 Open Access
 Total Downloads : 206
 Authors : Hassan Kubba
 Paper ID : IJERTV5IS040398
 Volume & Issue : Volume 05, Issue 04 (April 2016)
 Published (First Online): 16042016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Efficient and More Reliable Second Order Load Flow Solution Method with Cubic Interpolation Technique
Asst. Prof. Dr. Hassan Kubba*
Electrical Engineering Department, Engineering College, Baghdad University
AlJadreyia Complex, Baghdad, Iraq,
Abstract: This paper presents a fast, reliable, and new method for solving the load flow problem in electrical power systems. The proposed method is a second order load flow technique based on the "Taylor series expansion" of a multivariable function. This approach takes the first three terms of the Taylor series.
The method has advantages over Newton's method in terms of speed of solution (no. of iterations), and reliability of convergence. By inserting a minimization technique in this proposed method, the algorithm exhibits a control of the convergence. By means of this control, the method converges for cases when conventional Newton's method and some other popular methods diverge.
Also this paper presents a comparison between the proposed method and the NewtonRaphson method according to the major criteria, namely reliability of convergence, and speed of solution. Two test systems (five busbar typical test system and forty busbar typical test system) are used to examine the performance of each method.
Keywords: Load flow problem, Taylor series expansion, Second order load flow model, Cubic interpolation techniques.

INTRODUCTION
The load flow studies are the backbone in the planning of a power system. They are the means by which the future operation of the system is known ahead of time. A load flow study involves the determination of voltage, current, power, and power factor or reactive power at various points in an electrical network under existing or contemplated conditions of normal operation, subject to the regulating capability of generators, condensers, and tap changing under load transformers as well as specified net interchange between individual operating systems. This information is essential for the continuous evaluation of the current performance of a power system and for analyzing the effectiveness of alternative plans for system expansion to meet increased load demand. They are being increasingly used to solve very large systems, to solve multiple cases for such purposes as outage security assessment, and within more complicated calculation as optimization and stability.
The studies for the load flow calculations started with the Ward and Hale method[1] in 1956, and currently the Newton
Raphson method with the sparsity technique and suboptimal ordering are being widely used[2]. The conventional (NR) method has their relative advantages and disadvantages.
A load flow problem consists of solving a set of nonlinear equations. The conventional (NR) technique uses the first two terms of the Taylor series. This approach transforms the nonlinear load flow equations to a linear form before a solution is attempted. The proposed method formulate the load flow problem by using the first three terms of the Taylor series. In other words, second order terms, which are not insignificant, can be included in the algorithm and can be used during digital computation. The significant second order terms are found to be minor variations of the terms of the Jacobian matrix. It is shown in this paper that the coefficients of the second order terms are not required to be separately stored. In the proposed technique, the state vector is first calculated by an iteration of the conventional NewtonRaphson technique. Using the calculated state vector and elements of the Jacobian matrix, second order terms are estimated and subtracted from the residual vector. The modified residual vector obtained in this manner is then used to compute a new state vector. This procedure is repeated till the elements of the latest state vector are within permissible tolerance of those previously calculated. The magnitudes and phase angle of bus voltages are then updated. The total procedure is then repeated until a converged solution is obtained.
The next problem sought to be solved in the proposed method is how to solve illconditioned systems or determine the existence of load flow solutions the proposed method is a combination of second order load flow algorithm and the cubic interpolation technique to determine an optimal multiplier for improving the load flow calculations. The algorithm exhibits a control of the convergence process and a nondivergent characteristic for any problem. Also the existence of the solution from the initial estimate can be easily judged.

