TTemporal Motifs in Smart Grid
Rucha Bhalchandra Joshi [email protected]
Annada Prasad Behera [email protected]
Subhankar Mishra [email protected]
Machine Learning And Building (MLAB),School of Computer SciencesNational Institute of Science Education and Research, Bhubaneswar -752050, IndiaHomi Bhabha National Institute, Anushaktinagar, Mumbai - 400094,India
Abstract
A complex network can be characterized by patterns. Such frequently occur-ring significant patterns are called motifs and in a time dependent network, theyare called temporal motifs. One of the temporal networks where temporal motifsare observed and play a major role; is the Smart Grid. The energy consumptionpattern across the appliances, houses, communities and entire cities help energyutility companies and consumers plan their electricity generation and consump-tion. The temporal motifs for the smart grid constitutes of the consumers and pro-ducers and the edge or connection represents energy flow between two participantsof the network, these connections last till the power is being consumed/generated.This paper formally defines the temporal motifs for smart grid network and pro-poses a way to create such temporal motifs in the network. We also discuss howthe temporal motifs fit into the hierarchical structure of power distribution systemof Smart Grid.
Keywords—
Smart Grid, Temporal Motifs, Complex Systems, Cyber-physicalSystems
Many complex systems can be abstracted with the help of networks. The entitiesparticipating in the systems are modeled as nodes and the relations by which they a r X i v : . [ c s . S I] F e b re linked to each other are modeled as the edges of a network graph. Abstrac-tions help us study the complex system such as food chain, citation network. Timedependent systems can be abstracted as temporal networks. Some notable exam-ples of temporal network are Facebook, Email as well as recent networks such asBitcoin. Structure of the temporal network changes with time. Since the edges intemporal graphs depend on time, their presence is determined only at a given time.To understand the behaviour of the temporal network, it is essential to consider thetime of occurrence of temporal edges.One such system is smart grid network. A smart grid has various entities, suchas producers, consumers, transmitters of power, participating in the network. Thehierarchical structure of the smart grid has been discussed by (Aggarwal et al.,2010; Mishra et al., 2016) where the power distribution in the grid is according tothe voltage. (Rech and Harth, 2012) discussed the existence of the consumptionsector of the hierarchy wherein the lowermost layer consists of the last links ofdistribution grid connecting consumers to the main grid. A transformer suppliesthe power to the consumers that are connected to it. So, the second layer fromthe bottom consists of transformers. The third layer is of substations that supplypower to the transformers in second layer. This hierarchy goes up along with allthe participating entities in the grid network. Pattern of distribution of power con-sumption over the period of time form motifs. These motifs depicts the behaviourof the participating entities.In a smart grid, the meters can get the data pertaining to each of the appliancesused in the house that consumes amount of electricity. Smart meter keeps trackof the consumption of each of the rooms or any other infrastructure that consumeselectricity. Hence, at household level, the electricity consumption and distributionis known with the help of smart meters. The inferences about the usage of electric-ity at the lowermost layer in a smart grid can be drawn by taking this consumptiondata into consideration.Our contributions are as follows:• We formally define temporal motifs that occur in smart grid network.• We show a method for constructing such temporal motifs in smart grid net-work.• We discuss temporal motifs in smart grid with overlapping window and fit-ting the temporal motifs in the hierarchical structure of the smart grid.The remainder of the paper is organized as follows: Section 2 discusses therelated work. We discuss the background work in Section 3. Section 4 gives thedetailed explanation of our proposed model. We present a case study and discus-sion of temporal motifs in smart grid network are in Section 3 and 4 respectively.We present the conclusions in Section 6. Motifs (Milo et al., 2002) are the basic building blocks of a complex network.They are recurring, significant patterns of interconnections. The network motifsare defined to study the structural design principles of complex networks in variousfields such as biochemistry, neurobiology, ecology, engineering and so on.Motifs, defined as the frequently occurring and significant patterns in time, canbe used to characterize the time series data (Lin et al., 2003). Motifs in tempo- al networks have been defined in order to understand their role in the temporalnetworks such as the network of emails, phone calls, social media etc. (Paranjapeet al., 2017). However, they only consider the occurrence of the edges one at atime without any duration attached to the existence of the edges.Motif based pattern detection technique was proposed to discover regular be-haviour of smart meter users (Funde et al., 2018). The model proposed by (Fundeet al., 2018) considers one appliance at a time and detects the motifs formed by it.They develop temporal association rule mining to find the relation between usageof energy by various appliances in a particular time period. However, consideringonly one appliance at a time does not tell a complete story about the consumptionpattern of the members of the house. 