- freely available
Energies 2017, 10(8), 1097; doi:10.3390/en10081097
- We model a smart grid with RE generators and loads (prosumers) as an undirected graph so that each link allows for the bidirectional exchange of electric energy.
- We propose an objective function to be optimized that combines cost elements (related to the number and average length of links and also to the number of nodes with many links) and several properties that are beneficial for the SG (such as energy exchanges at local scale and high robustness and resilience). Our optimization problem includes some restrictions used in  and also others that help our EA find optimal synthetic structures for the SG, starting from scratch. This is a “greenfield” strategy, used by companies in those zones where they do not have infrastructure, deploying thus the new grid starting from scratch. This is another difference when compared to , in which the authors have just adopted a “brownfield” approach aiming at evolving the conventional low voltage power grid into a smart grid.
- We use an EA with a problem representation in which the chromosome , which encodes each potential graph (or individual), is the upper triangular matrix of its “adjacency matrix”, . In this formulation, is a square, symmetric and binary matrix in which any element encodes whether node i is linked to node j () or not () . Since there is no self-connected node, the adjacency matrix has zeros on its main (principal) diagonal (). These are the reasons why the connection information in graph is stored by its upper triangular matrix . Thus, chromosome encodes in a compact form the graph . As will be shown in detail in Section 2, this encoding is different from others found in the literature using EAs on graphs, such as, for instance, a chromosome formed by a one-dimensional array with N elements (the number of graph nodes) , N-length chromosome of two-dimensional elements  (where a node is specified by its location in the graph) or a set of vectors in which each allele (or gene value) represents a community . The mutation and crossover operators are fully adapted to our encoding. This approach could be generalized by considering the strength of the connection between node i and j in terms of its link weight .
2. Related Work
2.1. The Smart Grid as a Complex Network: Related Work
2.2. Evolutionary Computation in Graph Approaches: Related Work
3. Background: Complex Networks Concepts
3.1. Some Useful Definitions in Complex Network
- An “undirected” graph is a graph for which the relationship between pairs of nodes are symmetric, so that each link has no directional character (unlike a “directed graph”). Unless otherwise stated, the term “graph” is assumed to refer to an undirected graph.
- A graph is “connected” if there is a path from any two different nodes of . A disconnected graph can be partitioned into at least two subsets of nodes so that there is no link connecting the two components (“connected subgraphs”) of the graph.
- A “simple graph” is an unweighted, undirected graph containing neither loops nor multiple edges.
- The “order” of a graph is the number of nodes in set , that is the cardinality of set , which we represent as . We label the order of a graph as N, .
- The “size” of a graph is the number of links in the set , , and can be defined (≐) as:
- The “degree” of a node i is the number of links connecting i to any other node and is simply:
- The node degree is characterized by a probability density function giving the probability that a randomly-selected node has k links.
- A “geodesic path” is the shortest path through the network from one nodes to another; or in other words, a geodesic path is the path that has the minimal number of links between two nodes. Note that there may be and often is more than one geodesic path between two nodes .
- The “distance” between two nodes i and j, , is the length of the shortest path (geodesic path) between them, that is the minimum number of links when going from one node to the other.
- The “average path length” of a network is the mean value of distances between any pair of nodes in the network :
- The “clustering coefficient” is a local property capturing the density of triangles in a network. That is, two nodes that are connected to a third node are also directly connected to each other. Thus, a node i in a network has links that connects it to other nodes. The clustering coefficient of node i is defined as the ratio between the number of links that exist between these vertices and the maximum possible number of links (. The clustering coefficient of the whole network is :
- The “betweenness centrality” quantifies how much a node v is found between the paths linking other pairs of nodes, that is,
3.2. Small-World Property and Its Importance in Robustness
- A small-world network is a complex network in which the mean distance or average path length ℓ is small when compared to the total number of nodes N in the network: as . That is, there is a relatively short path between any pair of nodes [71,72]. The term “small-world networks” is often used to refer Watts–Strogatz (WS) networks, first studied in . It can be generated by the “rewiring” method shown in Figure 1a: Link , which was connecting Node 1 to Node 3, is disconnected (from Node 3) and rewired to connect Node 1 to Node 9. In the resulting network, going from Node 1 to Node 9 only requires one jump via the rewired link (and thus, ). However, in the original regular network, going from Node 1 to Node 9 through the geodesic or shortest path () involves four links (). This leads to networks with small average shortest path lengths between nodes ℓ, and high clustering coefficient . Figure 1b shows the aspect and of a WS we have generated with nodes and “rewiring probability” . It has a short mean distance, , and high clustering, . Most of the small-world networks have exponential degree distributions .
- Figure 1b ( and ) also illustrates that the architecture of real small-world networks is extremely heterogeneous: the vast majority of the elements are poorly connected, but simultaneously, few have a large number of connections . The robustness of small-world network has been explored in [75,76] leading to the conclusion that, in a non-sparse WS network (), simultaneously increasing both rewiring probability and average degree improves significantly the robustness of the small-world network.
- An interesting variation of the WS model is the one proposed by Newman and Watts  (NW small-world model) in which one does not break any connection between any two nearest neighbors, but instead, adds with probability p a connection between a pair of nodes. It has been found that for sufficiently small p and sufficiently large N, the NW model is basically equivalent to the WS model . At present, these two models are together commonly termed small-world models.
4. Background: Hybrid Approaches Combining Complex Networks and Electric Engineering Concepts
5. Discussion: Is the CN Approach Useful in Power Grids?
5.1. Power Grids: Is There a Dominant Topology?
5.2. Unweighted and Weighted Graphs: Which Is the Best?
6. Proposal: Metrics, Objective Function and Problem Statement
6.1. Metrics to Construct the Objective Function
- It is necessary for the SG to have a structure with reduces losses in the electric cables used to transport electric power from one node to another. This electrical restriction can be modeled using the condition:
- Since the node degree of a node i is the number of links connecting i to any other node, its maximum value gets an upper limit related to the maximum power that a node can support:The value of is related to the maximum power that a node is able to support and is directly related to its economic cost. In , average degree values ranging from to lead to a good balance between performance and cost.
- The clustering coefficient defined by Equation (4) of a smart grid should be higher than that of the corresponding random network (RN) with the same order (number of nodes) and size (number of links). This aims at assuring a local clustering among nodes because it is more likely that electricity exchanges occur in the neighborhood in a scenario with many small-scale distributed RE generators .
- We measure the network vulnerability by using the concept of multiscale vulnerability of order p of a graph [113,114],
- A coefficient of variation for betweenness ,
6.2. Proposed Objective Function
- Reducing in the effort of decreasing the economic cost and the electric losses in the links used to transport electricity from one node to another. Reducing makes the network less robust. This is because the minimum value of , , is reached for the “fully-connected network” or “completely-connected graph” in which any node is connected with all of the others. As the number of links decreases, the network becomes increasingly fragile and . Reducing to a great extent leads to an inexpensive, but very fragile structure ( [113,114]). Thus, the decrease of the number of links and the increase of the robustness have opposite tendencies. This is why we propose a balance between and via the weight parameter , which controls the linear combination between constituents with opposing trends.
- Reducing (approaching one from above) to increase robustness and also to improve resilience.
- Reducing ℓ along with maximizing leads to a small-world structure.
- Increasing aiming to stimulate the local electricity exchanges in scenarios with many small-scale distributed RE generators.
6.3. Problem Statement
7. Proposed Evolutionary Algorithm
7.1. Basic Concepts
7.1.1. Genotype-Phenotype Relationship
7.1.2. Natural Evolution
7.2. Evolutionary Algorithm Used
7.2.1. Encoding Method
7.2.2. Initial Population
- Fifty percent of are Watts–Strogatz random graphs (with small-world properties, including short average path lengths and high clustering) with rewiring probability ranging from to one.
- Fifty percent of are Erdős–Rényi (ER) random graphs with N nodes and links.
7.2.3. Implementation of Evolutionary Operators
- Select at random () two individuals from the population (father and mother).
- Select at random the same row in the parents.
- Exchange the selected rows between the father and the mother, which leads to two child chromosomes.
8. Experimental Work
8.2. Results: Optimizing the Structure
- The optimum value of in Figure 5a is achieved for . This corresponds to the optimum graph shown in Figure 5h. This graph has (Figure 5), which is an intermediate robustness between the one of (Figure 5f) and that corresponding to (Figure 5j). arises from the tradeoff between having a reasonable robustness and efficient power exchange at the local scale (high and low ℓ) with a limited number of links (, the number of links of ). represents a network with high number of links, very interconnected, and thus, potentially very expensive, and with very high robustness (smallest fragility, ). On the contrary, , which has , is thus very fragile: note in Figure 5j that the occurrence of abnormal conditions on the marked link will completely disconnect the power grid. In this limiting case (), the optimum network has only 51 links (very low economical cost), but at the expense of being very vulnerable to targeted attacks on hubs.
- A key point to note in Figure 5a is that, in the interval , the objective function has a slight variation , in which the corresponding optimum graphs , and exhibit some beneficial properties for the smart grid:
- Intermediate robustness, ranging from 50 to 80 in Figure 5b.
- Clustering coefficient considerably higher than that of the random graph (with the same number of nodes and links), (see Figure 5d). This is related to the local exchange of power between neighbor nodes . These two latter conditions are topological features that help the power grid be a smart grid. Furthermore, the two latter features show that the graphs in have the small-world nature. In particular, these graphs with approach non-sparse small-world networks (with ), which, according to , are the most robust.
- Additionally, in , , in good agreement with the properties described in .
- The average node degree has values , which has been proven to be beneficial in .
- ranges between and , which limits the existence of nodes with many links, and therefore, with high capacity (≈ more expensive).
8.3. The Benefits of Adding Links
8.4. Comparison with an Evolution Strategy
9. Summary and Conclusions
Conflicts of Interest
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|Symbol||Definition or Meaning|
|Adjacency matrix of graph .|
|Element of the adjacency matrix that encodes whether node i is linked to node j () or not ().|
|Mean value of betweenness or multi-scale vulnerability of order 1.|
|Mean value of of the multi-scale vulnerability of order 2.|
|Betweenness centrality of link l.|
|Multi-scale vulnerability of order p of a graph . It is defined by Equation (9)|
|Mean clustering coefficient of a network. It is defined by Equation (4).|
|Set of all chromosomes.|
|Betweenness centrality of node v. It quantifies how much a node v is found between the paths linking other pairs of nodes. It is defined by Equation (5).|
|Chromosome that encodes the graph .|
|Clustering coefficient of node i. It is defined as the ratio between the number of links that exist between these vertices and the maximum possible number of links (.|
|Clustering coefficient of a random graph|
|Node degree matrix: . It is the diagonal matrix formed from the nodes degrees.|
|Euclidean distance between any pair of nodes and in a spatial network.|
|Distance between two nodes i and j. It is the length of the shortest path (geodesic path) between them, that is, the minimum number of links when going from one node to the other.|
|Coefficient of variation for betweenness. It is defined by Equation (10).|
|objective function to be minimized. It is defined by Equation (11).|
|Mean value of the objective function .|
|Graph representing a network.|
|Set of all possible connected graphs with N nodes and links.|
|Set containing all of the candidate graphs.|
|Optimum graph that solves the objective function with combination parameter .|
|Average node degree: .|
|Degree of a node i. It is the number of links connecting i to any other node. It is defined by Equation (2).|
|Maximum node degree.|
|ℓ||Average path length of a network. It is the mean value of distances between any pair of nodes in the network. It is defined by Equation (3).|
|Set of links (edges) of a graph.|
|Laplacian matrix (or Kirchhoff matrix) of graph . It is defined by Equation (14).|
|Average path length of a random graph.|
|Algebraic connectivity of graph .|
|M||Size of a graph . It is the number of links in the set . It is defined by Equation (1).|
|Set of nodes (or vertices) of a graph.|
|N||Order of a graph . It is the number of nodes in set , that is the cardinality of set : .|
|Probability density function giving the probability that a randomly selected node has k links.|
|Normalized weight of the link between nodes i and j:|
|Average entropic degree.|
|Entropic degree of node i defined by Equation (6).|
|Standard deviation of betweenness.|
|Upper triangular matrix of graph .|
|Set of weight elements .|
|Weight of link . It models the strength of the connection between node i and j.|
|Parameter that controls the linear combination between components with opposing trends in the objective function to be minimized given by Equation (11).|
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