Source-linked AI summary
Resilient Disaster Recovery Logistics of Distribution Systems: Co-Optimize Service Restoration with Repair Crew and Mobile Power Source Dispatch
Shunbo Lei, Chen Chen, Yupeng Li, Yunhe Hou
TL;DR
Post-disaster DS recovery requires coordinated logistics for RCs and MPSs, but their dispatch has not been well investigated jointly with restoration. The paper formulates and linearizes a co-optimization model for routing, scheduling, microgrid formation, and DS reconfiguration, and case studies demonstrate improved service restoration and computational efficiency.
Problem
Joint logistics of RCs and MPSs with DS restoration is not well investigated, despite their importance for post-disaster outage management.
Method
A co-optimization model coordinates RC and MPS routing and scheduling with dynamic DS reconfiguration, spanning-forest radiality constraints, equivalent convexification, MILP linearization, and preprocessing.
Results
Case studies show that co-optimizing RC dispatch, MPS dispatch, and DS restoration has the best performance and effectively restores loads through dynamically formed microgrids.
Takeaways & Limitations
Recovery strategies can coordinate MPS-powered microgrids with RC repairs and DS switch actions, while preprocessing improves computational efficiency without sub-optimality in Case I.
Abstract
from arXiv · showhide
Repair crews (RCs) and mobile power sources (MPSs) are critical resources for distribution system (DS) outage management after a natural disaster. However, their logistics is not well investigated. We propose a resilient scheme for disaster recovery logistics to co-optimize DS restoration with dispatch of RCs and MPSs. A novel co-optimization model is formulated to route RCs and MPSs in the transportation network, schedule them in the DS, and reconfigure the DS for microgrid formation coordinately, etc. The model incorporates different timescales of DS restoration and RC/MPS dispatch, the coupling of transportation and power networks, etc. To ensure radiality of the DS with variable physical structure and MPS allocation, we also model topology constraints based on the concept of spanning forest. The model is convexified equivalently and linearized into a mixed-integer linear programming. To reduce its computation time, preprocessing methods are proposed to pre-assign a minimal set of repair tasks to depots and reduce the number of candidate nodes for MPS connection. Resilient recovery strategies thus are generated to enhance service restoration, especially by dynamic formation of microgrids that are powered by MPSs and topologized by repair actions of RCs and network reconfiguration of the DS. Case studies demonstrate the proposed methodology.
I. INTRODUCTION
The paper frames post-disaster distribution-system outage management as a coordinated logistics problem involving repair crews, mobile power sources, and network restoration. It proposes a co-optimization model that coordinates routing, scheduling, repair, and reconfiguration decisions.
- Motivation: Natural disasters cause severe power outages, making efficient outage management a critical requirement for resilient power grids.The paper identifies outage-related economic loss and resilience enhancement as key motivations.
- Motivation: Repair crews and mobile power sources support restoration by repairing damaged components and supplying critical loads disconnected from the main grid.Mobile power sources include truck-mounted mobile emergency generators and mobile energy storage systems.
- Research gap: DS restoration, RC dispatch, and MPS dispatch are interdependent, but their joint optimization is not well investigated.Existing work often simplifies RC-dispatch interdependence, while MPS dispatch is seldomly co-optimized with restoration.
- Contributions: The proposed model jointly routes and schedules RCs and MPSs, forms microgrids dynamically, and reconfigures the DS while accounting for coupled transportation and power networks.It also addresses different dispatch and restoration timescales, MESS state-of-charge variation, and radiality under changing physical structure and MPS allocation.
- RC routing and scheduling: RC routing uses vertex-wise time-indexed variables rather than conventional edge-wise TSP variables, allowing transportation-network coupling and different timescales to be modeled more simply.The formulation does not require crews to visit every damaged component, permitting a minimal repair set that can restore all loads.
- RC routing and scheduling: The RC scheduling model links crew locations, travel times, repair durations, resource capacities, and single-repair constraints within the co-optimization.The formulation can retrieve a feasible path from scheduling variables, while explicit single-repair constraints tighten the MILP relaxation and may reduce solution time.
III. ROUTING AND SCHEDULING OF MPSS
MPS dispatch jointly determines transportation, connection, and power-operation decisions. The formulation represents MEG output limits and MESS charging, discharging, and state-of-charge evolution over time.
- MPS routing and scheduling: MPS dispatch comprises interdependent routing and scheduling tasks, selecting routes among candidate connection nodes and managing travel and power-output behaviors.Scheduling covers MPS travel and power outputs, including charging inputs for MESSs.
