FORWARD: A Feasible Radial Reconfiguration Algorithm for Multi-Source Distribution Networks

TL;DR

FORWARD addresses the feasible radial reconfiguration problem in multi-source distribution networks, proving feasibility is weakly NP-complete and proposing a polynomial-time, decomposition-based algorithm. The method uses graph partitioning, dual-graph condensation, and capacity-aware edge swaps to guarantee feasible radial configurations while preserving optimality in a compositional framework. It demonstrates strong empirical performance on networks up to 400 nodes, outperforming commercial MINLP solvers in both speed and reliability, and provides a practical initialization approach for iterative solvers. The work has important implications for real-time distribution management and can inform extensions to other network flow domains.

Abstract

This paper considers an optimal radial reconfiguration problem in multi-source distribution networks, where the goal is to find a radial configuration that minimizes quadratic distribution costs while ensuring all sink demands are met. This problem arises in critical infrastructure systems such as power distribution, water networks, and gas distribution, where radial configurations are essential for operational safety and efficiency. Optimal solution for this problem is known to be NP-hard. In this paper, we prove further that constructing a feasible radial distribution configuration is weakly NP-complete, making exact solution methods computationally intractable for large-scale networks. We propose FORWARD (Feasibility Oriented Random-Walk Inspired Algorithm for Radial Reconfiguration in Distribution Networks), a polynomial-time algorithm that leverages graph-theoretic decomposition and random walk principles to construct feasible radial configurations. Our approach introduces novel techniques including strategic graph partitioning at articulation points, dual graph condensation to address greedy shortsightedness, and capacity-aware edge swapping for infeasibility resolution. We provide rigorous theoretical analysis proving feasibility guarantees and establish a compositional framework enabling parallel processing while preserving optimality properties. Comprehensive numerical evaluation on networks ranging from IEEE standard test systems to 400-node small-world networks demonstrates that FORWARD consistently outperforms commercial MINLP solvers, achieving optimal or near-optimal solutions in seconds where traditional methods require hours or fail entirely. The algorithm's polynomial-time complexity and scalability make it particularly suitable for real-time distribution network management and as an effective initialization strategy for iterative optimization solvers.

Figures (10)

• Figure 1: In the optimal reconfiguration problem, the highlighted nodes in dark are the sources and the remaining nodes are the sinks. In radial configuration, some nodes, e.g., sink node 10 may receive receive input from two different edge; despite that there is no cycle in the graph. The network used here is the IEEE 33 network ieee33.
• Figure 2: Example where radial distribution constructed from MST (plot (b)) is not the minimum radial configuration, $\mathcal{G}(\mathcal{V}_D,\mathcal{S}^\star)$, (plot(c)). This phenomenon is due to the quadratic nature of the cost; let the demand at each sink be $d$. The source node, highlighted in bold, can supply input $5d$. The edge weights are shown on the edges. In the feasible radial distribution network (b) the cost is $1\cdot(d)^2 +1\cdot(4d)^2+1\cdot(3d)^2+1\cdot(2d)^2+2\cdot(d)^2 =32d^2$, meanwhile in the feasible radial distribution network (c) the cost is $1\cdot(2d)^2 +4\cdot(1d)^2 +1\cdot(3d)^2+1\cdot(2d)^2+1\cdot(1d)^2=22d^2$.
• Figure 3: An example where MSF results is a better outcome than MST: sources highlighted provide an input of $2d$ each and the sink nodes demand each $d$.
• Figure 4: An example of a distribution scenario described in the proof of Theorem \ref{['thm::hardness']}. Capacities are shown as edge weights. Source nodes are shown in dark
• Figure 5: A distribution network $\mathcal{G}_D$ (plot (a)) and partitioned subgraphs with adjusted nodal value at separation nodes (plot (b)).