Model Based Spatial Data Mining for Power Markets

Jiangzhuo Chen ∗ V.S. Anil Kumar ∗ Achla Marathe ∗ Karla Atkins ∗

1 Introduction

In this paper, we use model-driven data mining tech-niques to address the issue of realistically assigning ser-vice areas to electrical power substations as well as plac-ing substations in a synthetic city. In model based anal-ysis, a system’s state variables and domain knowledgeare used to determine the details about the processesunderlying the system behavior. This research is mo-tivated by the problems encountered during the devel-opment of a simulation framework aimed at analyzingthe deregulated electricity markets [1]. In order to con-struct simulation based models for representing dereg-ulated electrical markets, one needs to assign serviceareas to electrical substations. In the absence of pub-lic availability of data on the service areas, one has toconstruct a model that maps the consumer nodes to thesubstations in a way that is economically and physicallyfeasible. Physical feasibility implies that the grid shouldbe able to deliver electricity from the source nodes to thesink nodes as defined by the market. If the economicallycleared contracts cannot be carried out on the grid, onemay have to reassign the service areas in such a way thatthe grid is able to perform physical clearing. This reas-signment of service areas requires domain knowledge aswell as acquired knowledge from running different mod-els. This research outlines some of the models that havebeen considered to determine the optimal assignment ofservice areas. A high level of service reliability is impor-tant in general but especially during disaster recoveryand restoration. When electrical systems go down, oneneeds to be able to estimate the time varying load inorder to carry out restoration tasks. Simulation frame-works that can dynamically change service areas duringdisaster preparedness and contingency management iscritical to emergency planners. Some of the work shownin this paper is also driven by the need to build a vir-tual city with all the critical infrastructures includingelectrical infrastructure. One of the tasks was to deter-mine the placement of the generators and substationswith respect to the buildings in the city. We use modelbased spatial data mining methods to examine Port-

∗Virginia Bioinformatics Institute, Vir-ginia Tech, Blacksburg, VA 24061. Email:{chenj,akumar,amarathe,katkins}@vbi.vt.edu.

land’s electrical infrastructure and obtain results thatcan be applied towards the building of the syntheticcity.

2 Summary of ResultsWe apply several model based mapping strategies to es-timate service areas of load serving substations. Weuse peak power demand data from FERC and a de-tailed synthetic population from Simfrastructure [5] fordeveloping and comparing these models. Our resultsshow that a simple strategy that probabilistically as-signs buildings to substations, such that the probabilityof assignment to a substation is inversely proportional tothe distance to the substation, outperforms more com-plicated strategies. We observe that real demand dis-tributions of several power grids are well matched bya Weibull distribution. We also analyze the proximityof buildings with respect to the substations and gener-ators in order to populate a virtual city with electricalassets. We find that the electrical distribution systemis located close to the consumers to minimize transmis-sion costs, however, the generation system is located farfrom the city to avoid environmental concerns. Finallywe analyze the geographic spread of the substations tohelp manage the electrical assets and perform strategicplanning.

3 Data and Methodology

First, we briefly describe the data used in this study.This research is based on the synthetic population dataof 1.6 million individuals/consumers residing in the cityof Portland, Oregon [4]. The 1.6 million consumers inPortland are distributed over 243,000 spatial locations.Given that electricity is demanded at a location, wecalculate the demand for electricity for each of the243,000 locations on an hourly basis. The GIS dataon each of the locations for the city of Portland, is usedin the analysis. There are a total of 662 substations inPortland, out of which about 50% are not load serving.For each of the substations in Portland, we extract theactual peak demand for the year 1999 from the FERC(Federal Energy Regulatory Commission) website1. We

1http://www.ferc.gov

also used FERC website to download peak demand forsubstations in Chicago, Houston, WSCC, northeast andERCOT.

The Simfrastructure tool created a regional popu-lation imitation of Portland with demographics statis-tically matching the real population [3, 5, 2]. The dataon the synthetic population, its demographics, locationsand activities used in this study has been made pub-licly available online [8]. Data about land use and de-mographics, combined with the survey data from thou-sands of households is employed to create the syntheticpopulation, where each person has a specific home ad-dress. A sequence of daily activities, and the locationswhere these activities are to be done, is determined foreach person based on the activity surveys, travel timeand land use data.

