topological and primitive impedance

IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 10, NO. 1, WINTER-SPRING 2011

1682-0053/11$20 2011 ACECR

10

AbstractIn this paper, an effective topological and primitive impedance based distribution power flow algorithm is developed for both balanced and unbalanced distribution systems. This method fully exploits the radial structure of the network and solves the distribution load flow directly. Using this concept and primitive impedances of the lines, only diagonal elements of the Distribution Load Flow (DLF) matrix are computed and stored in single dimension vectors to obtain the distribution load flow solution. Unlike other traditional methods, the proposed approach does not require any LU decomposition or Bus admittance matrix. It is robust, time-efficient and needs very less memory for any size of the distribution system. The proposed method is tested on a unbalanced practical Indian distribution system and also on various standard IEEE test systems including weakly meshed distribution system. Results are quite promising and the method has great potential application in the distribution automation.

Index TermsBalanced and unbalanced distribution load flow, radial and weakly meshed distribution system.

I. INTRODUCTION ISTRIBUTION Power Flow is an important tool for the analysis of distribution system and it is used in the

operational as well as in planning stages. Distribution system with their radial structure and wide ranging resistance and reactance values are inherently ill-conditioned and conventional load flow methods like Gauss-Seidel, Newton-Raphson and fast decoupled techniques are inefficient in solving such networks. The recent tendency towards the distribution automation (DA) has led the researchers to focus on robust and efficient load flow methods. Chiang [1] has presented a distribution load flow method by iterative solution of three fundamental equations representing real power, reactive power and voltage magnitudes. Das, et al. [2] have proposed a load flow method by writing an algebraic equation for bus voltage magnitude. The Gauss Implicit Z-matrix method [3] is one of the most commonly used methods, however this method does not effectively exploit the radial structure of the distribution system and therefore, needs to find the solution of a set of equations of the order of number of

Manuscript received August 25, 2009; revised January 4, 2011. This work was supported in part by the Faculty of Engineering-Sohar

University, Oman. K. Prakash is Research Scholar in National Institute of Technology,

Warangal, A. P., India. (e-mail: [email protected]). M. Sydulu is with the Department of Electrical Engineering, National

Institute of Technology, Warangal, A. P., India. (e-mail: [email protected]).

Publisher Item Identifier S 1682-0053(11)1831

buses. Some of the researchers proposed special load flow techniques based on topological characteristics of distribution system [4]-[5]. Most of these techniques require new data format or some data manipulations. Compensation based approach is proposed in [6] to solve the distribution load flow problems in which forward/backward sweep algorithm is adopted. A feeder- lateral based model is reported in [7] requires layer-lateral based format. Jabr [8] has formulated the distribution load flow problem as Conic Programming based Convex Optimization Problem. Hamounda and Zeher [9] have proposed a distribution load flow based on Kirchhoffs laws characterized by radial configuration and laterals. Singh, et. al. [10] presented a load flow solution for radial and weakly meshed distribution system formulated as an optimization problem solved by Primal dual Interior point method. In the paper [11] for a balanced radial distribution system, the proposed load flow algorithm requires formation of bus-injection to branch current (BIBC) matrix with 1s & 0s as elements and branch-current to bus-voltage (BCBV) matrix with primitive impedances as elements & distribution load flow (DLF) matrix. DLF matrix is obtained as product of (BCBV) and (BIBC) matrices. These three matrices require large memory space when the proposed method applied for bigger distribution system. Further, these three matrices contain more number of zero elements and hence memory space is not utilized economically, especially for large size distribution networks. Another negative aspect of it is that, to obtain a load flow solution it needs direct multiplication of BCBV & BIBC matrices and DLF & current Injection column vector matrices. This requires sufficiently large CPU time.

In this paper a classical but novel technique is proposed with effective data structure and implementation approach. The main aim of this paper is to develop a new formulation for load flow method, which exploits the topological characteristics of a balanced distribution system. A unique effective data structure is proposed to identify all those lines that are traced in the path connecting the feeding bus and any selected bus. This feature acts as a potential support in solving the distribution load flow equations derived in terms of primitive impedances of the lines. Unlike to other traditional methods, the proposed approach does not require any LU decomposition or Bus admittance matrix. Compared to the algorithm reported in [11], the proposed method does not require any direct matrix multiplications and no need of formation of BIBC, BCBV and DLF matrices. It only requires the calculation of diagonal

Topological and Primitive Impedance based Load Flow Method for Radial and Weakly

Meshed Distribution Systems K. Prakash and M. Sydulu

D

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PRAKASH AND SYDULU: TOPOLOGICAL AND PRIMITIVE IMPEDANCE BASED LOAD FLOW METHOD FOR

11

1 2 3

4 5

6

B1 B2

B3 B4

B5 I2 I3

I4 I5

I6

Sz12 z23

z34 z45

z36

Fig. 1. A Sample 6-Bus radial distribution system. elements of DLF matrix in terms of the primitive impedances of the lines. In this proposed algorithm, sparsity technique [12] is used to identify the adjacent buses and adjacent lines of any selected bus of the system. The proposed distribution load flow method is robust, time-efficient and needs very less memory even for a large size distribution system.