TAYLOR SERIES EXPANSION
A function can be evaluated by using the Taylor series expansion. The procedure is simple and well known for functions consisting of a single variable. Taylor series of multivariable functions can also be defined.
For an (n) variable function, the series is expressed as follows:
f x1 x1, x2 x2 ,……, xn x2 f x1, x2 ,……, xn
m 1
i
i! x
x1 x
x2 …… x
xn
f x1 , x2 ,….., xn Rm
(1)
i1 1 2
Where
1
n
m1
Rm m 1! x
x1 x
x2 ….. x xn
f x1 a1x1, x2 a2x2 ,….., xn anxn
1 2 n
And 0 < ai < 1 ; i = 1, 2, ,, n
Because the summation in equation (1) consists of (m) terms, a residue, Rm , is introduced to take care of the summation from (m+1)th term to infinity. This residue is not known exactly because definite values cannot be assigned to (ai's). Neglecting the third and higher terms in equation (1), equation (2) is obtained.
f (x1 x1 , x2 x2 ……. xn xn ) f (x1 , x2 ,….., xn ) x x1 x x2 …… x xn f x1 , x2 ,….., xn
1
1 2 n
2
2 x
x1 x
x2 …. x
xn
f x1, x2 ,…., xn
(2)
1 2 n
Equation (2) can be rearranged to give equation (3) which can be expanded as equation (4).
f
f x1 x1 , x2 x2 ,……, xn xn f x1 , x2 ,…., xn
1
2
f x x x x … x x
f x, x,…x
2 x
x1 x
x2 …… x
xn
f x1, x2 ,…., xn
1 2
n
(3)
f n
f x1, x2 ,…., xn x 1 n
2 f x , x ,…., x
2
2
i
i
1 2 n x
2
2
i 1
xi
i
i 1
x2 i
n
n
n 1
2 f x , x ,….., x
1 2 n x x
i j
i j
(4)
i 1 j i 1
xixj

THE PROPOSED METHOD

Second order load flow model
In a power system, real and reactive power injected into a bus, say K, is equal to the net flow in the elements connected to this bus. Power flowing in an element connecting two buses, say K and m, can be defined in terms of the magnitudes and phase angles of the voltages at these buses and the parameters of the element. A load flow is, therefore, a problem of solving a set of nonlinear equations consisting of the magnitudes and phase angles of the system bus voltages as variables and the parameters of the system elements as constant coefficients. Let the power mismatch at a bus, K, of an (n bus system be defined as the difference between the scheduled power injection into this bus and the sum of the calculated power flows in all the elements connected to this bus. Real power mismatch can be defined in terms of the above mentioned variables by equation (5). This second order equation is similar to the Taylor series expansion excluding third and higher order terms in equation (4).
Pk
n
n
m1
Pk
m
m
m
n
n
m1
Pk V
m
m
Vm
1
n
n
2 m1
2
P
P
k ( )2
2 m m
n
n
1
k V
n1
n
n
2
2
P
P
k
m r
m r
P
P
2
2
2 m1
V 2 m m
m1 r m1
m r
n n 2 P n n 2 P
m r
m r
k V k V V
(5)
m1
r 1
m
Vr
m r
m1
r m1
Vm
Vr
Where Pk is the real power injected into bus K,
m is the phase angle of the voltage at bus m, Vm is the magnitude of the voltage at bus m,
defines small changes in the variables.
n
n
m
m
It is interesting to note that the terms of first two series in equation (5) are similar to the terms of the Jacobian matrix used in the NewtonRaphson method. The remaining five series constitute the second order terms. In a similar manner, reactive power mismatch can be defined as follows:
n
n
Q
Qk
Qk V
1
2Q 2
k
m1
m
m
n
n
m1
Vm
2 m1
k
k
2
2
m
m
m
n
n
1
2Q
V
n1
n
n
2
2Q
m r
k
k
m r
k
k
2 m1
V m 2
m
m1 r m1
m r
n
n
n
n
2
Q
Q
k
n1
V
2
n
n
Q
Q
m r
m r
k V V
(6)
m1
r 1
m
Vr
m r
m1
r m1
Vm
Vr
Expressing equation (5) for all system buses except the slack bus and equation (6) for all load buses, a set of equations is obtained in the following form.
P
J1
J 2
i
Vh
S1
S2 S2
i J
V V
(7)
Q J
J S
S S h l
3
4 Vh 4 5
6
Vt
s
s
The submatrices S1 through S6 include series of second order terms of all buses similar to those given in equations (5) and (6). Suitable values are assigned to subscripts i, j, h, l, s, and t. Many elements of the second order coefficient matrix, [S], are very small and can be neglected. Also, it is not essential to compute the elements of matrix [S] and store it in computer memory. These two aspects will now be discussed.
Real and reactive power injection into a bus K of an (n) bus system can be mathematically expressed by equation (8) and (9) respectively. Second order coefficients can be derived from these equations and can be grouped into twenty categories. These coefficients are given in equations (10.1) through (10.20). a comparison with the elements of the Jacobian matrix indicates that the second order terms given in equations (10.1) through (10.10) are minor modifications of the Jacobian elements. These aspects are also indicated in these equations. The terms defined by equations (10.11) through (10.20) can be neglected as well be discussed later in this section.
n
Pk
m1
VkVmYkm Cos k
m