Finally, for each individual, for every hour, total of24 hours, a power demand profile is created based onits demographics and activity location. After the de-mand for each location has been computed, the simula-tion model determines which electrical substations canoptimally serve the demand at that location. A realisticmapping of buildings or locations to the electrical sub-stations is required [6]. Ideally, if the true mappings ofsubstations and locations are publicly known, the modelcould just use those for the assignment. In the absenceof any knowledge about the exact location-substationmapping, one needs to develop other estimation meth-ods that can lead to a realistic mapping. In this paper,we study different model based methods that can per-form the location-substation mapping. We study howthe geographical locational information can be utilizedto spatially map buildings to substations in such a man-ner that the assignment leads to a load distribution onsubstations that is very close to the actual load distri-bution at a substation level. The assumption is that amapping close to the real mapping will reduce the like-lihood of artificial grid congestion.

Our methodology consists of applying different geo-mapping strategies to the Portland data, then comput-ing the similarity measure between the load assignmentgiven by our methods and the real peak demand pro-vided by FERC, and choosing one of our methods thatgives the best similarity results. The similarity met-ric we use is the cosine similarity, which is defined asfollows.

Definition 3.1. The cosine similarity between two n-dimensional vectors x, y, denoted by sim(x,y), is givenby:

x · y‖x‖2‖y‖2

.

4 Model Driven Spatial Service AreaEstimation

For comparing different aggregation methods, we usethe following naive aggregation as a benchmark. Eachbuilding is randomly assigned to a substation accordingto a probability distribution that is given by the em-pirical distribution of the real substation demand. No-tice that this assignment does not take into account anyspatial information. Next, we make use of the locationinformation of buildings and substations in two ways:assign each building to its closest substation, or assigneach building randomly to a substation with probabilitydepending on the distance. Finally we create a hybridmodel that uses the spatial information as well as thesubstation demand distribution information. It assignseach building to a nearby substation such that the syn-thetically mapped demand distribution is close to thereal peak demand distribution.

4.1 Distribution-based assignment. We firststudy the real substation-level demand distributionsof different power grids. The empirical cumulativedistribution plots in Figure 1 show that all power gridshave similar demand distributions. We use MaximumLikelihood Estimation method to determine that aWeibull distribution with shape = 0.8203967 andscale = 20.8899117 can characterize the substationlevel demand for all the grids. To assign buildings tosubstations, we randomly generate probabilities foreach substation such that the probabilities follow theempirical Weibull distribution just obtained. Eachbuilding is then randomly assigned to a substationwith the prescribed probability. In Figure 2, we plotthe cumulative distribution functions of the (scaled)synthetic data and the real peak data; the dashedred curve is the distribution function of the aggregatedata obtained by the random assignment based onthe Weibull distribution, while the black curve is thedistribution function of the real data. The cosinesimilarity between the synthetic demand data and thereal Portland peak demand is 0.411199.

4.2 Closest assignment. In this assignmentmethod, we make use of the spatial information. Eachbuilding is assigned to the closest substation. We haveconsidered two different distance measures: Euclideanand Manhattan. The cosine similarity between thesynthetic data and the real Portland peak demand is0.357955 for the Euclidean measure and 0.355915 forthe Manhattan measure. These are worse than thesimilarity result of the naive assignment. The mainreason is that the aggregate demand from the closestassignment has a very different distribution compared

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Figure 1: Empirical cumulative distribution functions ofsubstation-level demand of different power grids.

to the real demand. Specifically, for the Portland powergrid, only 209 out of the 319 loaded substations areclosest (in Euclidean measure) to some buildings. Thatis, about one third of the substations are not the closestto any building and thus have zero aggregate demandin the synthetic data. The cumulative distributionfunction of the synthetic aggregate data from theEuclidean closest assignment is plotted in Figure 2(thick red curve).

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Figure 2: Empirical cumulative distribution functions ofsynthetic and real substation-level demand.

4.3 Distance-weighted random assignment. Toovercome zero load on substations that are not theclosest to any buildings, we assign each substation apositive probability and randomly choose a substationfor each building. The probability that building iis assigned to substation j is di,j/

∑j′ di,j′ , where

di,j is the Euclidean distance between building i and

substation j. This assignment method again uses spatialinformation. we find that the cosine similarity betweenthe synthetic data and the real data is 0.516280, whichis better than the similarity performance of the previoustwo assignment methods. The cumulative distributionfunction of the synthetic aggregate data is plotted inFigure 2 (blue curve).

