J. Cent. South Univ. Technol. (2011) 18: 190−195 DOI: 10.1007/s11771−011−0679−5
Power interconnected system clustering with advanced fuzzy C-mean algorithm WANG Hong-mei(王洪梅)1, KIM Jae-Hyung1, JUNG Dong-Yean2, LEE Sang-Min3, LEE Sang-Hyuk3 1. School of Mechatronics, Changwon National University, Changwon 641-773, Korea; 2. Daeho Tech Co. Ltd., Changwon 641-465, Korea; 3. Department of Electronics Engineering, Inha University, Incheon 402-751, Korea © Central South University Press and Springer-Verlag Berlin Heidelberg 2011 Abstract: An advanced fuzzy C-mean (FCM) algorithm was proposed for the efficient regional clustering of multi-nodes interconnected systems. Due to various locational prices and regional coherencies for each node and point, modified similarity measure was considered to gather nodes having similar characteristics. The similarity measure was needed to contain locational prices as well as regional coherency. In order to consider the two properties simultaneously, distance measure of fuzzy C-mean algorithm had to be modified. Regional clustering algorithm for interconnected power systems was designed based on the modified fuzzy C-mean algorithm. The proposed algorithm produces proper classification for the interconnected power system and the results are demonstrated in the example of IEEE 39-bus interconnected electricity system. Key words: fuzzy C-mean; similarity measure; distance measure; interconnected system; clustering
1 Introduction In the regional management of interconnected network systems, the efficient and economical operation of the networked systems in terms of system coherency was essential [1]. Hence, the research of system coherency has been made by numerous researchers [2−3]. However, most of the studies were emphasized on the dynamic grouping [4]. Hence, a novel approach to partition the total system into several regions considering locational information, such as locational cost, loss and regional distances, was needed. Grouping the locations in a networked system with similar locational prices has been proposed considering the regional coherency. Locational costs in a networked system imply the price at which the good was consumed at each location. Due to the physical characteristics of the transmission network of the systems, the good was lost as it was transmitted from supplying locations to consuming locations, and an additional supply must be provided to compensate for the loss. Also, the transmission network of the systems has a capacity limitation preventing full uses of cheap production. Therefore, locational prices at each point or node, was differently decided depending on the network topology
and supply−demand configuration. Similarity measure between data was introduced by LIU [5]. Later, explicit expression of fuzzy entropy and similarity measure were also followed [6−7]. Furthermore, relation between fuzzy entropy and similarity measure was shown continuously [8−9]. To consider locational prices and regional coherency simultaneously, it was needed to extend the conventional similarity measure. Hence, to design the measure of two properties not only locational prices but also regional coherency should be considered. In the previous researches, similarity measure was designed through distance measure or fuzzy entropy [10]. Well-known Hamming distance was also used to construct similarity measure. With only conventional similarity measure, unpractical results could be followed. Emphasizing the only way of locational prices or regional coherency, results were originated by the structure of single variable similarity measure. Hence, the addition of regional coherency made it possible to complete the modified similarity measure. Then, fuzzy C-mean (FCM) was introduced, and similarity measure was proposed. Proposed similarity measure was designed by the distance measure. Similarity measure was modified with additional regional coherency to apply for power interconnected system. And illustrative example was shown.
