## 2 Related Work and Background
The energy consumption issue of D2D communication is also treated in [15], where the authors propose a three-device topology (i.e., source, relay, destination) and exploit the data entropy (i.e., the data to be forwarded by the relays) for reducing the energy consumption. Through mathematical calculations, it is shown that including a relay, as an intermediate device between the source and the destination, brings energy efficiency advantages with respect to the direct source-to-destination communication. In summary, authors in [14; 15] find solutions to improve the energy efficiency of D2D communications without considering the cellular network. Another work aiming at the energy maximization of D2D communications is [16] that proposes a power control based on the users' channel conditions. Similarly to [15], the presence of relays is encouraged with respect to the single hop D2D direct communication. In this perspective, an improvement is discussed in [17], where an adaptive forwarding approach is proposed with the aim to choose the best relay devices which provide the greater energy efficiency in multi-hop D2D communications. Authors in [18] formulate the problem of energy consumption minimization as a convex optimization problem, and find an optimal solution for the transmission mode selection (macro cellular, D2D, or small cell cellular). In particular, the threshold is determined beyond which the D2D mode activation is preferable rather than the macro or the small cell modes. In [19], high energy efficiency is achieved in mmWave small cells through a multicast scheduling scheme that adjusts the D2D transmission power following a power control algorithm. In the literature, particular interest has been gained by Energy efficiency in D2D-aided multicast service, above all in the last years. In [20], the authors propose a device-to-device multicast (D2 MD) scheme to enhance the performance of content sharing by exploiting D2D communication established on the basis of the social relationship and the physical distance among users. Furthermore, the scheme also proposes resource allocation with the aim to maximize the throughput and to overcome the problem of interference by performing power control and channel assignment. The social relationship and the physical distance among users are also taken into account in [21], where the energy efficiency of D2D communications is maximized by properly selecting the optimal channel mode and the transmission power to each user with an adaptive genetic algorithm. Diversely from [20], the proposed solution in [21] does not consider multicast transmissions. In [22] D2D communications are exploited for the retransmission of data, received in multicast from the Base Station (BS), by using the same channel (rather than multiple channels) in TDMA mode. The scheme minimizes the energy consumption of retransmitters by optimizing their transmission power. Another work dealing with D2D multicasting is [23], where the authors formulate a heuristic gradient power allocation algorithm to solve a resource allocation problem in order to maximize the energy efficiency of multicast D2D communications with the aim to satisfy the constraints of spectral efficiency and maximum transmission power. Nevertheless, in the literature most of the works deal with D2D in the case of HTC. Instead, MTC-D2D and the evaluation of energy efficiency in a MTC scenario are little investigated topics. Indeed, to the best of our knowledge, an in-depth analysis is still missing on D2D-aided group-oriented service (i.e., software upgrade/update) delivery to MTC devices over 5G NR networks, which is the topic of this paper. Differently from [13], the proposed EED2D-SF aims to improve the overall network (not only D2D) energy efficiency with a new resource allocation technique. The analysis of the network energy efficiency is performed under different cellular frame configurations, which has not been done before.