Eigenvalue approach clustering algorithm for building equivalent models of distribution systems

S.S. Fouda M.M.A. Salama A. Vannelli A.Y. Chikhani

Indexing terms: Clustering algorithm, Distribution systems, Eigenvalue approach

Abstract: A method for representing a large dis- tribution system is developed. The method is based on utilising the clustering technique to build an equivalent distribution system. The behaviour of the original large system may then be studied by analysing the equivalent (simpler) system, with less computational time and with high accuracy. Such behavioural features may include voltage regulation, equipment loading and total system losses. The use of the equivalent system leads to saving in both computer and distribution systems operator@) time. The use of the clustering tech- niques significantly reduces the complexity of the problem yet, at the same time, provides very accu- rate results.

1 Introduction

The behaviour of large distribution systems may be studied by analysing the whole system or through indi- vidual feeders depending on the desired system features. For instance, studying equipment loading, system voltage regulation or total system losses requires that the system be analysed as a whole.

Analysing distribution systems involving hundreds of feeders computationally could be very expensive, espe- cially if different policies have to be evaluated. Methods of modelling distribution systems with the purpose of reducing their complexity are therefore, desirable. Study- ing an equivalent (simpler) system that maintains the characteristics of the original large distribution system results in a faster and more economical analysis of the whole system.

Willis et al. [l] proposed the use of a K-means algo- rithm for clustering large numbers of feeders into groups with similar characteristics. For each group, an average feeder is selected to represent the group in a representa- tive model. These feeders are then represented as con-

0 IEE, 1995 Paper 17546 (P9), first received 26th April and in revised form 21st November 1994 S.S. Fouda, M.M.A. Salama and A. Vannelli are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 A.Y. Chikhani is with the Department of Electrical and Computer Engineering, Royal Military College, Kingston, Ontario, Canada K7K 5LO

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nected to a single substation. The behaviour of the representative system is then studied instead of the whole system. It was demonstrated that such a procedure results in a reasonably accurate prediction of the behav- iour of distribution systems.

In this paper, the problem of the equivalent distribu- tion system, tackled in Reference 1 is treated differently. An alternative mathematical model is used for the analysis, namely, an eigenvalue approach clustering algo- rithm to reduce the complexity of distribution systems. The equivalent model produced by the clustering algo- rithm is then studied as being representative of the whole distribution system. The study in hand differs from that in Reference 1 in the following aspects:

(i) A two-level clustering approach is applied. This makes the modelling of any distribution system relatively easy.

(ii) An efficient clustering algorithm, namely, an eigen- value based approach, is used. Therefore, the proposed approach may result in a further reduction in computer time.

2 Problem definition

Consider the distribution system in Fig. 1. There are three sets of substations distributed according to their

l l O k V ? I TL I TL I TL I + + +

Fig. 1 Actual distribution system

voltage levels (i.e. 110 kV, 60 kV and 35.5 kV). Each substation has a number of feeders connected to it. Substations are connected via transmission lines.

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The system in Fig. 1 is reduced to that in Fig. 2, by clustering substations according to their voltage levels to three sets (in general an n-set, depending on the system

a 110 k V

- 0

34.5 k V 7

link 23 link 12

60kV - fe‘eder ’

Fig. 2 Equivalent distribution system L? actual system b equivalent system

voltage levels), grouping transmission lines between each set of substations into a single transmission line, and clustering the feeders of each voltage level into a number of sets depending on the feeder data. These groups of feeders are then connected to the substation with the same voltage level. In essence, we apply a two-step modelling method:

(i) grouping substations according to their voltage level; and

(ii) clustering feeders and transmissions lines according to the system’s data (e.g. length, load, voltage level, etc.).

By noting that the first step is deterministic (i.e. depends only on the voltage level), the problem then reduces to the clustering of feeders and transmission lines. In doing so, the voltage level for each feeder must he maintained.

To meet the above constraint, a procedure involving three steps of clustering is adopted

(i) clustering all feeders and transmission lines into three sets according to their voltage level

(ii) separating transmission lines from feeders by noting the different characteristic impedance of the line involved

(iii) clustering the feeders into subsets (groups) accord- ing to the system’s data.

