fuzzy logic in distribution network

Fuzzy Sets and Systems 142 (2004) 293–306www.elsevier.com/locate/fss

A fuzzy multiple linear regression based loss formula inelectric distribution systems

Ying-Yi Honga ;∗, Zuei-Tien Chaoa, Miin-Shen Yangb

aDepartment of Electrical Engineering, Chung Yuan Christian University, Chung Li 32023, TaiwanbDepartment of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan

Received 4 September 2002; received in revised form 22 February 2003; accepted 31 March 2003

Abstract

Estimation of energy loss is an essential task for both operation and planning in electric distribution systems.The energy losses are considered to be uncertain due to measurements. A new method based on fuzzy-c-numberclustering and fuzzy multiple linear regression analysis is proposed for developing energy loss formulas toestimate losses in this paper. The proposed method is implemented with three stages. A part of Taipowerdistribution system in Taipei is used for illustrating the performance of the proposed method.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Loss formula; Fuzzy multiple linear regression; Fuzzy clustering; Electric distribution systems

1. Introduction

The purposes of energy loss estimation in distribution systems include conductor size design,substation expansion, rate analysis, loss allocation to customers, energy conversion analysis, andcontrol strategy development, etc. (see [6]). Therefore, exploration of an energy loss (kWh) formulais very essential for a utility in the viewpoint of both operation and planning.

There were many existing literatures related to loss estimation for distribution systems (see[1–5,7–10,14]). A sophisticated model for loss estimation was proposed in [14]. An empirical lossfactor equation considering the load composition and various time periods was presented in [7]. Anenergy formula further considering constant voltages, power factors and resistors was developed in[8]. Capacitors and taps were considered as independent (control) variables in an energy loss formulafor the sake of developing proper control strategies in [1]. Unbalanced voltages and currents wereused as independent variables for a loss formula in [4]. A measurement based loss formula was

∗ Corresponding author. Tel.: +886-3-2652300; fax: +886-3-2652399.E-mail address: [email protected] (Y.-Y. Hong).

0165-0114/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0165-0114(03)00143-X

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294 Y.-Y. Hong et al. / Fuzzy Sets and Systems 142 (2004) 293–306

proposed for residential wiring system in [5]. A simpliCed loss model for distribution systems waspresented in [3]. Other papers [2,9,10] were also related to loss formulas.

On the other hand, regression is a method for Ctting variables to a function, which may be linear,nonlinear or multiple. Regression is the determination of coeEcients of the Ctting function. Simi-larly, fuzzy regression is developed for Ctting fuzzy variables into a fuzzy or real function becauseengineers usually use fuzzy linguistic sentences to describe engineering phenomena [15,16,18]. Thefuzzy multiple linear regression method was proposed in [15,16] for forecasting physical quantities.A new method based on fuzzy clustering incorporated with fuzzy regression to be the cluster-wisefuzzy regression was presented by Yang and Ko [18]. In the power engineering area, the peak loadwas forecasted using fuzzy regression in [13].

It is known that the loss values can be modeled with fuzzy numbers because the system load andenergy source values are uncertain due to measurements [12,13]. In this paper, therefore, a three-stage method based on [18] and fuzzy-c-number (FCN) clustering [17] is proposed for obtainingproper energy loss formulas. The reason for adopting [18] is that this method avoids linear pro-gramming used in [15,16]. The FCN algorithm is used Crst to partition the 24-h energy loss patterninto three clusters covering several segments where each cluster contains several time periods. Thethree clusters are designed for the peak-, medium-peak and oI-peak losses. Then each segment isfurther clustered into several sub-clusters, which are Ctted by fuzzy multiple linear regression. Theindependent variables are the kW h and kVARh loads in this paper.

A realistic distribution system in Taipower system of Taiwan Power Company is studied in thispaper. From the case study results, it is found that the proposed method can eEciently estimate thesystem kW h losses for engineers.

2. Characteristics of distribution system losses

The distribution losses are more nonlinear and diversiCed compared with the transmission losses.A part of the distribution (supply) system for Taipower, including two 161=69 kV and 12 69=11:95=11=6:9 kV transformers, is illustrated in Fig. 1 for depicting the above phenomenon. This supplysystem, Song-Shan area, is a part of the whole Taipei distribution system.

