###### Abstract
Previously, BS has removed as candidates CHs those nodes with an energy below the total average. The work in [20] describes another centralized method that employs a very complex technique. Specifically, it uses a convolutional neural network to determine the best CH. The authors in [21] present a centralized algorithm in which the BS uses artificial-intelligence as fuzzy c-means to determine the best location for the center of each cluster. The authors of [22] show a centralized algorithm that uses a Particle-Swarm based Optimization (PSO) for choosing the best CHs. Additionally, the BS determines the time that a CH remains active based on the residual energy of the system. Although the idea is very promising, the previous implementation is complex in real-time applications because its solution can be only achieved after the convergence of an iterative process. In this paper, we opt for a hybrid approach which could integrate the benefits of both centralized and distributed clustering algorithms. We propose a mainly distributed algorithm that is guided by the BS at specific instants. In our proposal, the BS sends setup information only three times during the whole lifetime of the network. Thus, those messages change only one characteristic of the distributed algorithm at the following instants: (i) at the beginning of the communication, (ii) when the first node depletes its battery, and (iii) when the network only keeps 50% of alive nodes. This occasional intervention of the BS is easy to implement and results in a better performance. In particular, the messages of the BS are used to dynamically tune the _skip_ value. The operation of a WSN can be divided into rounds. In a traditional approach, at the beginning of each round, the CHs are selected and then the nodes send their data to their corresponding CHs, which will aggregate and forward in a new message to the BS. However, it is possible to maintain the same structure of CHs for several rounds (referred to as the _skip_ value) so that the CH configuration messages are only sent at the first round of this group of \(skip\) rounds. This would lead to a reduction of the energy consumption of the CHs. In previous works, the \(skip\) value was a constant value [23], but, in this paper, we propose to dynamically adjust it as the inertia of the network varies during its lifetime. The approach presented in this paper has three main novelties when compared with previous works:  
* It is a distributed clustering algorithm guided by the BS in a simple way, with just three transmissions during the network lifetime. The transmissions help the nodes know about the status of the network. * It uses a dynamic \(skip\) value to keep the same CH structure in the network during a \(skip\) number of rounds. In this way, the configuration of the \(skip\) value is adapted to the dynamic inertia of the network during its operative time. As a consequence, the network can reduce the energy consumption for the CH configuration more efficiently. * The present paper constitutes a novelty concerning the type of fuzzy system used in the nodes. The nodes implement a new Type-2 Tagaki-Sugeno-Kang (TSK) model for the fuzzy system [24]. Previous fuzzy-logic based clustering algorithms rely on a Mandami model. However, the TSK has revealed itself more appropriate in real-time applications. Moreover, the input variables in the fuzzy system are carefully selected to extend the network lifetime as it is described in Section 2.1.1. Consequently, this clustering algorithm does not need complex or expensive hardware in the nodes because it is a distributed one. Moreover, it achieves good results without any optimization process in the BS, which could be a demanding process in terms of resources and time. This consumption could not be always negligible, even for the BS. The algorithm has been evaluated in a wide set of scenarios.