## 1 Introduction  
With the rapid development of the global economy, living standards have been improving. Simultaneously, energy consumption is increasing sharply. As the largest energy consumer in the world, China's energy consumption accounted for 23.2% of the global consumption in 2017, of which building energy consumption accounted for more than 30% of the total social energy consumption [1, 2]. Within building energy consumption, HVAC systems and domestic hot water systems consumed nearly 60% of the total energy [3]. Solar energy is a type of clean renewable energy with abundant reserves, the development and utilization of which can effectively resolve energy shortages [4]. In recent years, various countries have been actively developing solar energy technology [5, 6], and solar water heating systems are one of the most mature and widely used technologies. According to data from 2011, the total installed capacity of solar water heaters in operation in China accounted for 65.3% of the installed capacity worldwide, and it is clear that China currently dominates the global solar water heater market [7]; however, there are also some disadvantages in the application of solar energy, which is seriously affected by local climate conditions, changes in supply at night, etc. Therefore, for solar water heating systems, auxiliary heat sources are usually needed to maintain normal supplies of hot water in rainy weather and at night. Traditional auxiliary heat sources include coal boilers, oil boilers, gas boilers, electric boilers, and so on. The heat pump is a new energy-related technology that has received considerable attention in recent years [8]. The heat pump is an energy-saving device that uses high-level energy to make heat flow from a low-level heat source to a high-level heat source. During this process, only a small amount of electricity is consumed. The integration of a heat pump unit into a system can be strategizes to increase the performance level of the system and enhance the use of solar energy [9]. There are two main types of solar-assisted heat pump (SAHP) systems: direct expansion solar-assisted heat pump (DX-SAHP) systems and indirect-style solar-assisted heat pump (i-SAHP) systems. i-SAHP systems can be further divided into parallel-type, serial-type, and dual-source-type systems [10; 11]. The concept of combining solar energy with heat pumps was first proposed by [PERSON] and [PERSON] [12]. A solar-assisted, wastewater source, heat pump, hot water system was built in TRNSYS, which enabled a more stable hot water temperature in winter with a significant decrease in the energy consumption [13]. Following this, scholars in various countries carried out a large number of theoretical and experimental studies on different system forms. The DX-SAHP combines a solar collector and a heat pump evaporator into a unit, in which the refrigerant absorbs solar energy evaporation directly [14]. [PERSON] et al. [15] developed a numerical model of the DX-SAHP system. Then, based on the Typical Meteorological Year weather data of Hong Kong, an annual average coefficient of performance (COP) of 6.46 was obtained for the system, which meant it was clearly superior to the traditional heat pump system. [PERSON] et al. [16] developed a simulation model to predict the thermal performance of a DX-SAHP system, and the simulation results agreed well with the experimental data. [PERSON] et al. [17] conducted a detailed, long-term thermal economic analysis of the energy-saving potential and economic feasibility of a DX-SAHP system. The results indicated that the life cycle cost of the system could be minimized by optimizing the collector area. [PERSON] et al. [18] recommended and analyzed a DX-SAHP water heater experimental setup, the average COP of which was measured under the spring climate in Shanghai and was found to be 5.25. Based on this experiment, some methods to improve the thermal performance of each component and the entire water heater system were proposed.