Song-Shan Supply area is a P/S (primary system) fed from six 161=69 kV lines. There are 12distribution transformers (69=11=11:95=6:9kV; 25MVA) connected to four S/S (secondary systems),namely Nei-Hu, Ming-Shen, Chung-Nun, and Hsin-Ya. Generally, there are two peak loads dailyand therefore the energy loss pattern is a two-peak pattern. Table 1 shows the daily kWh loss rangefor diIerent periods in the Song-Shan supply system. As shown in Table 1, a single formula isinadequate to express the kW h losses in this system. Actually, there are diIerent load character-istics in various hours daily. For example, rotating machine loads are dominated in the peak-loadhours while there are TV and light loads in the oI-peak load hours (at night). There exist sometransition periods between the peak load and oI-peak load periods; therefore, medium-peak loadhours should be considered. Hence, a partition process is required for this loss pattern. As shownin Table 1, there are seven “segments” in a day: (1, 2), (22–24), (3–7), (12–16), (19–21), (8–11),and (17,18).

After the three periods (peak-load, medium-peak load, and oI-peak load) are identiCed(clustered), one should further examine the loss characteristics in each segment. The realistic

Y.-Y. Hong et al. / Fuzzy Sets and Systems 142 (2004) 293–306 295

Fig. 1. Song-Shan supply system in Taipei.

measurement data show that the kW h loss characteristics are still very nonlinearly diversiCed. Be-cause the system load and energy source values are uncertain due to measurements, this motivatesone to cluster the losses in each segment and then develop the loss formula with fuzzy multiple linearregression.


Table 1Daily kW h loss pattern for a supply system

kW h loss range Hours

OI-peak load ¡6 (1, 2), (22–24)Medium-peak load 6–13 (3–7), (12–16), (19–21)Peak load ¿ 13 (8–11), (17, 18)

3. The proposed method

There are three stages in the proposed method for developing the proposed kW h loss formula asfollows:

• Stage one: This stage includes a clustering process. The FCN algorithm is used to partition thedaily load loss pattern into three periods covering some segments (peak-load, medium-peak load,and oI-peak load) based on the observations in Section 2.

• Stage two: Each segment is further partitioned into several sub-clusters. This clustering will in-corporate with the fuzzy multiple linear regression.

• Stage three: An optimal sub-cluster number will be identiCed for each segment.

The fuzzy multiple linear regression in the proposed method is extended from the fuzzy linearregression proposed in [18]. For engineers, an adopted method should be simple, e2cient and feasiblefor the problem characteristics. The reasons for using FCN and developing fuzzy multiple linearregression based on [17,18] are as follows:

(i) Both FCN and the proposed fuzzy multiple linear regression method are based on Lagrangian.The Crst derivative to Lagrangian may lead to optimum. The proposed method is much simpler,compared with [15,16] using linear programming.

(ii) Both FCN and the proposed fuzzy multiple linear regression method are based on LR-type fuzzynumbers. It is feasible to model the kW h loss by the LR-type fuzzy number: the measuredvalue corresponds to the mean (center) value and the uncertain factors are implemented withleft/right spreads.

(iii) Both FCN and the proposed fuzzy multiple linear regression method are e2cient: the CPU timeis small and the accuracy is acceptable (as described in Section 4).

(iv) The most important factor for considering the proposed fuzzy multiple linear regression methodis that the kW h loss characteristics are still very nonlinearly diversiCed and scattered ineach segment (see Section 4). Further partitioning in each segment is needed. The proposedfuzzy multiple linear regression method is a “cluster-wise” approach that obtains the regressioncoeEcients and deals with sub-clustering in a single stage. This proposed method is very suitablefor dealing with the loss characteristics in the problem.

