Energy efficient power control for device to device communication in 5G networks

Received Sep 29, 2019 Revised Mar 3, 2020 Accepted Mar 12, 2020 Next generation cellular networks require high capacity, enhanced efficiency of energy and guaranteed quality of service (QoS). To reach these goals, device-to device (D2D) communication is a candidate technologie for future 5th Generation especially applications that require the reuse, the hop and the proximity gain. The present paper studies the energy efficient power control for the uplink of an OFDMA (orthogonal frequency-division multiple access) system composed of both regular cellular users and device-to-device (D2D) pairs. First, we analyze and model mathematically the prerequisites for D2D communications and classical cellular links in terms of minimum rate and maximum power requirement. Second, we use fractional programming in order to convert the initial problem into a concave one and we apply non-cooperative game theory in order to characterize the equilibrium. Then, we got the solution of the problem from the results of a water-filling power allocation. Moreover, we employ a distributed design for power allocation by means of three methods: a) Theory of fractional programming b) Closed form expression (the novelty is the use of Wright Omega function). c) Inverse water filling. Finally, simulations in both static and dynamic channel setting are realized to demontrate the enhanced gain in term of EE, SE (spectral efficiency) and time of execution of the iterative algorithm (Dinkelbach) than the closed form algorithms.


INTRODUCTION AND ARTICLE CONTRIBUTIONS
Nowadays, numerous applications of wireless communications has emerged: Mobile broadband, smart grid, intelligent transportation systems, smart home, E-Health and are using an increasing number of devices: 18 billions are expected by 2022 [1]. The growth of the mobile internet is estimated to 46 percent annually from 2017 to 2022 [2] forecasted to reach 77 exabytes (EB) corresponding to a seven-fold increase every 5 years. The 5G systems are expected by 2020 and according to the wireless industry and academic, the 1000x capacity of 5G should be obtained using the same amount of energy of the previous cellular generation. Therefore, 5G have to increase energy efficiency by a factor 1000 or more [3].
From the perspective of the end user, battery autonomy is a decisive element in mobile data experience [4]. To match with the growth of mobile data, further enhancement in battery technology is needed. However, battery technology does not develop as quickly [5]. Therefore, improving the EE (energy efficiency) of mobile communication is essential to lessen the difference between increasing data consumption The authors of [5] study interference-limited environment employing non-cooperative games, then a trade-off between spectral efficiency and EE is obtained. The authors of [23] apply the concept of repeated game in order to handle decentralized power control. The uplink of an OFDMA based HetNet (hetegenous network) composed of a macro cell and helped by a set of small cell was studied by the authors of [24] where the maximization of EE is done while respecting minimum QoS requirements. They define a solution based on Debreu Equilibrium that generalizes Nash equilibrium, which leads to a water filling like best response. The trade-off between spectral efficiency (SE) and energy efficiency (EE) is investigated for device-todevice (D2D) communications reusing uplink channel of cellular networks in [25].
The rest of the article is structured as following. The notations adopted are defined in the section 3. The system model is discussed in section 4. In the section 5 the power control game is described, then the Nash equilibrium solution is obtained using fractional programming theory in section 6. The practical aspect and the design of the algorithm are given in Section 7. The simulations results are commented in| the section 8. W conclude finally in section 9.

NOTATIONS
The notation in the sequel of the present of the present paper are as follows:  Bold letters are used for matrices and vectors The vector ( 1 , … , ) is written more compactly as ( , − ) with:  | | is used for the cardinality of a set

SYSTEM MODEL
The system we adopt is a single cell served by one BS (base station), cellular UEs and D2D transmitter/receiver pairs. The UEs of the system belong to the following set: = { 1 , 2 , … , } The set of D2D pairs in the system is denoted as: = { 1 , 2 , … , } Spectral resource allocation is based on the OFDMA technology. One sub-carrier is allocated to each cellular user. Therefore, no interference exists in the cellular tier. By contrary, every D2D transmitter has the freedom to transmit on all sub-carriers. We assume no sub-carrier allocation to D2D pairs is performed, but rather D2D transmit randomly their powers over the subcarriers. The Table 1 gives the main notations adopted.
With n k is an AWGN (Additive White Gaussian Noise) that is having a mean 0 and variance σ 2 . Similarly, the signal of the baseband is written for D2D k in the sub-carrier n as: with n k,n is an AWGN having the same mean and variance as n k . The channels are assumed complex Gaussian: the real part and imaginary part are both Gaussians. For each D2D pair i, its power vector is denoted (over all subcarriers): and the vector power of D2D pairs excluding the D2D pair is: The transmitting vector corresponding to all D2D pairs: The total power of D2D pair accounting the circuit related consumption: and the power vector of UEs except UEk is: The global power of cellular users is denoted: The total power of UEk accounting the circuit related consumption: The SINR (Signal to Interference plus Noise Ratio) of UEk at the n th sub-carrier: We denote the SINR of D2D pair at the n th sub-carrier: Let us define the effective channel gain of the cellular user UEk as following: The SE (Spectral Efficiency) utility function of UEk is: With Γ the SINR gap w.r.t BER as defined in [26]:Γ = − ln(5BER)

