Fuzzv classification and statistical methods for load ... - IEEE Xplore

FUZZY CLASSIFICATION AND STATISTICAL METHODS FOR LOAD PROFILING: A COMPARISON

C.S Ozveren

K.L. Lo, Z Zakaria

School of Electrical Engineering, University of Abertay, Dundee United Kingdom

Dept. of Electrical and Electronic Engineering. University of Strathclyde, United Kingdom

load profiles. For a long time, power utilities have been using load profiles as the means to facilitate the formulation of retail tariffs. In fact, the idea of using load profiles to hill an individual consumer is well established.

Abstract Load profiling is an estimated load shape for a group of customers, which can be developed from historical or current day data. Many techniques for load profiling have been reported in the past. The techniaues ranee from conventional method to more sophisticated ones based on Fuzzy Set theory, Fuzzy Mathematics and Artificial Neural Network.

The process ofestimation of load profiles is known as load profiling. Ideally, the most efficient method of determining electricity consumption would be the direct monitoring. This can be achieved by installing time interval meters, quarter-hourly, half-hourly or hourly at each point of consumption. However, this approach is cost-prohibitive due to the equipment and processing costs. Furthermore, a significant amount of time would be needed to develop such a system. Therefore, load profiling is seen as the alternative solution that would provide a satisfactory, costeffective approach.

This paper compares statistical method that is used in load orofiline with fuzzv, loeic - method. Multiole discriminant analysis is chosen as the statistical approach and will be compared with fuzzy classification approach. These two approaches have the same objectives i.e. to recognise similarities, clusters (identify different categories) and classify the individual load profiles of different customers to one of the identified categories. The paper starts with a description of the background theories concerning both methods. The load data used for the comparison consists of thirty different feeders’ daily load curve derived from a distribution network. U

This paper examined the ability of a fuzzy classification method based on fuzzy relations calculated using a cosine similarity metric to cluster and classify load data of different feeders in a particular distribution network. The classification produced by this technique was evaluated and compared to the classification made by a more conventional model based on multiple discriminant analysis.

The paper discusses the strength and weaknesses of the approach taken from the simulated results.

1.

INTRODUCTION

2.

Electric load in distribution system depends on the consumer demands that vary in accordance to their activities. As a result, power companies must produce sufficient energy as and when demanded to meet the requirements of the consumers. Since electrical energy cannot be stored, utility companies need load data for power production planning. Moreover, load data is also needed for tariff and pricing, distribution network planning and operation and finally for providing information to the consumer and regulatory reporting.

CLASSIFICATION TECHNIQUES

According to Arabie et al (I), classification is bath an ancient discipline and a modern one. The classification of similar objects into groups is an important human activity. Scientists believe there is an underlying structure in most of the phenomenon that need to be understood. By finding structure, the data will be classified according to similar pattern, attributes, features and other characteristics. Another generic term, which is commonly used, is cluster analysis. Cluster analysis can be defined as a process to classify a set of observations into two or more mutually exclusive unknown groups based on combinations of interval variables. In load profiling,

The most significant load data is load shape or also known as load ctuve. When load shape is produced for a group of consumers, which have common characteristics over a given period, it is known as

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A. Classification by Equivalence Relations

to achieve an effective classification is a very demanding task. There are several factors that determine customer load profiles, representing various technical, commercial and behavioural aspects.

The traditional crisp relation based on the concept that everything is either has complete relationship or no relationship at all. Thus, a crisp relation maps or relates elements of two or more sets. Fuzzy relations also map elements of one universe to another universe, through the Cartesian product of the tn’o universes. However, the ‘strength’ of the relation between the two universes is measured with a membership function that expressed various ‘degrees’ of strength of the relation on the unit interval [0,I].

In their paper, Chicco et al ( 2 ) discussed that the key concepts of classification techniques in electricity customers’ are feature selection, time domain approaches (unsupervised clustering algorithm, statistics application, neural networks, fuzzy system) and frequency domain approaches (harmonic analysis, wavelets). Research in these methods were reported in several papers, Muller (3), Birch et al (4), Ozveren et al(5) and Pitt (6).

