One of the fundamental requirements of a power system, especially in a small isolated power system (SIPS), is secure and reliable energy supply. In this paper, a novel Energy-Based System Well-being Analysis framework (EBSWBA) is presented to assess the reliability of systems containing renewable energy sources (RES), fossil-fueled sources and electricity storage. The framework combines the advantages of the well-known System Well-being Analysis (SWBA) and Expected Energy Not-Served (EENS) approaches, thus overcoming the weaknesses of the SWBA by considering all key factors that influence the system adequacy. Furthermore, two techniques are proposed for calculation of the EBSWBA indices: the Monte-Carlo Simulation (MCS) adopted for the EBSWBA framework, and the Contingency Enumeration for Small Isolated Power Systems (CESIPS) technique, which is derived from the Contingency Enumeration (CE) methodology. These techniques enable formation of probability distributions of the indices. The comparison of the results of classical SWBA and EBSWBA when applied to different SIPS architectures, operating schemes and load profiles demonstrates the superiority of EBSWBA over SWBA in terms of rendered information about the system state as well as in terms of computational efficiency, especially when the CESIPS technique is used for calculation of the EBSWBA indices.