# A Power Planning Algorithm Based on RPL for AMI Wireless Sensor Networks
## 1 Introduction
Obviously, the construction of DODAG will be heavily dependent on ETX when it is the metric used in the objective function. However, RPL does not define any protocol to measure ETX. A common approach is to measure it using information from the media access control (MAC) layer, as implemented by the Contiki operating system [11; 12]. The MAC-based quality estimation of a link can be used only when unicast data traffic is sent through this link. Because RPL builds the topology using multicast control messages, the DODAG should be built using default initial values for ETX [12]. This leads to the use of bad quality links, which can have numerous consequences (see Section 2). Another important point comes from the size of parent sets. A small parent set can cause instability to the DODAG if the quality of the links are highly variable. When a node identifies that it is not connected to any parent, it sends a DODAG Information Solicitation (DIS) message to ask for new parents, causing the reconstruction of the sub-tree below it. A large parent set can also cause problems to RPL because the rank of a node must be greater than the rank of any member of its parent set. Even though a node should select the best node in the parent set as the preferred parent, it computes its rank and the cost of the path to the root considering the worst element in the parent set. Thus, using an excessively large parent set may inhibit a child node to select the best parent and will likely result in sub-optimal routing [13]. There is a tradeoff concerning the size of parent sets, which can be neither too small nor too large. Although it is an important parameter for the resilience of the DODAG, RPL only allows the maximum number of parents to be defined. If a certain amount of parents is needed to meet resilience requirements, RPL does not have mechanisms to pursue this goal. To address this issue, we propose a power adjustment method, which can be used to plan DODAGs with predefined size for the sets of parents, without compromising the quality of the paths to the root. Nodes in wireless sensor networks can adjust their transmission power to increase the network connectivity. However, using the highest power level for all nodes can result in a significant reduction in average lifetime and capacity of the network. Because the nodes in AMI networks are not battery powered, the first issue does not arise, but the second should be addressed. In this sense, to improve the DODAG connectivity and resilience, we propose an algorithm that assigns power levels to the transmitters in RPL networks. Power level assignment selects the transmission power so that the resulting communication graph respects certain constraints and minimizes energy consumption. [PERSON] et al. (2014) proved that this problem is NP-hard for sets of three-dimensional points, and [PERSON] et al. (2015) proved the same for sets of two-dimensional points. [PERSON] and [PERSON] (2016) considered the problem for the case where each node can transmit in one of two possible power levels. They proved that this problem is also NP-hard, and provided two heuristic algorithms to solve it. [PERSON] and [PERSON] (2017) provide an almost optimal heuristic for the same case, which aims to minimize total power consumption while maintaining a strong level of connectivity. They also present a heuristic algorithm to minimize total power using three power levels. [PERSON] et al. (2018) studied the problem for the case where nodes can transmit at any power level. They provided an approximation algorithm for minimizing the total power for some monotone properties, which is not realistic because there is a limited amount of transmission levels. More recently, a few studies have attempted to determine the optimal set of power levels. [PERSON] et al. (2019) defined the optimal probability distribution associated with these sets, while [PERSON] et al. (2019) investigated the minimum cardinality that does not compromise the service life of the WSN.