## 1 Introduction  
In Japan, the energy consumption in the residential sector accounts for about 16% of the total one in all the sectors, and the energy consumption for hot water supply accounts for about 28% of that in the residential sector [1]. This is because hot water is used not only for washing, cooking, and showering, but also for baths. Thus, saving energy has been important for hot water supply in the residential sector. Under this situation, water heating systems (WHSs), each of which is composed of a heat pump (HP) using CO\({}_{2}\) as a natural refrigerant and a hot water storage tank, have been developed and commercialized widely [2]. In addition, their performances have been enhanced, while their functions have been expanded. Although HP WHSs have been introduced in other countries, their refrigerants are not CO\({}_{2}\), but R410a and R134a. In addition, since the hot water consumption is smaller, sizes of storage tanks are also smaller. It seems that the situation in Japan is not true for other countries. In earlier years, attention has been paid to analyzing and enhancing the performance of a CO\({}_{2}\) HP as the main component of a WHS. Some review papers on the performance analysis of COHPs have been published: [PERSON] has made a survey of activities in the CO\({}_{2}\) HP area [3]; [PERSON] et al. have reviewed studies on CO\({}_{2}\) HP and refrigeration cycles [4]. In addition, many studies have been conducted to analyze and enhance the performance of CO\({}_{2}\) HPs theoretically and experimentally, as follows. Prototype CO\({}_{2}\) HPs have been developed, and their performance characteristics have been investigated experimentally: [PERSON] et al. have prototyped an HP water heater (WH) and have investigated its characteristics [5]; [PERSON] et al. have made an HP for water heating and have predicted its performance by a computer model [6]; [PERSON] et al. have made a comparison in the performance between prototyped CO\({}_{2}\) and conventional HPs [7]. On the other hand, simulation models have been developed to analyze the performance: [PERSON] and [PERSON] have made a comparison in the performance between CO\({}_{2}\) and conventional cycles [8]; [PERSON] et al. have verified a simulation model with measurements for vapor compression systems [9]; [PERSON] et al. have investigated how ambient and operating conditions affect the performance of an HP [10]; [PERSON] et al. have shown the influence of operating parameters on the performance of HP WHs [11]; [PERSON] et al. have made a comparison between simulated and experimental results for an HP system with an expander [12]; [PERSON] et al. have verified simulated results for an HP WH using an existing machine [13]; [PERSON] et al. have validated the consideration of geometrical variations in a simulation model for an HP system with experimental measurements [14]. Some studies have also been conducted to enhance the performance: [PERSON] et al. have analyzed a two-stage cycle for its optimization theoretically and experimentally [15]; [PERSON] and [PERSON] have investigated the optimal heat rejection pressure for a split system experimentally [16]; [PERSON] et al. have studied the performance of a system with ejector under optimum high-side pressure experimentally [17]; [PERSON] et al. have examined the optimal rejection pressure for a WH experimentally [18]; [PERSON] et al. have analyzed subcooling methods to improve the performance of CO\({}_{2}\) refrigeration [19]. Other studies have been conducted to apply CO\({}_{2}\) HPs to a simultaneous cooling and heating: [PERSON] et al. have optimized and simulated HP systems [20; 21]; they have also analyzed the performance of a system experimentally [22]. Other studies have been conducted to apply CO\({}_{2}\) HPs to cooling: [PERSON] et al. have analyzed a 4-bed based adsorption cooling system using CO\({}_{2}\) as the refrigerant [23]; [PERSON] et al. have proposed a CO\({}_{2}\)-assisted compression-adsorption hybrid for cooling and desalination [24].