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.

1. Introduction

It is well known that buildings are the main energy consumers in the world. In the European Union (EU), buildings are responsible for almost 40% of energy consumption and 36% of carbon dioxide (CO${}_2$) emissions. This is due to the fact that 75% of buildings in the EU are energy-inefficient [1]. To reverse this trend, the latest Directive of the European Parliament is 2018/844, which modifies the previous Directive 2010/31 on energy performance of buildings and Directive 2012/27 on energy efficiency. This new directive extends the so-called “20-20-20 goal”, which includes a 20% increase in energy efficiency, a 20% reduction in CO${}_2$ emissions from 1990 levels and a 20% of energy production is from renewable energy. The new goals (32.5%-40%-32%) are more demanding and will be required by 2030 [2,3].

Renewable energy production becomes the only viable solution to obtain buildings with null or nearly null consumption. Among renewable energy resources, sunlight is the most extended, accessible and available [4,5], which makes it the perfect candidate to cover buildings’ energy demands. Solar energy integration in domestic buildings decreases the grid dependency and carbon emissions [6]. Besides, due to the rapid reduction in PV costs, more recent forecasts predict that tens of terawatts of photovoltaic capacity will be implemented by 2050 [7]. This represents an investment of several tens of billions of dollars, which is the same order of magnitude of the world’s annual economic production. Thus, a breakthrough in research that would allow for a small increase in PV yield would have tremendous economic impacts.

Contrary to the case of utility-scale solar PV plants, space availability for PV systems in residential buildings becomes a highly limiting factor. This limitation entails the need for more efficient technologies and the performance optimization of the existing ones in order to enhance production or minimize losses without exceeding available space.

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.

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 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.

2. Graph Theory Background

Networks are made up of elements and the connections between them. The relationship between elements can be undirected or directed, depending on whether the communication between them is two-way or only one-way. A mixed graph $M=(V,E,A)$ consists of a nonempty set V of elements called vertices, a set E whose elements (edges or undirected edges) are non-ordered pairs of elements of V and a set A of ordered pairs of vertices, called arcs (or directed edges). An example is shown in Figure 2.

From this point of view, an undirected graph (or simply graph) has all its edges undirected and a digraph (or directed graph) has all its edges directed. The term ’graph’ will sometimes be used loosely to refer to undirected, directed or mixed graphs indistinctly, when it does not generate confusion.

The GT definitions needed for the study are shown below. For more definitions and other details see [27].

(1)

The number of vertices is known as the order of the graph. The number of edges and/or arcs is known as the size of the graph.

(2)

If u and v are vertices of graph G ($u,v\in V\left(G\right)$) we write $uv$ to represent the edge $\{u,v\}=\{v,u\}$ and we say that u is adjacent to v or vice-versa. If u and v are vertices of digraph D we write $u\to v$ to represent the arc $(u,v)$, and we say that u is adjacent to v, or v is adjacent from u.

(3)

The underlying graph U of a digraph D is the one obtained when arcs $u\to v$ are replaced by edges $uv$.

(4)

Given a graph G and $u\in V\left(G\right)$, the degree of u written $d\left(u\right)$ is the number of vertices adjacent to (or from) u. Given a digraph D and $u\in V\left(D\right)$, the outdegree of u written $d^{+}\left(u\right)$ is the number of vertices adjacent from u and the indegree of u written $d^{-}\left(u\right)$ is the number of vertices adjacent to u.

(5)

A graph H is a subgraph of graph G if $V\left(H\right)\subseteq V\left(G\right)$ and $E\left(H\right)\subseteq E\left(G\right)$. When $V\left(H\right)=V\left(G\right)$, H is said to be a spanning subgraph of G.

(6)

A path is a graph whose vertices can be labeled by $v_1,v_2,\dots ,v_n$ such that its edges are $v_i{v}_{i+1}$ for $i\in [1,n-1]$. A directed path from $v_1$ to $v_n$ is a digraph whose vertices can be labelled by $v_1,v_2,\dots ,v_n$ such that its arcs are $v_i\to v_{i+1}$ for $i\in [1,n-1]$.