4.4 Close assignment. Finally, we create a hybridmodel that combines the spatial and distribution in-formation. We first assign each building to its near-est substation; then for each building, with probabilityp, we assign it to the nearest unloaded substation. Ifthere are still unloaded substations, we will do the ran-dom re-assignment again. The probability p is chosensuch that the distribution of the synthetic aggregate de-mand is close to the Weibull distribution obtained inSection 4.1. We test for p = 0.1, 0.2, . . . , 0.9 and findthat p = 0.1 gives the best result. Again the cumu-lative distribution function of the synthetic aggregatedata from the Euclidean close assignment is plotted inFigure 2 (green curve). The cosine similarity betweenthe synthetic data and the real Portland peak demandis 0.276636 for the Euclidean measure and 0.267565 forthe Manhattan measure. The performance of this map-ping method is worse than the previous three methods.

5 Analysis of Spatial Proximity betweenBuildings, Substations and Generators

In this second set of experiments, we analyze thespatial proximity between buildings, substations andgenerators in Portland. The aim is to determine therelative placement of electrical assets in a real cityand then use that information to develop models thatcan place substations and generators in a syntheticcity. Electrical facilities must be constructed suchthat the spatial location is efficient from an electricalstandpoint and it is economical to the end users. Thetransmission grids span across regions and countries toderive the benefits of load balancing. The loads areoften correlated locally which implies that on a hotday everyone will turn on air conditioner and on a coldday, everyone will turn on the heat. It is often cheaperto import power from long distance especially on dayswhen deviations from normal are high. A mix of longand short distance interconnections between generatorsand consumers also makes the transmission grid morerobust. However, the long distance transmission ofelectricity is expensive. The longer the transmissionline, the higher are the transmission losses. The lossesare attributable to the heating of the power lines thatare carrying the electrical current to the consumernodes. As a result, there is economic pressure to locate

the power plants near the consumers that they serve.Most electricity is generated by burning fossil fuels

which produces waste and chemical substances andpollutes the atmosphere. Some of these pollutantscan seriously affect the health of the population whichmakes it desirable to have the generation plants set upfar from the densely populated areas in the cities. Theseobservations suggest that the electrical distributionsystem should be located close to the cities whereas thegeneration system should be built far from the cities.We test these hypotheses for the city of Portland.

5.1 Proximity Analysis We use two different mea-sures i.e. Euclidean distance and cosine similarity,to compare the geographical distributions of buildings,substations and generators in terms of their cardinal-ity and capacity. There are three kinds of substations:generators, transformers and distributors. The trans-formers are the non-load serving substations whereasthe distributors are the load serving substations. Inthis study, we focus only on the generators and the loadserving substations. Usually about 60% of substationsin a city are load serving. We divide the city of Portlandinto a k by k grid. For each grid cell, we compute thecardinal density of buildings, and the capacity densityof substations and generators. This produces three k2-dimension vectors, which we call density vectors. Wescale the densities such that they are fractions of thetotals. We then compare the distributions between (i)buildings and substations, (ii) buildings and generatorsand (iii) substations and generators, by computing theEuclidean distance between the two k2-dimension nor-malized density vector.

The lower the Euclidean distance, the higher is thesimilarity between the matrices. Figure 3 shows theresults. We find that the building distribution andthe substation distribution are the most similar. Notethat these substations are load serving. In order tomaximize the economic efficiency of the distributionsystem, one would expect the substations to be locatedclose to the buildings whose loads they are serving. Theresults also show that the building distribution and thegenerator distribution is the least similar. This againconfirms our hypothesis that to avoid pollution andother environmental concerns, most of the generationcapacity is built far away from the cities. These resultswill be useful in constructing a virtual generic city withelectrical infrastructure.

In the following experiment, we compare the nor-malized density vectors of the buildings, substations,and generators using the cosine similarity measure. Theresults are shown in Figure 4. The higher the cosinemeasure, the higher is the similarity between the distri-

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Figure 3: Euclidean distance between normalized densityvector.

butions. Again, the results are the same as shown byEuclidean distance measure, i.e. the building distribu-tion is most similar to the substation distribution andthe least similar to the generator distribution.