km
(8)
n
n
Qk
m1
VkVmYkm Sink m km
(9)
Where Ykm is the admittance of the element connecting buses K and m and is given by
YKm
Km
k
k
2 P
V Y sin
J1k ,k
(10.1)
n
n
k Vk
m1
m km
k m km
V
V
k
V
V
2 P
k
k
J1k ,m
k
k
k
Vm
Vk Ykm sin k m km
m
(10.2)
2 P
k
k
J1k ,m
V
V
m
k
k
Vk
VmYkm sin k m km
k
(10.3)
V
V
2 P
k
k
J1k ,m
m
k
k
Vm
Vk Ykm sin k m km
m
(10.4)
k
k
2Q
k
k
2
n
n
m1
mk
VkVm
Ykm
sin k
m

km
J1
k, k
(10.5)
k
k
2Q
m
m
2
VkVmYkm sin k
m
km
J1 k, m
(10.6)
k
k
2Q
k
k
V 2
2Ykk sin kk
2
V
V
2 J 4 (k, k)
k

Qk
(10.7)
2Q
k
k
k
k
k m
VkVmYkm sin k m
km J1 k, m
(10.8)
k
k
2Q
V V
Ykm sin k m km
J1 k, m
V V
(10.9)
k
P
P
2
k
k
k
V 2
2 P
m
2Ykk
Cos kk
2 J
V
V
2
2
2
2
k
k, k
k m
Pk
(10.10)
k 0
m
m
V 2
2Q
k 0
m
m
V 2
(10.11)
(10.12)
k
k
2 P
k
k
2
n
n
m1
mk
VkVm
Ykm
cos k
m

km
(10.13)
P
P
2
k
m
m
2
VkVm
Ykm
cos k
m

km
(10.14)
P
P
2
k V V Y
cos
(10.15)
cos
cos
(10.16)
(10.16)
k
m k
m km
k m km
k Y
k Y
2 P
Vk Vm
k
k
2Q
km
n
n
V Y
k m
cos
km
(10.17)
k
Vk
m1 mk
m km
k m km
k
k
n
n
2Q
V Y cos
(10.18)
k
Vm
m1 mk
m km
k m km
2Q
k
k
m
k
k
Vk
VmYkm cos k
m km
(10.19)
2Q
k
k
m
k
k
Vm
Vk Ykm cos k
m km
(10.20)
The elements defined by equations (10.11) and (10.12) are equal to zero. Equations (10.13) through (10.20) define the second order elements which include cos(k m km) as a multiplier. Since (km) is close to (90o) and (k m) is usually small, cos(k m km) is quite small and therefore, these coefficients can be neglected. This assumption and the simple relationship of the second order coefficients with the elements of the Jacobians matrix makes the application of second order load flow mode straightforward and with minimal additional computing effort. The double summation is equation (5) and (6) have Ykm as multipliers. For any bus K, Ykm has nonzero values only for the buses (m) connected to it; for all other values of m, Ykm is zero. Therefore, the double summations for a bus can be reduced to single summation which include only a few terms depending on the number of busses connected to that particular bus. Deleting the significant terms from equations (5) and (6) and converting the double summations to single summation, equation (11) and (12) are obtained.
n
n
P
P
2 P 2
Pk
m1
k
n
n
m
m
k
k
m1
k V
Vm
k v
k
k
n
n
m
m
V 2 K
n
n
p
p
2
k V
2
n
n
P
P
k
V
2
P
P
k V
(11)
k V
k V
m1 m K
k
Vm
K m
m1
mk
m
Vm
m m m1
m
Vk m
n
n
Q
Qk
Qk V
1 2Q 2
k
k
k
m1
m
m
n
n
m1
Vm
m 2 V 2 k
n
n
1
2Q
2
k
k
2
2
2Q
n
n
k m
k m
2Q
V V
(12)
2 m1
m
m
n
n
m1
mk
k
m
k m
m1
mk
k
k
Vk Vm