Foundation item: Work supported by the Second Stage of Brain Korea 21 Projects; Work(2010-0020163) supported by Priority Research Centers Program through the National Research Foundation (NRF) funded by the Ministry of Education, Science and Technology of Korea Received date: 2010−03−28; Accepted date: 2010−06−24 Corresponding author: LEE Sang-Hyuk, PhD, Professor; Tel: +82−55−213−3884; E-mail:
[email protected]
J. Cent. South Univ. Technol. (2011) 18: 190−195
191
2 Fuzzy C-mean clustering Fuzzy C-means clustering was proposed by BEZDEK in 1973 as an improvement over hard C-mean (HCM) [10]. FCM played a roll of partitioning arbitrary n vectors into c fuzzy groups, also finding a cluster center for each group such that a cost function of similarity measure was maximized, or dissimilarity measure was minimized. Well-known facts about FCM and HCM indicated that FCM employed fuzzy partitioning such that a data point could belong to several groups with the degree of membership grade between 0 and 1. 2.1 Preliminaries Fuzzy C-mean was introduced briefly as follows. These results could be found in Ref.[11]. Membership matrix U was defined as c
∑ uij = 1, ∀ j = 1,
L, n
i =1
(1)
The cost function for FCM was constructed by c
c
n
J (U , c1 ,L , cc ) = ∑ J i = ∑∑ uijm dij2 i =1
(2)
i =1 j =1
where uij is denoted between 0 and 1, ci is also the center of fuzzy group i, dij=|ci−xj| satisfies the Euclidean distance between i-th cluster center and the j-th data point xj, and m is defined by the weighting value. With Lagrange multiplier, the necessary conditions for Eq.(2) to reach a minimum were found in Ref.[11] as n
∑ ci =
j =1 n
uijm x j
∑
j =1
and uij = uijm
1 c
dij
∑(d k =1
)
2.2 Similarity measure with distance function To define the similarity between data sets, it was essential to consider similarity measure. Similarity measure was proposed by LIU [5]. Axiomatic properties of the similarity measure were proposed in Definition 1. Of course, there were lots of similarity measures that satisfied Definition 1 [12−13]. Definition 1: A real function s: P2→R+ or F2→R+ is a similarity measure if s has the following properties:
(S1) s(A, B)=S(B, A), ∀ A, B ∈ F ( X ) (S2) s(D, DC)=0, ∀ D ∈ P( X ) (S3) s(C, C)= max A, B∈F s ( A, B), ∀ C ∈ F ( X ) (S4) ∀ A, B, C ∈ F ( X ), if A ⊂ B ⊂ C, then s(A, B)≥ s(A, C) and s(B, C)≥s(A, C) where F(X) denotes a fuzzy set and P(X) is a numeric set. Similarity measure was proposed with the distance measure as follows. Theorem 1: For any set A, B ∈ F(X) or P(X) if d satisfies Hamming distance measure, then s(A, B)=4−2d((A I B), [1]X)−2d((A U B), [0]X)
(3)
is a similarity measure between set A and B. Proof 1: Proof was followed by proving that Eq.(3) satisfied the similarity definition. (S1) means the commutativity of set A and B, hence it was clear from Eq.(3) itself. From (S2), s(A, AC)=4−2d((A I AC), [1]X)−2d((A U AC), [0]X) then 2d((A I AC), [1]X) and 2d((A U AC), [0]X) were the maximum values between A and arbitrary set. For arbitrary sets A and B, inequality of (S3) was obtained by s(A, B)=4−2d((A I B), [1]X)−2d((A U B), [0]X)≤ 4−2d((D I D), [1]X)−2d((D U D), [0]X)=s(D, D) Inequality was satisfied from
2 /( m −1)
kj
With these results, well-known FCM algorithms could be listed as Step 1: Initialize membership matrix U. Step 2: Calculate ci, i=1, …, c. Step 3: Computate Eq.(2). Stop if either it is below a certain tolerance. Step 4: Computate a new U. Now for minimizing Eq.(2), the less the distance dij=|ci−xj| was, the smaller the cost function became. Hence, distance values have a relation to the similarity between two data points. Finding similarity could be determined from the types of data, time series signal, image, sound, etc. Then, similarity measure for the fuzzy sets was introduced, and the proposed similarity measure could be applied to the power interconnected problem.
d((A I B), [1]X)≥d((D I D), [1]X) and d((A U B), [0]X)≥d((D U D), [0]X) Finally, ∀A, B, C ∈ F ( X ), if A ⊂ B ⊂ C, s(A, B)=4−2d((A I B), [1]X)−2d((A U B), [0]X)= 4−2d(A, [1]X)−2d(B, [0]X)≥ 4−2d(A, [1]X)−2d(C, [0]X)=s(A, C) Similarly, s(B, C)≥s(A, C) was obtained easily through d(B, [0]X)≤d(C, [0]X) and d(B, [1]X)≤d(A, [1]X). Therefore, proposed similarity measure Eq.(3) satisfied similarity measure. Similarly, numerous similarity measures could be obtained, and it was also designed with the distance measure. Theorem 2: For any set A, B ∈ F(X) or P(X), if d satisfies Hamming distance measure, then s(A, B)=2−2d((A I BC), [0]X)−2d((A U BC), [1]X) is also similarity measure between set A and B.