To formulate the problem, the original distribution system and its equivalent system, Figs. 1 and 2, respec-

a b C

d 0 b C

d

0 b C

d

a

b

Fig. 3

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Graph model of distribution system

\

tively are modelled by a graph of modules and nets as shown in Fig. 3. In Fig. 3a, the 110 kV substation is con- nected via three separate transmission lines to three dif- ferent substations of 69 kV each. In turn, each 69 kV station is connected to a 34.5 kV station via a single transmission line, and to some feeders. Finally, each 34.5 kV substation is connected to a number of feeders. In Fig. 3b, an equivalent graph obtained by the pro- cedure above is shown, where substations of the same voltage level are grouped together. Note that transmis- sion lines connecting classes of substations are grouped together (e.g. the lines between modules 1 and 2). Feeders (shown connected to module 3) may be clustered into two or more groups.

It has to be noted that the proposed method is used for a radial distribution system. However, this method could be applied to the interconnected distribution system. In this case, an extra group of feeder types will be introduced, which will represent the interconnected link between the busses of the same voltage.

Since the clustering of substations is deterministic, the problem of clustering the feeders is discussed in more details in the following Section.

3 System model

To effectively represent the distribution feeder by an equivalent model, two aspects have to be dealt with: the data model and the clustering algorithm (an eigenvalue based approach in our case). These are explained in the following Subsections.

3.7 Data model The characteristics of a feeder can be represented by a set of variables, such as the feeder resistance (ohm), feeder reactance (ohm), active power (kW) and reactive power (kvar). These variables are of different units and could hardly be related in their original form to a clustering method.

A standard way of expressing similarity between differ- ent variables is through a set of distance functions between pairs of objects. Sometimes the choice of the appropriate distance function is no less important than the choice of variables. The distance function known as the Euclidean distance [2] is used in this paper. In the next Section, the application of the Euclidean distance to the feeder problem is presented. Let N be the number of feeders (cases) and M be the number of variables charac- terising the N cases. Define A(I, J) as the value of the Jth variable for the Ith case

variables

1 rR(ohm) x(ohm) ~ ( k y ~ ( k v a r ) I

The Euclidean distance between case I and K is defined as the square root of the sum of the squared error between the two cases and is given by

The measure in eqn. 2 is the raw distance between the vectors corresponding to the Ith and Kth cases. If the variables are of different units, as is the case here, it is

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necessary to prescale the variables to make their values comparable. This can be achieved by requiring the use of a weight function for each variable, as follows

where W(J) is a weight function for variable (J). There are several viable weight functions defined in the

literature [2]. Determining the most suitable weight func- tion for a certain application may prove to be a compli- cated process. The coefficient of variation is chosen to be the weight function. Such a choice is motivated by the following

(a ) the coefficient of variation is dimensionless (has no units)

(b) the coefficient of variation is not dependent on the magnitude of the data, but rather on the data peaked- ness.

The coefficient of variation is defined as the ratio of the standard deviation (SD) to the mean value (m) and is given by

By choosing the weight function W(J) = CV(J), we can normalise the effect of every variable J on the distance function by dividing it by a measure of the variability of J , CV(J).

The weighted Euclidean distance of the data matrix (eqn. I), given by eqn. 3, is the symmetric ( N x N ) matrix

d 1 2 d13 ” ’ d I M

. . . . . . . . d,, . , . . . . . . .

Note that D(f, f ) = 0 for all I, as it is the distance between an identical case.

3.2 Clustering algorithm (eigen value approach) In general, a graph partioning problem for minimising the number of cut nets between K blocks, where block i has exactly mi modules, is N P complete [3]. To reduce the complexity of the problem, heuristic techniques were developed in the literature [4]. This technique is going to be used in this analysis as explained in the next Section.

The clustering algorithm used in this paper utilises: (a ) the constraint clustering approximation [3] and (b) the linear transportation approximation [4] to find

an equipartition of a netlist matrix Q where,

1 if module i is in net j

The next Section describes the approximations made to carry out the algorithm.