3.

B. Methodology The proposed algorithm is summarised as below: Produce the fuzzy relations (r ij ) matrix using the cosine amplitude method Produce the fuzzy equivalence matrix using m a min composition. Obtain the clusters of data (crisp-equivalence matrix) using the classification by equivalence relations method using the lambda-cut method.

MODEL

Both classification techniques were tested to a set of load data of diiTerent feeders. The data is the electricity consumption usage taken fiom 30 different feeder derived from a distribution network in Malaysia. These feeders are connected to four types of consumers i.e. domestic (D), commercial (C), industrial (1) and mixed consumers (D& C).

Feeder No

Original Group

Type of consumer

i) Cosine Amplitude. This method makes use of a collection of data samples, n. If these data samples are collected, they form a data array, X.

Approximate number of

x = {x,,x2).....,

X“}

Each of the elements, ‘c, in the data array X is itself a vector of length m, that is xi={X,,.xi* ).,...,x i m }

Thus, each of the data samples can he thought of as a point in m-dimensional space where each point needs m coordinates for a complete description. Each element of rii results from a pair wise comparison of two data samples, say xi and xi where the strength of the relationship between them is given by the membership value that is 0 5 b, 2 1. The cosine amplitude method calculates rij in the following manner:

TABLE 1 -Feeders data description The time interval of sampling load curve data is 30 minutes. Therefore the load profile is represented by 48 load values throughout the day. Table 1 illustrates the summary of the load data.

4.

FUZZY CLASSIFICATION

One of the popular methods of fuzzy classification is using equivalent relations, which was proposed by Zadeh in 1971, followed by Bezdek and Harris later in 1978. This approach makes use of certain special properties of equivalent relations and the concept of defuzzification known as lambda-cuts on the relations as described in Ross ( 7 ) .

where i j = 1,2 ...n The relation matrix will be of size n x n and will he reflexive and symmetric -hence a tolerance relation.

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into those used for interpreting the group differences and those employed to classify cases into categories or groups.

ii) Max-min Composition. This method is used to reform a tolerance relation into a fuzzy equivalence relation by employing the familiar operators of fuzzy set, inax ( v ) and min ( A ) . If R is a relation that relates elements from X to Y and S is the relation that relates elements from Y to Z , a composition will k defined as relation, T that relates the same elements in X that R contains to the same elements in 2 that S contains. The max-min composition is defined by the set-theoretic and membership function-theoretic expressions, U , (!

xj:,)

= IE.Y V {P R( - ,:x,

) A P ! ( .q.xj)}

Although the initial study of discriminant analysis involved the biology and medical sciences, statisticians and researches found that this technique is also useful in areas of study such as business, education, engineering and psychology Multiple discriminant analysis (MDA) is an extension of discriminant analysis. It is used when the number of categories is more than two. In this paper, MDA is used for the classification purpose.

(2)

Several compositions may be necessary until the three properties previously mentioned is realised. A. Classification Procedures

iii) Lambda-Cut for Fuzzy Relations. Lambda-cut is one of the methods to defuzzify fuzzy results to crisp results. In classification of data, this method was used to cluster the data. This involved taking the fuzzy equivalence matrix and setting every element above the cut-off point, h (between 0 and 1) to a ‘ I ’ and every element k l o w it to a ‘0’. The crisp equivalence matrix, which has a certain number of classes, will he produced. The numbers of classes depend upon the value of h. As this value gets larger, more classes will be obtained.

There are several means in which classification can k achieved. Normally, these methods involve defining some concept of “distance” between the case and each group centroid with the case k i n g classified into the “closest” group. i) Classification Functions. This function can k used to determine to which group each case most likely belongs. The function has the following form:

s.1 =c.I + wI17 * + w1 *x + ... +W’.u n*xm 22

C. Classification Process

where i = respective group ; I , 2,.... rn = m variables; c, =constant for the r‘th group, ii’, =weight for the .fth variable in the computation of the classification score for the I’th group x, =is the observed value for the respective case for thej‘th variable.