(7)

A graph G is connected if for every given pair of vertices $u,v\in V\left(G\right)$ there exists a subgraph P, which is a path starting at u and ending at v. A digraph D is strongly connected if, for every given pair of vertices $u,v\in V\left(G\right)$, there exists a subdigraph P, which is a directed path from u to v. A digraph is weakly connected if its underlying graph is connected.

(8)

Two graphs $G_1$ and $G_2$ are isomorphic if there exists a bijective function $\varphi :V\left(G_1\right)\to V\left(G_2\right)$ such that if $uv\in E\left(G_1\right)$ then $\varphi \left(u\right)\varphi \left(v\right)\in E\left(G_2\right)$. Function $\varphi$ is called isomorphism from $G_1$ to $G_2$.

3. Methodology

3.1. Modeling

In this study, PV arrays will be modeled using digraphs in the following manner:

• Panels are represented by arcs, which point in the direction that the photo-current is created. This characterization shows the fact that panels have a determined polarity. The head represents the positive pole and the tail the negative one.
• Vertices represent nodes in the network structure. If different panel poles represented by heads or tails of arcs are incident to or from a given vertex, there exists a cable connection between them.

An example is found in Figure 3. The number of panels p of a given structure determines the size of the associated digraph, and imposes a restriction in its order, which can go from 2 to $p+1$. The first case being the parallel association of all p panels, while the second corresponds to all panels in a series connection.

An array-digraph is defined as a digraph that represents a PV panel structure, this is, a digraph D such that:

(1)

There exists only one vertex $s\in V\left(D\right)$, called the source, such that $d^{-}\left(s\right)=0$, i.e., there is only one vertex s, such that there is no arc adjacent to s.

(2)

There exists only one vertex $t\in V\left(D\right)$, called the target, such that $d^{+}\left(v\right)=0$, i.e., there is only one vertex t, such that there is no arc adjacent from t.

(3)

For every arc $a\in A\left(V\right)$ there exists at least one path C from s to t, such that $a\in A\left(C\right)$, i.e., every arc in the digraph belongs to a path from s to t.

It can be observed that every array-digraph D is weakly connected, as it is directly deduced from condition three.

3.2. Construction

The goal in this section is to develop an algorithm that builds all digraphs (of size p) that satisfy all three array-digraph conditions. To do so, it is desirable for all size p digraphs to be obtained as subgraphs of some kind of underlying “complete graph” that contains them all. We define the complete mixed structure of order $2p$, denoted as $G_{2p}$, as the mixed graph whose vertex set is $\{0,1,\dots ,2p-1\}$, such that:

(1)

Vertex $2k$ is adjacent to $2k+1$ through an arc $(2k\to 2k+1)$, and adjacent to all other vertices through an edge for all $k\in \left[0,\lfloor (p+3)/2\rfloor \right]$.

(2)

Vertex $2k+1$ is adjacent from $2k$ through the arc $(2k\to 2k+1)$, and adjacent to all other vertices through an edge for all $k\in \left[0,\lfloor (p+3)/2\rfloor \right]$.

Here, $\lfloor m\rfloor$ means the floor function of $m\in \mathbb{Z}$, i.e., the greatest integer not greater than m. The complete mixed structure is the one that contains all possible edge connections between all heads and tails of all panels, as shown in Figure 4 for the case $p=4$.

Given a spanning mixed subgraph H of $G_{2p}$, let ∼ be an equivalence relation in the vertex set such that, given u,$v\in V\left(H\right)$, we say that u is related to v ($u\sim v$) if, and only if there exists a path from u to v (made up of edges) contained in H. The quotient set $V\left(H\right)/\sim$ defines a partition on the vertex set given by the equivalence classes. For every spanning mixed subgraph H we build a digraph D such that:

(1)

Every vertex is an equivalence class of $V\left(H\right)/\sim$.