II. STRUCTURE OF VARIOUS MATRICES IN PROPOSED LOAD FLOW METHOD

It has been mentioned in the earlier sections about the BIBC, BCBV and DLF matrices, which were explained in the paper [11]. Consider a sample radial distribution system as shown in the Fig. 1 for the formation of the above matrices

A. Bus Injection to Branch Current (BIBC) matrix For distribution systems, at a bus i the complex load is

expressed by (1) And the corresponding equivalent current injection is given By (2)

( ) i = 1,2,...,ni i iS P jQ (1)

( )i i iI S V

(2)

The vector of current injections for the above sample system is given as below

Bus No. 2 3 4 5 6 Current Injection 2I 3I 4I 5I 6I For the system shown in Fig. 1, apply Kirchhoffs

current law (KCL), the branch currents can be expressed in terms of equivalent current injections as

1 2 3 4 5 6B I I I I I (3)

2 3 4 5 6B I I I I (4)

3 4 5B I I (5)

4 5B I (6)

5 6B I (7)

1 2

2 3

3 4

4 5

5 6

1 1 1 1 10 1 1 1 10 0 1 1 00 0 0 1 00 0 0 0 1

B IB IB IB IB I

The above branch current equations can be rearranged in the generalized form as below

[ ] [ ][ ]B BIBC I (8)

B. Branch Current to Bus Voltage (BCBV) Matrix The relationship between the branch currents and bus

voltages are expressed as

2 1 1 12V V B z (9)

3 2 2 23V V B z (10)

4 3 3 34V V B z (11)

5 4 4 45V V B z (12)

6 3 5 36-V V B z (13) On Substitution of (9) & (10) in (11), the voltage at bus

4 is given by

4 1 1 12 2 23 3 34- - -V V B z B z B z (14) Similarly, the other bus voltages can be rewritten as

3 1 1 12 2 23V V B z B z (15)

5 1 1 12 2 23 3 34 4 45V V B z B z B z B z (16)

6 1 1 12 2 23 3 34 4 45 5 36 - - - - -V V B z B z B z B z B z (17) Equations (9), (14), (15), (16), (17) can be rearranged as

below

12 11 2

12 23 21 3

12 23 34 31 4

12 23 34 45 41 5

12 23 36 51 6

0 0 0 00 0 0

0 00

0 0

V V z BV V z z BV V z z z BV V z z z z BV V z z z B

[ v] = [BCBV] [B]' (18) Now, substitute (8) in (18) and the resulting equation is

expressed as [ v] = [DLF][I]' (19)

where [DLF] represents distribution load flow matrix given as

12 12 12 12 12

12 12 23 12 23 12 23 12 23

12 12 23 12 23 34 12 23 34 12 23

12 12 23 12 23 34 12 23 34 45 12 23

12 12 23 12 23 12 23 12 23 36

z z z z zz z z z z z z z zz z z z z z z z z z zz z z z z z z z z z z zz z z z z z z z z z

From the above DLF matrix, the following useful observations are used in developing the proposed topological and primitive based distribution load flow method

1. All elements of DLF matrix ( -1) ( -1)n nu are complex non-zero and symmetric.

2. Diagonal elements are given by the sum of the primitive impedances of all those lines in the path connecting the substation bus and any selected bus.

3. Each bus-p of the network can have one unique path from substation bus.

4. Off-diagonal p-q elements are given by the sum of the primitive impedances of those lines which appear common to the paths of p and q buses from substation bus.

These observations are effectively used in proposing the

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Start

set counter cntr=1; ipathf[1]=1; ipathto[1]=1;

set the bus count i=2

Using sparsity technique [9], pick up the first adjacent bus-q of bus-i

if q=1

set counter j=1

no

cntr=cntr+1 pzc=pzc+1

ipathf[i ] = cntr pathsize[i]=pzc

j= j+1

ipathto[i ]= ipathf[i ]

yes

cntr =cntr+1; set pzc=1

i=i+1

j pathsize[q ]

yes

no

If i < n

yes

no

stop

pathsize[i] = pzc ipathto[i ] = cntr

i=i+1

Start

set pathline[1 ] =0

Set the bus count i=2.

Set j = ipathf [q]

pathline [cntr] = pathline[j ]

j ipathto [q]

yes

j = j+1

no

pathline[cntr] = k pathlinesize = cntr

If i < n

i = i+1

yes

stop

cntr= cntr+1

Set counter, cntr = ipathf [ i ]

Pick up the first adjacent bus-q and first adjacent line-k of the bus-i using sparsity

technique

no

If q=1

yes

no

Fig. 2. Flowchart of [ ]ipathf i and [ ]ipathto i vectors.

