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This paper studies the optimal transmission switching (OTS) problem for power systems, where certain lines are fixed (uncontrollable) and the remaining ones are controllable via on/off switches. The goal is to identify a topology of the power grid that minimizes the cost of the system operation while satisfying the physical and operational constraints. Most of the existing methods for the problem are based on first converting the OTS into a mixed-integer linear program (MILP) or mixed-integer quadratic program (MIQP), and then iteratively solving a series of its convex relaxations. The performance of these methods depends heavily on the strength of the MILP or MIQP formulations. In this paper, it is shown that finding the strongest variable upper and lower bounds to be used in an MILP or MIQP formulation of the OTS based on the big- $M$ or McCormick inequalities is NP-hard. Furthermore, it is proven that unless $P=NP$ , there is no constant-factor approximation algorithm for constructing these variable bounds. Despite the inherent difficulty of obtaining the strongest bounds in general, a simple bound strengthening method is presented to strengthen the convex relaxation of the problem when there exists a connected spanning subnetwork of the system with fixed lines. With the proposed bound strengthening method, remarkable improvements in the runtime of the mixed-integer solvers and the optimality gaps of the solutions are achieved for medium- and large-scale real-world systems. [ABSTRACT FROM AUTHOR]

System operators are required to serve the load in the most cost-effective way while maintaining the integrity of the system and heeding reliability requirements. In the day-ahead market, operators acquire reserves in an attempt to guarantee N-1 reliability; yet, reserve deliverability is not guaranteed. Prior research has shown that the use of transmission switching, or topology control, may help improve reserve deliverability. In this paper, transmission switching is used as a corrective mechanism to help the system achieve N-1-1 reliability, where not only has the system lost a single element, but also it experiences the loss of a second major element after an adjustment period. In an attempt to preserve N-1-1 reliability, for this paper, a day-ahead unit commitment model that acquires supplementary reserves is solved. The day-ahead market solution is then tested for N-1-1 reliability using contingency analysis models with and without transmission switching. The methodology can be employed at the day-ahead time stage to ensure the system has acquired sufficient supplemental reserves. The results demonstrate that not only can corrective transmission switching be beneficial post-contingency without inhibiting the ability to return to N-1 reliability, but it can also help obtain an N-1-1 reliable solution. [ABSTRACT FROM AUTHOR]