3.2.1 Constraint clustering problem approximation: This approximation depends on generating a measure- ment matrix (A) from the netlist matrix (Q) and mini- mising the sum of the elements of each of the K blocks of

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A, according to eqn. 7 K m - - 1 m

I = 1 i = l j = i + l min 1 1 a$)xilxjr

5 xi j = 1 for i = 1,2, ..., m

1 x i l = mj

j = 1

m for i = 1, 2, ..., K

i = l

xij E (0, 1) Eqn. 7 is equivalent to

K m - 1 m max 1 1 -a$)x i ,x j I

1 = 1 i = 1 j = i + 1

K

x i j = 1 for i = 1, 2, ..., rn j = 1

m C x i l = m j f o r j = 1 , 2 , ..., K

i = l

xij E { O , 1) The problem in eqn. 8 is still N P complete. To reduce. the conmplexity of the problem to polynomial time, a second approximation is applied, namely linear transportation.

3.2.2 Linear transportation problem approximation: Barnes [4] has shown that, by applying linear transpor- tation, an approximate polynomial time solution to the graph partioning problem can be obtained. Applying this to eqn. 8, yields the following

(9)

K

1 x i j = 1 i = 1, 2, ..., rn j = l

1 x . . = m. i = 1

j = l , 2 ,..., K

x i j 2 0

where uij is the ith element of the eigenvector correspond- ing to the largest eigenvalue in block j of the measure- ment matrix A (i.e. the weighted Euclidean matrix in our case), K is the number of blocks and mj the number of modules in block j.

Eqn. 9 can be solved in polynomial time, and has a 0-1 optimal solution. For the two-block equipartition case, the transportation problem above can be written as

,,, r .. 1

i = 1 I

“ m 2

E X i = -

X i l + x i2 = 1

xi E {O, 1) i = 1, ..., rn Eqn. 10 is solved by sorting the objective function coetli- cients (i.e. the eigenvector) from the largest to smallest, and setting x i to 1 for the first m, cases in the sorted list and 0 to the remaining ones.

The procedure proposed by Barnes [4] results in a good partition for a netlist. To find an optimal solution, we use the interchange method [SI.

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3.2.3 Interchange rnerhod: The interchange method [ 5 ] is a heuristic technique that moves modules from one group to another to reduce the number cutnets. The use of the interchange methods allows finding an optimal or a near optimal clustering of a netlist. A complete descrip- tion of the method and its applications can be found in Reference 5.

4 Procedure

Starting with a set of data variables for a number of feeders (modules), the modules were clustered into K clusters through a number of two-block clustering pro- cedures, as shown below

(i) Generate the weight vector W(J), for each variable ( J ) .

(ii) Generate the weighted Euclidean distance measure-

(iii) Find the eigenvalues and corresponding eigen-

(iv) Find the smallest and second smallest eigenvalues

(v) Define

ment (D).

vectors for the matrix D.

and their corresponding eigenvectors (i.e. v l , v2).

where

m, = number of modules (feeders) in block 1, and

m2 = number of modules (feeders) in block 2.

(vi) Sort {uI - u2} and {ul + u 2 } and generate a parti-

(vii) Select the case with the minimum number of cuts. (viii) Use the interchange method to find an optimal

(or near optimal) partition. (ix) Compute the average values for the data corre-

sponding to each block. (x) Use the average value of the data as input to the

load flow package to calculate the total system losses.

The results of the equivalent system are then compared to those of the actual distribution system. Our clustering approach is successful if both results are reasonably close. A flow chart of the procedure above is shown in Fig. 4.

Then obtain u1 - u 2 , as well as u1 + U,. tion for each case.

5 Results

Two test distribution systems were examined, one with 34 feeders and the other with 70 feeders. The data available for the feeders are the feeder resistance (R), feeder reac- tance (X), power (P) and rective power (Q). The pro- cedure in Section 4 is applied for three different conditions:

(a) variables used in the weighted measurement (b) number of blocks (equivalent feeders) (c) qumber of modules (feeders) in each block.

5.1 Variable selection The case of partitioning the 34-feeder distribution system into three blocks is first considered. The results are pre- sented as a percentage of the total losses of the equivalent distribution system. This percentage of the total losses is compared to the actual distribution system to calculate the percentage error of the total losses. Fig. 5 shows the percentage error of total losses of the equivalent distribu- tion system obtained from different weight measurements

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for both the {U, - U,} and {U, + u 2 } cases (uI, u2 are as defined in Section 4, step 5). The following observations are made

(i) Using X for the weight or both R and X results in the same percentage of the total losses. This implies that R is not a dominant variable.

