration objective considers total restored demand, restoration duration, and post-restoration reliability while seeking maximum critical-load restoration.The formulation jointly optimizes restoration reliability and duration for critical loads under a given disaster condition.
  • 1) Restoration Path Reliability (RP: Restoration path reliability is the probability that the restored subtree remains operational, requiring a source-to-load path through the DER-restored network.For each DER, reliability is based on the success probabilities of the distribution-line edges included in its restoration paths.
  • 1) Restoration Path Reliability (RP: For n(M) available DERs, the resulting restored network’s restoration path reliability is defined collectively across the DER-based restoration network.The paper gives this network-level reliability in equation (2).
  • 2) DER Availability:: DERs are expected to operate in islanded mode during disasters, and their availability depends on lifeline performance and microgrid configuration.Failure and repair rates can be used with minimal cut-set or Markov-based methods to calculate DER availability.
  • 2) DER Availability:: For DER k, restoration time T_k is the duration it can continuously supply its critical loads, determined from reserve energy E_k and time-varying load demand P_i,t.The resulting T_k is a non-linear function.
  • 3) Critical Load Restoration Time:: Assuming constant expected load profiles, the formulation seeks approximately equal restoration times across restored loads while maximizing DER-capacity utilization.When all DERs jointly restore all critical loads, T_net defines the maximum duration, and each DER restoration time T_k is constrained close to T_net.

III. RESILIENT RESTORATION PROBLEM FORMULATION · A. Objective Function

The formulation linearizes restoration path reliability and incorporates DER availability into an objective that minimizes effective restoration unavailability while maximizing restored critical loads. It thereby forms restoration service networks robust to post-restoration failures.

  • III. RESILIENT RESTORATION PROBLEM FORMULATION: The restoration problem is formulated as a mixed-integer linear program subject to feeder operational and connectivity constraints.The MILP formulation is presented for the proposed restoration problem.
  • A. Objective Function: The objective is derived from restoration path reliability and DER availability, with the nonlinear reliability function transformed into linear form.The transformation uses a logarithmic reformulation of restoration path reliability.
  • A. Objective Function: Because pe < 1, maximizing restoration path reliability is transformed using the monotonically decreasing function logpe.Taking logarithms of the total restoration path reliability enables the linearized objective transformation.
  • A. Objective Function: Effective path unavailability, UP, quantifies the restoration service networks’ failure process after the restoration plan is executed.UP indicates the possibility that restoration service networks fail after restoration, and differs from conventional continuously operated system unavailability.
  • A. Objective Function: Effective restoration unavailability, UR, is a weighted sum of UP multiplied by the unavailability of the respective DERs.UR incorporates DER unavailability into the post-restoration failure probability measure.
  • A. Objective Function: The final objective minimizes UR while restoring the maximum number of critical loads, producing a maximally reliable restoration service network.The combined metric URC represents both effective restoration unavailability and critical-load restoration.
  • A. Objective Function: The weighted objective forms restoration service networks using available DERs that are robust to post-restoration failures.The second objective prioritizes restoring the maximum number of critical loads, while the first measures effective restoration unavailability.

B. Restoration Problem Constraints

The restoration problem defines constraints for a distribution circuit modeled as a connected graph, with available DERs supplying critical-load restoration; these constraints are given in equations (16)–(31).

  • Network representation: The distribution circuit is represented as a connected graph G = (V, E) containing n nodes and l distribution lines.Nodes are denoted n ∈ V, while distribution lines are denoted l ∈ E.
  • DER availability: m DERs are assumed available to restore critical loads.The available DERs are denoted m ∈ M.
  • Constraint formulation: The restoration constraints are categorized and defined in equations (16)–(31).The section introduces the several constraints associated with the proposed restoration problem.

1) Connectivity Constraints:: · 2) Power Flow Constraints::

The connectivity constraints ensure that restored subnetworks remain connected and radial while enforcing DER, load, switch, line, and alternate-path relationships. The power-flow constraints model branch flows and voltages, select one restoration path where alternatives exist, and linearize resulting bilinear terms.