(4)
J. Cent. South Univ. Technol. (2011) 18: 190−195
192
Proof 2: Proofs were also followed similarly as Theorem 1.
3 Similarity consideration with additional information To apply FCM with dij=|ci−xj|, it was required that dij had to satisfy the similarity property. Hence, dij was considered the modified similarity measure in Eq.(2). The proposed similarity measure has to be gathered for the point that has similar characteristic values. However, the large power connected system has multiple characteristics for each node. For example, locational prices and geometric information (location) were the representative characteristics. It could happen that two nodes had very similar locational prices even they were located far away. Then, it was not realistic to gather even though they had similar valued measure. To overcome this drawback, another similarity measure had to be required to gather data properly. With Eqs.(3) and (4), similarity measure was considered: s2(A, B)=2/(1+d)
(5)
where d denotes the geometrical distance value. Combining similarity measure as s(A, B)=ω1s1(A, B)+ω2s2(A, B)
(6)
where s1(A, B)=4−2d((A I B), [1]X)−2d((A U B), [0]X), ω1 and ω2 are the weighting values. Usefulness of Eq.(6) was verified directly, and properties of s1(A, B) were proved in Theorem 1 and Theorem 2. Usefulness for the similarity of s2(A, B) could be verified as follows: Commutative property of distance was the same, hence (S1) was easily shown. From (S2), distance of A and AC was the longest, hence s2(A, AC) satisfied the minimum value. For all A, B ∈ P(X), inequality of (S3) was proved by s2(A, B)=2/(1+d(A, B))≤2/(1+d(D, D))=s2(D, D) In the above, d(D, D) is the smallest value, i.e., zero. So (S3) could be verified. Finally, ∀A, B, C ∈ F ( X ) and A ⊂ B ⊂ C, s2(A, B)=2/(1+d(A, B))≥2/(1+d(A, C))=s2(A, C) where d(A, C) was longer than d(A, B). Similarly, s2(B, C)=2/(1+d(B, C))≥2/(1+d(A, C))=s2(A, C) was also satisfied. Hence, it could be verified that s2(A, B) satisfied Definition 1. Then, similarity measure extension for the multiple facts could be possible. Furthermore, more flexible design for multiple facts could also be possible by adjusting the weighting values:
s2(A, B)=ω1s1(A, B)+ω2s2(A, B)+…+ωnsn(A, B)= n
∑ ωi si ( A, B) i =1
As mentioned before, for the power interconnected system clustering problem, multiple considerations have to be followed. Because each node has multiple representative values such as geometric information (location), locational prices or other values could be followed. Hence, the extension of similarity measure would be useful to evaluate the similarity of complex node information. Then, Eq.(6) could be used as the similarity measure for the particular points which have locational prices and geometric information at the same time.
4 Illustrative example As an illustrative application, the interconnected electricity system was considered. The IEEE reliability test system which was prepared by the reliability test system task force of the application of probability methods subcommittee in 1996 [14] is considered a test system. In the test system, 39 nodes (buses) and 10 generators are contained and each bus has its own locational price and information. In networked electricity systems, due to the physical characteristics of the electricity transmission network, electricity is lost when it is transmitted from supplying nodes (i.e., supplying buses) to consuming nodes (consuming buses), and additional generation must be supplied to provide energy in excess of that consumed by customers. Moreover, the capacity limitation of the transmission network of electricity systems prevents full uses of system wide cheap electricity. Therefore, the electricity price at each node, i.e., the price at which the electricity is consumed at each node is differently decided depending on the network topology and energy configuration [15]. The electricity prices at each node are defined as locational prices at each node and the locational prices represent the locational value of energy, which includes the cost of electricity and the cost of delivering it, i.e, the delivery losses and network congestion. In Table 1, each locational price is illustrated for the 39 buses, and per unit geometrical information for nodes is also shown. Locational prices of each nodes were from 24.98 $/(kW·h) to 55.00 $/(kW·h), and 39 locational information is represented through 2-dimensional plane at which the plane is assumed to be flat. In order to apply FCM for the clustering problem of Fig.1, replacing s(A, B) in Eq.(6) into dij=|ci−xj| in Eq.(2), then the similarity of locational prices and locations can be assigned by s1(A, B) and s2(A, B), respectively. Combined similarity measures are constituted as follows with the proposed
J. Cent. South Univ. Technol. (2011) 18: 190−195
193
Table 1 Locational prices and location (normalized geometric information) for each bus Bus No.