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 ∈ V and a timestamp t isassociated with the edge. Temporal motif is a collection of edges in a particularsequence that form a particular structure in a given time window δ . Since thetimestamps are attached with each of the edges in temporal network, the motif isthe structure occurring within time δ from the occurrence time of first edge. Thistime window of size δ is slid over the time as we consider the next motif. Whenwe consider the temporal edges in smart grid, the edges occur at different timesbut they last for a duration of time. Smart grid is a specific application of temporalnetwork where the edges have a time of occurrence and it remains in the networktill the appliance that caused it to occur was switched off or no longer draws anyenergy from the meter. This differentiates our work from the previous work asedges stay alive for a certain duration. We say that an edge occurs at a giventime when the appliance corresponding to the node connecting the edge is turnedon. 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 theconsumption. The energy consumption by same appliance may vary at differenttimes. Figure 1: Topology of Power Grid
The power distribution grid is arranged according to the voltage (Aggarwalet al., 2010). The various levels of the hierarchy are connected using the voltage etworks, here power plants are connected via high voltage networks and the levelof household appliances is connected using a voltage network. The smart grid isdefined to consist of nodes N and interconnected edges E ; where nodes representthe actors and are connected to b other nodes ( b is branching factor). The levelsof this hierarchy is denoted by L (Rech and Harth, 2012). We propose a model (Figure 2) for creation of temporal star motifs with associ-ated symbol corresponding to the energy consumption. We also show how thesemotifs help to draw inferences from the hierarchy of the participants of a smartgrid network. In our proposed model the edges of the motifs have been taggedwith symbols associated with corresponding energy consumption levels and timewindow. These timestamps are indicatives of window frame numbers. They implythe order of occurrences of the motif structures.
Figure 2: Overview of motif creation process for smart grids
Consider m i to be meter reading (total) at a given time i . We consider a timeseries T = m , m ,..., m t . Each m i consists of the values corresponding to inter-nal distribution of energy among all the appliances utilizing the energy at a giventime i . Let A be the set of appliances. Let c ji be the energy consumed by the j th appliance at time i . Therefore, m i = ∑ j ∈ A c ji .1 Motif Creation A star motif is defined as a graph of k nodes in total, out which one node (the centernode) has k − Figure 3: Example of a star motif. There are 5 nodes in total. One center node and fourneighbors to the center node. The direction of the edges between the center node andthe neighboring nodes depends on the relationship between them.Figure 4: Example of a static star motif in a house
We consider a particular consumer where there are various appliances in ahouse that contribute to overall consumption of energy in the house. All of whichare essentially connected to the main smart meter. We create a star motif of theseappliances along with the smart meter. Fig 4 shows a star motif within a housewhere we consider the smart meter in the house to be the center node and all theappliances 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 usingan edge from meter (center node) to the appliance (corresponding neighboringnode). For any other equipment that generates energy (e.g. solar panel), the edgegoes from the equipment to the center node. Data Preprocessing and Min-max Normalization . Time series data pre-processing is done to normalize the data. We consider smart meter datawhich keeps track of consumption of each of the appliance. We performmin-max normalization so that the values after normalization lies between [ , ] . The normalized value y , of a data point x , is given by y = ( x − min )( max − min ) u Supplier of power (any appli-ance that produces or suppliespower) v Consumer of power (any appli-ance that consumes power) t w Timestamp of the time windowcorresponding to the edge x Symbol corresponding to thelevel of energy consumption as-signed to the edge n the givenwindowTable 1: Description of variables associated with a temporal edge in smart grid where min is the minimum and max is the maximum data value in the dataset.2.