- MPS routing and scheduling: Binary connection variables indicate whether each MPS is connected to candidate DS node i at time t.Candidate-node capacities limit the number of MPSs that may connect at each node.
- MPS routing and scheduling: Auxiliary variables distinguish MPS travel from DS connection during each time period.A traveling-state variable equals 1 while an MPS travels and 0 when it is connected to the DS.
- MEG dispatch: MEG real and reactive power outputs are restricted by their capacities and forced to zero when the MEG is not connected to the DS.Fuel limits are excluded because tanker trucks can refuel MEGs during long-term blackouts.
- MESS dispatch: MESS constraints enforce mutually exclusive charging and discharging, connection-dependent power limits, state-of-charge evolution, and allowable state-of-charge ranges.The state-of-charge update uses the duration of each time period and charging or discharging efficiencies.
IV. DYNAMIC NETWORK RECONFIGURATION AND POWER DISPATCH OF THE DS
The paper models dynamic distribution-system reconfiguration and power dispatch under changing physical structure and mobile-power-source allocation. Feasible topologies are represented as spanning forests to support radial microgrid formation and coordinated power restoration.
- Radiality modeling: Variable repairs and mobile-power-source allocation make existing distribution-system radiality formulations inapplicable.The physical network and source-node distribution vary with the repair plan and mobile-power-source placement.
- Radiality modeling: Each feasible topology is a spanning forest whose subgraphs are spanning trees, rather than a single spanning tree connecting all nodes.This structure represents radial microgrid formation with multiple source components.
- Radiality modeling: A fictitious undamaged network is constrained to form a spanning tree, while the operating distribution system may close only a subset of its branches.These constraints produce a spanning forest for the distribution system at each time period.
- Power dispatch: The model defines restored-load status, nodal demand and mobile-power-source output, squared voltage magnitude, voltage bounds, and branch power-flow variables.These variables support time-indexed distribution-system power-flow and restoration constraints.
- Power dispatch: The dynamic power-dispatch constraints coordinate load restoration, nodal real and reactive power balance, voltage limits, branch-capacity limits, and DistFlow equations.Open branches carry zero real and reactive power, while closed branches follow the DistFlow formulation.
V. THE CO-OPTIMIZATION MODEL
The co-optimization model links repair-crew dispatch, mobile-power-source dispatch, and distribution-system restoration. It maximizes weighted restored load over time while coordinating repairs, mobile supply, and network reconfiguration.
- Objective: The objective maximizes the weighted sum of restored loads over time and minimizes repair-crew and mobile-power-source travel.A small ε makes service restoration dominant over the travel-minimization objective.
- Interdependence: Repair-crew dispatch determines when damaged branches become operable, linking repair actions to subsequent distribution-system restoration.A damaged branch becomes operable only if repaired by a repair crew in the previous time period.
- Interdependence: Mobile-power-source dispatch supplies nodal real and reactive power through sources connected to candidate distribution-system nodes.The nodal mobile-power-source outputs are aggregated into the corresponding real and reactive power terms.
- Interdependence: Repair-crew and mobile-power-source dispatch are intrinsically interrelated with distribution-system restoration and therefore require joint optimization.The model coordinates these decisions to attain better service restoration.
A. Linearization Techniques
The non-convex MINLP is transformed into an equivalent mixed-integer linear program through bounded McCormick-envelope linearization and further linearization of branch constraints.
- Linearization: Nonlinear non-convex terms in the co-optimization model are linearized using McCormick envelopes.Because the binary and continuous variables have explicit bounds, the linearization is equivalent.
- Linearization: The model is convexified into a mixed-integer second-order cone program and then linearized into a MILP solvable by off-the-shelf solvers.The paper cites Gurobi as an example of a suitable solver.
B. Pre-Assigning a Minimal Set of Repair Tasks to Depots
A preprocessing MILP assigns a minimal set of repair tasks to depots before solving the full co-optimization. The assignment accounts for travel distance, depot resources, and complete load restoration without mobile power sources.
- Preprocessing formulation: Preprocessing clusters and pre-assigns repair tasks to depots to reduce the MILP co-optimization model’s computational complexity.Assignment variables indicate whether each damaged component is assigned to a depot, while distances and depot crew sets define the preprocessing model.
- Preprocessing formulation: The preprocessing model minimizes damaged-component-to-depot distances, enforces adequate depot resources, and assigns each damaged component to at most one depot.These constraints aim to reduce repair-crew transportation time and support feasible task allocation.