(2)

Arc $\left[u\right]\to \left[v\right]$ is drawn for every $w\in \left[u\right]$ and $z\in \left[v\right]$ such that $w\to z$ is an arc of H.

Here, $\left[v\right]$ represents the equivalence class to which vertex v belongs. An example of this construction is found in Figure 5.

Given the association between array-digraphs and partitions, we propose a method to obtain all array-digraphs with p arcs that exploits this relation. In Figure 6, we present a sketch of the algorithm. It constructs all possible partitions of the set $\{0,1,\dots ,2p-1\}$, builds its associated graphs and tests the array-digraph conditions, and also checks isomorphisms with the saved digraphs.

This algorithm is not computationally efficient since, on one side, the number of partitions of $2p$ elements increases exponentially. These are known as Bell numbers. On the other side, checking isomorphy between graphs is an NP problem [28].

3.3. Simulation

The algorithm exposed in the previous sections provides all non-isomorphic partitions whose associated digraphs satisfy the three array-digraph conditions. In this section, every array-digraph is associated to an NgSpice file to simulate the behavior of the associated PV structure. The one diode model (ODM) will be used to simulate solar panels. Every arc will be introduced in the NgSpice file through the associated circuit of Figure 7.

In this simulation, all arcs are modeled using all the same parameters for the current source ($I_{pv}$), the type of diode (D) and the series ($R_s$) and shunt ($R_{sh}$) resistances. Namely, $R_s=35.6$ m$\mathsf{\Omega }$, $R_{sh}=312.55$ $\mathsf{\Omega }$, $I_{pv}=8.17$ A and the diode is model 1n4007. As a result of the simulation, we obtain the $IV$ curve and the power curve for every structure found by the filter algorithm. We will compare the performance of different structures by looking at the maximum value that reaches their power curve.

3.4. Reliability

Once all possible p panel array-digraphs have been found, and their performance has been tested, it is time to develop a classification in terms of its reliability. By reliability we understand the capacity of a given structure to keep producing power when some of the panels fail, or have a failing probability. To measure this, we will associate each array-digraph D to a polynomial $P_D\left(x\right)$ in the domain of $x\in [0,1]$. The variable x represents the probability of every single panel working properly. For instance, if $x=0.25$, every panel in the structure has a 25% probability of working correctly. The polynomial $P_D\left(x\right)$ is such that:

$$P_D\left(x\right)=\sum _{i=0}^p{\delta }_D\left(i\right)x^i{(1-x)}^{p-i}.$$

(1)

where ${\delta }_D\left(i\right)$ is the number of subdigraphs of D having size i, holding the array-digraph conditions, and keeping the same source and sink as D.

For the array-digraph in Figure 8, the subgraphs holding these conditions are presented in Figure 9. Thus, for this array-digraph, the associated polynomial is:

$$P_D\left(x\right)=2x^2{(1-x)}^3+x^3{(1-x)}^2+3x^4(1-x)+x^5.$$

(2)

Every term in the summation of $P_D\left(x\right)$ has two contributions. One is $x^i{(1-x)}^{n-i}$, which corresponds to the probability of i panels working properly in the structure. The other, ${\delta }_D\left(i\right)$, tells how many working substructures with i panels there are. By adding these terms, the polynomial $P_D\left(x\right)$ expresses the probability of an array-digraph working properly for every value of x.

This polynomial will act as a merit function which will allow us to classify array-digraphs in terms of their probability of working for every value of $x\in [0,1]$.

3.5. Expected Value of Power (EVP)

In the previous section, a reliability analysis of all valid array-digraphs of p panels has been carried out. However, it is not enough to choose a panel structure depending on its reliability since it is possible that a very reliable structure performs very poorly in terms of power production. This is why we should combine the probability of a structure working properly with the power that it can provide. To do so, the EVP given by every array-digraph D will be computed in terms of x (again $x\in [0,1]$ represents the probability of every single panel working properly).