Piecewise Aggregate Approximation . On the normalized data, we applyPiecewise Aggregate Approximation (PAA) to discretize the data. By se-lecting the right parameters in PAA, it can be altered to suit the needs ofthe application at hand. The normalized time series data is divided into w windows. The average of values in every window is calculated.3. Matching Symbols to Energy Levels . After PAA, we represent the energyconsumption of each of the appliance with a symbol. Number of energylevels and their corresponding symbols is another parameter that can be setaccording to use case. The symbol values represent the levels of energyconsumption 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 energyconsumption corresponding to each symbol. The number of energy levelsvary application to application.
The edges in static motif that we created in first step, for a house is assigned times-tamps associated with the time window in which we are determining the associatedsymbol. In the static motifs, edges are between the main line and an appliance. Thetemporal edge occurs in smart grid in the window in which an appliance is turnedon. 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 theoperations described in the previous steps on the data, to get the symbols associatedfor each of the appliance in the same set of time windows. Then, for a particularwindow, consider all the appliances along with meter as nodes, while the energyproduction and consumption among appliances determine the directions betweenthe edges and the level of energy consumption is given by the symbol associatedwith the edge. The time window is in which we are determining the consumptionlevel is given by the timestamp corresponding to the time window. The complete data of consumption in a grid can be represented as the collectionof 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 thesymbols assigned to the edges represent the level of consumption in a particulartime period. The motif helps us to look at the consumption and distribution ofenergy among all appliances in a time slot in a house. The collection of suchmotifs over a time windows of size δ is defined to be a temporal motif for energyconsumption data. We take an example to demonstrate the steps to create temporal motifs in smartgrid network. We take a part of Pecan Street Dataport (Dataport, ) from 15 minutedataset. We consider a house with data-id 27, which is located at New York. Forthis house, we consider the consumption values for the time period 04:00:00 to07.00.00 on 2019-05-01.
The underlying star motif for this house ID is shown in Fig.6. This is derived basedon the appliances used in the house. Data Preprocesssing and Min-max Normalization . Preprocessing andmin-max normalization of data is done according to formulas mentioned inSection 1 to get the normalized data.2.
Piecewise Aggregate Approximation . Since the data has a duration of 3hours, with windows size of 1 hour, the number of windows w =
3. For anygiven time window there is a value attached to it which is average of thevalues corresponding to the timestamps in the window. Matching Symbols to Energy Levels . We consider 4 symbols a , b , c and d , they correspond to four consumption levels. Very low consumption isrepresented by a symbol a , b represents average energy consumption, c rep-resents more than average consumption of energy while very high energyconsumption is represented by symbol d .We define the range for each of the symbols as shown in Table.2. Symbol Range a ≤ value < . b . ≤ value < . c . ≤ value < . d . ≤ value ≤ Each edge in the static motif created in Section 5.1 has a symbol that corresponds tothe average of the values of consumption on each edge in a window. The symbolsare assigned to the edges. An example of a motif in House ID 27 for time windowlabelled t for duration on 1 hour is shown in Fig.7a. . A sequence of motifs created in Section 5.3 may have different symbols asso-ciated with their edges in different time windows, since the energy consumptionvaries 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 con-sumption level is indicated by symbols inside upward pointing triangles, whereasthe decrease in consumption is indicated by symbols inside triangles facing down-ward in the temporal motifs. a) Motif at time t (b) Motif at time t (c) Motif at time t Figure 7: Temporal motifs in smart grid network9
DISCUSSION
Overlapping Temporal Motifs
While we only consider the window for finding the symbols associated with edgesto be non-overlapping, other possibilities to be considered are overlapping windowand the absolute consumption values. Considering absolute values is too specificbecause the consumption of energy on one day may not be exactly the same as thaton the next day. Hence assigning symbols based on absolute values at a given timemay not help infer anything useful about the data. In a time period, if the averagedata consumption is considered then it gives the approximate consumption level ofthe appliance.Overlapping window can be considered if it is needed for the application underconsideration. We slide the window over the time duration with some predeter-mined time overlapping in two consecutive windows. This would help in main-taining the information related to continuity of the data to some extent dependingon how much the overlap is.