Fig. 3. Flowchart for formation of pathline vector.

TABLE I FORMATION OF PATHLINE VECTOR OF THE 6-BUS RADIAL SYSTEM

Bus no. 1 2 3 4 5 6 Location count =j 1 2 3 4 5 6 7 8 9 10 11 12 13 14

pathline (j) 0 1 1 2 1 2 3 1 2 3 4 1 2 5

algorithm with the help of sparsity technique that exploits the topological structure of the network.

III. PROPOSED SOLUTION METHODOLOGY The proposed method directly determines the

distribution load flow solution without formation of any one of the matrices like Bus-Injection to Branch Current (BIBC), branch-current to bus-voltage (BCBV) and distribution doad flow (DLF) matrices, but simply uses primitive Impedances of the lines. This new algorithm determines the diagonal elements of the DLF matrix in terms of the primitive impedances of the lines. Split multiplication concept is used not only to account the off-diagonal elements of the DLF matrix but also direct determination of the [ v]' elements of the equation [ v]=[DLF] [I]' without performing the multiplication of [DLF] and [I] matrices. The proposed approach offers very significant saving in computational burden as it avoids the

formation of BIBC, BCBV and DLF matrices without any sacrifice in the end results. The proposed method is very effective for small to large size distribution systems.

A. Formation of Proposed Single Dimensional Vectors A radial distribution network has a typical tree structure

and the root of the tree would be the feeding substation node 1. There would be a connecting path between node-1 and any other selected node. Trace all the ( -1n ) paths and identify those lines associated with each path. Store all such lines of each path in a single dimensioned path-line vector, as shown in the Table I. For example, the path between node-1 & node-6 is having three branches 1, 2 and 5 and these are stored in reserved locations 12, 13 &14 respectively. The location reservation information of each path is indicated by another two integer vectors [ ]ipathf i & [ ]ipathto i . For a bus i [ ]ipathf i and [ ]ipathto i vectors indicate the start and end reserved location numbers for path- i . The elements

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1 2 3

4 5

6

B1 B2

B3 B4

B5 I2 I3

I4 I5

I6

SB

z12 z23

z34 z45

z36

B6

: Tie Switch

: Current Injection

Fig. 4. Simple distribution system with one loop.

TABLE II FORMATION OF IPATHF AND IPATHTO VECTORS OF THE 6-BUS RADIAL

DISTRIBUTION SYSTEM

Bus no. [i] [ ]ipathf i [ ]ipathto i 1 1 1 2 2 2 3 3 4 4 5 7 5 8 11 6 12 14

of these vectors for the given 6-bus system are presented in Table II. Figs. 2 and 3 shows the flow charts for the formation of [ ]ipathf i [ ]ipathto i and pathline vectors.

IV. ALGORITHM FOR PROPOSED DISTRIBUTION LOAD FLOW METHOD

1. Read the distribution system data and form sparsity vectors [12].

2. Form the [ ]ipathf i , [ ]ipathto i and pathline vectors.

3. For a bus - i , add the primitive impedances of all those lines that are stored in the pathline vector to obtain the diagonal element ( [ ]Zpp i ) of DLF matrix. Similarly, calculate the diagonal elements ( [ ]Zpp i ) for all other buses.

4. Initialize the Bus Voltages to 1 0j p.u. 5. Calculate the Power Injections and Current Injections

[ ]I i at all the buses. 6. Initialize the iteration count 1k . 7. Assign [ ] [ ]oldI i I i for all the buses. 8. Using the vectors determined in the step 2, calculate

the [ ]v' elements of the equation [ ] [ ][ ]v DLF I' , for a bus i using the equation given below

2,

[ ]* [ ]+ [( 1)* [ ] ( 2)* [ ] ...]n

j j iZpp i I i Zline I j Zline I j

z

where Zpp is Diagonal element of DLF matrix, [ ]I i , [ ]I j are Current Injections at bus i & bus j ,

Zline1, Zline2 are primitive impedances of set of those lines which are common in the path-i and path-j

respectively. 9. Update the bus voltages at all the buses.

[ ] [ ] [ ][ ]V i V i v i ' 10. Calculate the current Injections [ ]I i with the

updated bus voltages. 11. If max 1(| [ ] | - | [ ] |)k kI i I i >tolerance, then advance

the iteration count and go to step 7. 12. Print the converged load flow solution and Stop.

V. PROPOSED SOLUTION METHODOLOGY FOR WEAKLY MESHED NETWORKS

Some of the distribution feeders are serving high density load areas created by closing tie-switches which are normally kept in open position. The proposed method can also be extended to weakly- meshed distribution feeders. Existence of loops in the system does not effect the bus current injections, but new branches will need to be added to the system.