- Minimal repair set: Unlike assigning every repair task, the method selects only a minimal set that can fully restore all distribution-system loads without mobile-power sources.Additional distribution-system constraints ensure the selected repairs can make the required damaged branches operable and supply all loads.
C. Selecting Candidate Nodes for MPS Connection
The method reduces MPS connection candidates by identifying candidate nodes within each physical island and selecting a distance-efficient representative from each eligible island.
- Candidate-node reduction addresses computational time because the co-optimization model depends partly on the number of possible MPS connection nodes.
- Algorithm 1 uses intact-branch adjacency to identify the candidate-node set Iσ within each physical island.Its input is the DS node set, adjacency matrix, and all eligible candidate nodes; its output is Iσ for every physical island.
- The algorithm iteratively expands each island through intact-branch connectivity before retaining only candidate nodes in that island.
- The final integer program minimizes distances among selected nodes while requiring one node from every physical island containing at least one candidate.Products φi · φj are linearized using McCormick envelopes.
VII. CASE STUDIES
Case studies evaluate the co-optimization method on distribution systems with damaged branches, repair crews, and mobile power sources. The results show coordinated restoration through repair, dispatch, and dynamic microgrid formation.
- Case I: IEEE 33-Node Test System: Case I models 8 damaged branches, 2 depots, 2 repair crews, and a 500 kW/400 kVar MEG plus a 300 kW/300 kWh MESS.Repair requirements and times vary across damaged components, while load priorities and travel data are randomly generated.
- Case I: IEEE 33-Node Test System: 4 repaired branches are sufficient to fully restore all loads because normally open backup branches provide alternative network paths.The other 4 damaged branches can be repaired later for objectives other than maximizing restored load.
- Case I: IEEE 33-Node Test System: The preprocessing and co-optimization models are solved in 0.13 s, 0.14 s, 0.04 s, and 2.26 s, respectively, in Case I.The sequence is repair-task pre-assignment, candidate-node detection, candidate-node selection, and the final co-optimization.
- Case I: IEEE 33-Node Test System: The proposed strategy has the best restored-load performance among strategies that co-optimize both RC and MPS dispatch with DS restoration.The comparison includes strategies considering only RCs, only MPSs, or sequential rather than joint co-optimization.
- Case I: IEEE 33-Node Test System: At t = 3, the MESS forms microgrid 1 at node 29, while at t = 4 the MEG forms microgrid 2 at node 33.Subsequent repairs and switch actions extend or connect these microgrids to the main grid.
- Case I: IEEE 33-Node Test System: At t = 11, repairing branch 5-6 enables restoration of nodes 20 and 28, after which both MPSs disconnect because they are no longer necessary.
B. Case II: IEEE 123-Node Test System
Case II tests the method on an IEEE 123-node system with more damaged branches, resources, and remotely controlled switches. The reported results again indicate effective coordination of RCs, MPSs, and DS reconfiguration for load restoration.
- Case II: IEEE 123-Node Test System: Case II includes 20 damaged branches, 2 depots with 2 RCs each, 4 MPSs, 12 remotely controlled switches, and a 16-period horizon.The MPS fleet consists of 2 MEGs and 2 MESSs.
- Case II: IEEE 123-Node Test System: Pre-assignment, candidate-node detection, candidate-node selection, and co-optimization take 0.43 s, 0.45 s, 0.28 s, and 1183 s, respectively.
- Case II: IEEE 123-Node Test System: The candidate-node selection procedure chooses nodes {2, 27, 38, 59, 95, 103, 116, 123} for the second test system.
- Case II: IEEE 123-Node Test System: The proposed method again demonstrates effectiveness and superiority compared with alternative logistics strategies in restored-load performance.The comparison includes joint, single-resource, and sequential co-optimization strategies.
- Case II: IEEE 123-Node Test System: Dynamic microgrids are powered by MPSs and reconfigured or extended through DS switch actions and RC repair actions.
C. Computational Efficiency
The proposed preprocessing preserves solution quality while improving computational efficiency for the co-optimization model. In the larger case, it enables solving within 2 hr by pre-assigning only a minimal set of repair tasks.
- For Case I, all solution methods attain the same solution, showing that the proposed preprocessing does not cause sub-optimality.
- For Case II, only the proposed method solves the co-optimization problem in 2 hr.
- Pre-assigning a minimal set of repair tasks improves computational efficiency compared with pre-assigning all repair tasks.
- The co-optimization method reduces computational complexity by transforming the MINLP into a MILP and applying preprocessing.