3.1. Stage one: partitioning the daily loss pattern

The FCN algorithm is used to partition a set of fuzzy numbers into several clusters [17]. TheFCN algorithm can be summarized as follows:


According to Yang and Ko [17], we consider a set of LR-type fuzzy numbers �LR= {X1; X2; : : : ; Xn}([19], also see Appendix A). A FCN clustering objective function on the data set �LR is as follows:

J (�;W ) =n∑j=1

c∑i=1

�mi (Xj)d2LR(Xj;Wi); (1)

where

d2LR(Xj;Wi) = (mxj − mwi)2 + ((mxj − l�xj) − (mwi − l�wi))2

+((mxj + r�xj) − (mwi + r�wi))2; i = 1; : : : ; c; j = 1; : : : ; n (2)

and Xj = (mxj; �xj; �xj)LR; j= 1; 2; : : : ; n are LR-type fuzzy numbers with mxj, �xj and �xj representingthe mean value, left spread and right spread, respectively. Wi = (mwi; �wi; �wi)LR, i= 1; 2; : : : ; c, arethe cluster centers. �= (�1; �2; : : : ; �c) is a fuzzy c-partition with �ij = �i(Xj), the membership of Xjin ith cluster. Since the fuzzy c-partition (�1; �2; : : : ; �c) has the restriction

∑ci=1 �i(Xj) = 1 for all

j= 1; : : : ; n, the following Lagrangian should be considered:

L(�;W; �) =n∑j=1

c∑i=1

�mi (Xj)d2LR(Xj;Wi) − �

(c∑i=1

�i(Xj) − 1

): (3)

The symbols l and r in Eq. (2) are deCned in Appendix A. The weighting exponent m is largerthan 1. One can take the Crst derivatives of Eq. (3) with respect to all parameters and set them equalto zero. The necessary conditions for achieving the minimization of the FCN objective function Jare obtained as follows:

mwi =

∑nj=1 �

mi (Xj)(3mxj + l(�wi − �xj) + r(�xj − �wi))

3∑nj=1 �

mi (Xj)

; (4)

�wi =

∑nj=1 �

mi (Xj)(mwi − mxj + l�xj)l∑nj=1 �

mi (Xj)

; (5)

�wi =

∑nj=1 �

mi (Xj)(mxj − mwi + r�xj)r∑nj=1 �

mi (Xj)

(6)

and

�ij = �i(Xj) =(d2LR(Xj;Wi))

−1=(m−1)∑ck=1(d

2LR(Xj;Wk))−1=(m−1)

: (7)

One may examine the value of �ij = �i(Xj) to determine the relation between Xj and the ith cluster.An example about the kW h loss classiCcation of the Song-Shan supply system is provided inAppendix B for FCN.

In this paper, the fuzzy numbers �LR= {X1; X2; : : : ; Xn} are the kW h losses. The value of n is24. The clustering number is 3 for the peak, medium-peak and oI-peak loss levels. Once the dailylosses are clustered, a two-peak loss pattern may be partitioned into several segments. Table 1 showsseven segments in three clusters.


3.2. Stage two: fuzzy multiple linear regression

Only one segment is discussed in this section since the same treatment is implemented for allsegments. The purpose of the fuzzy multiple linear regression is to Ct the system energy loads invarious hours to the energy losses in a segment including several hours. In each segment, 1, 2 or 3loss formulas (sub-clusters) are developed depending the performance of regression.

Yang and Ko [18] had proposed a cluster-wise (simple) fuzzy regression analysis. Let G be the setof observations (Xj; Yj); j= 1; : : : n, where Xj = (mxj; �xj; �xj)LR and Yj = (myj; �yj; �yj)LR are LR-typefuzzy numbers. Suppose that these observations are heterogeneous and come from c simple linearregression models,

Yj = a0i + a1iXj; i = 1; : : : ; c; j = 1; : : : ; n; (8)

where a0i and a1i are unknown real coeEcients. Let �ij = �i(Xj) be the membership of Xj in ithcluster. A cluster-wise fuzzy regression objective function proposed by Yang and Ko [18] is asfollows:

JR(a0i ; a1i) =n∑j=1

c∑i=1

�mijd2LR(a0i + a1iXj; Yj); (9)

where d2LR is the same as deCned in (2) of Section 3.1. However, in this paper the kWh and kVARh

loads are used to be the independent variables. Thus, the objective function (9) needs to be extendedto the multiple linear regression model with K= 2:

Yj = a0i +K∑k=1

akiXkj; i = 1; : : : ; c; j = 1; : : : ; n: (10)

