Because of tight reactive power balance, receiving-end power grid is confronted with serious short-term voltage stability problems. One of the most effective solutions is installing static synchronous compensator (STATCOM). This paper minimizes the investment of STATCOM installation with considerations of network dynamics, power flow constraints, security constraints, and upper and lower limits. The planning of location and capacity for STATCOMs is incorporated into a mixed-integer programming. A convex relaxation method is proposed to deal with the model. In this method, linear superposition method is used to make the model convex with respect to binary variables. Then, a tight relaxation is conducted to the substitution variables to prevent solving mixed-integer programming directly. The example test shows that the investment of the planning in this paper is smaller than that of the conventional method. Additionally, the convex relaxation method has a higher efficiency compared with commercial solvers.