Locational price/($·kW−1·h−1)
Location
Bus No.
Locational price/($·kW−1·h−1)
Location
Bus No.
Locational price/($·kW−1·h−1)
Location
1
29.21
( 0.9, 9 )
14
41.74
(6.6, 6)
27
51.45
(4.6, 3.5)
2
28.53
(0.6, 6.2)
15
43.79
(6.6, 4.9)
28
55.00
(2.7, 1.5)
3
31.40
(3, 7.5)
16
45.84
(6.5, 4)
29
55.00
(2.7, 0.8)
4
32.78
(4.7, 7.5)
17
47.90
(5, 4.5)
30
28.53
(0, 6.2)
5
37.57
(7, 7.6)
18
46.40
(4.2, 6)
31
38.26
(8.3, 6.6)
6
38.26
(8.5, 7.6)
19
45.84
(6.9, 2.8)
32
40.00
(11.3, 5.8)
7
37.81
(9.6, 8.4)
20
45.84
(6.9, 1.7)
33
45.84
(8, 1.7)
8
37.35
(8.5, 9.1)
21
45.84
(8.7, 2.8)
34
45.84
(5.5, 1)
9
30.56
(6.1, 9.5)
22
45.84
(10, 2.8)
35
45.84
(10, 1.6)
10
40.00
(10.8, 5.8)
23
45.84
(11.1, 2.8)
36
45.84
(11.1, 1.6)
11
39.42
(9.7, 6.3)
24
45.84
(8.2, 4.3)
37
24.98
(0.7, 3.7)
12
40.00
(11.1. 7.1)
25
24.98
(1.4, 4.7)
38
55.00
(2.7, 0)
13
40.58
(8.5, 5.5)
26
55.00
(2.7, 3)
39
29.88
(3.4, 9.5)
Fig.1 Networked electricity system with 39 nodes (buses) and 10 generators
measure. Consider power interconnected systems in Fig.1. At first, 39 buses are partitioned to the 3 groups, and the result is illustrated in Fig.2. 39 buses are shown in 3 dimensional space. X-Y plane represents the locational information and height means the locational price. Three dimensional 39 vectors are projected to the X-Y plane. In Fig.2 the result with only locational information is
obtained. Three groups are marked by “○”, “×” and “ ”. This indicates that three groups are considered with only locational information because weighing values are considered as ω1=0 and ω2=1. This strict condition is satisfied for only location consideration. Hence, to satisfy more flexible request, the locational price and locational information have to be considered simultaneously. Next, with the variations of weighting
J. Cent. South Univ. Technol. (2011) 18: 190−195
194
values ω1 and ω2, clustering results are obtained in Fig.3. By varying both values of ω1 and ω2, 9 nodes change their clusters. Two “ ” were changed into group “○”. One “×” is changed into group “○”. Finally, 6 “×” are changed into group “ ”. Clustering results are illustrated in Fig.3. It is very interesting to find the changes in groups. Their changes all happen at the boundary areas. However, other changes do not happen for further variation of weighting values ω1 and ω2 till 0.27 and 0.73, respectively. The result is shown in Fig.4, which shows exactly the same result of Fig.3. Then, the difference of clustering result between Fig.3 and Fig.4 when the weighting values are ω1=0.27 and ω2=0.73 is shown in Fig.4.
Fig.2 Clustering by only location consideration (ω1=0, ω2=1)
Fig.3 Clustering by two consideration (ω1=0.2, ω2=0.8)
As the value ω1 approaches to 0.27 and ω2 to 0.73, even any special changing cannot be noticed in Fig.4, there are so many changes near the cluster boundaries at each iteration. Explicitly, when ω1 approaches from 0.2 to 0.27 and ω2 from 0.8 to 0.73, data clustering is considered by unstable status, i.e., not determined. However, when ω1 and ω2 are over 0.27 and 0.73, respectively, node clustering is convergent 3 groups without changing any node.
