The functionality to both charge and discharge energy from and to the power grid to a Battery Electric Vehicle (BEV) is referred to as Vehicle-to-Grid (V2G). This allows the customer to buy energy when the spot price is low and sell energy when the price is high to make a profit, called energy arbitrage. However, when the battery is charging, discharging, or idling for storage, battery degradation occurs due to chemical properties and reactions. This thesis developed a mathematical optimisation model in Python, using the modelling language Pyomo. Mathematical equations are used to integrate energy arbitrage and degradation data to reduce the total cost in terms of degradation and energy by finding an optimised charge and discharge pattern. The model allows different user scenarios to be analysed by changing inputs such as charger power, battery cost or daily driving distance. When using V2G technology, the State-of-Charge (SoC) level of BEVs battery packs can be adjusted to find SoC levels which minimise the battery degradation, while allowing the user to make a profit from energy arbitrage. The result shows that the V2G charging protocol, compared to protocols without a bidirectional charger could be beneficial for the simulated time periods, by both reducing degradation and the total energy cost. The results also indicate that the degradation cost of the battery is often the determining factor in the decision of when to charge or discharge, i.e., the substantial cost-saving strategy is to control the storage and cycle degradation to reduce the total degradation, rather than controlling the energy arbitrage. The model and the result of this thesis can be used by car manufacturers to learn more about how battery cell types behave in V2G mode and influence further work on V2G control.