Electric buses (EBs) are undergoing rapid development because of their environmental friendliness. Different from private electric vehicles, EBs are scheduled by a public transport company and required to be charged as soon as possible during operation hours. Consequently, the implementation of fast charging stations (FCSs) is essential to support the operation of EBs. Usually, the size of an FCS is constrained by the residual capacity of the connected distribution network and the investment budget. An energy storage system (ESS) is considered as a potential supplement not only to reduce the network integration cost for FCSs but also to reduce the charging cost of EBs through electricity price arbitrage. To quantify the value of ESS in an electric bus FCS, a mixed integer nonlinear programming (MINLP) formulation for optimal sizing and control of FCS and ESS is built. A simplification method is proposed to convert the MINLP problem to a linear one without any loss in optimality, which accelerates the solving process. Simulations are performed based on historical data of a practical FCS. Numerical results indicate that ESS can significantly help reduce the overall investment and charging cost of the FCS. Meanwhile, the parameters of investment cost, lifespan and time-of-use electricity price have significant influences on the overall value of ESS for an FCS.