Microsofl Excel was chosen as the spreadsheet program to implement the algorithm for fuzzy classification. The data is entered into the spreadsheet in matrix form with 48 columns (half-hourly data) and 30 rows (number of feeders). From the simulation. a lambda value of 0.873 is found as a suitable value because it gives a good classification results compared with other value. Using this value, the algorithm classified the feeder data into six major clusters.

5.

(3)

The classification functions can k used directly to compute classification scores for some new observations. Generally, the case is classified as belonging to the group for which it has the highest classification score.

MULTIPLE DISCRIMINANT ANALYSIS

Klecka (8) defined discriminant analysis as a statistical technique, which allows the researcher to study the differences between two or more group of objects with respect to several variables simultaneously. It was first developed by Fisher in 1936, who was seeking to solve problems in physical anthropology and biology. It is also a broad term, which refers to several closely related statistical activities. Generally, these activities can be divided

ii) Mahalanobis Distances. The concept of “distance” is not well defined if the variables are correlated and do not have the same measurement units and standard deviations. In 1963, Mahalanobis has proposed a generalized distance measure, which solves this problem. Mahalanobis distances are used in analysing cases in discriminant analysis. For example, in analysing a new, unknown set of cases in

I49

comparison to an existing set of known cases. Mahalanobis distance is the distance between a case and the centroid for each group (of the dependent) in attribute space (n-dimensional space defined by n variables). A case will have one Mahalanohis distance for each group, and it will be classified as belonging to the group for which its distance is smallest. Thus, the smaller the Mahalanobis distance, the closer the case is to the group centroid and the more likely it is to be classed as belonging to that group.

From the fuzzy method, six clusters were obtained. Each of these clusters is shown in Figure 1 - 5. Cluster 1 consists most of the domestic, commercial and D&C feeders. However, there are only three feeders in cluster 2 i.e. combination of domestic and commercial feeders. Most of the industrial feeders are grouped in cluster 3. Cluster 4, 5 and 6 consists of unique load patterns that could not be categorize into their original group. Cluster 2

B. Classification Process

Statistical package SPSS was used to carry out the MDA method. The inputs were the 48 variables (half-hourly) 'for the 30 feeders and their predefined groups. In this study, MDA was performed by assuming equal prior probabilities of classification among the groups.

Half-hourly &Feeder5

6.

RESULTS AND DISCUSSION

Feeder 12 +Feedet

I4

Figure 2: Cluster 2 (results from fuzzy method)

Cluster 1

-

1501

f 2

100

$

0

50

2

r

a

- ' D z % z R ; %

-

Half-hourly -4-Feeder 19 -36- Feeder 22

+Feeder +Feeder

21 23

?igure 3: Cluster 3 (results from fuzzy method) - a -

' D E W z R ; P Half-hourly Cluster 4

I tFeeder 1 '

-0- Feeder 2

+Feeder

Feeder8

-Feeder 9

Feeder 11 +Feeder

+Feeder

16

4Feeder 24

-

Feeder 3

Fr mm

6 t Feeder 7

Feeder 4

b

g

Feeder 10

13 +Feeder

-Feeder 18

t Feeder 25

Feeder 26

ID

+ .

-

Feeder 27 -5- Feeder 29 -A-

30

a m

15

Feeder 17

40

Half-hourly

Feeder 30

-Feeder 9 +Feeder

25

jigure 4: Cluster 4 (results from fuzzy method)

Figure 1: Cluster 1 (results from fuzzy method)

150

Group 5 & 6

Cluster 2

I

'MO

1

i? t 0

P 0 a

1 4 ?-Feeder

20

.f

I

Half-hourly

+Feeder 28

+.