Since the loss characteristics in the three clusters (7 segments) are still diversiCed and are foundto be scattered, it is necessary to cluster each segment into sub-clusters. The proposed method incor-porates the fuzzy number clustering with the multiple fuzzy regression in a stage. Thus, the proposedcluster-wise multiple fuzzy regression objective function for K= 2 is then deCned as follows:

JMR(�; a0; a1; a2)

=n∑j=1

c∑i=1

�mijd2LR(a0i + a1iXij + a2iX2j; Yj)

=n∑j=1

�mij{(a0i + a1imx1j + a2imx2j − myj)2

+ S1i[(a0i + a1imlx1j + a2imlx2j − mlyj)2 + (a0i + a1imrx1j + a2imrx2j − mryj)2]

+ S2i[(a0i + a1imlx1j + a2impx2j − mlyj)2 + (a0i + a1imrx1j + a2imqx2j − mryj)2]

+ S3i[(a0i + a1impx1j + a2imlx2j − mlyj)2 + (a0i + a1imqx1j + a2imrx2j − mryj)2]

+ S4i[(a0i + a1impx1j + a2impx2j − mlyj)2 + (a0i + a1imqx1j + a2imqx2j − mryj)2] ; (11)

where m¿1 is the weighting exponent as the index of fuzziness; a0i ; a1i and a3i are unknown realnumber coeEcients; S1i = 1; S2i = S3i = S4i = 0 if (a1i¿0 and a2i¿0), S2i = 1; S1i = S3i = S4i = 0 if


(a1i¿0 and a2i¡0), S3i = 1; S1i = S2i = S4i = 0 if (a1i¡0 and a2i¿0), and S4i = 1; S1i = S2i = S3i = 0 if(a1i¡0 and a2i¡0). Let L(�; a0; a1; a2; �) be the Lagrangian with L(�; a0; a1; a2; �) = JMR(�; a0; a1; a2)+∑nj= 1 �j(

∑ci=1 �ij − 1). Set the Crst derivatives of L with respect to all parameters equal to zero.

Then, the n fuzzy numbers can be partitioned into c clusters and the coeEcients a0i, a1i and a2i

of the fuzzy multiple linear regression can be also obtained. The minimum of JMR can be obtainedthrough the Crst derivative of L and through 4 sets of solutions for (i) a1i¿0 and a2i¿0, (ii) a1i¿0and a2i¡0, (iii) a1i¡0 and a2i¿0, and (iv) a1i¡0 and a2i¡0, respectively. In case of a1i¿0 anda2i¿0, the necessary conditions for a minimizer (�ij; a0i ; a1i ; a2i) of JMR are obtained as follows:

a0i =

∑nj=1 �

mij(mtyj − a1imtx1j − a2imtx2j)

3∑nj=1 �

mij

; (12)

a1i =

∑nj=1 �

mij(mllrr1j − a0imtx1j − a2imlxrxj)∑n

j=1 �mijm

2tx1j

; (13)

a2i =

∑nj=1 �

mij(mllrr2j − a0imtx2j − a1imlxrxj)∑n

j=1 �mijm

mtx2j

; (14)

�ij =(d2LR(a0i + a1iXj + a2iX2j; Yj))−1=m−1∑c

k=1(d2LR(a0k + a1kX1j + a2kX2j; Yj))−1=m−1

; i = 1; : : : ; c; j = 1; : : : ; n: (15)

The symbols m’s with diIerent subscripts in the above equations are deCned in Appendix C. Theupdate equations for the cases (ii), (iii) and (iv) can be obtained as similar to Eqs. (12)–(15).

For example, suppose that there are 30 energy loss patterns (30 days) available and they aresimilar. Hence, there are 30×5 sets of {X1j; X2j; Yj} for developing c loss formulas in the 4thsegment as illustrated in Table 1. We mention that the proposed Stage two is extended from [18].The diIerences between them can be described as follows:

(i) The proposed fuzzy multiple-linear regression method has three unknowns a0i ; a1i and a2i forthe coeEcients; the method in [18] is related to the fuzzy single linear regression only and theunknowns are a0i and a1i.

(ii) There are four diIerent conditions for the solutions in the proposed method due to the signsof a1i and a2i. However, there are only two conditions, positive or negative values for a1i, in[18].

