Probabilistic OPF Model for Unbalanced Three-Phase Electrical Distribution Systems Considering Robust Constraints

Juan S. Giraldo*, Juan Camilo Lopez, Jhon A. Castrillon, Marcos J. Rider, Carlos A. Castro

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

7 Citations (Scopus)


This paper proposes a probabilistic optimal power flow (POPF) for unbalanced three-phase electrical distribution systems considering robust constraints. Wind velocity, solar irradiation, and demands are considered as exogenous random variables with dissimilar probability distribution functions. The proposed two-stage POPF model determines the optimal generation dispatch that minimizes the average production cost while satisfying probabilistically robust constraints. The robustness of the solution is adjusted through a parameter involving the first two moments of the random state variables. The 2m+1 point estimate method (PEM) provides the scenarios for the proposed POPF model, which can be solved using commercial solvers. The model's accuracy is evaluated by comparing the results obtained with the PEM and with a classical scenario generator technique. The model is validated through two IEEE distribution feeders with 13 and 123 nodes, comprising dispatchable and renewable generation. Results show that the model can accurately estimate the mean and standard deviation of the state random variables when compared with results obtained through Monte Carlo simulations. Various levels of conservatism are assessed, demonstrating the model's flexibility and applicability.

Original languageEnglish
Article number8681440
Pages (from-to)3443-3454
Number of pages12
JournalIEEE Transactions on Power Systems
Issue number5
Publication statusPublished - Sep 2019
Externally publishedYes


  • 2m+ 1point estimate method
  • Probabilistic optimal power flow
  • probabilistically robust constraints
  • unbalanced three-phase electrical distribution systems


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