Residential energy demand varies widely in terms of time-series behaviors, amounts consumed between families, and even within one family. Residential energy demand profiles have a high degree of uncertainty in their essentials because the demand profile is entirely based on the occupant-driven load. When evaluating residential energy systems like co-generation systems, hot water and electricity demand profiles are critical. In this paper, in order to clarify rational energy system selection guidelines and rational operation strategies, authors aim to extract basic demand time-series patterns from two kinds of measured demand (electricity and domestic hot water), measured over 26307 days of data in Japan. Authors also aim to reveal the relationship between primary energy consumption and demand patterns. Demand time-series data are categorized by means of a kind of "unsupervised" learning, which is a hierarchical clustering method using a statistical pseudo-distance. The statistical pseudo-distance is calculated from the generalized Kullback-Leibler divergence with the Gaussian mixture distribution fitted to the demand time-series data. The classified demand patterns are built using a hierarchical clustering and then a comparison is performed between the optimal operation of the two systems (a polymer electrolyte membrane fuel cell co-generation system, and a CO2 heat pump system) and the operation of a reference system (a conventional combination of a condensing gas boiler and electricity purchased from the grid) using the demand profiles appropriately built. Our results show that basic demand patterns are extracted by the proposed method. The demand patterns, the amount of daily demand and heat-to-power ratio of demand affect the primary energy reduction ratio of the polymer electrolyte membrane fuel cell co-generation system.