When a new branch say 6B is added to the sample system in Fig. 1 forms a mesh as shown in the Fig. 4, the new current injections at bus-5 and bus- 6 are given by

5 5 6I I Bc (20)

6 6 6I I Bc (21) Due to the above new current injections at bus-5 and

bus-6, the BIBC, BCBV and DLF matrices are modified. As discussed earlier, the proposed method does not require to form the modified BIBC and BCBV matrices, only requires to modify the DLF matrix.

A. Modification of Proposed DLF Matrix To account the possible modifications for a new branch

kB between i and j buses, the following is to be carried out based on the observations of the elements of the matrix shown in (22). If a new branch kB is added between the bus i and bus j , then subtract the elements of the j th column from the elements i th column in the existing DLF matrix (19) and fill these resultant values in the off-diagonal positions of the new k th column and k th row of the DLF matrix respectively. Finally add all those primitive impedances of the branches which are involved in forming the weakly mesh and fill it as the diagonal element of the k th row of the DLF matrix (22).

The modified DLF matrix for the sample system with one loop shown in the Fig. 4 is expressed in (22).

The general form of the load flow equation using the modified DLF matrix is expressed as

> @mod0 kIV

DLFB

' (23)

12 12 12 12 12

12 12 23 12 23 12 23 12 23

12 12 23 12 23 34 12 23 34 12 23 34

12 12 23 12 23 34 12 23 34 45 12 23 34 45

12 12 23 12 23 12 23 12 23 36 36

34 34 45 36 34 4

00

0 0

z z z z zz z z z z z z z zz z z z z z z z z z z zz z z z z z z z z z z z z zz z z z z z z z z z z

z z z z z

5 36 56z z z

(22)

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Fig. 5. Three phase line section model.

0.75

0.8

0.85

0.9

0.95

1

1.05

1 2 3 4 5 6 7 8 9 10Bus Number

Volta

ge in

p.u

Fig. 6. Load flow result of 10-bus radial distribution system.

In (23), the element concerned to the newly added kB branch in the [ ]V' portion is taken as zero since there is no extra node added in the system. Now by Applying, Krons reduction technique [11], the additionally added k th row and k th column of the above modified DLF matrix, can be eliminated and the resulting equation is given below as

> @ > @ > @modV DLF I' (24) In this proposed solution technique for weakly meshed

systems requires minor modification in the DLF matrix. Therefore, the proposed method can offer the load flow solution very effectively for weakly meshed distribution systems.

VI. PROPOSED UNBALANCED THREE PHASE DISTRIBUTION LOAD FLOW METHOD

Fig. 5 shows a three-phase line section between bus- i and bus- j . The primitive impedance of this line can be solved using method developed by Carson and Lewis [13]. A 4 4u matrix which takes into account the self and mutual coupling effects of the unbalanced three phase line section is expressed as

> @aa ab ac an

ba bb bc bnabcn

ca cb cc cn

na nb nc nn

z z z zz z z zZz z z zz z z z

(25)

After Krons reduction technique [13] is applied, the above 4 4u matrix is then reduced to phase impedance 3 3u matrix which includes the effects of the neutral or ground wire as shown as

> @aa n ab n ac n

abc ba n bb n bc n

ca n cb n cc n

z z zZ z z z

z z z

(26)

The relation between the bus voltages and branch currents of the three phase line can be expressed as

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Bu Number

Volta

ge in

p.u


0.75

0.8

0.85

0.9

0.95

1

1.05

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Bus Number

Volta

ge in

p.u


TABLE III RESULTS OF THE PROPOSED METHOD TESTED ON STANDARD SYSTEMS

Test System Active Power Loss (kW) Reactive Power Loss

(kVar) 10 bus 783.756 1036.45 15 bus 61.78 57.28 31 bus 1104.03 1038.57 34 bus 221.67 65.09 69 bus 224.97 102.15 85 bus 315.60 198.29 119 bus 1296 978.04

aa n ab n ac n Aaa A

ba n bb n bc n Bbb B

ca n cb n cc n Ccc C

V V z z z IV V z z z IV V z z z I

(27)

In a three-phase line if any of the phases are not present, then its corresponding row and column elements in the matrix contains zeros.

The above proposed algorithm can be extended to a multi phase line section or bus easily. For example, if a three-phase line connected between bus- i and bus- j , then its corresponding branch iB of [ ]B matrix will be a 3 1u vector, +1 element in the BIBC matrix is replaced by a 3 3u identity sub matrix and ijz element in the BCBV matrix is replaced by 3 3u primitive line impedance matrix. Similarly the elements at each bus of DLF matrix are 3 3u sub matrices.

VII. TEST RESULTS The proposed radial distribution load flow program is

implemented using Turbo C++ language and tested on Windows-XP based Pentium-IV computer with 2.40GHz and 256 MB RAM. Two methods were used in the tests, and the convergence tolerance was at 0.001 p.u. Method 1: The method in reported by Jen- Hao Teng [11]. Method 2: The proposed algorithm.

