# Graph Theory-Based Characterization and Classification of Household Photovoltaics
## 5 Conclusions  
A simple model to characterize PV array structures through graph theory has been developed by assigning arcs to panels and electrical connections to vertices. This modeling, along with a found relationship between PV array structures and partitions of a given set, has been used to build all possible valid structures, not only combinations of series and parallels, but also other less-symmetrical options.  
The developed algorithm is not computationally efficient. This is due to its filter structure since the creation of all possible partitions of a given set is an NP problem, and so it is checking isomorphy. It would be interesting for further work to improve its behavior.  
For every structure between 3 and 7 panels, a simulation has been run to obtain a classification in terms of the maximum power we can extract from them. Classical parallel and series combination structures perform the best as expected, but some other structures are found to perform nearly as well.  
A classification method has been developed in terms of reliability. As expected, the structures with all \(p\) panels connected in parallel are ahead in this classification no matter which failure probability is assigned to panels, and something similar happens with  
Figure 14: For a single working probability of \(x=0.75\), this table contains the top five configurations for every \(p\in[3,7]\). The percentage below every graph shows the normalized EVP, and it is represented in the last column. Bars and graphs are in the same order, and the coloring of the vertices remarks this fact.  
series structures, which are always the least reliable. Most of the structures, though, are preferred to others depending on the failure probability. A possible continuation of this reliability classification could be to consider different probabilities of failure for each panel.  
Power obtaining and reliability can be simultaneously considered by computing the expected value of the power in terms of the probability of every single panel working properly. A classification in terms of the expected value of power has been carried out for all structures of p panels (\(p\in[3,7]\)). Structures of all \(p\) panels associated in parallel are optimal in terms of the expected value of power.  
Results found in this study emerge from the first incursion made through the proposed characterization, which does not mean they are the only ones that can be deduced. They only prove that the model is worth investigating, so plenty of work is still to be done in this direction. Potential future applications of the characterization may include the study of reconfiguration structures under different conditions such as total or partial shading, the determination of the best PV array configuration under current and voltage limitations of an inverter or an efficient computation of the Expected Value of Power.  
Conceptualization, J.M.C., D.C., N.L.; methodology, J.M.C., D.C., N.L.; simulation and results: J.M.C., D.C., N.L.; writing--original draft preparation, writing--review and editing, J.M.C., D.C., N.L. All authors have read and agreed to the published version of the manuscript.  
The authors would like to thank \"Ministerio de Economia y Competitividad\" and \"Ministerio de Ciencia e Innovacion\" of Spain for the funding (grant reference EN2016-81040-R and PID2019-111536 RB-I00). Research of [PERSON] was supported by Secretaria d'Universitats i Recerca del Departament d'Empresa i Coneixement de la Generalitat de Catalunya (grant 2020 FISDU 00596). [PERSON] thanks \"Institucio Catalana de Recerca i Estudis Avancats (ICREA)\" for the ICREA Academia award. Research of [PERSON] was supported in part by grant MTM2017-86767-R and PID2020-115442 RB-I00 (Spanish Ministerio de Ciencia e Innovacion).  
Not applicable.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} PV & PhotoVoltaic \\ GT & Graph Theory \\ ODM & One Diode Model \\ EVP & Expected Value of Power \\ \end{tabular}