Formulating the Restoration of Distribution Networks as a Multiple Traveling Salesman Problem

TL;DR

This paper reframes electrical distribution restoration after severe weather as a cost-constrained, reward-maximizing multi-TSP on a doubly weighted graph ($CCRM-mTSP-DW$), capturing repair times, travel times, and electrical continuity within a rolling-horizon MILP. By transforming the network into a complete line-graph representation and introducing two working graphs, the authors enable a TSP-like solution that respects per-crew time budgets and precedence constraints that guarantee energization from the source. A single-commodity flow based subtour elimination scheme and directed electrical-continuity constraints underpin feasible schedules, with validation on the IEEE 123-node feeder showing near-optimal performance under realistic uncertainty. The approach blends classic combinatorial optimization with power-system constraints, offering a scalable, adaptable framework for rapid, constraint-aware restoration planning in practice.

Abstract

Severe weather events can cause extensive damage to electrical distribution networks, requiring a multi-day restoration effort. Optimizing the dispatch of repair crews minimizes the severe socio-economic consequences of such events. Considering both repair times and travel times, we use graphical manipulations to transform this multiple crew scheduling problem into a type of traveling salesman problem(TSP). Specifically, we demonstrate that the restoration problem bears major resemblance to an instance of a cost constrained reward maximizing mTSP (multiple TSP) on node and edge weighted (doubly weighted) graphs (a variant we dub the CCRM-mTSP-DW), where the objective is to maximize the aggregate reward earned during the upcoming restoration window, provided no crew violates its time budget and electrical continuity constraints are met. Despite the rich history of research on the TSP and its variants, this CCRM-mTSP-DW variant has not been studied before, although its closest cousin happens to be the "Selective TSP" (S-TSP). This reinterpretation of the restoration problem not only opens up the possibility of drawing on existing solution methods developed for the TSP and its variants, it also adds a new chapter in the annals of research on "TSP-like'' problems. In this paper, we propose a "TSP-like'' mixed integer linear programming (MILP) model for solving the restoration problem and validate it on the IEEE PES 123-node test feeder network.

Figures (7)

• Figure 1: Timeline of the restoration process.
• Figure 2: (a) Graph of a radial distribution network, $G = (\mathcal{N},\mathcal{E})$. (b) Modified version of graph $G$, which we denote by $G_m$. (c) Line graph of $G_m$, which we denote by $L(G_m)$. In this figure, we adopt a two-index notation for the node labels in the line graph simply for clarity. (d) The graph $L(G_m)$ converted to a complete symmetric directed graph, which we denote by $L_K(G_m)$. We refer to the node $(0,r)$ as the root of $L_K(G_m)$.
• Figure 3: (a) Directed version of the radial distribution network shown in Figure \ref{['fig:Fig1_2']}, which we denote by $G^d$. Arrow directions on the edges represent the direction of power flow. (b) Modified version of graph $G^d$, which we denote by $G_m^d$. (c) Line graph of the directed graph shown in panel (b), which we denote by $L(G_m^d)$.
• Figure 4: Illustrating the "graph collapse" procedure for a partially damaged network. (a) $L(G_m^d)$ as obtained after application of the procedure illustrated in Figure \ref{['fig:Fig5']}. (b) $L(G_m^d)$ after the undamaged nodes (other than the root) are deleted.
• Figure 5: MILP formulation for $m$-crew repair scheduling during the $k^{th}$ restoration window. Note that the electrical continuity constraints, eqn. (\ref{['eq:const11']}), are valid only for single source, radial power distribution networks.

Theorems & Definitions (1)

• Remark 4.1