# Graph Theory-Based Characterization and Classification of Household Photovoltaics
###### Abstract  
With the clear goal of improving photovoltaic (PV) technology performance towards nearly-zero energy buildings, a graph theory-based model that characterizes photovoltaic panel structures is developed. An algorithm to obtain all possible configurations of a given number of PV panels is presented and the results are exposed for structures using 3 to 7 panels. Two different classifications of all obtained structures are carried out: the first one regarding the maximum power they can produce and the second according to their capability to produce energy under a given probability that the solar panels will fail. Finally, both classifications are considered simultaneously through the expected value of power production. This creates structures that are, at the same time, reliable and efficient in terms of production. The parallel associations turn out to be optimal, but some other less expected configurations prove to be rated high. graph theory; network reliability; PV systems; expected value of power; household photovoltaics +
Footnote †: journal: Applied Physics Section  
1 Applied Physics Section of the Environmental Science Department, Universitat de Lleida,  
25001 Lleida, Spain; [EMAIL_ADDRESS]  
1 Applied Physics Section of the Environmental Science Department, Universitat de Lleida,  
25001 Lleida, Spain; [EMAIL_ADDRESS]  
1  
[PERSON], N. Graph Theory-Based Characterization and Classification of Household Photovoltaics. _Appl. Sci._**2021**, _11_, 10999. [[https://doi.org/10.3390/app11221099](https://doi.org/10.3390/app11221099)]([https://doi.org/10.3390/app11221099](https://doi.org/10.3390/app11221099))  
Academic Editor: [PERSON]  
19 October 2021  
Accepted: 15 November 2021  
Submitted: 20 November 2021  
The most common PV system installed in buildings consists of a small number of PV panels located on the rooftop (see Figure 1). Improvement in production will result from both increasing the PV panel efficiency and finding the interconnection configuration that maximizes production. Finding the configuration that maximizes energy production is not a static problem in the sense that the optimal configuration may change with time depending on different factors such as partial or total shading or failure/degradation of some panels [8; 9; 10; 11; 12]. The PV system layout should provide the maximum power possible and should be flexible enough to keep providing the maximum possible power in case some components fail. In order to face this problem, a mathematical characterization of these PV systems that accurately reproduce their physical behavior becomes essential. One tool that has proved to be effective for characterization, modeling and optimization purposes in many disjoint fields of science is Graph Theory (GT). From molecule modeling in chemistry to traffic light controlling, GT provides great mathematical insight wherever it is applied. This is also the case for electrical networks [13; 14; 15] and PV systems [16; 17; 18; 19; 20; 21]. Even more, GT could be applied to model and optimize different processes involved in the same field. Let us provide examples in which GT is used in the topic of PV systems. At the very heart of PV performance, electronic and semiconductor physics are found. In semiconductors, due to the so-called recombination processes, part of the current that is generated by the PV panels is lost. Therefore, by minimizing the recombination current, the total current delivered by the PV is maximized for a certain photogeneration. In order to control these phenomena, accurate electronic modeling should be developed. In [16], the authors build a GT-based model of electronic states and transitions so that generation and recombination processes in semiconductors and electronic state concentrations can be obtained exclusively through GT elements, proving that GT is a suitable tool for the description of generation and recombination processes.