## 1 Introduction  
Refrigeration is responsible for a great part of the worldwide energy consumption [1], as well as for global warming because of the use of harmful refrigerants [2]. In recent years, regulations such as the EU F-Gas Regulation 517/2014 [3] have been created in order to establish the use of clean and sustainable refrigerants, such as CO\({}_{2}\), NH\({}_{3}\), propane, butane, and, generally, refrigerants with lower Global Warming Potential (GWP) values [4]. However, there is not a global optimum refrigerant from all the points of view (performance, environmental impact, and safety level) [4; 5]. CO\({}_{2}\) is the unique refrigerant which is non-toxic, non-flammable (A1 ASHRAE safety group), and has an extremely low global warming potential (GWP = 1), but its refrigeration systems suffer from reduced efficiency [6]. So, a lot of research has been focused on the refrigeration systems with CO\({}_{2}\) as the main or the only refrigerant [7], in order to design systems with higher performance. One important reason for the lower performance of the CO\({}_{2}\) refrigeration systems is the relatively low critical temperature of CO\({}_{2}\), which is 31.1 \({}^{\circ}\)C. So, the CO\({}_{2}\) refrigeration systems cannot operate with latent heat rejection to the ambient, especially in warm climates conditions, and they operate in transcritical mode [8]. This fact is the key parameter that leads to a decreased coefficient of performance (COP) and, consequently, to higher electricity consumption. In this direction, a lot of researchers have studied the use of alternative methods for enhancing the COP of the refrigeration cycle with CO\({}_{2}\)[9]. The use of an internal heat exchanger, after the condenser and after the evaporator, is a simple way for subcooling the CO\({}_{2}\) refrigerant. [PERSON] et al. [10] studied this idea and they determined the optimum system high pressure with the heat exchanger. Moreover, [PERSON] et al. [11] found a 12% enhancement with the use of an internal heat exchanger for subcooling purposes. The next studied idea regards the use of a two-stage compression system. [PERSON] et al. [12] found that this idea increases the system coefficient of performance (COP) about 25%. In this direction, the use of a parallel compressor has been studied by [PERSON] et al. [13] and they found maximum possible enhancement, up to 47%, when there is a high-temperature difference between the heat rejection and the refrigeration temperature levels. The parallel compression has been found to increase the COP about 19%, according to [PERSON] et al. [14], while [PERSON] et al. [15] found 30% improvement experimentally. The incorporation of ejector devices in the system for increasing the COP of the refrigeration cycles has been also studied in the literature. [PERSON] et al. [16] found that the COP can be improved up to 26% with the use of an ejector, while [PERSON] et al. [17] found enhancements in the ranges of 25% to 30%. An alternative idea regards the use of an expander in the system in order to produce a small work amount and to decrease the network demand. [PERSON] et al. [18] found that this idea is able to increase the COP of a CO\({}_{2}\) refrigeration system up to 11%, while similar enhancements, about up to 12%, have been found by [PERSON] et al. [19]. The utilization of cascade refrigeration systems with CO\({}_{2}\) is also common in the literature. [PERSON] et al. [20] studied an R134a/CO\({}_{2}\) cascade refrigeration system and they stated that the direct cascade system is more efficient than the indirect system with a deviation of about 11%. [PERSON] et al. [21] examined a cascade system with CO\({}_{2}\)/N\({}_{2}\)O and ejectors, and they concluded that these refrigerants have similar performances in the cascade systems because of their similar critical temperatures. Moreover, the use of NH\({}_{3}\)/CO\({}_{2}\) cascade systems is common in the literature [22].