  • 1) Connectivity Constraints::: Connectivity constraints ensure restored networks are connected and operate in a radial topology.
  • 1) Connectivity Constraints::: DER-connected nodes must belong to the corresponding DER-supplied RSN, while non-critical-load nodes can belong to at most one RSN.
  • 1) Connectivity Constraints::: A critical load is supplied by an RSN if and only if its critical-load pickup variable s_i equals 1.
  • 1) Connectivity Constraints::: When a remote-control switch is unavailable, parent and child nodes must belong to the same RSN or to no RSN.
  • 1) Connectivity Constraints::: Unavailable distribution lines prevent their endpoint nodes from belonging to the same RSN, even if those nodes remain active or supply critical loads.
  • 1) Connectivity Constraints::: Radial topology requires a node without an alternate path to belong to an RSN only when its parent belongs to that RSN; alternate paths use node-path assignments and energize all required parents.
  • 2) Power Flow Constraints::: Power-flow constraints use a linearized DistFlow approximation for branch flows and voltages, with each tree-structured RSN having a DER at its root and one in-flow per node.
  • 2) Power Flow Constraints::: For nodes with multiple restoration paths, power-flow and voltage equations are written along each path, one path is selected, and Big M transforms bilinear constraints into integer linear constraints.

3) Operational Constraints::

The operational constraints define desired attributes for the restored network, including valid bus voltages, DER capacity limits, and equitable restoration times for critical loads.

  • Voltage constraints: Bus voltage must remain within a specified range at nodes belonging to restoration service networks and be zero otherwise.The logical bus-voltage constraints are eliminated as expressed in (29).
  • DER capacity constraints: Critical loads served by each DER must not exceed its maximum active and reactive power capacity.This requirement is imposed by constraint (30).
  • Restoration-time constraints: Restoration times are constrained to promote equitable allocation of DER capacity across critical loads.Constraint (31) uses ϵ to represent the acceptable difference between restoration times, ideally setting ϵ to zero; a small value is specified for differing DER capacities and load demands.

IV. RESULTS AND DISCUSSION

The proposed restoration framework is evaluated on IEEE 123-node and 906-bus feeders through multiple disaster, DER-availability, scalability, and comparative case studies. Exact OpenDSS power-flow analyses and comparisons with state-of-the-art methods assess solution applicability and optimality.

  • Evaluation setup: The framework is tested on IEEE 123-node and 906-bus distribution feeders using multiple case studies.The MILP is solved with CPLEX 12.6 after formulation in MATLAB R2016a.
  • IEEE 123-node feeder: The 123-node studies examine lesser-impact and higher-impact disasters, respectively involving a few lines and multiple faults.The cases assess restoration while accounting for feeder damage.
  • IEEE 123-node feeder: The experiments also vary DER availability and capacity, including equal and unequal availabilities.These configurations are used to demonstrate restoration of a maximum number of critical loads.
  • Validation: Exact OpenDSS power-flow analyses report significantly small losses for each restoration service network, validating the linear power-flow approximation.Each RSN is simulated with detailed line and load models.
  • Comparative evaluation: The approach is compared with two state-of-the-art methods, showing optimality through mathematical optimization rather than heuristic search.The cited heuristic method searches all possible solutions and therefore obtains an optimal result.

A. Case Study I: IEEE 123-Node feeder System · 1) Restoration during Minor Damage in Distribution Network:

The IEEE 123-node study evaluates restoration under minor damage using five DERs and 11 critical loads. Three cases examine equal DER availability, unequal availability, and critical-load restoration-time constraints.

  • 1) Restoration during Minor Damage in Distribution Network:: The minor-damage tests assume lines 18-13 and 52-152 are faulted, while remote-controlled switches preserve radial topology in the restored networks.The proposed approach operates the remaining switches as needed during restoration.
  • 1) Restoration during Minor Damage in Distribution Network:: Case I assumes equal DER availability of 0.95, reducing the objective to minimizing the total number of nodes in restored subtree networks.Five restored subtree networks are formed, each energized by one DER, and restoration paths are reported for each critical load.
  • 1) Restoration during Minor Damage in Distribution Network:: Case II assigns unequal availability to DERs at nodes 44 and 60, with values of 0.92 and 0.90, respectively, producing a different restoration topology.The objective minimizes restored-subtree nodes weighted by each DER’s respective unavailability.
  • 1) Restoration during Minor Damage in Distribution Network:: Case III adds the critical-load restoration-time constraint to the unequal-availability scenario, requiring restoration-time considerations beyond minimizing unavailability.The constraint is described by equation (31).
  • 1) Restoration during Minor Damage in Distribution Network:: Case II yields restoration times of 43.99 hours for the RSN formed with DER 60 and 5.78 hours for the RSN formed with DER 86.The disparity arises when energy sharing among DERs is ignored while minimizing restoration unavailability.
  • 1) Restoration during Minor Damage in Distribution Network:: Equal restoration times increase effective restoration unavailability because equitable DER-capacity allocation can require longer restoration paths for some RSNs.Cases I and II have relatively smaller effective restoration unavailability than Case III, according to Table II.

