High penetration of renewable energy sources (RES) in distributed Microgrid (MG) systems provides significant economic and environmental benefits. Nevertheless, the intermittent nature of RESs is the greatest challenge in the efficient utilisation of these resources. Using Battery Energy Storage Systems (BESS) in MGs is proposed as an effective solution to mitigate the fluctuations of RESs power generation. However, BESSs are expensive components with a short lifetime and therefore, they should be optimally sized according to the characteristics and requirements of MGs. This paper proposes a BESS sizing approach to determine the capacity and power rating of the BESS in a Grid-Connected MG, taking economic and technical criteria as well as the uncertainty of dispatchable generators (DG) and RESs into consideration. The technical requirements include the degree of reliability, BESS degradation, and operating constraints of the DGs. On the other hand, the cost factors associated with the utility grid including peak cost, valley cost, and Time of Use tariff (TOU) are taken into account. In this paper, a multi-objective mixed-integer quadratic model is developed to include the aforementioned parameters in the BESS capacity optimisation algorithm. Inspecting the impact of the considered parameters on the optimal size of the BESS and total cost of the MG, four case scenarios are studied. The first two scenarios evaluate the effect of the BESS depreciation, whereas the third and fourth scenarios investigate the impact of grid volatility (GV) and reliability indices on the optimal solution. The obtained results from the case studies verify the effectiveness of the proposed BESS sizing method.