## 2 Energy-Efficient Optimization Model  
In the LTE/5G system using CA, we study the energy-efficient optimization allocation for CC, RB, and power considering MCS. The system adopts an adaptive MCS according to the CQI feedback by the user and allocates CC, RB, and power. CQI is usually obtained by measuring the signal-to-interference-noise ratio (SINR) in the downlink. Suppose that a base station eNB (evolved Node B) provides communication services for K users through n component carriers (CCs) and different users can support different bandwidths, C\({}_{k}\) represents the number of component carriers that user K can support and \(\widetilde{C}=[C_{1},C_{2},\ldots,C_{k}]\) is the vector representing the number of CCs that can be supported by different users. One CC contains M resource blocks (RBs) where n \(\in\{1,2,\ldots,N\}\) represents n represents the nth CC, k \(\in\{1,2,\ldots,K\}\) represents the kth end user, and m \(\in\{1,2,\ldots,M\}\) represents the mth resource block. Assuming that the CQI feedback of the user is available on RBs of all CCs, C\({}_{n,m,k}\) represents the CQI index value when the mth RB in the nth CC is allocated to user k and C\({}_{n,m,k}\in\{1,2,\ldots,J\},J=31\). When the mth RB in the nth component carrier is allocated to user k, the rate that user k can achieve in time slot t is r\({}_{n,m}^{k}(t)\), its expression is:  
\[r_{n,m}^{k}(t)=B_{\rm RB}\log_{2}\left(1+p_{n,m}^{k}(t)r_{n,m}^{k}(t)\right) \tag{1}\]where \(\rm p_{n,m}^{k}(t)\) represents the transmission power allocated on the resource block when the mth RB in the nth CC is allocated to user k in time slot t; \(\rm B_{RB}\) represents the bandwidth of the resource block; \(\rm\gamma_{n,m}^{k}(t)\) represents the channel state information when the mth Rb in the nth component carrier of time slot t is allocated to user k (it is assumed that the base station can obtain all the channel state information in current time through CQI feedback),  
\[\rm\gamma_{n,m}^{k}(t)=H_{n,m,k}^{2}(t)/N_{0}B_{RB} \tag{2}\]  
where \(\rm H_{n,m,k}^{2}(t)\) represents the channel power gain when the mth RB in the nth CC in time slot t is allocated to user k and \(\rm N_{0}\) is the single-sided power spectral density of additive white Gaussian noise. For simplicity of description, the user k of the corresponding MCS when the CQI index is j is simply referred to as the user k using MCSj. When the mth RB in the nth component carrier is allocated to user k, the rate \(\rm r_{n,m}^{k,j}(t)\) that can be reached by user k using MCSj in time slot t, the expression is:  
\[\rm r_{n,m}^{k,j}(t)=\left\{ \begin{matrix}0,C_{n,m,k}<j\\ \rm r_{n,m}^{k}(t),C_{n,m,k}\geq j\end{matrix} \right. \tag{3}\]  
Equation (3) indicates that when the CQI value of RB is lower than the CQI value corresponding to the target MCS, the RB cannot be used. Because according to TR 36.912 [25], on each CC, all the RBs allocated to designated users must use the same MCS and satisfy the block error rate \(\rm BLER\leq 10\%\), otherwise the RB cannot be used. Let \(\rm\eta_{n,m}^{k,j}(t)\) represent the resource block allocation identifier. When the mth RB on the nth CC in time slot t is allocated to user k using MCSj, \(\rm\eta_{n,m}^{k,j}(t)=1\), otherwise, \(\rm\eta_{n,m}^{k,j}(t)=0\). At the same time, Formula (4) needs to be satisfied:  
\[\rm\sum_{k=1}^{K}\sum_{j=1}^{J}\rm\eta_{n,m}^{k,j}(t)\leq 1,\forall n,\forall m \tag{4}\]  
Let \(\rm\chi_{n}^{k,j}(t)\) represent the CC allocation identifier. When the nth CC in time slot t is allocated to user k using mcsj, \(\rm\chi_{n}^{k,j}(t)=1\), otherwise, \(\rm\chi_{n}^{k,j}(t)=0\). At the same time, the constraint condition of Formula (5) needs to be satisfied:  
\[\rm\sum_{n=1}^{N}\sum_{j=1}^{J}\rm\chi_{n}^{k,j}(t)\leq C_{k},\forall k \tag{5}\]  
where \(C_{k}\) is the carrier aggregation capability of user k, including narrowband users and broadband users. The carrier aggregation capability of narrowband users is equal to 1 and the carrier aggregation capability of broadband users is an integer in the interval [2,5].