Cubic Interpolation Technique [3, 4, 5]
It is wellKnown that the load flow calculation can be regarded as a nonlinear programming problem [6], which determines the direction and magnitude of the solution so that a certain function F(x) may be minimized. The F(x) is the squares of the active and reactive mismatch power. By employing this formulation, the valuable property can be obtained that the solution never diverges. The value of the function F(x) becomes eventually zero if there is a solution from the initial estimate, and stays at a positive value if no solution exists. In nonlinear programming approach (FletcherPowell method), (x) is modified by a correction factor () which can be considered as an acceleration factor. The computation of () is made such that minimize F(x). The function to be minimized is:
FV ,
kPQ,PV
P2
k
k
kPQ
Q2
(13)
k
k
Where the first summation in (13), k includes load and generator buses, the second summation, k includes load buses only.
The minimization of F(x) with respect to () in the direction of (x) is a onedimensional problem. The object is to determine the correction factor () given (x) and the point (x). The problem can be stated as that of finding a value of () that will minimize:
Fxo x
And therefore the derivative of (Z) with respect to (x) is :
' 2 f yf yx
y
(14)
(15)
Where
Y xo x
A cubic interpolation technique is used to find (x) as follows:

A step "a" is chosen as:
a = min (1, 2 (Fx Fo ) / Z'x) (16)
Where (Fo) is an estimate of (F) at the problem optimum, (Fx) is the value of function (F) at point (x), and (Z'x) is the derivative of (Z) with respect to (x) evaluated at point (x).

A step of size (a) is taken to arrive at point (y), Y = x +x, and (Zy') is evaluated to determine whether a change of sign has occurred with respect to (Zx'). Such a change of sign, from negative (Zx') to positive (Zy') would indicate that the minimum is enclosed within these two points. If there is no change of sign, successive step of size (a) are taken until two adjacent points that enclose the minimum are found. Let these two adjacent points be called (w) and (y) which are located at
distances ( w ) and ( y
) from the original point (x).

The distance () from (x) to the minimum point is:
y
y

w
y'r s
(17)
Where
y y'Zw'2r
1
1
r s 2 Zw' Zy'2
(18)
and
S 3 Zw Zy Zw'Zy'
y w
(19)


