(119f) Learning-Assisted AC Optimal Power Flow

Electric power systems are facing important operational challenges due to the growing integration of weather-dependent renewable energy sources and the increasing electrification in the transportation and industry sectors. For decades, the power industry has used simplified models of the underlying physical governing laws resulting in computationally more tractable problems. However, the growth in electric demand and renewable production has driven the operation of power systems to conditions where such models can be highly inaccurate [1]–[3], which has caused important financial losses and an unreliable supply of electricity.

One of the fundamental problems in power system operations is the AC optimal power flow (ACOPF), which aims at determining the dispatch of power generating units distributed over the power grid to supply the demand at the minimum cost while satisfying engineering constraints and the governing laws [4]. Despite the well-documented benefits of using more accurate models of the governing laws, known as power flow equations, to operate the power grid efficiently and reliably, there are still concerns in the power industry about their computational performance. In the last decade, important efforts have been devoted to addressing the main challenges of the ACOPF problem including (i) developing better models based on linear and convex relaxations and approximations of the power flow equations [5]–[11], and (ii) exploiting historical information from previously solved instances or synthetic datasets and end-to-end machine learning techniques to predict optimal solutions [12-14].

This works presents a model of the ACOPF using a convex approximation of the power flow equations, which relies on the prediction of a subset of primal and dual variables. We formulate the ACOPF in rectangular coordinates as a bilinear problem whose nonconvex constraints are expressed as a difference of convex functions. The concave terms are linearized using a first-order Taylor series approximation around a given point. Such convexification renders a second-order conic problem. To avoid infeasibilities due to the convexified constraints, we add a slack variable and penalize it in the objective function. The linearization point as well as the penalty weight to the slack variable are predicted using a neural network.

The proposed model is tested on several realistic test systems under a wide range of operating conditions. The performance of the convexified model is studied in terms of solution time, optimality gap, and approximation error with respect to the solution of the original nonconvex problem using an interior point solver. Our numerical experiments show the benefits of the proposed model with respect to well-documented linear approximations and convex relaxations.

[1] M. Bynum, A. Castillo, J. P. Watson, and C. D. Laird, “Strengthened SOCP Relaxations for ACOPF with McCormick Envelopes and Bounds Tightening,” Comput. Aided Chem. Eng., vol. 44, pp. 1555–1560, Jan. 2018, doi: 10.1016/B978-0-444-64241-7.50254-8.

[2] M. Bynum, A. Castillo, J. P. Watson, and C. D. Laird, “Tightening McCormick relaxations toward global solution of the ACOPF problem,” IEEE Trans. Power Syst., vol. 34, no. 1, pp. 814–817, Jan. 2019, doi: 10.1109/TPWRS.2018.2877099.

[3] H. Zhang, G. T. Heydt, V. Vittal, and J. Quintero, “An improved network model for transmission expansion planning considering reactive power and network losses,” IEEE Trans. Power Syst., vol. 28, no. 3, pp. 3471–3479, 2013, doi: 10.1109/TPWRS.2013.2250318.

[4] A. Gomez-Expósito, A. J. Conejo, and C. Cañizares, Electric Energy Systems: Analysis and Operation. CRC Press, 2008.

[5] R. A. Jabr, “A conic quadratic format for the load flow equations of meshed networks,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2285–2286, Nov. 2007, doi: 10.1109/TPWRS.2007.907590.

[6] J. Lavaei and S. H. Low, “Convexification of optimal power flow problem,” 2010 48th Annu. Allerton Conf. Commun. Control Comput. Allerton 2010, pp. 223–232, 2010, doi: 10.1109/ALLERTON.2010.5706911.

[7] J. Lavaei and S. H. Low, “Zero duality gap in optimal power flow problem,” IEEE Trans. Power Syst., vol. 27, no. 1, pp. 92–107, Feb. 2012, doi: 10.1109/TPWRS.2011.2160974.

[8] C. Coffrin and P. Van Hentenryck, “A Linear-Programming Approximation of AC Power Flows,” Inf. J. Comput., vol. 26, no. 4, pp. 718–734, Nov. 2014, doi: 10.1287/ijoc.2014.0594.

[9] C. Coffrin, H. L. Hijazi, and P. Van Hentenryck, “The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow,” IEEE Trans. Power Syst., vol. 31, no. 4, pp. 3008–3018, Jul. 2016, doi: 10.1109/TPWRS.2015.2463111.

[10] A. Castillo, P. Lipka, J.-P. Watson, S. S. Oren, and R. P. O’Neill, “A successive linear programming approach to solving the iv-acopf,” IEEE Trans. Power Syst., vol. 31, no. 4, pp. 2752–2763, 2016, doi: 10.1109/TPWRS.2015.2487042.

[11] D. K. Molzahn and I. A. Hiskens, “A Survey of Relaxations and Approximations of the Power Flow Equations,” Found. Trends® Electr. Energy Syst., vol. 4, no. 1–2, pp. 1–221, 2019, doi: 10.1561/3100000012.

[12] “Machine Learning for Optimal Power Flows,” Inf. Tutor. Oper. Res., doi: 10.1287/educ.2021.0234.

[13] Kilwein, Z., Boukouvala, F., Laird, C., Castillo, A., Blakely, L., Eydenberg, M., ... & Batsch-Smith, L. (2021). AC-Optimal Power Flow Solutions with Security Constraints from Deep Neural Network Models. In Computer Aided Chemical Engineering (Vol. 50, pp. 919-925). Elsevier.

[14] Jalving, J., Eydenberg, M., Blakely, L., Kilwein, Z., Boukouvala, F., & Laird, C. (2021). Physics-Informed Machine Learning Surrogates with Optimization-Based Guarantees: Applications to AC Power Flow (No. SAND2021-14467C). Sandia National Lab.(SNL-NM), Albuquerque, NM (United States).