9

I

Figure 5: Cluster 5&6 (results from fuzzy method)

Half-hourly 1-0-Feeder

On the other hand, four clusters were obtained from MDA. These clusters are shown in Figure 6 - 9 below. All except one domestic feeder is classified in cluster I together with several commercial and D&C feeders. Cluster 2 consist of three commercial feeder while all industrial feeders is grouped in Cluster 3. Finally, cluster 4 consists of two load patterns, which are in their own unique group

16 +Feeder

15 +Feeder

19

Figure 7: Cluster 2 (results from MDA)

Cluster 3

Cluster 1

Half-hourly 20

Feeder 23

U F e e d e r 21

Feeder 22

+Feeder

Figure 8: Cluster 8 (results from MDA)

Cluster 4

-+-Feeder

1 U Feeder 2

+Feeder

5

-

Feeder8 Feeder 11

+Feeder

14

-&-Feeder21

-

Feeder 26

+Feeder

-

6

Feeder 9

Feeder 4 +Feeder

7

Feeder 10

Feeder 12 -X-Feeder 13

-

Feeder 17

-

Feeder 18

4 F e e d e r 2 4 +Feeder25 Feeder 27 4Feeder 29

Half-hourly

-b- Feeder 30

1 +Feeder

25 -0-Feeder 28

Figure 6: Cluster 1 (results from MDA) Figure 9: Cluster 9 (results from MDA)

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Figure I O compares the classifications accuracies of the original groups for both methods. As this figure shows, the MDA outpcrrorms the corresponding fuzzy method across all original grouped cases. However, 6om fuzzy classification, there were two additional clusters emerged. This is a very helpful feature in isolating a unique load pattern

may not be in their original grouping and assign suitable tariff accordingly. P. Arabie, L. J. Hubert, and G. De Soete, "Clustering and Classification." New Jersey: World Scientific, 1996, pp. 490.

G. Chicco, R. Napoli, F. Piglione, P. Postolache, M. Scutariu, and C. Toader, "A Review of Concepts and Techniques for Emergent Customer Categorisation," presented at TELMARK Discussion Forum European Electricity Markets, London, 2002.

Classification Accuracies 12n 1

H. Muller, "Classification of Daily Load by Cluster Analysis," presented at 8th Power System Computation Conference, 1984.

1-

Original group

A. P. Birch, C. S. Ozveren, and A. T. Sapeluk, "A Generic Load Profiling Technique using Fuzzy Classification," presented at 8th International Conference on Metering & Tariff for Energy Supply, 1996.

Figure 10: Accuracies comparison between fuzzy and MDA From both analyses, the results show that most of the commercial load patterns were grouped together with the domestic pattern. This may be due to the fact that these commercial consumers consists of small shops which operated similar to domestic usage style but with more higher loads.

C. S. Ozveren, L. Fyall, and A. P. Birch, "A Fuzzy clustering and classification technique for customer profiling," presented at University Power Engineering Conference, 1997.

Another interesting result is that most of the mixed feeder (D &C) was grouped together with domestic feeders. One reason for this result is that the number of commercial consumers in this group is relatively small compared with the domestic users. Thus, the domestic load pattern has suppressed the commercial load pattern.

B. D. Pitt, "Application of Data Mining Techniques to Load Profiling," M I S T , 2000. T. J. Ross, Fuzzy Logic with Engineering Applications: McGraw Hill International Editions, 1997. W. R. Klecka, Discriminant Analysis, vol. 19. Iowa City: Sage Publications, Inc, 1980.

CONCLUSION

This study evaluated the ability of fuzzy classification and MDA to classify the predefined groups of feeders. Results from both methods were compared based on accuracies and sensitivity. In general, MDA produced greater accuracies than those produced by fuzzy method. However, fuzzy approach is more flexible and sensitive in selecting the cluster boundary. These characteristics enable it to single out cases that have unique or slightly different pattern and amihutes. Another advantage of the fuzzy method is that power companies can reclassify some of patterns, which

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