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472604 Efficient Global Optimization for a Mixed AC-DC Power Distribution System

ACOPF problem plays a central role in optimal design and operation of electrical power networks, and the formal study of this problem can date back to 1962 [2]. The vast majority of ACOPF literature focuses on operational ACOPF problems that are nonconvex continuous optimization problems, and only a few of the existing methods (e.g., [3]) can guarantee finding a global optimal solution. This is because operational ACOPF problems usually need to be solved many times a day and none of the existing global optimization methods can solve the problem fast enough. However, global optimization seems to be more attractive for ACOPF design problems, as it is reasonable to make a design decision in a day or a week. Frank and Rebennack [1] successfully applied a decomposition-based global optimization method, called nonconvex generalized Benders decomposition (NGBD [4]), to some mixed AC-DC power distribution systems, and demonstrated the computational advantage of NGBD over state-of-the-art global optimization solvers.

The performance of NGBD is highly dependent on tightness of the convex lower bounding problem used. When the size of the distribution system becomes larger and the convex lower bounding problem becomes less tight, the performance of NGBD degrades quickly, as identified by Frank and Rebennack [1]. Here we propose to integrate domain reduction techniques into NGBD, in order to enhance the tightness of the convex lower bounding problem progressively during the decomposition procedure. Two types of domain reduction techniques are considered for NGBD. One is to solve customized convex bound contraction problems whose sizes are independent of the number of time periods. These bound contraction problems are also strengthened with valid linear cuts derived from the previously solved convex NGBD subproblems. The other is to perform range reduction calculations [5] for nonconvex NGBD subproblems, according to the primal and dual solutions of the previously solved convex NGBD subproblems. The domain reduction techniques not only can reduce the number of NGBD iterations, but also can speed up the solution of nonconvex NGBD subproblems. It will be shown in the case study that, the proposed method is significantly faster than the NGBD used by Frank and Rebennack [1] for the mixed AC-DC power distribution system.

In the case study, the convex lower bounding problem used for NGBD is obtained through linear relaxation, which is known to be weaker than a second-order cone programing (SOCP) relaxation or semi-definite programming (SDP) relaxation for ACOPF problems (e.g., [6]). Therefore, our method can be further improved via the use of a tighter nonlinear convex lower bounding problem.

**References**

[1] S. M. Frank and S. Rebennack, “Optimal design of mixed AC-DC distribution systems for commercial buildings: A nonconvex generalized Benders decomposition approach,” *European Journal of Operational Research*, vol. 242, no. 3, pp. 710 – 729, 2015.

[2] J. Carpentier, “Contribution to the economic dispatch problem,” Bulletin de la Socit Franaise des Electriciens, vol. 8, no. 3, p. 431447, 1962.

[3] C. Chen, A. Atamturk, and S. S. Oren, “Bound tightening for the alternating current optimal power flow problem,” IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–8, 2015.

[4] X. Li, A. Tomasgard, and P. I. Barton, “Nonconvex generalized Benders decomposition for stochastic separable mixed-integer nonlinear programs,” *Journal of optimization theory and applications*, vol. 151, no. 3, pp. 425–454, 2011.

[5] Ryoo, H. S., Sahinidis, N. V., “A branch-and-reduce approach to global optimization,” *Journal of Global Optimization*, vol. 8, no. 2, pp. 107-138, 1996.

[6] X. Bai, H. Wei, K. Fujisawa, and Y. Wang, “Semidefinite programming for optimal power flow problems,” International Journal of Electrical Power and Energy Systems, vol. 30, no. 67, pp. 383–392, 2008.

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