1.5
The SE (Spectral Efficiency) utility function of the k th D2D pair is: The EE (Energy Efficiency) utility function of the uplink communication of UEk is: The EE (Energy Efficiency) utility function of the k th D2D pair: Each UE will increase its energy efficiency and at the same time respect the minimum rate requirements. The UEk will try to maximize its global energy efficiency while respecting minimum rate constraint. Consquently, the following problem has to be solved for UEk: Similarly, the following problem has to be solved for k th D2D pair: Hence, there are ( + ) optimization problems to be jointly solved in the system. The multivariate optimization cannot be applied; as there is no central entity that is controlling all variables of the system, instead each has partial control on his own variables, which leads to the application of Game Theory.

GAME THEORY FORMULATION
The non-cooperative game theory appears as the natural tool to tackle to problem as we have independent, rational players that control their variables. The following triplet defines the non-cooperative game formally as: In order to get a solution, we have to find the equilibrium of the game. Indeed, the importance of the equilibrium is game is important as agents will become predictable and stable. The network operates effectively when reaching equilibrium [27]. The power strategy of each user must be feasible by respecting minimum QoS (quality of service) conditions. This leads us to use a different Game theory solution that allows each player to choose its strategy knowing the strategy of other players while respecting feasibility conditions: the GNE (generalized Nash equilibrium). By supposing that each player's strategy may depend on the other players', the GNE goes beyond the common NE (Nash equilibrium) [28]. Not only utility functions are dependent but also the players' action sets are couple, which is different from classical non-cooperative games [29].
We will denote by ( − ) the set of user UEk actions respecting the minimum rate and the maximum power requirements with respect to other players' strategy: Definition. Generalized Nash equilibrium The matrix of power * = ( * , − * ) is a Generalized Nash equilibrium if: − * ∈ ( − ) and: Which means that no player has the interest to deviate unilaterally from a GNE to another feasible point (respecting the constraints).

FRACTIONAL PROGRAMMING FORMULATION AND GAME SOLUTION
The EE objective function in (18) and (19) are fractional and non-convex. The most suitable mathematical framework to tackle the optimization of such functions is the fractional programming, which provide polynomial complexity algorithms when the numerator is concave and denominator is convex [29]. First, let us give a formal definition of Fractional programming: Definition. Fractional programming Let ∈ ℝ convex subset and let → ℝthe following functions: : → ( ) and g: → ( ) A fractional program is the optimization problem: The fractional programming allows us to solve equivalent form of the problem that is less complex than the original one: Proposition 1. The vector * ∈ solves the (22) if and only if: ( ( * ) − * ( * )) = 0, * the zero of the function: For a given value, the UEk will solve the following transformed fractional programming problem instead of (18): For the optimal value, , * we get: This means that the best response power , * given other transmitters ( − , )verifies the following expression: Similarly, for a given value the equivalent problem of optimization that the k th D2D pair will solve instead of (19): For the optimal value, λ k d, * we get: This means that the best response power given other transmitters ( − , )verifies the following expression: Practically, the solution of (24) and (27) is got when the players are giving their best response to each other. As follows, the k th user best response function which is the best action to adopt when the strategy of the other players is − ∈ − ( ): The necessary conditions are more easily obtained in the case of a single carrier. Proposition 2. In the single carrier case, the uniqueness and existence of Generalized Nash equilibrium are guaranteed, as the BR functions of the user k are standard and verify the following properties:  Concavity: Proof. The proof is given in [30] In the case of multi-carrier, the equilibrium can be charactrized by giving the expression of power for each sub-carrier and for each user as in the following proposition. Proposition 3. The best power of the UEk is given by the following when the game admits a GNE: The optimal value , * : With: = log ( ) is calculated when the rate requirements are active: Proof. The proof is given in [31] applied to our model.

