Thermal energy storage is key in making solar-thermal power plants more economically competitive compared to conventional plants. In this work, a new algebraic solution for thermocline thermal energy storage tanks, allowing for any initial temperature profile, is developed and presented. The model, called the Algebraic IC model, is successfully validated by comparing with experimental data and numerical solution of the governing partial differential equations. Additionally, the algebraic solution is extended to incorporate heat losses from the thermocline tank walls to the environment. The algebraic solution is significantly less computationally expensive than other one-dimensional models, since algebraic, rather than differential, equations are solved. An explicit formula for optimal fluid velocity is developed and validated through a parametric study of a thermocline tank. The effect of the dimensionless heat transfer coefficient is also investigated. Finally, the operation of thermocline tanks under multiple consecutive charging and discharging cycles is studied. The tank efficiency depends on the amount of thermocline allowed to exit during each cycle, and was found to decrease initially and subsequently reach steady state in less than 10 cycles.