(iii) Two sets of closed form solutions can be derived in [18]; in other words, the solutions can beobtained without iterations. On the contrary, the relation among a0i ; a1i and a2i shown in Eqs.(12)–(15) are coupling and iterative steps are required to obtain the solutions. In this paper,the Gauss–Seidel method is used.

3.3. Stage three: determination of optimal c

The determination of c value in stage one is trivial. It can be based on the daily loss patternand c= 3. However, the determination of optimal c in each segment of individual segment (peak,


medium-peak, and oI-peak) should be further explored. This paper uses the approach in [11] toobtain an optimal c. The R2(c) value deCned in [11] is used to identify the optimal c for eachsegment. The optimal c has the property of R2(c) being the closest to unity. This concept is thesame as that adopted %2 in the traditional regression techniques. The optimal c is the one withthe maximum of R2(c). R2(c) is deCned by the weighted mean of the determination coeEcientsR2i ; i= 1; : : : ; c, as follows:

R2(c) =c∑i=1

wiR2i ; i = 1; : : : ; c; (16)

where

Ri =[Ci(Y; X )]2

Vi(X )Vi(Y ); wi =

∑nj=1 �

mij∑c

i=1

∑nj=1 �

mij; (17)

Ci(Y; X ) =

∑nj=1 �

mijmllrrj

3∑nj=1 �

mij

−∑nj=1 �

mijmtyj

∑nj=1 �

mijmxj

3(∑n

j=1 �mij

)2 ; (18)

Vi(Y ) =

∑nj=1 �

mijm

2tyj

3∑nj=1 �

mij

−(∑n

j=1 �mijmtyj

3∑nj=1 �

mij

)2

: (19)

It is noted that the criterion R2(c) for the determination of optimal c could be a good validity indexin the cluster-wise fuzzy regression because of the determination coeEcients R2

i being commonlyused as the measure of goodness in the regression model Ctting.

4. Case study results

The Song-Shan regional supply system described in Section 2 is studied in this section. Theweekday measurement data from January 10 to 14, 2000, are used for obtaining the loss formulaswhile those from January 17 to 21, 2000, are used for verifying the accuracy of the loss formulas.

The FCN algorithm is used Crst to partition the daily energy loss pattern into three clusters(peak, medium-peak and oI-peak levels). As shown in Table 1, these three clusters include 7segments. However, diIerent segments in the same cluster are found to have discriminatory losscharacteristics. More speciCcally, 8th–11th (6th segment), and 17th and 18th (7th segment) hours arein the same cluster (peak loss); however, these two segments have discriminatory loss characteristicsresulting from diIerent customer kW h consumption behavior. Consequently, the loss formulas foreach segment should be developed individually.

The 6th segment covering the 8th–11th hours is used to demonstrate the loss fuzziCcation here.There are 20 sets of data for regression because there are 5 weekdays and 4 operation conditions(hours) in the 6th segment. The maximum (minimum) value among these 20 energy losses is theright (left) bound for each measured value. The measured value is considered as the mean value.


Fig. 2. Energy loss w.r.t. real and reactive loads.

Table 2Peak-loss formulas for the 6th and 7th segments

Segments Hours Peak-loss formulas (kW h) Optimal c

6 8, 9 Loss = 2:3076 + 0:1763 kW h + 0:0010 kV A R h 210, 11 Loss = 45:6893 − 0:2667 kW h + 0:0017 kV A R h

7 17, 18 Loss = 21:8293 − 0:0478 kW h + 0:0011 kV A R h 1

Fig. 2 shows the kW h losses with respect to the kW h and kV A R h loads for the 6th segment.It can be found the losses are not clustered together in a group: one sub-cluster is for hours 8 and9 and the other sub-cluster is for hours 10 and 11. Stage two and Stage three, therefore, are usedto develop the kW h loss formula for each segment. The peak loss formulas are shown in Table 2.

There are at least three reasons to indicate that the results shown in Table 2 are feasible toengineers:

(i) It is expected that the number of the loss formulas is as small as possible and the form ofthe formula is as simple as possible to engineers. In the results, there are 16 (4, 9 and 3for the oI-peak, medium-peak and peak losses, respectively; rather than 24) concise formulasobtained for the 7 segments. Each segment is veriCed by R2(c) to develop optimal number ofloss formulas. The engineer can locate a corresponding formula for each hour easily.