5 Conclusions 1) Fuzzy C-mean clustering method is modified to apply for the clustering of interconnected power system. Conventional Euclidean distance used in fuzzy C-mean is changed into the modified similarity measure. The proposed similarity measure is applicable to the multisimilarity problem. Furthermore, the usefulness of the proposed multi similarity measure is verified. 2) For grouping of the power interconnected networked system in terms of an appropriate similarity measure, not only locational prices but also locational information should be properly considered in the formulation of the similarity measure. Hence, a modified similarity measure accompanied with regional information is proposed, followed by example on the IEEE reliability test system to verify the usefulness of the proposed idea of the modified similarity measure. From the results, the coherency between the degree of similarity level and the number of clusters can be checked. 3) The convergence of clustering is verified by the variation of weighting values of each similarity measure. Locational price and geometric information weighting values converge to 0.27 and 0.73, respectively. It is also found by simulation result that some nodes change near the boundary of each group.
References [1]
LI W, BOSE A. A coherency based rescheduling method for dynamic security [J]. IEEE Transactions on Power Systems, 1998, 13(3): 810−815.
[2]
JOO S K, LIU C C, JONES L E, CHOE J W. Coherency and aggregation techniques incorporating rotor and voltage dynamics [J]. IEEE Transactions on Power Systems, 2004, 19(2): 1068−1075.
[3]
GALLAI A M, THOMAS R J. Coherency identification for large electric power systems [J]. IEEE Transactions on Circuits and Systems, 1982, CAS-29(11): 777−782.
[4]
WU F F, NARASMITHAMURTHI N. Coherency identification for power system dynamic equivalents [J]. IEEE Transactions on Circuits and Systems, 1983, CAS-30(3): 140−147.
Fig.4 Changes happened continuously with group boundary (ω1=0.27, ω2=0.73)
[5]
LIU Xue-cheng. Entropy, distance measure and similarity measure of fuzzy sets and their relations [J]. Fuzzy Sets and Systems, 1992, 52:
J. Cent. South Univ. Technol. (2011) 18: 190−195 305−318. [6]
195 [12]
FAN J L, XIE W X. Distance measure and induced fuzzy entropy [J].
dissimilarity measures for fuzzy sets on the basis of distance measure
Fuzzy Set and Systems, 1999, 104: 305−314. [7]
FAN J L, MA Y L, XIE W X. On some properties of distance
[J]. International Journal of Fuzzy Systems, 2009, 11(2): 67−72. [13]
measures [J]. Fuzzy Set and Systems, 2001, 117: 355−361. [8]
[9]
1783−1786. [14]
reliability test system task force of the application of probability
LEE S H, KIM J M, CHOI Y K. Similarity measure construction
methods subcommittee [J]. IEEE Transactions on Power Systems,
Artificial Intelligence, 2006, 4114: 952−958.
[11]
The IEEE Reliability Test System-1996. A report prepared by the
4114: 134−139. using fuzzy entropy and distance measure [J]. Lecture Notes in [10]
LEE S H, RYU K H, SOHN G Y. Study on entropy and similarity measure for fuzzy set [J]. IEICE Trans Inf & Syst, 2009, E92/D(9):
LEE S H, CHEON S P, KIM J. Measure of certainty with fuzzy entropy function [J]. Lecture Notes in Artificial Intelligence, 2006,
LEE S H, PEDRYCZ W, SOHN G Y. Design of similarity and
1999, 14(3): 1010−1020. [15]
ALLAN R N, BILLINTON R, ABDEL-GAVAD N M K. The IEEE
BEZDEK J C. Fuzzy mathematics in pattern classification [D].
reliability test system - extensions to and evaluation of the generating
Applied Math Center, Cornell University, Ithaca, 1973: 50−55.
system [J]. IEEE Trans on Power Systems, 1989, PWRS-1: 1−7.
JANG J S R, SUN C T, MIZUTANI E. Neuro-fuzzy and soft computing [M]. Upper Saddle River, Prentice Hall, 1997: 425−427.
(Edited by YANG Bing)