The point (xo + x) is accepted as the minimum point if the function F(xo +x) = Z(x) is smaller than both (Zw) and (Zy). If this is not true, the interpolation is repeated using the point (xo +x) and either (w) or (y), chosen so that the minimum is enclosed. This decision is based on the sign of Z'(). The cubic interpolation process is shown in fig (1).
Fig (1) Cubic interpolation to minimize F(x) with respect to ( ) in the direction of ( x )
The interpolation process can be simply implemented with the second order load flow approach.
Equation (15) must be evaluated for several values of ()
When 0
f(y) = f(x) (20)
And
f y
y
f x
x
(21)
Therefore
2 f yf yx 2 f xf xx
(22)
y x
From the equation f(x) = J x Equation (21) becomes:
Z ' 2 f x2 2Fx
(23)
The minus sign of equation (23) shows that the direction at minimization (x) always points in a direction which reduces F(x). The terms f(x), which represent the mismatch power, are calculated from the second order algorithm. In the
proposed method, a simple alteration of the second order algorithm's program will allow a reevaluation of the Jacobian at
f y
point (y) in order to obtain the term
, also f(y) represents the mismatch power at point (y).
y
Then the interpolation process calculates the optimum () and the new point(x1 = x0 +x) is used for the start of the next iteration. The additional requirement of this method is the reevaluation of the Jacobian matrix which is small, using sparsity techniques. There is no need for extra GaussElimination and back substitutions, just additional evaluation of the Jacobian.


ITERATIVE ALGORITHM OF THE PROPOSED METHOD
The basic iterative algorithm for solving the load flow problem by the proposed method is as follows:

Set the initial voltage magnitudes of the busbars equal to that of the slack bus and usually (VK = V1 = 1 pu.) . All voltage angles are set equal to the slack bus ( K= 1 = 0), where bus (1) represents the slack bus.

Compute the real and reactive power mismatches and elements of the Jacobian matrix using the specified loads and
p p
p p
generations, system parameters and estimated magnitudes and phase angles of bus voltages.
p n
p n
P p jQ p V pe j k
Y V pe j k V pe j k
(24)
k k k
m1 mk
km k
m
k,m=1,2..,n
Where (*) complex conjugate
And (n) no. of busses, and (p) represent the iteration index.
Pp Psp Pp
k k k
Qp Qsp Qp
(25)
k k k
(sp) indicates the specified value.

Neglect the second order terms, now the problem is the same as in Newton Raphson load flows. Using Gauss elimination technique, evaluate the state vector [ V/V ]T.

For each P and Q of the residual vector, compute the second order terms (defined in equations 11 and 12) using
's and V's evaluated in step 3.
At this stage the residual vector is known and the sum of second order series for each residual term has been estimated. Transferring the second order terms to the left hand sde, equation (7) becomes
P S S
S i j J
J i
1 2
3 V V 1
2 Vk
(26)
Q S
S S k
L J
J
4
5 6
V 3
4 Vk
s t
As a second order series is evaluated, it is subtracted from the corresponding residual term which procedure provides a modified residual vector defined by the left hand side of equation (26). It is important to realize that the elements of the matrix [S] are not computed and therefore, no additional storage is used for this matrix.

Using the modified residual vector and equation (26), recalculate the state vector [ V/V]T The triangularized Jacobian used in step (3) is reused at this stage.

(p+1) and (Vp+1) are modified by a correction factor () which can be considered as an acceleration factor. The computation of () is made such that it minimize F(x), which is the sum of the squares of the active and reactive mismatch power. An optimal correction is determined by finding an optimal () following the cubic interpolation technique.

Calculate the new values of the voltage magnitudes and angles for all busbars except the slack as follows:
p1 p p1
k k k
And,
Vk Vk V
Vk Vk V
p1 p p1 k

An iterative process is required until the expressions
(27)
k
k
P p1
k
k
Q p1
(28)
are satisfied for all busses except the slack, where ( ) is a small power tolerance value. When the expression (28) are satisfied, the problem is solved. If not, the procedure is repeated with the next iteration, computing the elements of the Jacobian matrix and proceed to step (3).
5. ANALYSIS BASED ON NUMERICAL EXAMPLES
A 5bus system taken from reference [7], and 40 bus system based on Iraqi National Grid were chosen to test the load flow solution methods. Tables (1) and (2) shows that the load flow problem was solved by Newton's method in 4 iterations to an accuracy of 103 for each individual mismatch power. In the proposed method 3 iterations were required with the same accuracy. The values of optimum () obtained for each iteration in the proposed method were close to one, and in the final iteration of a convergent load flow case, the value of () would be very close to one or exactly equal to 1.0. My experience has shown that optimum () is either close to 1.0 or very close to 0.0. The cubic interpolation formula will produce an appropriate value of () even in the case where the optimum is near, but outside the interpolating limits. So optimum () may sometimes be slightly greater than one. If the interpolation is performed between zero and one, the correction () value would be determined for all cases without any extra Jacobian calculation per iteration, thus saving computation time.
The principle value of the proposed method lies in the control of the convergence process for both illconditioned and data error cases, whereas using the conventional NewtonRaphson method alone during the iterations of a load flow problem may result in poor solution or divergence.
A study of the maximum power mismatches for both (NR) method and the proposed method indicates what the maximum mismatches are smaller when second order algorithm (proposed method) are used. This indicates that the solution descends into the final state at a faster rate in the second order load flows than in the conventional (NR) cases. Also, the contributions of the second order terms were noticed to be negligible after the first iteration.
The performance of the proposed method with an illconditioned system was tested on the 5busbar typical test system with insertion of series capacitors in four lines connecting the busbars for two different cases. The value of the capacitor is about 0.01 pu. on the same base of the other admittances. The solution by conventional NewtonRaphson method in the case, in which the diagonal elements of the [Y] matrix are less effective by the insertion of series capacitors, converged, while in the other case, the solution diverged. The proposed method converged for all cases.
Also, a 40busbar typical test system with insertion of series capacitors in ten lines connecting the busbars, including the slack, was chosen for this test. The solution by conventional (NR) method diverged, while the proposed method converged with the value of optimum () as in table (3). The convergence characteristics of the 40bus illconditioned system was shown in fig (2). From the figure, the following observation is made. In the case of optimum () is not applied, the solution continues to oscillate. However, if the optimum () is used the system converged easily.
Table (1) Comparison of the proposed method and the conventional (NR) method
with the values of optimum ( ) for 5 bus bar typical test system, 4 load busbar and 1 slack .
Method
No. of iterations
Optimum / iteration
Proposed method
NewtonRaphson Proposed method
4
3
1
2
3
1.02
1.02
1.003
Notes: 1 Power tolerance = 103 for each individual mismatch power .

This system is the same as that used in reference [7].

Polar coordinates are used for both the proposed method and NR method.

Table (2) Comparison of the proposed method and the conventional (NR) method
with the values of optimum () for 40 busbar typical test system, 30 load busbar, 9 generator busbar, and 1
slack basbar[8].
Method 
No. of iterations 
Optimum / iteration Proposed method 

NewtonRaphson Proposed method 
4 3 
1 
2 
3 
0.965 
1.01 
1.000 
Notes: 1 Power tolerance = 103 for each individual mismatch power .

This system is based on the Iraqi National Grid system.

Polar coordinates are used for both the proposed method and NR method.
Table (3): Values of optimum () "Proposed method"
Iteration Count 
Optimum () 

5bus illconditioned system 
40bus illconditioned system 

1 
1.034 
0.784 
2 
1.000 
1.071 
3 
0.955 
1.000 
Fig (2) Convergence characteristics of 40 bus illconditioned system
102
101
100
101
102
103
Proposed method N – R
0 1 2 3 4
Fig (3). Convergence patterns of load flows of the 40 bus test system by (a) the Newton – Raphson approach
and, (b) proposed method.

CONCLUSION
A very simple mathematical model and method for solving load flow problems has been developed. It would be concluded that the combined use of second order algorithm and minimization method by cubic interpolation technique has many attractive characteristics:

The second order load flow model has some advantages over the conventional (NR) approach. It is obvious that the proposed technique has better convergence characteristic, the magnitudes and phase angles of bus voltages descend into the final solution at a faster rate than that observed in (NR) method. Therefore, in many cases, the second order approach requires lesser iterations than (NR) technique. Moreover, theadditional computing effort is only slightly greater compared to a (NR) load flow. It has also been shown that the elements of the second order coefficient matrix need not be stored separately

A more rapid convergence and a nondivergent characteristic for any problem. A control of the convergence process for both wellconditioned and illconditioned systems by the optimum
correction factor ().

