# Energy-Efficient Connected-Coverage Scheme in Wireless Sensor Networks
## 3 System Model
### Problem Formulation  
Monitoring the MTP consumes the energy of the sensor node. Because the battery capacity of the node is limited, once its energy is exhausted, the sensor node will die and the performance of the network will be influenced. Let \(\tau\) be the lifetime of the WSN. Once there exist a MTP that cannot be covered by any sensor node, the network operation is assumed to be interrupted. The total working time of the network is the value of \(\tau\).  
When many sensor nodes are applied to detect one MTP, there will be redundant coverage, as shown in Figure 2. If these sensor nodes work at the same time, the network energy will be consumed too quickly. If one sensor node is used to monitor the MTP, when its energy is exhausted, the neighbor sleeping node can replace it. Such a cycle can extend the working time of the network. Therefore, as long as the minimum sensor nodes are found to cover all MTPs, the network lifetime can be maximized. Furthermore, it can be seen from (5) that the energy that is consumed by data transmission is related to the distance, an appropriate CH choosing policy can reduce the energy consumed on the long rang data transmission, and then the network lifetime can be further extended.  
According to the above description, sensor nodes are distributed in a high density, which may lead to the redundant coverage. Thus, much energy will be wasted if all sensor nodes work simultaneously. Subsequently, the lifetime of the network cannot be optimal. At the same time, the collected data of sensor nodes are sent to the sink through single-hop or multi-hops, which also consumes the limited energy of sensor nodes. The energy efficiency of data transmission is closely related to the routing selection. Hence, the network life maximization problem can be divided into two sub-problems with regards to the sensor nodes scheduling and route protocol optimization.  
First, the sensor nodes scheduling can be described as the set coverage problem (SCP), which is defined, as follows. Considering the limited energy of the sensor node, the redundant coverage should be avoided to prolong the lifetime of the network. The problem is how to find the minimum number of sensor nodes to cover all MTPs in the WSN, so that, the energy can be utilized more efficiently. Second, to optimize the route protocol and save more energy consumed on the communication, the energy saved maximization problem is proposed and described, as follows. Given a minimum coverage set, how to optimize the route protocol so that the network lifetime can be further extended. By considering these two sub-problems, the lifetime maximization problem can be described as  
\[\max \tau\] (7a) s.t. \[s_{j}\in\mathsf{s}_{\min Cover} \tag{7b}\] \[\sum_{j}G_{j,k}\times v_{j}\geq 1,k\in[1,M]\] (7c) \[e(s_{j})\times v_{j}>e_{th},v_{j}\in\{0,1\}\] (7d) \[p_{ath}(s_{j},sink)\geq 1 \tag{7e}\]  
Figure 2: Network structure diagram.  
where \(\mathbf{s}_{minCover}\) is the set of sensor nodes, which is the solution of the optimization problem with the target of \(\min\sum\limits_{i=1}^{N}v_{i}\) under constraints (7c) and (7d). \(v_{j}=1\) means that the sensor node \(s_{j}\) is in the active state, while \(v_{j}=0\) means that the sensor node \(s_{j}\) is in the sleep state. Constraint (7b) indicates that the number of activated sensor nodes is minimized. (7c) ensures that each MTP \(P_{k}\) can be covered by at least one node. The network works in rounds, (7d) shows that the residual energy of the activated node should be greater than the energy threshold \(e_{th}\). Subsequently, the work time of each cycle can be longer. \(p_{ath}(s_{j},sink)\) represents the path of the sensor node \(s_{j}\) in \(\mathbf{s}_{minCover}\) to the sink, and (7e) indicates that all of the sensor nodes have at least one path to the sink. By optimizing routing, the energy saved of each path are maximized, so the lifetime of the network can be further prolonged.