ALGORITHMS DESIGN AND ANALYSIS 7.1. Practical design
From practical point of view, each player give its best response to solve EE maximization problem knowing other players' strategy. We consider a scheme that is asynchronous as the user chooses its power strategy at time based on the other players' strategy of − 1. Indeed, an instantaneous feeback is is very difficult in practice. We consider that each player knows his individual CSI (channel state information) whithin the SINR on each subcarrier. As intial value, the cellular users choose and D2D users choose . In each iteration, the user UEk estimates the effective channel gain based on the previous value and based on the previous SINR: The user UEk estimates the EE as following and tries to increase it w.r The challenge here is to find , * ( )and , * ( ), for that we use three methods:  The first method that was used by [24] relies on Dinkelbach algorithm: algorithm 3.  The second method is the proposed in this article relies on the closed form expression as in algorithm 4.  The third method is the IWF (Inverse Water Filling) is applied using algorithm 2 which aims to have exactly the minimum normalized SE (Rmin).

Complexity analysis
The different algorithms comprise two loops: the inner and the outer loop. Superlinearty convergence characterize the outer loop iteration; hence, the inner loop dominate the two-loop power control algorithm [32]. The complexity of the Dinkelbach algorithm is calculatedas (( + ) 3.5 ) the algorithm IWF has superlinear convergence properties both in the inner and in the outer loops. For the Closed-Form algorithm we use the Wright-Omega [33] and the Lambert-W functions embedded [34] in MATLAB Software.

Convergence analysis
The methods used to calculate the value of λ are iterative (algorithm 1), which leads to an increasing serie of λ and converge to an optimal value. The algorithms got the optimal power value for a given λ, which is obtained from previous power values. The algorithms stop running whenever the resultant is relatively small [31]. The algorithms used are asynchronous: using the past strategy of the competitor in order to react to take a present action. The repeated games framework guarantees the convergence in the long-term. Indeed, players decide about the future based on their experiences in the previous stages of the game [23].

SIMULATION AND COMMENTS
For simulations, we choose a single cell system having a radius of 300 meters radius and Kc=3 cellular users and Kd=5 D2D pairs. The Algorithm 4 was implemented using two different ways using Lambert-W function and Wright-Omega function. to initiate power for all users

Static channel: number of iterations
The Table 2 presents main parameters adopted for the simulations to compare algorithm 3 (Dinkelbach) and algorithm 4 (Closed-form). The channel gain is expressed using the parameter to calculate the path-loss: ℎ = We perform 10 iterations to be sure that the users will end by learning equilibrium by executing the interative algorithms 3 and 4. We average the energy efficiency, spectral efficiency and power per subcarrier of the reference user to obtain the results of simulation. The position of the reference user was averaged 100 times. The simultations show that equilibrium of the game exist and converge before the 10th iteration. We do not have a saturated equilibrium, which does not reach the maximum power. For the subsequent simulations, static and flat fading channel model was choosen. The algorithm 3 and 4 both respect minimum SE and consequently converge to a solution that respect the minimum rate requirement.

4129
The IWF (inverse water filling) algorithm respects the minimum rate requirement by its construction to achieve the minimum rate exactly. Hence, the conditions in the maximization problem (C1) and (C2) are respected. The flat fading assumption leads to the users to choose the same power over all the sub-carriers. We give below our comments on results obtained: EE: The Energy efficiency corresponds to the EE utility times the bandwidth.  Cellular: The simulations in Figure 1 that represent the EE of a reference cellular user shows that the DBK algorithm is 800% better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm. Unsurprisingly, the IWF algorithm has the least EE.  D2D: The simulations in Figure 2 that represent the EE of a reference D2D user shows that the DBK algorithm is 20% better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm.
The IWF algorithm has the least EE.  Comparison Cellular and D2D: The D2D communication brings proximity ga in, the EE of D2D is at least 2.5 times the EE of Cellular.  Figure 3 that represent the SE of a reference cellular user shows that the DBK algorithm is 10% better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm.
The IWF algorithm has the least EE.  D2D: the simulations in Figure 4 that represent the EE of a reference D2D user shows that the DBK algorithm is slightly better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm.
The IWF algorithm has the least EE.  Comparison Cellular and D2D: The D2D communication brings proximity gain; the SE of D2D is at least 8.3 times the SE of Cellular. Power: the instant consumed power  Cellular simulations in Figure 5, that represent the power of a reference Cellular user shows that the CF algorithm consumes 75% more power than DBK algorithm. The IWF algorithm consumes the least power.  D2D simulations in Figure 6, that represent the power of a reference Cellular user shows that the CF algorithm consumes 25% more power than DBK algorithm. The IWF algorithm consumes the least power. Time of execution: simulations in Figure 7 shows the cumulative time of execution expressed in millisecond. In the 6 first iterations the DBK algorithm consumes more time than the CF algorithm. The CFOM algorithm based on Wright Omega function has slightly less complexity than Lambert W function starting from the seventh iteration the DBK algorithm join the closed form. The least time execution belongs to IWF algorithm.

