QPADL: Quadratic Programming for Allocation of Distributed Energy Resources to Minimize Power Loss in Distribution Networks

Distributed Energy Resources (DERs) offer significant potential for reducing power losses, improving voltage stability, and enhancing resilience in distribution networks. To effectively address network-specific operational constraints and maximize DER performance, it is crucial to strategically optimize both their siting and sizing. Existing works primarily adopt analytical or search-based approaches for DER placement aimed at minimizing power losses. However, analytical methods, while computationally efficient, frequently yield suboptimal solutions at higher DER penetration levels, whereas search-based methods, despite their robustness, become computationally impractical for large-scale networks due to exponential complexity growth. To overcome the limitations, this paper proposes a novel analytical framework that establishes an exact quadratic relationship between power losses and DER injections, enabling a precise analytical estimation and optimization. The proposed approach explicitly relates nodal power demands to their respective contributions to system line losses, providing detailed theoretical insights into the root causes of power losses. Practically, the proposed framework facilitates real-time, large-scale DER allocation optimization while maintaining high accuracy. Furthermore, our theoretical analysis quantifies the impact of the DER power factor on optimal placement for loss reduction. This insight provides a direct, simplified method for integrating power loss considerations into complex, multi-objective optimization models. We validate our method on 33, 69, 123 and 533-bus distribution networks. It significantly outperforms feature-based analytical approaches and matches or exceeds traditional search-based methods. On the largest 533-bus system, our algorithm completes the allocation in just 0.5 s, confirming its effectiveness and practicality for real-world applications.