The above methods are compared on the basis of their accuracy and performance parameters like execution time and iterations.

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0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Bus Num ber

Vol

tage

in p

.u


0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67Bus number

Volta

ge in

p.u


0.8

0.85

0.9

0.95

1

1.05

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85Bus Number

Vol

tage

in p

.u


TABLE IV COMPARISON OF EXECUTION TIME AND NUMBER OF ITERATIONS

Test System

Method-1 Execution Time (ms) and No. of Iterations

Method-2 Execution Time (ms) and No. of Iterations

10 bus 3.861 6 2.178 6 15 bus 5.742 3 2.673 3 31 bus 6.039 5 4.554 5 34 bus 7.530 3 5.445 3 69 bus - - 6.732 4 85 bus - - 7.634 4 119 bus - - 7.864 5

A. Test Results of Balanced Radial Distribution Systems The proposed distribution load flow method was tested

on IEEE 10 bus [14], IEEE 15 bus [2], IEEE 31bus [15], IEEE 34 bus [16], IEEE 69 bus [17], IEEE 85 bus [2], and 119 bus [18] radial distribution systems and the load flow results obtained are shown in Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11 and Fig. 12 respectively. Table III shows the active power and reactive power losses occurred in the various test systems. Voltage profile of various buses and losses in both methods are identical. Table IV shows the execution time and number of iterations taken by the proposed method and the method reported in [11] on various balanced test systems. It is to be noted that the proposed method is faster than the other method [11]. There was a compiling problem while using the method in [11] to obtain the load flow solution for 69 bus, 85 bus, and 119 bus systems, indicating the large size of two dimensional array declarations of BIBC, BCBV and DLF


Fig. 13. IEEE 33-Bus distribution system with 5 tie-lines.

TABLE V LOAD FLOW RESULTS OF 33-BUS DISTRIBUTION SYSTEM

Bus Number

Bus Voltage magnitude(p.u )for

radial system

Bus Voltage magnitude (p.u ) for weakly meshed

system 1 1 1 2 0.997033 0.997088 3 0.982943 0.986074 4 0.975465 0.982169 5 0.968071 0.978491 6 0.949678 0.969843 7 0.946194 0.968875 8 0.941352 0.967647 9 0.935086 0.963308

10 0.929274 0.961969 11 0.928414 0.96183 12 0.926915 0.961698 13 0.920805 0.958822 14 0.91854 0.95789 15 0.917128 0.957766 16 0.915761 0.956115 17 0.913735 0.953059 18 0.913128 0.952117 19 0.996505 0.995388 20 0.992927 0.981337 21 0.992223 0.977419 22 0.991585 0.974856 23 0.979358 0.980707 24 0.972687 0.97026 25 0.969363 0.96319 26 0.94775 0.968753 27 0.945187 0.967359 28 0.933753 0.961553 29 0.925539 0.957572 30 0.921983 0.954292 31 0.917824 0.950887 32 0.916909 0.950235 33 0.916626 0.950306

Active power loss(KW) 202.659 118.379578

Reactive power loss (KVar) 135.13 81.407607

matrices which contain complex elements. But this problem is not present in the proposed method for the same above test systems. In fact, the sparsity based vectors and ipathf , ipathto , and pathline vectors are acting as potential support in the proposed algorithm to reduce execution time & memory requirement for large distribution systems. This aspect would go as strength of

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1a

1b

1c

2a

2b

2c

3a

4b

4c

5b

6c

7c

8c

sub-stn bus

Fig. 14. 8-Bus three phase distribution system.

Fig. 15. Convergence characteristic of 8-bus system using proposed method.

TABLE VI COMPARISON OF TEST RESULTS OF IEEE 33-BUS SYSTEM WITH 5 TIE LINES

IEEE 33-bus [20] Distribution System

Primal Dual Interior Method [10]

Proposed Method

Execution Time (s) 0.26 0.0005 No of Iterations 10 3

Convergence Criteria 0.001 0.001

TABLE VII RESULT OF THE PROPOSED METHOD TESTED ON IEEE 8-BUS SYSTEM

Bus Number Voltage Magnitude(p.u.) Angle (rad) Phase

1 1.000000 0.00000 A 1 1.000000 -2.094408 B 1 1.000000 2.094408 C 2 0.983917 0.003197 A 2 0.971066 -2.090163 B 2 0.969667 2.093914 C 3 0.983178 0.003137 A 4 0.965161 -2.089749 B 4 0.966824 2.093230 C 5 0.963999 -2.089832 B 6 0.964839 2.093065 C 7 0.968319 2.093794 C 8 0.967042 2.093661 C

the proposed algorithm without any convergence problems even on large distribution systems.

