A concentrated solar power (CSP) plant typically has thermal energy storage (TES), which offers advantages of extended operation and power dispatch. Using dual-media, TES can be cost-effective because of the reduced use of heat transfer fluid (HTF), usually an expensive material. The focus of this paper is on the effect of a start-up period thermal storage strategy to the cumulative electrical energy output of a CSP plant. Two strategies—starting with a cold storage tank (referred to as “cold start”) and starting with a fully charged storage tank (referred to as “hot start”)—were investigated with regards to their effects on electrical energy production in the same period of operation. An enthalpy-based 1D transient model for energy storage and temperature variation in solid filler material and HTF was applied for both the sensible heat storage system (SHSS) and the latent heat storage system (LHSS). The analysis was conducted for a CSP plant with an electrical power output of 60 MWe. It was found that the cold start is beneficial for both the SHSS and LHSS systems due to the overall larger electrical energy output over the same number of days compared to that of the hot start. The results are expected to be helpful for planning the start-up operation of a CSP plant with a dual-media thermal storage system.