The Constrained Layer Tree Problem and Applications to Solar Farm Cabling

Abstract

Motivated by the cabling of solar farms, we study the problem Constrained Layer Tree. At its core, it asks whether there exists a tree that connects a set of sources (the leaves) to one sink (the root) such that certain capacity constraints at the inner nodes are satisfied. Our main algorithmic contribution is a dynamic program with various optimizations for Constrained Layer Tree. It outperforms the previously used MILP by multiple orders of magnitude. Moreover, our experiments show that the somewhat abstract problem Constrained Layer Tree is actually the core of the cabling problem in solar farms, i.e., the feasible solution produced by our dynamic program can be used to bootstrap an MILP that can then find good solutions for the original cabling problem efficiently.

Figures (8)

• Figure 1: The $k$-combination of two trees $T_a$ and $T_b$ with branching layer $k_a$ and $k_b$, respectively.
• Figure 2: Any valid tree $T_c$ is a $k$-combination of two valid trees $T_a$ and $T_b$.
• Figure 3: The setting before and after rebalancing. We $k_b$-combine $T_a$ and $T_d$ first, before $k$-combining the result with $T_e$.
• Figure 4: A possible solar farm $S$ of an instance $I'$ of SoFaCLaP Embedding with an embedding of some forest $F$. Since the embedding has total length of at most $k = mT(T+1)$, the depicted instance is a YES-instance of SoFaCLaP Embedding.
• Figure 5: A comparison of the running times between Gurobi and our dynamic program. The instances are separated by feasibility and sorted by the running time of Gurobi.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof