Temporal Motifs in Smart Grid

In a temporal graph $G=(V,E)$ where $V$ is set of vertices and $E$ is set of edges, the temporal edges are represented as $(u,v,t)$ where $u,v\in V$ and a timestamp $t$ is associated with the edge. Temporal motif is a collection of edges in a particular sequence that form a particular structure in a given time window $\delta$. Since the timestamps are attached with each of the edges in temporal network, the motif is the structure occurring within time $\delta$ from the occurrence time of first edge. This time window of size $\delta$ is slid over the time as we consider the next motif. When we consider the temporal edges in smart grid, the edges occur at different times but they last for a duration of time. Smart grid is a specific application of temporal network where the edges have a time of occurrence and it remains in the network till the appliance that caused it to occur was switched off or no longer draws any energy from the meter. This differentiates our work from the previous work as edges stay alive for a certain duration. We say that an edge occurs at a given time when the appliance corresponding to the node connecting the edge is turned on. The smart meter captures the energy at particular time interval in a sequence, hence we have to assign a window time to it rather than the actual start time of the consumption. The energy consumption by same appliance may vary at different times.

The power distribution grid is arranged according to the voltage [Aggarwal et al., 2010]. The various levels of the hierarchy are connected using the voltage networks, here power plants are connected via high voltage networks and the level of household appliances is connected using a voltage network. The smart grid is defined to consist of nodes $N$ and interconnected edges $E$; where nodes represent the actors and are connected to $\#b$ other nodes ($b$ is branching factor). The levels of this hierarchy is denoted by $L$ [Rech and Harth, 2012].

A star motif is defined as a graph of $k$ nodes in total, out which one node (the center node) has $k-1$ neighbors, which is the center node while all the other nodes have only one neighbor each. An example of star motif is shown in Fig. 3.

We consider a particular consumer where there are various appliances in a house that contribute to overall consumption of energy in the house. All of which are essentially connected to the main smart meter. We create a star motif of these appliances along with the smart meter. Fig 4 shows a star motif within a house where we consider the smart meter in the house to be the center node and all the appliances to be the rest of the nodes which are only connected to the center node. The nodes for the appliances that draw energy from the meter are represented using an edge from meter (center node) to the appliance (corresponding neighboring node). For any other equipment that generates energy (e.g. solar panel), the edge goes from the equipment to the center node.

1. 1.

   Data Preprocessing and Min-max Normalization. Time series data preprocessing is done to normalize the data. We consider smart meter data which keeps track of consumption of each of the appliance. We perform min-max normalization so that the values after normalization lies between $[0,1]$. The normalized value $y$, of a data point $x$, is given by

   $$y=\frac{(x-min)}{(max-min)}$$

   where $min$ is the minimum and $max$ is the maximum data value in the dataset.

2. 2.

   Piecewise Aggregate Approximation. On the normalized data, we apply Piecewise Aggregate Approximation (PAA) to discretize the data. By selecting the right parameters in PAA, it can be altered to suit the needs of the application at hand. The normalized time series data is divided into $w$ windows. The average of values in every window is calculated.

3. 3.

   Matching Symbols to Energy Levels. After PAA, we represent the energy consumption of each of the appliance with a symbol. Number of energy levels and their corresponding symbols is another parameter that can be set according to use case. The symbol values represent the levels of energy consumption over normalized data. The data values range between 0 and 1, so we decide on number of symbol to be used and range of each of energy consumption corresponding to each symbol. The number of energy levels vary application to application.

We define the temporal edge for grid network as quadruple represented as $(u,v,t_{w},x)$. Table 1 describes each component of the quadruple. We perform the operations described in the previous steps on the data, to get the symbols associated for each of the appliance in the same set of time windows. Then, for a particular window, consider all the appliances along with meter as nodes, while the energy production and consumption among appliances determine the directions between the edges and the level of energy consumption is given by the symbol associated with the edge. The time window is in which we are determining the consumption level is given by the timestamp corresponding to the time window.

The complete data of consumption in a grid can be represented as the collection of the temporal edges we defined earlier.

As shown in Fig. 5, along with assignments of symbols to the edges in static motif, note that this motif structure occurs in a particular time duration, since each of the symbols assigned to the edges represent the level of consumption in a particular time period. The motif helps us to look at the consumption and distribution of energy among all appliances in a time slot in a house. The collection of such motifs over a time windows of size $\delta$ is defined to be a temporal motif for energy consumption data.

The underlying star motif for this house ID is shown in Fig.6. This is derived based on the appliances used in the house.

1. 1.

   Data Preprocesssing and Min-max Normalization. Preprocessing and min-max normalization of data is done according to formulas mentioned in Section 1 to get the normalized data.

2. 2.

   Piecewise Aggregate Approximation. Since the data has a duration of 3 hours, with windows size of 1 hour, the number of windows $w=3$. For any given time window there is a value attached to it which is average of the values corresponding to the timestamps in the window.

3. 3.

   Matching Symbols to Energy Levels. We consider 4 symbols $a,b,c$ and $d$, they correspond to four consumption levels. Very low consumption is represented by a symbol $a$, $b$ represents average energy consumption, $c$ represents more than average consumption of energy while very high energy consumption is represented by symbol $d$.

   We define the range for each of the symbols as shown in Table.2.

   Symbol	Range
   $a$	$0\leq value<0.25$
   $b$	$0.25\leq value<0.5$
   $c$	$0.5\leq value<0.75$
   $d$	$0.75\leq value\leq 1$

Each edge in the static motif created in Section 5.1 has a symbol that corresponds to the average of the values of consumption on each edge in a window. The symbols are assigned to the edges. An example of a motif in House ID 27 for time window labelled $t_{1}$ for duration on 1 hour is shown in Fig.7(a).

. A sequence of motifs created in Section 5.3 may have different symbols associated with their edges in different time windows, since the energy consumption varies over time. Such a sequence is the temporal motif in a smart grid network. The final temporal motif with 3 time windows is shown in Fig.7. Increase in consumption level is indicated by symbols inside upward pointing triangles, whereas the decrease in consumption is indicated by symbols inside triangles facing downward in the temporal motifs.