(ii) The algorithm is implemented in a PC with Pentium III 450. The CPU time for computing thecoeEcients of a formula is less 5 s on the average in MATLAB 5.3.

(iii) The accuracy of the proposed method is acceptable. The approach for calculating the accuracyis described as follows:


Table 3Errors for the 7 segments

Segments Min. error Ave. error Max. error

1 0.0188 0.0453 0.11162 0.0020 0.0138 0.03583 0.0059 0.0174 0.04834 0.0069 0.0266 0.06585 0.0229 0.0290 0.03776 0.0203 0.0404 0.07487 0.0043 0.0176 0.0336

The realistic data from January 17 to 21, 2000, are served for veriCcation of the accuracy forall loss formulas. Let (T2; T1; T3)LR and (F2; F1; F3)LR be the true (measured) and estimated fuzzyenergy losses, respectively. The assessment index is deCned as follows:

Error ≡√

(T1 − F1)2 + (T2 − F2)2 + (T3 − F3)2√(T1)2 + (T2)2 + (T3)2

: (20)

Table 3 shows the minimum, average and maximum errors for the 7 segments. It is found thatthe maximum error occurred in the 1st segment. The average errors for the rest of the segmentsare less than 5%; most of them are close to 2%. The accuracy could be improved by taking moremeasured data into account for the fuzzy multiple linear regression.

5. Conclusions

A three-stage method based on the FCN algorithm and the fuzzy multiple linear regressionis proposed for developing energy loss formulas in distribution systems. The FCN is adoptedto partition the daily loss pattern into some segments. The three-stage method is presented be-cause the loss pattern is scattered. The optimal cluster number in a segment can be veriCed byR2(c) values. A typical distribution system with realistic operational data in the Song-Shan regionalsupply area, Taipei, Taipower is used to validate the performance of the proposed method. WeCnd that the proposed three-stage method can well estimate the system kW h losses in electricdistribution systems.

Appendix A.

An LR-type fuzzy number can be expressed with (m; �; �)LR where m; �, and � represent themean, left spread and right spread, respectively [19]. If an LR-type fuzzy number is expressed with


the triangle membership function �T, then

�T(x) =

L(x) = 1 − m− x

�for x 6 m;

R(x) = 1 − x − m�

for x ¿ m(A.1)

and the variables l and r used in Eq. (2) are deCned as follows [17]:

l =∫ 1

0L−1(x) dx; (A.2)

r =∫ 1

0R−1(x) dx; (A.3)

where L−1(x) and R−1(x) are the inverse functions for L(x) and R(x), respectively.

Appendix B.

This appendix provides an example for the solution process of FCN. Let �LR= {X1; X2; : : : ; Xn}be a set of LR-Type fuzzy numbers. �LR includes the fuzziCed kW h loss X1; X2; : : : ; Xn and n= 24.

�LR =

0:533 1:693 1:6935:206 1:454 1:4546:059 1:411 1:4116:341 1:373 1:3736:571 1:331 1:3319:381 1:368 1:36810:347 1:467 1:46716:812 1:604 1:60419:237 1:931 1:93116:803 2:184 2:18415:850 2:221 2:2219:776 2:376 2:3768:546 2:291 2:29112:117 2:272 2:27212:006 2:290 2:29012:061 2:313 2:31315:742 2:381 2:38115:253 2:599 2:5998:638 2:621 2:6218:455 2:571 2:5717:528 2:519 2:5194:634 2:377 2:3771:545 2:145 2:1451:684 1:936 1:936

LR

:


Suppose that C = 3; m= 2; ,= 0:001. Let h= 0 and the element values of �(0) are given randomlyas follows:

�(0) =

0:2900 0:2756 0:43440:2296 0:2080 0:56240:1977 0:1755 0:62680:1832 0:1612 0:65560:1694 0:1479 0:68270:5262 0:2991 0:17470:2244 0:7661 0:00950:3686 0:3966 0:23470:3617 0:3816 0:25670:3688 0:3966 0:23460:3724 0:4055 0:22210:7742 0:2048 0:02100:0434 0:0317 0:92490:3881 0:4935 0:11840:3876 0:4997 0:11270:3879 0:4964 0:11560:3729 0:4066 0:22050:3751 0:4121 0:21270:0937 0:0671 0:83920:0363 0:0270 0:93670:0804 0:0664 0:85320:2442 0:2232 0:53260:2836 0:2678 0:44860:2825 0:2666 0:4509