Fitting Temporal Motifs into Smart Grid hierarchy
We propose this model for residential type of locality. This is easily scalable toother types of localities as well, such as industrial, commercial etc and can beextended to fit other hierarchical levels in the smart grid as well as other complexnetworks. To study the role that these motifs play in smart grid network, it isessential that we consider the hierarchy of the participants of the network. Thehierarchy described in (Rech and Harth, 2012) is discussed below to suit for therequirement of our model.• The very basic level in the hierarchy consists of the appliances in a house-hold. As discussed earlier the motifs which are formed at this layer are basedon electricity consumption of each of the appliances at various times.• The layer above the layer of appliances is of houses in a locality or a commu-nity residing at a particular location. The motifs would consist of the housesin the locality and the point of supply of electricity to all these houses asnodes. The consumption of each of the houses at various times would deter-mine the edges and the direction of the edges. These motifs can be definedin similar fashion as we did earlier by determining the suitable parametervalues.• Similarly, the next layer consists of communities which together form a city.• More layers can be thought of and considered on top of the previously men-tioned layers so as to build the model of motifs that will help enable us tostudy and determine various aspects of a smart grid network.
Temporal motifs play an important role in characterizing networks. The changein usage of power over time helps to study the behaviour of the consumers. Wehave formally defined the temporal motifs in smart grid network. We have alsodescribed the construction of such temporal motifs in smart grid network, without verlapping windows, how to fit them in the hierarchical structure of the network.In future we will use motifs to draw inferences about participants of the grid net-work. We will also look at impact of this model on the energy distribution policies. ACKNOWLEDGEMENTS
This research was partially supported by NRDMS/UG/S.Mishra/Odisha/E-01/2018
References
Aggarwal, A., Kunta, S., and Verma, P. K. (2010). A proposed communicationsinfrastructure for the smart grid. In , pages 1–5, Gaithersburg, MD, USA. IEEE.Dataport. https://dataport.pecanstreet.org/. Pecan Street Inc. Dataport, accessedon 24 January 2020.Funde, N., Dhabu, M., and Balande, U. (2018). Motif-Based Pattern DetectionMethod for Smart Energy Meter Data. In , pages 1–5, Pune. IEEE.Lin, J., Keogh, E., Lonardi, S., and Chiu, B. (2003). A symbolic representationof time series, with implications for streaming algorithms. In
Proceedingsof the 8th ACM SIGMOD Workshop on Research Issues in Data Mining andKnowledge Discovery , DMKD ’03, page 2–11, New York, NY, USA. Asso-ciation for Computing Machinery.Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., and Alon, U.(2002). Network Motifs: Simple Building Blocks of Complex Networks.
Science , 298(5594):824–827.Mishra, S., Li, X., Pan, T., Kuhnle, A., Thai, M. T., and Seo, J. (2016). Pricemodification attack and protection scheme in smart grid.
IEEE Transactionson Smart Grid , 8(4):1864–1875.Paranjape, A., Benson, A. R., and Leskovec, J. (2017). Motifs in temporal net-works.
Proceedings of the Tenth ACM International Conference on WebSearch and Data Mining - WSDM ’17 .Rech, D. and Harth, A. (2012). Towards a decentralised hierarchical architecturefor smart grids. In
Proceedings of the 2012 Joint EDBT/ICDT Workshops on- EDBT-ICDT ’12 , page 111, Berlin, Germany. ACM Press., page 111, Berlin, Germany. ACM Press.