B. Test Results of Weakly Meshed Systems The proposed solution methodology for weakly meshed

networks is tested on the standard IEEE 33-bus distribution system having five tie-lines [19] shown in Fig. 13. Table V shows the obtained load flow results of 33-bus distribution system for both radial and weakly meshed systems. The obtained results are found similar to the results as reported in [10]. Table VI shows the comparison of the primal dual interior method [10] and the proposed method in terms of execution time and iterations taken by the methods. It clearly shows that the proposed method is much faster than the other method. The obtained load flow results for the same system are also compared with the results of Loop Based Load Flow method [20] but the authors have not mentioned the execution time and iterations taken by the system.

Fig. 16. A Practical distribution feeder in India.

TABLE VIII COMPARISON OF METHOD-1 AND PROPOSED METHOD-2

For 8-bus system Method -1 Method -2 Execution Time (ms) 3.168 0.891

No of Iterations 3 3 Convergence Criteria 0.0001 0.0001

TABLE IX

LOAD DATA FOR THE FEEDER SHOWN IN FIG. 16

Bus No.

Phase-A Phase-B Phase-C kW kVar kW kVar kW KVar

1 0 0 0 0 0 0 2 34.6 16.7 17.3 8.4 34.6 16.7 3 36.7 17.8 17.3 8.4 32.4 15.7 4 13.5 6.5 18.9 9.2 21.6 10.5 5 21.6 10.5 17.3 8.4 15.1 7.3 6 14 6.8 10.3 5 9.7 4.7 7 32.4 15.7 27 13.1 27 13.1 8 24.8 12 17.8 8.6 11.3 5.5 9 41 19.9 49.7 24.1 44.3 21.4 10 11.3 5.5 14 6.8 8.6 4.2 11 24.8 12 24.8 12 36.7 17.8 12 32.4 15.7 27 13.1 27 13.1 13 14.6 7.1 17.8 8.6 21.6 10.5 14 10.3 5 10.3 5 13.5 6.5 15 14.6 7.1 16.2 7.8 23.2 11.2 16 25.9 12.6 34.6 16.7 25.9 12.6 17 21.6 10.5 16.2 7.8 16.2 7.8 18 17.8 8.6 17.8 8.6 18.4 8.9 19 29.2 14.1 33.5 16.2 23.8 11.5

C. Test Results of Three Phase Distribution System The proposed method is tested on IEEE 8-bus three

phase unbalanced radial distribution system as shown in Fig. 14 and the load flow results obtained are shown in Table VII. The load flow results are found similar and accurate when compared to the results obtained using the method in [11]. Fig. 15 shows the convergence characteristic of 8-bus distribution system using proposed three phase load flow method. Table VIII shows the execution time and number of iterations taken by the proposed method and the method reported in [11] on the test system. It is observed that the proposed method is faster than the other method [11]. In fact, the sparsity based vectors and ipathf , ipathto , and pathline vectors are acting as potential support in the proposed algorithm to reduce execution time & memory requirement.

Fig. 16 shows the practical unbalanced three phase distribution feeder emanating from Pathardhi 132/11KV-Grid substation in India [21]. The Load data, Network data, and Line data are given in Table IX, Table X, and Table XI, respectively. Using the proposed method, for this practical system, shown in Fig. 16, the three phase load flow solution has taken only three iterations for

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TABLE X LINE DATA FOR THE FEEDER SHOWN IN FIG. 16

Line From Bus To Bus Length (km) 1 1 2 3 2 2 3 4 3 2 4 1.5 4 4 5 1.5 5 4 6 1 6 6 7 2 7 6 8 2.5 8 8 9 3 9 9 10 5 10 10 11 1.5 11 11 14 1 12 11 13 5 13 14 17 3.5 14 14 18 4 15 10 12 1.5 16 12 16 6 17 12 15 5 18 15 19 4

TABLE XI

PHASE IMPEDANCE MATRIX OF THE LINES FOR FEEDER SHOWN IN FIG. 16

Line No.

Phase Impedance Matrix (p.u.) For Base kVA=1000,BASE KV=11

1,8, 16

0.0387+j0.0167 0.0129+j0.0056 0.0129+j0.0056 0.0129+j0.0056 0.0387+j0.0167 0.0129+j0.0056 0.0129+j0.0056 0.0129+j0.0056 0.0387+j0.0167

2,7, 12

0.0323+j0.0139 0.0108+j0.0046 0.0108+j0.0046 0.0108+j0.0046 0.0323+j0.0139 0.0108+j0.0046 0.0108+j0.0046 0.0108+j0.0046 0.0323+j0.0139

3,10, 15

0.0193+j0.0083 0.0064+j0.0028 0.0064+j0.0028 0.0064+j0.0028 0.0193+j0.0083 0.0064+j0.0028 0.0064+j0.0028 0.0064+j0.0028 0.0193+j0.0083

