# Graph Theory-Based Characterization and Classification of Household Photovoltaics
###### Abstract
PV systems can suffer from different kinds of faults during operation, leading to a decrease in energy production and possible system damage. This is why diagnostic algorithms are applied to constantly determine whether the system is operating properly or some kind of fault has arisen. GT can be used both to develop diagnosis algorithms that detect and classify malfunctions of the system and to elaborate a response that overcomes the failure in the most effective manner [17]. Regarding diagnosis, we find some classification algorithms in which GT takes place, but very few are found regarding PV systems. In [18], the authors present a diagnosis method using graph-based semi-supervised learning, which uses GT to represent data that has been partially classified. The algorithm decides which label assigns unclassified data, showing how GT is also useful for machine  
Figure 1: Example of a rooftop PV installation made up of 6 solar panels all connected to the same inverter. learning algorithms that can be applied to a vast quantity of fields, such as PV systems. For a system to overcome a detected failure, a GT-based method is proposed in [19] to find the optimal connection configuration that maximizes energy production given that some of the panels in the PV array fail. The authors provide an algorithm that finds the shortest path from a given panel to the inverter, using the so-called Manhattan distance, and propose pseudocode that optimizes production; however, no application of this pseudocode is shown. Finally, in [20,21], a GT-based algorithm is proposed to find the optimal switch-set, which minimizes the number of switches while allowing different desired configurations. Switch-set configurations are modeled through graphs in which one of their edges represents a panel, and the others simulate switches. Although the algorithm is very flexible concerning module interconnections, the related efficiency for building a PV system model could be limited due to the high number of nodes and arcs involved for representing each module. Configurations allowed by the optimal switch set include series, parallel and series-parallel associations, but these do not cover all possibilities. GT has been applied to different fields in PV production but not very prodigiously. Bearing in mind the very few studies reported in the literature apply GT to PV systems, and considering the identified weaknesses of those, further research is needed. In this regard, the present investigation poses a characterization based on GT that models a PV installation of up to seven panels, considering this as representative of a family-house rooftop PV system's topology. The manuscript is structured as follows. In Section 2, a summary of the GT definitions and concepts needed for this paper are provided. In Section 3, the GT characterization of PV arrays is exposed, an algorithm is proposed to build all possible configurations of \(p\) panels \(p\in[3,7]\) and simulations are carried out to obtain the power of each PV array, its reliability polynomial (see [22,23]) and the expected value of the power produced. Network reliability has been thoroughly studied for distribution networks, see for instance [24, 25, 26], but its application to PV arrays constitutes a novelty. Results are presented, explained and discussed in Section 4. Finally, the main conclusions are stated in Section 5.