Energy storage systems (ESSs) are increasingly used in power system optimization by deriving different ESS mathematical models. The most widely-used model is the piecewise linear ESS model which utilizes non-convex constraints to represent the ESS power losses, resulting in challenging optimization problems. To reduce the problem complexity, convex relaxation models are often derived but may compromise the solution quality of the underlying problems. This work investigates the exact and relaxed versions of three different mathematical representations of the piecewise linear ESS model, in terms of their solution quality and execution efficiency. Towards this direction, the three ESS models are incorporated into the unit commitment problem which often violates the ESS relaxation exactness under ramping constraints. Simulation results present (a) the execution times of the exact and relaxed ESS models and (b) the optimality gap of the relaxed models when the relaxation exactness is violated.