We develop three novel enhanced mixed integer-linear representations of the power limit of the battery and its efficiency as a function of the charge and discharge power and the state of charge of the battery, which can be directly implemented in large-scale power systems models and solved with commercial optimization solvers. Using these battery representations, we conduct a techno-economic analysis of the performance of a 10 MWh lithium-ion battery system testing the effect of a 5-min vs. a 60-min price signal on profits using real time prices from a selected node in the MISO electricity market. Results show that models of lithium-ion batteries where the power limits and efficiency are held constant overestimate profits by 10% compared to those obtained from an enhanced representation that more closely matches the real behavior of the battery. When the battery system is exposed to a 5-min price signal, the energy arbitrage profitability improves by 60% compared to that from hourly price exposure. These results indicate that a more accurate representation of li-ion batteries as well as the market rules that govern the frequency of electricity prices can play a major role on the estimation of the value of battery technologies for power grid applications.