(ii) This is further supported by noting that using X and P for the weight results in the same percentage of the total losses as using X, P and R.

(iii) Using P and Q only is similar to using R, X, P and Q (i.e. P and Q are dominant variables in this particular example).

(iv) The most accurate total losses calculated by using the equivalent distribution system is the case where (ul - u2) is used and the four variables are used in the weight measurement.

To investigate this matter further, a 70-feeder distribution system was tested. Fig. 6 shows the percentage error of total losses obtained from different weight measurements,

eliminate assigned compute eignvectors modules from the

reconfigure system by assigning modules to clusters

partitioning using interchange method

colculote overages of variables in each cluster

of equivalent system using load flow

Fig. 4 Flow chart of clustering method

a, Ln

0 - - 3 4 - I e ?5

$ 2 C b

R X P Q R,X X,P P,Q R.X.PX.P.Q RP.Q RXRQ $ 0

variables Fig. 5 block case 0 (U, - U,) - ( U * + 5)

Percentage error for different weight measurements, for three-

285

- 0 L e b a 2 m c

0, Y $0

R X P Q RX X,P P,Q RAP XP,Q R,P,Q RX.P.Q variables

Fig. 6 feeder distribution system 0 (U, -U*) - (5 + 4

Percentage error for different weight measurements, for 70-

P Q R. i

for three blocks. The results indicate that observations (i) and (ii) do not always hold. But observations (iii) and (iv) are the same as for the 34 feeders. This shows that, the more data available for the feeder, the more accurate the clustering is, and, subsequently, the more accurate is the total losses of the equivalent system.

5.2 Number of blocks Figs. 7 and 8 show the performance of the 34-feeder system and the 70-feeder distribution system under differ- ent number of blocks, respectively. From the Figs. 7 and 8, the following observations are made

(i) The case of {ul + U,} results in a worse calculation of total losses for the equivalent system.

P P.0 R,X variables

Fig. 7 Percentage error for different number of blocks, for 34-feeder distribution system

L Fig. 8 two blocks three blocks four blocks six blocks 0 za (", - " 2 ) - (U, +U,)

Percentage errorfor different number of blocks case, for 70-fi

286

n I .P XP.

.P P,Q R.X,P X.P,Q RRQ R.X.P,Q ibles ' distribution system

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(ii) The observation made regarding the dominance of the P and Q measures still hold in the two-block case.

(iii) The three-block case results in a clustering that is more representative of the actual distribution system. This is because variables P and Q, demonstrated to be the dominant ones, had three distinct ranges, and there- fore, three clusters seem the more natural choice.

For the 70-feeder distribution system, and as shown in Fig. 8, the six-feeder equivalent system provides the most accurate measurement of the total losses of the distribu- tion system. However, since the three-feeder equivalent system results in reasonably accurate measurements com- pared with the six-feeder system, the three-feeder equiva- lent system has to be chosen to represent the distribution system. This is because the aim of the clustering method is to reduce the complexity of the problem. Thus, if the three-feeder equivalent system gives a reasonable accu- rate measurement compared with the six-feeder equiva- lent system, it has to present the distribution system.

5.3 Number of modules Figs. 9 and 10 show the effect of the number of modules in each block for the 34- and the 70-feeder cases, respec- tively. Again, the two- and the three-block case, as well as the cases of (uI - u2) and (ul + uz) are shown.

As the feeder data seem to fall into three different

X P R.X X.P voriables

6 Complexity of the problem

In this Section, the effectiveness of using the clustering algorithm to reduce problem complexity is demonstrated.

Consider the problem of finding the total system losses in a distribution system. Typically, the Newton-Raphson method can be used to obtain the load flow, and then the system losses.

The computation time for the N-R method is of O(N3) [SI. The load flow goes through a number of iterations, I, of the N-R method until results converge. The complexity of the problem is, therefore, O(N31).

The application of the clustering algorithm described in the paper involves the ten steps in Section 4 (also see Fig. 4). Obtaining the weight vector (step 1) is of O(N). Generating the weighted Euclidian distance matrix is of O(N2). In general, finding the eigenvectors of a matrix is of O(Nz). However, if the measurement matrix is sparse, this can be done in O(N) [4].