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Figure 4: Cosine Similarity.

To provide the reader with a better understandingof the spatial density and locations of the buildings, sub-stations and generators, we divide the grid into squarecells. We define a cell as a dense cell if there are at least500 buildings in it. In Figure 5 we plot the city of Port-land and highlight the dense cells. There are 73 cellsthat qualify as dense. In Figure 6, we plot the substa-tions and the dense cells. We compute the quantity andcapacity of substations in and near dense cells. We findthat out of 319 load serving substations and 9315.5MWof total capacity, 30 substations (9.404%) with 773.7MW (8.305%) capacity are located in the dense cells.We also find that 44 substations (13.79%) with 1367MW (14.67%) capacity are located in cells next to the

dense cells. These findings also suggest that the dis-tribution of the substation capacity may not be verydifferent for the subset of substations located in denselypopulated areas and the subset located in sparsely pop-ulated areas given that the percentage of substationsand the percentage of capacity is almost the same. Fig-ure 7 shows all the generators and the dense cells. Wecompute the quantity and capacity of generators in andnear dense cells. The results show that out of 41 gen-erators and 3296.4 MW of total capacity, none lies inthe dense cells. Only one generator (2.43%) with 260MW (7.88%) capacity is in cells next to the dense cells.The most dense cell with a substation in it has 1068buildings, while the most dense cell with a generator init has only 388 buildings in it.

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Figure 5: Buildings and Dense Cells.

In summary, the results of the similarity measuresshow that the capacity distribution of the substationsand generators is similar to the cardinality distributionof the buildings. Also, all graphical and quantitativeanalysis shows that the building distribution is mostsimilar to the substation distribution and least similarto the generator distribution.

6 Asset Management and Security

Each substation represents a major security asset be-cause of the key power system components within itsphysical structure. A loss of substation power deliverycomponent can result in blackout affecting thousands ofpeople. Substations are responsible for stepping downand routing the voltage delivered from generators to theconsumers. In this part of the analysis, we examine thevulnerability of the substations.

We cover all substations with squares of a given

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Figure 6: Substations and Dense Cells.

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Figure 7: Generators and Dense Cells.

size. Given that Covering-By-Squares is NP-hard [7],we apply a greedy algorithm, in which we find the left-most uncovered substation and locate the left-side of asquare along it; then we find the bottom-most uncov-ered substation and align the bottom-side of the squareto it. The lesser the number of the squares it takes tocover, the higher is the concentration of the electricalassets. This may or may not be desirable because ahigher concentration of physical assets would make itmore prone to a successful attack, however it may alsoimply easier protection. For example, one would needa fewer number of field RTUs (Remote Terminal Units)and SCADAs (Supervisory Control and Data Acquisi-tion) spread over the geographical area to monitor and

control the electrical equipment. We compare the vul-nerability/robustness of the real substation distributionwith that of a random distribution and a regular (grid)distribution using the above algorithm.

Figure 8 shows the results. It plots the numberof squares needed to cover all the substations for agiven square size. Three different curves are plotted.The bottom most curve represents the actual spatialdistribution of the substations. The middle green curveshows the number of squares needed if the substationsare uniformly randomly distributed over the geographicarea. The top blue step line shows the number ofsquares needed if the substations were located in aregular grid like manner over the area. The results showthat for almost any given square size, the number ofsquares needed are the least for the actual distributionof the substations as compared to the uniform randomor regular distribution.

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Figure 8: Covering by Squares.

7 Conclusions

We use model based data mining methods to map build-ings to substations and compare the similarity betweendistributions of the synthetically mapped aggregate de-mand and the real aggregate demand. We find thatmore complicated methods do not necessarily performbetter. Moreover, different ways to make use of thesame information may perform significantly differently.For the same distance information, if we use it to assignprobabilities in a random mapping, we obtain much bet-ter synthetic aggregation than if we use it to do closestassignment. We also analyze the proximity of buildingswith respect to the substations and generators in or-der to populate a virtual city with electrical assets. Wefind that the electrical distribution system is locatedclose to the consumers and the generation system is lo-cated farther from the city. Finally we examine the

geographic spread of the substations for collecting dataand information required for electrical asset manage-ment and strategic planning.

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