4

0.0097+j0.0042 0.0032+j0.0014 0.0032+j0.0014 0.0032+j0.0014 0.0097+j0.0042 0.0032+j0.0014 0.0032+j0.0014 0.0032+j0.0014 0.0097+j0.0042

5,6, 11

0.0129+j0.0055 0.0043+j0.0018 0.0043+j0.0018 0.0043+j0.0018 0.0129+j0.0055 0.0043+j0.0018 0.0043+j0.0018 0.0043+j0.0018 0.0129+j0.0055

9,17

0.0645+j0.0278 0.0215+j0.0093 0.0215+j0.0093 0.0215+j0.0093 0.0645+j0.0278 0.0215+j0.0093 0.0215+j0.0093 0.0215+j0.0093 0.0645+j0.0278

13

0.0451+j0.0194 0.015+j0.0065 0.015+j0.0065 0.015+j0.0065 0.0451+j0.0194 0.015+j0.0065 0.015+j0.0065 0.015+j0.0065 0.0451+j0.0194

14

0.0258+j0.0111 0.0086+j0.0037 0.0086+j0.0037 0.0086+j0.0037 0.0258+j0.0111 0.0086+j0.0037 0.0086+j0.0037 0.0086+j0.0037 0.0258+j0.0111

18

0.0516+j0.0222 0.0172+j0.0074 0.0172+j0.0074 0.0172+j0.0074 0.0516+j0.0222 0.0172+j0.0074 0.0172+j0.0074 0.0172+j0.0074 0.0516+j0.0222

convergence with a tolerance value of 0.00001 p.u and the execution time is 4.37 ms. The load flow results are presented in the Table XII.

VIII. CONCLUSIONS A simple and efficient load flow technique for balanced

and unbalanced distribution networks has been proposed. It has the support of proposed effective data structure for solving the radial and weakly meshed distribution networks. It has been found that the method has good and faster convergence characteristics compared with other existing methods. Unlike other traditional methods, the

TABLE XII LOAD FLOW RESULT OF THE PRACTICAL 19-BUS DISTRIBUTION SYSTEM

Bus no. aV (p.u) bV (p.u) cV (p.u) 1 1 0.999978 0.999978 2 0.986013 0.987819 0.986566 3 0.984878 0.987589 0.985651 4 0.980327 0.982023 0.981028 5 0.980132 0.981887 0.980925 6 0.976896 0.978537 0.977725 7 0.976522 0.978263 0.977454 8 0.969671 0.970754 0.970364 9 0.961967 0.961997 0.961738

10 0.951174 0.950279 0.949733 11 0.94977 0.948947 0.947836 12 0.94953 0.948364 0.948145 13 0.949437 0.948468 0.947167 14 0.949205 0.948486 0.947304 15 0.947254 0.945514 0.945551 16 0.948754 0.947082 0.947359 17 0.948273 0.947913 0.946739 18 0.948819 0.9481 0.946894 19 0.945969 0.94388 0.944666

proposed approach does not require any LU decomposition or Bus admittance matrix. The proposed algorithm is robust, time-efficient and needs very less memory for any size of the distribution system and find great potential application in the distribution automation.

ACKNOWLEDGMENT The authors gratefully acknowledge the support and

facilities extended by the Department of Electrical Engineering, National Institute of Technology, Warangal and Vaagdevi College of Engineering, Warangal (A.P) India.

REFERENCES [1] H. D. Chiang, "A fast decoupled load flow method for distribution

power network: Algorithms analysis and convergence study," Int. J. Electr. Power Systems, vol. 13. no. 3, pp. 130-138, 1991.

[2] D. Das, D. P. Kothari, and A. Kalam, "Simple and efficient method for load flow solution of radial distribution networks," Electrical Power & Energy Systems vol. 17. no. 5, pp. 335-346, 1995.

[3] T. H. Chen, M. S. Chen, K. J. Hwang, P. Kotas, and E. A. Chebli, "Distribution system power flow analysisA rigid approach," IEEE Trans. Power Delivery, vol. 6, no. 3, pp. 1146-1152, Jul. 1991.

[4] D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo, "A compensation-based power flow method for weakly meshed distribution and transmission networks," IEEE Trans. Power Syst., vol. 3, no. 2, pp.753-762, May 1988.

[5] G. X. Luo and A. Semlyen, "Efficient load flow for large weakly meshed networks," IEEE Trans. Power Syst., vol. 5, no. 4, pp. 1309-1316, Nov. 1990.

[6] C. S. Cheng and D. Shirmohammadi, "A three-phase power flow method for realtime distribution system analysis," IEEE Trans. Power Syst., vol. 10, no. 2, pp. 671-679, May 1995.

[7] R. D. Zimmerman and H. D. Chiang, "Fast decoupled power flow for unbalanced radial distribution systems," IEEE Trans Power Syst., vol. 10, no. 4, pp. 2045-2052, Nov. 1995.