Rayleigh channel: number of iterations
In this subsection, we perform simulations with Rayleigh channel instead of static channel considered for the previous simulations. For the subsequent simulations, static and flat fading channel model was choosen. The algorithm 3 and 4 both respect minimum SE and consequently converge to a solution that respect the minimum rate requirement.
The IWF (inverse water filling) algorithm respects the minimum rate requirement by its construction to achieve the minimum rate exactly. Hence, the conditions in the maximization problem (C1) and (C2) are respected. The flat fading assumption leads to the users to choose the same power over all the sub-carriers. We give below our comments on results obtained: EE: The Energy efficiency corresponds to the EE utility times the bandwidth.  Cellular the simulations in Figure 8 that represent the EE of a reference cellular user shows that the DBK algorithm is 800% better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm. Unsurprisingly, the IWF algorithm has the least EE.  D2D the simulations in Figure 9 that represent the EE of a reference D2D user shows that the DBK algorithm is 20% better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm.
The IWF algorithm has the least EE.  Comparison cellular and D2D: The D2D communication brings proximity gain, the EE of D2D is at least 2:25 times the EE of Cellular.  Figure 10 that represent the SE of a reference cellular user shows that the CF algorithm is 20% better than DBK algorithm. CFOM algorithm realizes the same EE as CF algorithm.
The IWF algorithm has the least EE.  D2D The simulations in Figure 11 that represent the EE of a reference D2D user shows that the DBK algorithm is slightlly better than CF algorithm. CFOM algorithm realizes the same EE as CF algorithm.
The IWF algorithm has the least EE.  Comparison cellular and D2D: The D2D communication brings proximity gain, the EE of D2D is at least 8.33 times the EE of Cellular.
Time of execution: simulations in Figure 12 shows the cumulative time of execution expressed in millisecond.
In this case the DBK algorithm has clearly better complexity than the CF algorithms. Robustness: Although the algorithms DBK and CF do not converge but they show a good robustness as the power, EE, SE are smooth, stable, and slowly varying after few iterations.

Impact of minimum rate
In this subsection, we examine the impact of the minimum rate constraint on the SE and EE of the D2D pair. We take a fixed minimum rate constraint for cellular users = 0.2 / / and we vary the minimum rate for the D2D users from 0 to 30bit/s/Hz As in Figure 13 and Figure 14:

CONCLUSION AND FUTURE WORKS
In this work, we compare closed-form expression of the power allocation to Dinkelbach algorithm applied to a D2D communication underlaying a cellular network. Results shows that Dinkelbach algorithm provides more EE than the Closedform algorithm. Furthermore, in term of SE Dinkelbach algorithm is equivalent to closed form when it comes to D2D communication type. However, closed form algorithm outperforms the Dinkelbach for cellular type communication, which is in line with our previous work [31]. In term of time of execution, we remark that the Dinkelbach Algorithm suffers from a slow start in the first iterations then it becomes faster than closed form algorithms. Due to to its simplicity, the Inverse Waterfilling algorithm has excellent convergence, lowpower and complexity properties but its EE is very limited and its SE is just the minimum rate. This algorithm (IWF) can be applied to fixed rate application that are not ambitious in term of rate: for instance VoNR (voice over new radio). Despitethe non-convergence in the dynamic channel conditions, the proposed algorithms show good stability and very similar results to the static case in terms of EE, SE, and power. However, the time of execution in the dynamic case of Dinkelbach algorithm is much lower than closed form. Although the usage of the Wright Omega function compared to Lambert Wfunction brought some reduction of time execution, but it was not enough to get better results than Dinkelbach algorithm was. Whatever the used algorithms, thanks to proximity and hopgain, D2D communication bring significant improvement of energy efficiency compared to cellular communication. Yet, more investigation needs to be done in order to increase Energy Efficiency. Another possible future work is the application to more sophisticated scenarios such as MIMO and Small Cells.