Optimal Kron-based Reduction of Networks (Opti-KRON) for Three-phase Distribution Feeders

TL;DR

This work extends Opti-KRON to unbalanced three-phase distribution networks, introducing a structure-preserving Kron-based reduction that combines a three-phase clustering framework with an exhaustive GPU-accelerated search to identify optimal node aggregations. By operating directly in the complex domain and incorporating a radialization step, the method achieves large reductions (up to ~90%) with negligible voltage deviation across multiple loading scenarios, validated on real feeders with thousands of nodes. The exhaustive-search approach outperforms the MILP formulation in accuracy and scalability, with GPU implementations delivering substantial speedups (up to ~15×) on large networks. The resulting reduced models enable efficient steady-state analysis and OPF studies, and the framework lays groundwork for extending to transmission systems and lifting results back to the full network for control tasks.

Abstract

This paper presents a novel structure-preserving, Kron-based reduction framework for unbalanced distribution feeders. The method aggregates electrically similar nodes within a mixed-integer optimization (MIP) problem to produce reduced networks that optimally reproduce the voltage profiles of the original full network. To overcome computational bottlenecks of MIP formulations, we propose an exhaustive-search formulation to identify optimal aggregation decisions while enforcing voltage margin limits. The proposed exhaustive network reduction algorithm is parallelizable on GPUs, which enables scalable network reduction. The resulting reduced networks approximate the full system's voltage profiles with low errors and are suitable for steady-state analysis and optimal power flow studies. The framework is validated on two real utility distribution feeders with 5,991 and 8,381 nodes. The reduced models achieve up to 90% and 80% network reduction, respectively, while the maximum voltage-magnitude error remains below 0.003 p.u. Furthermore, on a 1000-node version of the network, the GPU-accelerated reduction algorithm runs up to 15x faster than its CPU-based counterpart.

Figures (7)

• Figure 1: Examples of allowable and unallowable (infeasible) reductions.
• Figure 2: Example of the assignment process, i.e., each reduced node transfers its injection to its assigned super-node.
• Figure 3: Example of assignment evolution under different reduction limits: Assume the best reduction configuration is when nodes 1 and 2 are assigned to node 3. Because of \ref{['eq: assign_adj']}, node 1 cannot be directly reduced to node 3 in a single iteration. However, when $q=2$, the algorithm behaves greedily and attempts to reduce two nodes simultaneously. Consequently, it fails to reach the optimal reduction.
• Figure 4: Change of topology with Kron reduction. While (a) shows the full network, (b), (c), and (d) represent the reduced networks, when nodes 1, 2, and 3 are Kron reduced, respectively. Kron reduction of node 3 with degree three results in a meshed network.
• Figure 5: Comparison of reductions based on MILP and exhaustive search on 100-node sub-feeders: maximum voltage magnitude error versus reduction.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof