FORWARD: Feasibility Oriented Random-Walk Inspired Algorithm for Radial Reconfiguration in Distribution Networks

TL;DR

This paper proposes a novel algorithm called FORWARD, which leverages graph theory to efficiently identify feasible configurations in polynomial time, and employs a combination of network preprocessing, intelligent partitioning, and strategic sampling to construct radial configurations that meet flow requirements.

Abstract

We consider an optimal flow distribution problem in which the goal is to find a radial configuration that minimizes resistance-induced quadratic distribution costs while ensuring delivery of inputs from multiple sources to all sinks to meet their demands. This problem has critical applications in various distribution systems, such as electricity, where efficient energy flow is crucial for both economic and environmental reasons. Due to its complexity, finding an optimal solution is computationally challenging and NP-hard. In this paper, we propose a novel algorithm called FORWARD, which leverages graph theory to efficiently identify feasible configurations in polynomial time. By drawing parallels with random walk processes on electricity networks, our method simplifies the search space, significantly reducing computational effort while maintaining performance. The FORWARD algorithm employs a combination of network preprocessing, intelligent partitioning, and strategic sampling to construct radial configurations that meet flow requirements, finding a feasible solution in polynomial time. Numerical experiments demonstrate the effectiveness of our approach, highlighting its potential for real-world applications in optimizing distribution networks.

Figures (5)

• 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 consumer be $d$ and the generator node, highlighted in bold, can supply input $5d$, in network (b) the cost is $1\cdot(d)^2 + 5\cdot(4d)^2=81d^2$, meanwhile in network (c) the cost is $5\cdot(2d)^2 + 3\cdot(3d)^2=47d^2$.
• Figure 3: Example where MSF results is a better outcome than MST.
• Figure 4: Demonstration of how Pre-processor,Islander and Net-Concad function. Filled solid nodes represent sources while others represent the sinks. The number next to the nodes show corresponding $p_i$. In plot (b) $p_9$ and $p_8$ are adjusted to reflect, respectfully, excess input to deliver through node 9 to $\mathcal{G}_p$ and extra demand at node $8$ to supply to the removed pendent node $11$. In plot (c), after partitioning the graph, node $1$ assumes different roles in each sub-graph. In plot (d) and (f) dashed circles represent super nodes. In plot (e), $\mathcal{T}_1=\mathcal{G}(\{9\},\{\})$ and $\mathcal{T}_2=\mathcal{G}(\{2,8,10\},(2\to8),(8\to10))$; note that here, Sampler has removed edge $(2,10)$ to avoid cycle.
• Figure 5: Illustration of the sampling procedure. In b) and c), note that priority queue in Sampler avoids the flow blockage.

Theorems & Definitions (11)

• Definition 1: Set of radial configurations
• Definition 2
• Lemma II.1
• proof
• Remark II.1: Complexity of FOWARD
• Lemma III.1
• proof
• Lemma III.2
• proof
• Lemma III.3