Let D be a size p array-digraph, and let $S_D\left(i\right)$ be the set of all subdigraphs of D with order i that hold the array-digraph conditions. Let $pow\left(d\right)$ be the maximum power extracted from $d\in S_D\left(i\right)$. Then, the EVP obtained from D is:

$$E_D\left(x\right)=\sum _{i=0}^p\left(\sum _{d\in S_D\left(i\right)}pow\left(d\right)\right)x^i{(1-x)}^{p-i}.$$

(3)

The probability of any array-digraph $d\in S_D\left(i\right)$ is $x^i{(1-x)}^{p-i}$. Thus, $E_D\left(x\right)$ is the summation of the product of $pow\left(d\right)$ times the probability of having $d\in S_D\left(i\right)$, which is precisely the definition of an expected value.

4. Results and Discussion

4.1. Construction and Simulation

The construction algorithm presented in Section 3.2 and sketched in Figure 6 was run to obtain for every $p\in [3,7]$:

(1)

The number of partitions of $2p$ elements.

(2)

The number of partitions that pass the filter, i.e., that are associated with an array-digraph.

(3)

The number of partitions that represent different (non-isomorphic) array-digraphs.

Results are shown in

Table 1. Results of the filter algorithm.

Panels	Num of Partitions	Pass the Filter	Non-Isomorphic
3	203	19	5
4	4140	195	15
5	115,975	2911	49
6	4,213,597	59,223	181
7	190,899,322	1,572,019	725

.

For every array-digraph, an NgSpice file is built containing the equivalent circuit using the ODM and we obtain its IV curve. The maximum power that can be produced by every array-digraph is extracted from the IV curves, and different array-digraphs of the same number of panels are compared through the histograms shown in Figure 10. Power is normalized so that the structure with maximum power is labeled with 1, and thus every bin shows how many array-digraphs exist that can produce the fraction of the maximum power that indicates its label.

Results show that the structures that perform best are made up of m rows connected in parallel strings of n panels connected in series ($n,m\in \mathbb{N}$). The number of array-digraphs depends on the number of divisors of the number of panels p. For prime numbers 3, 5 and 7, there are only two optimal array-digraphs (those with $m=1$ and $n=p$ or $m=p$ and $n=1$). The more divisors p has, the more optimal array-digraphs there are. The contribution of less-symmetrical structures should not be neglected, since some of them perform over 90% range, which is highly remarkable. Some of these array-digraphs are shown in Figure 11.

4.2. Reliability

For every array-digraph from 3 to 7 panels, the associated reliability polynomials are shown in Figure 12.

For every array-digraph D it holds that $P_D\left(0\right)=0$ and $P_D\left(1\right)=1$, this is due to the fact that for $x=0$ ($x=1$) no (every) panel is working properly.

Although for every $p\in [3,7]$ there are two structures, the all-parallel and all-series associations (whose associated polynomial is greater/smaller than any other for every value of $x\in (0,1)$), most of the polynomials cross each other at some point. Developing a ranking consists on fixing a given value of $x=x_0$ and sorting $P_D\left(x_0\right)$ for every array-digraph D.

4.3. Expected Value of Power (EVP)

The EVP extracted from every array-digraph of p panels ($p\in [3,7]$) for every value of $x\in [0,1]$ is shown in Figure 13.

Power is normalized so that the maximum expected value is one. The maximum power extracted from every subdigraph is computed in the same manner as that in Section 3.3, through an NgSpice circuit, using the ODM with the same parameters.

The EVP constitutes a valid variable to select a structure that is at the same time reliable and efficient in terms of production. Again, the array-digraphs consisting in all p panels associated in parallel prove to maximize the expected value. Figure 14 gathers the five best configurations for a single working probability of $x=0.75$, for array-digraphs from 3 to 7 panels.

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 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.