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This paper examines coalition stability among prosumers and consumers in a local energy community. Using cooperative game theory, we model how coalitions can minimise collective disutility by sharing distributed energy resources (DERs) and optimising battery energy storage system (BESS) operation schedules. For disutility allocation among coalition members, we apply the center-of-gravity of imputation set value (CIS-value), ensuring equitableness and computational feasibility. The analysis focuses on stability of coalition structures, where all possible player's deviations are considered. We implement a blocking mechanism, where deviations can be blocked by a coalition if one member is made worse off by the deviation. A case study involving ten players is presented, showing that stable coalitions lead to more effective outcomes in terms of cost, emissions, and demand peak reduction. These findings demonstrate that cooperative game theory with coalition stability can enhance community self-consumption (CSC) within local energy communities. (with Khaled Abedrabboh, Luluwah Al-Fagih, Elena Parilina).