2) Restoration during Major Damage in Distribution Network: · B. Case Study II: IEEE 906-bus Low-Voltage Test Feeder · C. Performance Comparison

The proposed strategy is evaluated on a heavily damaged IEEE 906-bus feeder and compared with two state-of-the-art restoration methods. The case study forms three DER-energized restored subtrees while accounting for faults, DER availability, and capacity constraints.

  • 2) Restoration during Major Damage in Distribution Network:: Multiple faults disconnect the IEEE 906-bus feeder from the main supply, creating a major-damage restoration scenario.The simulated faults affect lines 26-27, 13-18, 51-151, 91-93, 54-57, and 67-97.
  • 2) Restoration during Major Damage in Distribution Network:: The major-damage objective prioritizes restoring the maximum number of critical loads over equitable allocation.The strategy is tested under disaster conditions with DER availability varying by location.
  • B. Case Study II: IEEE 906-bus Low-Voltage Test Feeder: The low-voltage feeder has 906 nodes, three DER locations, and 17 critical loads.DERs are connected at nodes 125, 568, and 742, while the feeder operates at 416 V.
  • B. Case Study II: IEEE 906-bus Low-Voltage Test Feeder: The disaster configuration includes open or faulted switches 378-384 and 762-770, plus a normally open 618-881 switch that creates a loop.The feeder is disconnected from the main supply after the disaster.
  • B. Case Study II: IEEE 906-bus Low-Voltage Test Feeder: Three restored subtree networks are formed, each energized by one DER, while CL-860 remains unsupplied because of line faults and DER capacity constraints.The average simulation time is 7.69 seconds.
  • B. Case Study II: IEEE 906-bus Low-Voltage Test Feeder: The framework is generic and can represent real-world systems and operating scenarios by replacing its parameters and network model.Results vary with DER penetration and automation capabilities, but the formulation remains applicable and produces a feasible restoration plan.
  • C. Performance Comparison: The proposed approach is compared with methods and using IEEE test feeders and algorithms’ ability to restore critical loads.Method uses heuristic search, whereas method formulates critical-load restoration as an MILP problem.

1) Comparison with Heuristic Method in [12]: · 2) Comparison with MILP Formulation in [13]: · V. CONCLUSIONS

The proposed MILP-based restoration approach avoids search-based path enumeration, matches the heuristic method’s restoration result with much lower reported simulation time, and restores more critical loads than in major-damage cases. Its generic framework accounts for network failure probability and DER availability while minimizing effective unavailability for robust post-restoration operation.

  • 1) Comparison with Heuristic Method in [12]:: 13.8 minutes is the average simulation time for storing and updating the heuristic strategy table, compared with 1.23 seconds for optimization.The strategy table initially contains 16,431 restoration paths and is reduced to 54 feasible paths after filtering.
  • 1) Comparison with Heuristic Method in [12]:: The proposed MILP models path selection, topology, and power-flow constraints for multiple restoration paths, avoiding search-based methods.A binary variable represents path selection and restoration within the MILP formulation.
  • 1) Comparison with Heuristic Method in [12]:: The matching result occurs because the implementation of includes minimizing energized nodes, selecting a robust minimum-node restoration among multiple maximum-load solutions.The original formulation does not include this energized-node minimization condition.
  • 2) Comparison with MILP Formulation in [13]:: Tie-switches provide additional restoration-path flexibility in the proposed approach but are excluded from the formulation in.The paper notes that tie-switches can benefit critical-load restoration under multiple distribution-system damages.
  • 2) Comparison with MILP Formulation in [13]:: The proposed approach restores more critical loads than for both the modified IEEE 123-node feeder and the IEEE 906-bus test feeder.For the 123-node feeder, does not pick CL-94; for the 906-bus feeder, does not restore CL-858 or CL-906.
  • V. CONCLUSIONS: The proposed generic framework can be adapted to real-world systems and damage scenarios, while its restoration plan accounts for network failure probability and DER availability.The conclusions state that minimizing effective unavailability produces a plan robust to post-restoration failures from a second disaster strike while restoring a maximum number of critical loads.
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