The transportation problem (steps 4-7) involves search and sort operations. Without assuming any a priori knowledge of the data, this can be done in O(N log N) operations [7], whereas, the interchange method (step 8), which involves moving modules between different clusters is O(N log N). Finally, finding the average values of the variables in each cluster (step 9) is of O(N).

P. 0 R.X,P X,P,Q R.P,Q R.X.P.Q

Fig. 9 Percentage error of total lossesfor different number ofmodules,/or 34-feeder distribution system diKerent blocks equal blocks

two blacks three blocks two blacks three blocks 0 (", lBJBa sm3 ( V I +U,)

unequal sets, different block sizes cases are more repre- sentative of the actual system in hand. By having an equi- partition, we are forcing modules (feeders) that would have naturally fitted in another block into a wrong block.

Therefore, the choice of the number of blocks and the number of modules (feeders) in each block should be made in accordance with the distribution system to be modelled.

Steps 1-9, therefore, are of O(N2). For the case where the number of clusters, K > 2, the above procedure is repeated K - 1 times. The assigned models are removed from the distance matrix during each iteration. Hence the complexity of the clustering algorithm is of O(N2 log k).

The application of the load flow for the equivalent system using the Newton-Raphson method is of O(K31). Since K < N, we can assume that the complexity of the

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problem is O ( N Z log k) which is less than O(N31) of the of this simaler svstem is efficient and accurate. The aer- original system.

~I * , centage error of the total losses of the equivalent system

R,X R A P

Fig. 10 different blocks equal blocks

two blocks three blocks two blocks three blocks

Percentage error of total losses for different number of modules, for 7O-feeder distribution system - (U, - 4 - a m irnppDs - m ( U x + u.)

The computational efficiency of the proposed method is examined on the basis of the number of calculation required rather than the actual CPU time to make the comparison general and not restricted to a specific com- puter system.

7 Conclusions

This paper presents a clustering technique for represent- ing a large distribution system by an equivalent less com- plicated system using an eigenvalue approach.

The accuracy of the proposed eigenvalue clustering approachdependson

(i) The choice of variables in the weighted measure- ment. As the number of variables representing the actual system (R, X, P, Q) increases, so does the accuracy.

(ii) The number of blocks (equivalent feeders) used and the number of modules (feeders) in each block. The choice of the number of blocks (equivalent feeders) and the number of modules (feeders in each block was demonstrated to be best made in accordance with the dis- tribution system to be modelled.

(iii) The choice of adding or subtracting the eigen- vectors (i.e. {ul -U,} or {ul + U,}). It seems that sub- tracting the eigenvectors is more accurate for most cases.

It was found that representing the distribution system by an equivalent simpler system and studying the behaviour

to the actual distribution system was in the range of 1-3%. Yet, the complexity of the whole procedure is of the order O(N2 log k), which is about one order of magni- tude less than analysing the whole system using the Newton-Raphson method.

The authors believe that this method can be used not only to calculate the total losses of the actual distribution system but also to study other performance character- istics, such as maximum voltage drop for the entire system.

8 References

1 WILLIS, H.L., TRAM, H.N., and POWELL, R.W.: ‘A computerised, cluster based method of building representative blocks of distribution systems’, I E E E Trans., 1985, PAS104, (12), pp. 3469-3474

2 HARTIGAN, J.A.: ‘Clustering algorithms’ (John Wiley & Sons, 1975) 3 HADLEY, S.W., MARK, B.L., and VANNELLI, A.: ‘An efficient

eigenvector approach for finding netlist partitions’, I E E E Trans.,

4 BARNES, E.R.: ‘An algorithm for partitioning the nodes of a graph‘, S I A M J . Alg. Disc. Math., 1982, 3, (4), pp. 541-550

5 VANNELLI, A.: ‘An adaptation of the interior point method for solving the global routine problem’, I E E E Trans., 1991, CAD-10, (2). pp. 193-203

6 GERALD, C.F., and WHEATLEY, P.O.: ‘Applied numerical analysis’ (Addison-Wesley, 1985)

7 AHO, A.V., HOPCROFT, J.E., and ULLMAN, J.D.: ‘Data struc- tures and algorithms’ (Addison-Wesley, 1983)

1991, CAD-11, (7), pp. 885-892

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