;

�(0)w = [2:201 1:729 1:957];

�(0)w = [1:891 2:169 2:060]:

Then (m(1)w ; �

(1)w ; �

(1)w ) will be:

m(1)w = [11:147 11:790 7:322];

�(1)w = [2:180 1:837 2:078];

�(1)w = [1:974 2:131 2:147]


and

�(1) =

0:2328 0:2039 0:56330:1064 0:0845 0:80910:0596 0:0456 0:89480:0435 0:0328 0:92370:0312 0:0232 0:94560:4497 0:2270 0:32330:7373 0:2105 0:05220:3738 0:4910 0:13520:3743 0:4514 0:17440:3739 0:4911 0:13500:3699 0:5157 0:11450:5808 0:2443 0:17490:1647 0:1000 0:73530:0832 0:9131 0:00370:0526 0:9455 0:00190:0701 0:9271 0:00290:3692 0:5188 0:11200:3649 0:5351 0:10000:1948 0:1166 0:68860:1407 0:0869 0:77240:0051 0:0035 0:99150:1322 0:1069 0:76090:2176 0:1880 0:59450:2152 0:1855 0:5993

; When h= 32; �(32) =

0:0573 0:0162 0:92650:3159 0:0356 0:64850:5643 0:0445 0:39120:6448 0:0438 0:31130:7062 0:0419 0:25190:9874 0:0062 0:00640:9134 0:0545 0:03210:0053 0:9930 0:00170:0738 0:8977 0:02850:0044 0:9943 0:00140:0038 0:9951 0:00110:9686 0:0172 0:01420:9935 0:0024 0:00410:5867 0:3468 0:06660:6132 0:3204 0:06640:6001 0:3334 0:06650:0063 0:9920 0:00170:0261 0:9673 0:00660:9943 0:0022 0:00350:9885 0:0041 0:00740:9035 0:0232 0:07340:1784 0:0242 0:79750:0207 0:0052 0:97410:0163 0:0040 0:9797

and

|�(31) − �(32)| = 0:0007¡ ,; the algorithm stops:

It can be found that X3–X7, X12–X16 and X19–X21 have strong relation in the 1st cluster (column)due to their larger membership values compared with the values of the other columns. Moreover,the membership values of X8–X11 and X17; X18 are close to unity in the second column. Finally,X1; X2; X22–X24 are found to be in the third cluster because they have membership values of 0.9265,0.6485, 0.7975, 0.9741 and 0.9797, respectively. The result is summarized in Table 1.

Appendix C.

mx1 − l�x1 = mlx1; mx1 + l�x1 = mpx1;mx1 + r�x1 + mrx1; mx1 − r�x1 = mqx1;mx1 + mlx1 + mrx1 = mtx1; mx1 + mpx1 + mqx1 = mox1;m2x1 + m2

lx1 + m2rx1 = m2

tx1; m2x1 + m2

px1 + m2qx1 = m2

ox1;


mx2 − l�x2 = mlx2; mx2 + l�x2 = mpx2;mx2 + r�x2 = mrx2; mx2 − r�x2 = mqx2;mx2 + mlx2 + mrx2 = mtx2; mx2 + mpx2 + mqx2 = mox2;m2x2 + m2

lx2 + m2rx2 = m2

tx2; m2x2 + m2

px2 + m2qx2 = m2

ox2;mx1my + mlx1mly + mrx1mry = mllrr1;mx1my + mpx1mly + mqx1mry = mplqr1;mx2my + mlx2mly + mrx2mry = mllrr2;mx2my + mpx2mly + mqx2mry = mplqr2;mx1mx2 + mlx1mlx2 + mrx1 + mrx2 = mlxrx;mx1mx2 + mpx1mlx2 + mqx1 + mrx2 = mplqr;mx1mx2 + mpx1mpx2 + mqx1mqx2 = mpxqx:

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fuzzy logic in distribution network

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