[8] R. A. Jabr, "Radial distribution load flow using conic programming," IEEE Trans. on Power Systems, vol. 21, no. 3, pp. 1458-1459, Aug. 2006.

[9] A. Hamouda and K. Zehar, "Efficient load flow method for radial distribution feeders," J. of Applied Sciences, vol. 6, no. 13, pp. 2741-2748, 2006.

[10] R. Singh, B. C. Pal, R. A. Jabr, and P. D. Lang, "Distribution system load flow using primal dual interior point method," in Proc. IEEE Power System Technology and IEEE Power India Conf., 5 pp., New Delhi, India, 12-15 Oct. 2008.

[11] J. H. Teng, "A direct approach for distribution system load flow solutions," IEEE Trans. Power Delivery, vol. 18, no. 3, pp. 882-887, Jul. 2003.

[12] S. Maheshwarapu,, "An effective root based algorithm for power system topological observability," in Proc. of IEEE Region 10 Int.

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Conf. on Global Connectivity in Energy, Computer Communication and Control, TENCON98, vol. 2, pp. 596-600, New Delhi , India, Dec. 1998.

[13] W. H. Kersting, "Distribution feeder line models," IEEE Trans. Industry Applications, vol. 31, no. 4, pp. 715-720, Aug. 1995.

[14] J. J. Grainger and S. H. Lee, "Capacity release by shunt capacitor placement on distribution feeders: a new voltage dependent model," IEEE Trans on Power Apparatus and Systems, vol. 101, no. 5, pp. 1236-1244, May 1982.

[15] J. J. Grainger and S. Civanler, "Volt/Var control on distribution systems with lateral branches using shunt capacitors and voltage regulators: Part I, Part II and Part III," IEEE Trans on Power Apparatus and Systems, vol. 104, no. 12, pp. 3278-3297, Nov. 1985.

[16] M. Chis, M. M. A. Salama, and S. Jayaram, "Capacitor placement in distribution system using heuristic search strategies," IEE Proc.-Gener., Trans., Distrib., vol. 144, no. 3, pp. 225-230, May 1997.

[17] M. E Baran and F. F. Wu, "Optimal capacitor placement on radial distribution systems," IEEE Trans. On Power Delivery, vol. 4, no. 1, pp. 725-734, Jan. 1989.

[18] D. Zhang, Z. Fu, and L. Zhang, "An improved TS algorithm for loss-minimum reconfiguration in large-scale distribution systems," Electric Power Systems Research, vol. 77, no. 5-6, pp. 685-694, Apr. 2006.

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[20] S. Sivanagaraju, J. Viswanatha, and M. Giridhar, "A loop based load flow method for weakly meshed distribution network," ARPN J. of Engineering and Applied Sciences, vol. 3, no. 4, pp. 55-59, Aug. 2008.

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APPENDIX A The various vectors used in [12] for sparsity technique

are shown in Tables A-I, and A-II. The reserved locations information for bus-i

are indicated by ITAGF[i] and ITAGTO[i] vectors. ADJQ vector is used to store adjacent bus number-q and ADJL vector is used to store adjacent line number of bus-i.

TABLE A-I FORMATION OF ITAGF[I] AND ITAGTO[I] VECTORS FOR THE 6-BUS

SYSTEM

Bus no.-i ITAGF[i] ITAGTO[i] 1 1 1 2 2 3 3 4 6 4 7 8 5 9 9 6 10 10

TABLE A-II

FORMATION OF ADJQ[J] AND ADJL[J] VECTORS FOR THE 6-BUS SYSTEM

Bus no. 1 2 3 4 5 6 Location count =j 1 2 3 4 5 6 7 8 9 10

ADJQ[j] 2 1 3 2 4 6 3 5 4 3 ADJL[j] 1 1 2 2 3 5 3 4 4 5

K. Prakash received his B.E. (Electrical and Electronics Engineering, 1999) from University of Madras, M. Tech. (Power Systems, 2003) from National Institute of Technology, Warangal, India. He is currently pursuing his Ph.D. (Power Systems Engineering) in National Institute of Technology, Warangal, Andhra Pradesh, India. His areas of interest include Distribution system studies, meta-heuristic techniques and economic operation of power systems. He is a member of the Institute of Electrical and Electronics Engineers (IEEE). M. Sydulu received his B. Tech. (Electrical Engineering, 1978), M. Tech. (Power Systems, 1980), Ph.D. (Electrical Engineering-Power Systems, 1993), all degrees from Regional Engineering College, Warangal, Andhra Pradesh, India. His areas of interest include Real Time power system operation and control, ANN, fuzzy logic and genetic algorithm applications in power systems, distribution system studies, economic operation, reactive power planning and management. Presently he is working as a Professor in the Department of Electrical Engineering, National Institute of Technology, Warangal (formerly RECW), India.

Dr. M. Sydulu is a member of the Institute of Electrical and Electronics Engineers (IEEE).

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