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Modern power systems are experiencing the challenge of high uncertainty with the increasing penetration of renewable energy resources and the electrification of heating systems. In this paradigm shift, understanding electricity users' demand is of utmost value to the retailers, aggregators, and policymakers. However, behind-the-meter (BTM) equipment and appliances at the household level are unknown to the other stakeholders mainly due to privacy concerns and tight regulations. In this paper, we seek to identify residential consumers based on their BTM equipment, mainly rooftop photovoltaic (PV) systems and electric heating, using imported/purchased energy data from utility meters. To solve this problem with an interpretable, fast, secure, and maintainable solution, we propose an integrated method called Interpretable Refined Motifs And binary Classification (IRMAC). The proposed method comprises a novel shape-based pattern extraction technique, called Refined Motif (RM) discovery, and a single-neuron classifier. The first part extracts a sub-pattern from the long time series considering the frequency of occurrences, average dissimilarity, and time dynamics while emphasising specific times with annotated distances. The second part identifies users' types with linear complexity while preserving the transparency of the algorithms. With the real data from Australia and Denmark, the proposed method is tested and verified in identifying PV owners and electrical heating system users. The performance of the IRMAC is studied and compared with various state-of-the-art methods. The proposed method reached an accuracy of 96% in identifying rooftop PV users and 94.4% in identifying electric heating users, which is comparable to the best solution based on deep learning, while the speed of the inference model is a thousand times faster. Last but not least, the proposed method is a transparent algorithm, which can tackle the concerns regarding the agnostic decision-making process when policies prohibit some machine learning methods. [ABSTRACT FROM AUTHOR]