# Coordination of Macro Base Stations for 5G Network with User Clustering
## 3 Problem Reformulation with User Clustering and Benders Decomposition
### User Clustering Strategy for (\(\mathbf{P}_{1}^{t}\))  
To solve the problem in which the scale of (\(\mathbf{P}_{1}^{t}\)) grows with the number of users, user clustering is adopted in this work. Each cell \(m\) is evenly divided into six sub-cells \(m_{i}\) (\(i=1\), \(2\),..., \(6\)), as shown in Figure 4. All users in the same sub-cell \(m_{i}\) belong to user cluster \(m_{i}\) and are deployed as a whole. In this way, the number of binary variables in (\(P_{1}^{t}\)) is greatly reduced and does not increase with an increase in the number of users in the network.  
After the user clustering strategy is adopted, all users in cluster \(m_{i}^{\prime}\) may be connected to one of the \(\left|\mathbf{N}_{m_{i}^{\prime}}\right|\) adjacent BSs.  
The data rate from BS \(m\) to cluster \(m_{i}^{\prime}\) is shown in Equation (43) when they are connected.  
\[L_{tr,t}^{m,m_{i}^{\prime}}=\sum_{u_{k}\in\mathbf{U}_{m_{i}^{\prime}}}L_{tr}^ {u_{k}} \tag{43}\]  
Figure 4: Schematic of the user clustering method.  
where \(L_{tr,t}^{m,m_{i}^{\prime}}\) denotes the data rate from BS \(m\) to cluster \(m_{i}^{\prime}\), and \(\mathbf{U}_{m_{i}^{\prime}}\) denotes the set of users in cluster \(m_{i}^{\prime}\).  
Meanwhile, BS \(m\) needs to allocate transmit power \(P_{tr,t}^{m,m_{i}^{\prime}}\) to cluster \(m_{i}^{\prime}\) to meet user needs, as shown in Equation (44).  
\[P_{tr,t}^{m,m_{i}^{\prime}}=\sum\limits_{u_{t}\in\mathbf{U}_{m_{i}^{\prime}}}P_{tr }^{m,u_{t}} \tag{44}\]  
where \(P_{tr,t}^{m,m_{i}^{\prime}}\) denotes the transmit power from BS \(m\) to cluster \(m_{i}^{\prime}\).  
The constraints expressed by Equations (11)-(16) are converted to Equations (46)-(51).  
\[a_{t}^{m,m_{i}^{\prime}}\in\{0,1\} \tag{45}\]  
\[a_{t}^{m,m_{i}^{\prime}}\leq U_{bs}^{m,t},m\in\mathbf{N}_{m_{i}^{\prime}} \tag{46}\]  
\[\sum\limits_{m\in\mathbf{N}_{m_{i}^{\prime}}}a_{t}^{m,m_{i}^{\prime}}=1 \tag{47}\]  
\[L_{tr}^{m,t}=\sum\limits_{m_{i}\in\mathbf{N}_{m_{i}^{\prime}}}a_{t}^{m,m_{i}^{ \prime}}L_{tr,t}^{m,m_{i}^{\prime}} \tag{48}\]  
\[L_{tr}^{m,t}\leq L_{tr}^{\max,m} \tag{49}\]  
\[P_{tr}^{m,t}=\sum\limits_{m_{i}\in\mathbf{N}_{m_{i}^{\prime}}}a_{t}^{m,m_{i}^{ \prime}}P_{tr,t}^{m,m_{i}^{\prime}} \tag{50}\]  
\[P_{tr}^{m,t}\leq P_{tr}^{\max,m} \tag{51}\]  
where \(a_{t}^{m,m_{i}^{\prime}}\) denotes the connection relationship between BS \(m\) and cluster \(m_{i}^{\prime}\); \(\mathbf{N}_{m_{i}^{\prime}}\) denotes the set of adjacent BSs of cluster \(m_{i}^{\prime}\); and \(\mathbf{N}_{m}^{cl}\) denotes the set of clusters in cell \(m\) and its adjacent cells.  
Thus, (\(\mathbf{P}_{1}^{t}\)) is shown as follows.  
\[(\mathbf{P}_{1}^{t})\text{min}F_{1}^{t}=\text{min}\sum\limits_{m \in\mathbf{M}}P_{com}^{m,t}\Delta t\] \[s.t.(2),(5),(7),(10),(17)-(24),(43)-(51)\]  
The proposed user clustering method according to user locations has obvious practical value. According to Equation (3), the transmit power needed by user \(u\) is positively correlated with the distance between \(u\) and its connected BS \(m\). It is easy to observe that the set of adjacent BSs \(\mathbf{N}_{u}\) of each user \(u\) in cluster \(m_{i}^{\prime}\) is the same. The size relationship of the geographic distance between user \(u\) and each BS \(m\) (\(m\in\mathbf{N}_{u}\)) is basically uniform. Therefore, the size relationship of the required transmitting power of each user \(u\) in the same cluster \(m_{i}^{\prime}\) is basically uniform when connected to each BS \(m\) (\(m\in\mathbf{N}_{u}\)). This property ensures that the optimal solution of (\(\mathbf{P}_{1}^{t}\)) with or without the user clustering strategy is similar, while considerable calculation time is saved.