This article considers false data injection attacks constructed based on partial feedback of generator frequencies in a cyber-physical power system. The goal of the attacker is to destabilize the system, by compromising a subset of frequency control signals with false data injection, without consuming much energy. In this context, two attack design problems are studied, considering both location-fixed attacks and location-switching attacks based on measurable generator bus frequencies. They are further modeled as switched control problems, for which closed-form solutions can be attained. Leveraging the maximum principle, the diagonal partial feedback matrix is optimized by solving a convex optimization problem. The convexified switching variables describing the switching behavior are solved in a quadratic optimization problem and a fractional optimization problem, respectively. As a result, optimal switching conditions to select the best attack locations are obtained, along with optimal partial feedback attack matrices. Case studies on the IEEE 9-bus test system validate the practical merits of theory and numerical effectiveness of the proposed attack schemes.