## 2 Energy-Efficient Optimization Model
\(\rm P_{T}\) represents the total power transmitted by the base station. The base station allocates the power transmitted by the system to each resource block, \(\rm P_{s}\) is the sum of transmission power consumed on all the resource blocks and the following power constraint conditions need to be satisfied:  
\[\rm P_{s}=\rm\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{k=1}^{K}\rm\sum_{j=1}^{J}\rm \chi_{n}^{k,j}(t)\rm\eta_{n,m}^{k,j}(t)\rm p_{n,m}^{k,j}(t)\leq P_{T} \tag{6}\]  
where \(\rm p_{n,m}^{k,j}(t)\) indicates the transmit power that the mth resource block on the nth CC is allocated to user k using MCSj in the time slot. Let \(R_{k}(t)\) denote the total rate obtained by user \(k\) in time slot \(t\) and the expression is Formula (7):  
\[R_{k}(t)=\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{j=1}^{J}\chi_{n}^{kj}(t)\eta_{n,m}^{ kj}(t)r_{n,m}^{kj}(t) \tag{7}\]  
The energy efficiency (EE) of the system is defined as the ratio of the user's transmission rate to the total energy consumption (unit is bits/J), which has been used by many scholars as a performance criterion for energy-efficient resource allocation [24] and which reflects the system performance gain brought by energy consumption. The larger the EE value, the greater the system performance gain brought by the energy consumption, otherwise, the smaller the value is. We define the EE in the downlink of the LTE/5G system using CA technology as \(\eta_{EE}\) and its expression is:  
\[\eta_{EE}=\frac{R}{\zeta P_{s}+P_{C}} \tag{8}\]  
where, \(\zeta\) is the reciprocal of the efficiency of the transmission power amplifier; \(R\) is the total user transmission rate; \(P_{C}\) is the static power consumed by the circuit and is set as a fixed value. The optimization model \(P_{1}\) for maximizing the energy efficiency of the system is defined as follows (the time slot variable \(t\) is omitted for simplicity):  
\[P_{1}:\ \eta_{EE}^{*}\triangleq\max\frac{\sum_{k=1}^{K}R_{k}}{\zeta\sum_{k=1}^{ K}\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{j=1}^{J}\chi_{n}^{kj}\eta_{n,m}^{kj}P_{n,m}^{kj}+P_{C}} \tag{9}\]  
\[\sum_{k=1}^{K}\sum_{j=1}^{J}\eta_{n,m}^{kj}\leq 1,\forall n,\forall m \tag{9a}\] \[\sum_{n=1}^{N}\sum_{j=1}^{J}\chi_{n}^{kj}\leq C_{k},\forall k\] (9b) \[\sum_{k=1}^{K}\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{j=1}^{J}\chi_{n}^{kj}\eta_{n, m}^{kj}P_{n,m}^{kj}\leq P_{T}\] (9c) \[\sum_{j=1}^{J}\chi_{n}^{kj}\leq 1,\forall n,\forall k\] (9d) \[\rm\sum_{n=1}^{N}\sum_{j=1}^{J}\rm\chi_{n}^{k,j}(t)\leq C_{k},\forall k \tag{5}\]0 (9e) \[\rm\sum_{n=1}^{N}\sum_{j=1}^{J}\rm\chi_{n}^{k,j}(t)\leq C_{k},\forall k \tag{5}\]1  
where the constraint condition (9a) indicates that one RB can only be allocated to one user in the same time slot; the constraint condition (9b) indicates that the number of carriers allocated to different users must satisfy the limit condition of the maximum number of CCs supported by the users that have different carrier aggregation capabilities; the constraint condition (9c) indicates that the transmission power actually allocated should be less than or equal to the total transmission power; the constraint condition (9d) indicates that for each CC, all the RBs allocated to the specified user must use the same MCS; the constraint condition (9e) indicates that the CC identifier and the RB identifier are 0 or 1 and the power allocated to each RB is non-negative; and the constraint condition (9f) indicates that the value of CQI is an integer between 1 and 31.