# Joint Energy Supply and Routing Path Selection for
## Guideline Supplements  
Figure 1 shows the rechargeable wireless sensor network, which comprises a central base station (the sink node) that supervises the energy status of the whole network and formulates the charging schedule, a mobile charger carrying the WCD that is a programmable automatic aircraft with travel speed \(v_{m}\), and \(n\) sensor nodes equipped with a wireless power receiver. The entire sensor network forms an undirected graph \(G(P,E)\), where \(P\) denotes the set of nodes and \(E\) denotes the set of edges.  
We consider a sensor network with \(n\) nodes randomly deployed within a square area of area \(S\), where the maximum communication radius of each node is configured to be \(R\), and two nodes cannot establish communication links beyond this distance. Meanwhile, each sensor is equipped with a rechargeable battery with the same capacity \(E_{0}\), and the node is considered as dead if it has a battery energy that decays to 0. Furthermore, the central location of the square area is selected as the sink node, which is responsible for summarizing and processing data.  
The WSNs studied in this paper mainly focus on the typical data acquisition applications, in which sensor nodes periodically sense the relevant information in the monitoring area and transmit the monitoring data to sink node in a multi-hop manner. For purpose of facilitating further study, we need to simplify the sensor network model and give some reasonable assumptions as follows:  
Figure 1: The rechargeable wireless sensor network.  
1. The location of sensor nodes will not be changed after being deployed, and the distance from one sensor node to another can be estimated based on the received signal strength.
2. All sensor nodes are isomorphic with the same function of routing, issuing and collecting, and they also have the same initial state. Besides, the residual energy of each node at any time is accessible to the base station, so that it can make decision. Moreover, the energy of sink is not limited.
3. Each node collects data at an equal and constant rate \(r\), and the sensed data is not aggregated with the received data. Besides, they adopt the first-in-first-out approach to packet processing, and the initial load is 0, the length of the cache queue \(BF\) is also limited.  
Simultaneously, in order to describe the algorithm more clearly, our paper also makes the following relevant definitions.  
**Definition 1**.: _The neighbor node set of node \(i\) is given by:_  
\[N(i)=\left\{j|d_{ij}\leq R\right\} \tag{1}\]  
_where \(d_{ij}\) is the Euclidean distance between node \(i\) and node \(j\)._  
**Definition 2**.: _The forward neighbor node set of sensor node \(i\) is \(NF(i)\), which can be expressed as:_  
\[NF(i)=\left\{j\Big{|}j\in N(i),d_{j,sink}\leq d_{i,sink}\right\} \tag{2}\]  
**Definition 3**.: _The network lifetime refers to the persistent period from the deployment of the network nodes to the appearance of the first dead node due to energy exhaustion._  
**Definition 4**.: _The forward distance \(d_{f}(i)\) of node \(i\) is equal to the distance of node \(i\) to the sink node, namely:_  
\[d_{f}(i)=d_{i,sink} \tag{3}\]  
**Definition 5**.: _Charging capability is a measure of the amount of energy the sensor node receives from the charging device WCD per unit time._  
**Definition 6**.: _The energy supply capacity is the measure of the amount of energy that the WCD can replenish the network during a charging duration \(T_{c}\). Here, the charging duration represents the sum of the charging time of all charged nodes in a charging schedule._