## 3 System Model and Problem Formulation  
Figure 2 illustrates the downlink transmission of hybrid-powered HetNet infrastructure considered in this work, which consists of a macrocell overlaid by multiple small cells. Each cell is served by a BS capable of harvesting RE; therefore, it can be powered by both grid power and an RE source. The power lines of all the BSs are connected to a smart grid that facilitates the energy exchange among the BSs. More specifically, if there is excess HE at any BS, the excess HE will be transferred to the smart grid to share. It is also assumed that the BSs coordinate among each other to eliminate interference through CoMP-JP technique to serve the users.  
For simplicity, Table 2 lists the notations of the main variables and parameters used throughout the paper. The data rate of user \(i\) and the total grid power consumed are denoted by \(R_{i}\) and \(P^{Total}\), respectively, which can be represented as follows:  
\[R_{i} =B_{o}\log_{2}\left(1+\frac{\sum_{j}a_{ji}b_{j}p_{ji}g_{ji}}{B_{o} N_{o}}\right) \tag{1}\] \[P^{Total} =\sum_{j}\left[\frac{b_{j}}{\lambda_{j}}\sum_{i}a_{ji}p_{ji}+b_{ j}p_{j}^{crit}-b_{j}\rho E_{j}\right] \tag{2}\]  
where \(a_{ji}\) is a Boolean value indicating association of user \(i\) with BS \(j\) where \(a_{ji}=1\) if the user \(i\) is associated with BS \(j\), and otherwise. Similarly, \(b_{j}\) is also a Boolean value representing the on/off status of BS \(j\) where \(b_{j}=1\) if BS \(j\) is on, and otherwise. \(E_{j}\) represents the HE of BS \(j\) and \(\rho\) denotes the HE efficiency. The EE is maximised by optimising the transmit power allocation and BS switching of SBSs. It is noteworthy that MBS is always ON, and therefore, \(b_{1}\) is always 1.  
Figure 1: Cellular network power consumption profile [5].  
The system EE is maximised by formulating an optimisation problem aiming to reduce total GPC. In this case, EE is defined as ratio of system total throughput to the total grid power consumption. Hence, the objective function and the constraints are shown as follows:  
\[\max_{\{p_{ji}\},\{b_{j_{k}\}\setminus\{1\}}}EE=\frac{\sum_{i}R_{i}}{P^{Total}} \tag{3a}\] \[s.t. \sum_{i}a_{ji}b_{j}p_{ji}\leq b_{j}P_{j,max},\quad\forall j,\] (3b) \[\frac{\sum_{i}a_{ji}b_{j}p_{ji}\delta_{ji}}{B_{o}N_{o}}\geq SNR_{i, thtr},\quad\forall i,\] (3c) \[\sum_{j}a_{ji}b_{j}\geq 1,\quad\forall i,\] (3d) \[\frac{b_{j}\rho e_{j}}{\sum_{i}a_{ji}b_{j}p_{ji}}\geq E^{R}_{j, thtr},\quad b_{j}=1 \tag{3e}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Descriptions** \\ \hline \(R_{i}\) & Data rate \\ \(B_{o}\) & Bandwidth \\ \(N_{o}\) & Noise power \\ \(p_{ji}\) & Transmit power from BS \(j\) to user \(i\) \\ \(g_{ji}\) & Channel gain from BS \(j\) to user \(i\) \\ \(E_{j}\) & Harvested renewable energy at BS \(j\) \\ \(P_{Total}\) & Net on-grid power consumption \\ \(p_{j}^{tr}\) & Circuit power of BS \(j\) \\ \(\rho\) & Energy transfer efficiency \\ \(\lambda_{j}\) & Power amplifier efficiency \\ \(E_{M}^{HC}\) & Energy harvested by MBS \\ \(E_{S}^{HC}\) & Energy harvested by SBSs \\ \(a_{ij}\) & User \(i-\) BS \(j\) association \\ \(b_{j}\) & BS \(j\) on/off status \\ \(P_{j,max}\) & BS / max transmit power \\ \(SNR_{i,thtr}\) & UE \(i\) SNR threshold \\ \hline \hline \end{tabular}
\end{table}
Table 2: List of main variables and parameters.  
Figure 2: CoMP-enabled hybrid-powered MBS and SBSs connected to a smart grid in HetNet architecture.  
Constraint (3b) limits the transmit power of the BS to \(P_{j,max}\). Constraint (3c) guarantees that every user can achieve a minimum SNR denoted by the threshold \(SNR_{i,thr}\). Constraint (3d) ensures that each UE is served by at least one BS. Constraint 5 enforces the BS-Sw condition by evaluating the ratio of harvested energy to BS transmit power. Only if the SBS fulfils the condition, where the ratio equals to at least the threshold \(E_{j,thr}^{R}\) value, it will be allowed to be switched on.