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Integrating unimodality into distributionally robust optimal power flow

Abstract

To manage renewable generation and load consumption uncertainty, chance-constrained optimal power flow (OPF) formulations have been proposed. However, conventional solution approaches often rely on accurate estimates of uncertainty distributions, which are rarely available in reality. When the distributions are not known but can be limited to a set of plausible candidates, termed an ambiguity set, distributionally robust (DR) optimization can reduce out-of-sample violation of chance constraints. Nevertheless, a DR model may yield conservative solutions if the ambiguity set is too large. In view that most practical uncertainty distributions for renewable generation are unimodal, in this paper, we integrate unimodality into a moment-based ambiguity set to reduce the conservatism of a DR-OPF model. We review exact reformulations, approximations, and an online algorithm for solving this model. We extend these results to derive a new, offline solution algorithm. Specifically, this algorithm uses a parameter selection approach that searches for an optimal approximation of the DR-OPF model before solving it. This significantly improves the computational efficiency and solution quality. We evaluate the performance of the offline algorithm against existing solution approaches for DR-OPF using modified IEEE 118-bus and 300-bus systems with high penetrations of renewable generation. Results show that including unimodality reduces solution conservatism and cost without degrading reliability significantly.

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Notes

  1. To handle uncertain loads, we denote \(P_L^f\) and \({\tilde{d}}\) as the load forecast and the uncertain load forecast error, respectively. We substitute the vector of loads \(P_L\) with \(P_L^f+{\tilde{d}}\) and the system uncertainty \({\tilde{w}}\) with \({\tilde{w}} + {\tilde{d}}\) in formulation (12). Then, we can use the same reformulation method and solution algorithm as described in Section 4 to solve this extended model.

  2. Note that each intermediate solution comes from a relaxed approximation and so the objective cost is lower than the true objective cost. Here we use a negative optimality gap to illustrate this relation.

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Acknowledgements

This research was supported by the U.S. National Science Foundation Awards CCF-1442495 and CMMI-1662774.

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Correspondence to Johanna L. Mathieu.

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Li, B., Jiang, R. & Mathieu, J.L. Integrating unimodality into distributionally robust optimal power flow. TOP (2022). https://doi.org/10.1007/s11750-022-00634-4

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  • DOI: https://doi.org/10.1007/s11750-022-00634-4

Keywords

  • Optimal power flow
  • Chance constraints
  • Distributionally robust optimization
  • \(\alpha\)-Unimodality

Mathematics Subject Classification

  • 90C15
  • 90C22
  • 90C34