Flexible AC transmission systems (FACTSs) can maximize capacity utilization under time-varying grid usage patterns by actively controlling the power flow of the transmission lines, e.g., with phase-shifting transformers (PST). In this paper, we propose an algorithm to determine the minimum number of PSTs and their location such that the grid can operate robustly for any realization of the (active) power set points from a known, continuous uncertainty set. As we show in our experiments, only considering a few extreme grid scenarios cannot provide this guarantee. The proposed algorithm considers the trade-offs between PST placement and operational decisions, such as PST control and redispatch. By minimizing the worst-case redispatch cost, it yields two affine linear control policies for these as a byproduct. Power flow is modeled as a constrained linear system, and the control design and actuator minimization tasks are formulated as a mixed-integer linear program (MILP). We also design a greedy algorithm, whose optimal value differs less than 20% from the MILP solution while being one to two orders of magnitude faster to compute. The proposed algorithm is evaluated for a small demonstrative 3-bus example and the IEEE 39 bus test system.

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