Systems and methods for fault classification in photovoltaic arrays using graph signal processing
Various embodiments of a system and associated method for detecting and classifying faults in a photovoltaic array using graph-based signal processing.
1. A system for determining a classification of one or more nodes of a physical array, comprising:
a physical array including a plurality of nodes, wherein each node of the plurality of nodes includes a plurality of measurable features and wherein a plurality of labeled nodes are a subset of the plurality of nodes and are labeled according to a respective classification of each labeled node; and
a processor in communication with a memory module and being operable to execute stored instructions, the stored instructions, when executed, cause the processor to:
obtain a feature matrix including the plurality of measurable features associated with each respective node of the plurality of nodes;
generate a graph shift matrix based on the feature matrix, the graph shift matrix being representative of similarity between each node of the plurality of nodes;
generate a node target class matrix denoting labels for each node of the plurality of nodes, the node target class matrix including a plurality of rows associated with a respective plurality of classifications and wherein each node is represented within each row of the plurality of rows of the node target class matrix;
determine a set of optimized graph filter coefficients for a graph filter representative of a mapping between the plurality of features of each node of the plurality of nodes and a respective classification for each node of the plurality of nodes based on the graph shift matrix and the node target class matrix; and
determine a respective classification for each unlabeled node of the plurality of nodes using the graph filter with the set of optimized graph filter coefficients.
2. The system of claim 1 , wherein the physical array is a photovoltaic array and wherein the plurality of nodes are each representative of a respective photovoltaic panel of the photovoltaic array.
3. The system of claim 1 , wherein the plurality of measurable features include one or more:
an open circuit voltage;
an short circuit current;
an maximum voltage;
an maximum current; and
an irradiance.
4. The system of claim 1 , wherein a classification corresponds to a respective fault type of an associated photovoltaic panel.
5. The system of claim 1 , wherein the graph shift matrix is generated by determining a similarity between each node of the plurality of nodes based on the Euclidean distance ρ(·) between each node of the plurality of nodes, given by
A i,j =ρ( x i ,x j )
where x i and x j are respective i th and j th rows of the feature matrix.
6. The system of claim 1 , further comprising stored instructions, the stored instructions, when executed, further cause the processor to:
obtain a plurality of graph signals that each include a binary value associated with each node of the plurality of nodes, wherein each graph signal of the plurality of graph signals corresponds to a respective classification of the plurality of classifications.
7. The system of claim 6 , wherein the node target class matrix is generated from the plurality of graph signals.
8. The system of claim 6 , wherein a value of “1” in a row of the node target class matrix associated with a particular classification is indicative that the associated node has a label corresponding to the classification.
9. The system of claim 6 , wherein a value of “0” in a row of the node target class matrix associated with a particular classification is indicative that the associated node does not have a label corresponding to the classification.
10. The system of claim 1 , wherein the node target class matrix includes both labeled nodes and unlabeled nodes.
11. The system of claim 1 , wherein the set of optimized graph filter coefficients are determined by solving a convex objective function with respect to the graph shift matrix and the node target class matrix such that:
=argmin h ∥RΣ l=0 L h l A l S−S∥ F ,
subject to h∈Θ h , Σh l =1
wherein ∥·∥ F represents Frobenius norm and R is an N×N diagonal matrix, wherein R i,i =1 if i th sample is labeled, otherwise R i,i =0, wherein A is the graph shift matrix, and wherein S is the node target class matrix.
12. The system of claim 11 , wherein the convex objective function is solved using an interior-point solver.
13. The system of claim 1 , wherein respective classification for each unlabeled node of the plurality of nodes is determined by applying the graph filter to each graph signal of the plurality of graph signals to generate a set of fully labeled graph signals.
14. A method for determining a classification of one or more nodes of a physical array, comprising:
obtaining a feature matrix including a plurality of measurable features associated with each respective node of a plurality of nodes of an array, wherein a plurality of labeled nodes are a subset of the plurality of nodes and are labeled according to a respective classification of each labeled node;
generating a graph shift matrix based on the feature matrix, the graph shift matrix representative of similarity between each node of the plurality of nodes;
generating a node target class matrix denoting labels for each node of the plurality of nodes, the node target class matrix including a plurality of rows associated with a respective plurality of classifications and wherein each node is represented within each row of the plurality of rows of the node target class matrix;
determining a set of optimized graph filter coefficients for a graph filter representative of a mapping between the plurality of features of each node of the plurality of nodes and a respective classification for each node of the plurality of nodes based on the graph shift matrix and the node target class matrix; and
determining a respective classification for each unlabeled node of the plurality of nodes using the graph filter with the set of optimized graph filter coefficients.
15. The method of claim 14 , further comprising:
obtaining a plurality of graph signals that each include a binary value associated with each node of the plurality of nodes, wherein each graph signal of the plurality of graph signals corresponds to a respective classification of the plurality of classifications.
16. The method of claim 15 , further comprising:
generating the node target class matrix from the plurality of graph signals.
17. The system of claim 15 , wherein a value of “1” in a row of the node target class matrix associated with a particular classification is indicative that the associated node has a label corresponding to the classification and wherein a value of “0” in a row of the node target class matrix associated with a particular classification is indicative that the associated node does not have a label corresponding to the classification.
18. The method of claim 14 , further comprising:
solving a convex objective function with respect to the graph shift matrix and the node target class matrix such that:
=argmin h ∥RΣ l=0 L h l A l S−S∥ F ,
subject to h∈Θ h , Σh l =1
wherein ∥·∥ F represents Frobenius norm and R is an N×N diagonal matrix, wherein R i,i =1 if i th sample is labeled, otherwise R i,i =0, wherein A is the graph shift matrix, and wherein S is the node target class matrix.
19. The method of claim 14 , further comprising:
applying the graph filter to each graph signal of the plurality of graph signals to generate a set of fully labeled graph signals.
20. The method of claim 14 , further comprising:
determining a similarity between each node of the plurality of nodes based on the Euclidean distance ρ(·) between each node of the plurality of nodes, given by
A i,j =ρ( x i ,x j )
where x i and x j are respective i th and j th rows of the feature matrix.