The method can be applied efficiently to large systems.
APPENDIX
Nodal admittance matrix elements for 40 busbars "Typical Test System", ykm=GjB Per unit quantity = 132 MVA, 132 KV all data in P.U.
Bus to Bus
G(P.U),
conductance
B(P.U),
susceptance
11
73.20302
193.05308
17
43.763
109.66543
136
21.26757
53.28579
138
8.17245
30.10186
22
15.35563
32.0213
23
1.05799
4.58466
220
5.31241
11.95284
33
9.62425
26.04736
315
3.07535
7.70527
328
5.4909
13.75742
44
32.25366
95.89834
414
8.001
29.47035
416
4.80817
17.70843
419
12.8506
32.19863
429
6.59387
16.52092
55
16.97918
59.86251
57
8.30093
29.2611
524
8.67825
30.59639
66
13.90746
51.44682
615
6.53796
24.08149
617
7.4295
27.36532
77
13.81057
46.88945
722
1.76607
4.4249
735
3.74355
13.19844
88
5.66517
19.64528
820
2.48434
8.63067
821
3.18083
11.0146
99
6.94882
12.10935
911
5.14539
8.9666
930
1.80343
3.14275
1010
18.01201
45.70547
1037
9.62425
26.04736
1039
5.31241
11.95284
1040
3.07535
7.70527
1111
11.37208
24.5679
1129
6.22669
15.60129
1212
22.89817
24.3483
1220
3.20594
8.03246
1223
19.69224
34.31584
1313
14.12767
52.03695
1315
4.97453
18.32287
1317
9.15314
33.71408
Bus to Bus
G(P.U.),
conductance
B(P.U.),
susceptance
1414
14.65299
53.97187
1415
6.65199
24.50151
1516
32.89171
110.77638
1515
8.17245
30.10186
1534
3.47942
6.06339
1616
12.98062
47.81029
1717
93.63839
244.98614
1727
5.05963
8.81716
1733
6.96555
12.1384
1818
6.60826
12.71954
1825
0.90372
2.03338
1827
4.23597
7.38181
1831
1.46855
3.30425
1919
34.11817
85.48443
2020
11.00269
28.61598
2121
15.63407
32.7162
2132
3.56803
6.21779
2222
5.26614
12.30004
2230
1.59093
3.57961
2231
1.90912
4.29553
2323
27.24222
53.23229
2326
7.54998
18.91645
2424
9.75381
33.01641
2425
1.07556
2.42001
2525
11.97929
4.4534
2626
9.43165
32.55575
2628
1.88166
4.6393
2727
9.29561
16.19897
2828
7.37256
18.39672
2929
12.82056
32.12222
3030
3.39437
6.72236
3131
3.37768
7.59978
3232
3.56803
6.21779
3333
6.96555
12.1384
3434
3.47942
6.06339
3535
3.74355
13.19844
3636
21.26757
53.28579
3737
9.62425
26.04736
3838
21.02305
62.30049
3840
12.8506
32.19863
3939
5.31241
11.95284
4040
15.92595
39.9039
Bus to Bus
G(P.U),
conductance
B(P.U),
susceptance
11
73.20302
193.05308
17
43.763
109.66543
136
21.26757
53.28579
138
8.17245
30.10186
22
15.35563
32.0213
23
1.05799
4.58466
220
5.31241
11.95284
33
9.62425
26.04736
315
3.07535
7.70527
328
5.4909
13.75742
44
32.25366
95.89834
414
8.001
29.47035
416
4.80817
17.70843
419
12.8506
32.19863
429
6.59387
16.52092
55
16.97918
59.86251
57
8.30093
29.2611
524
8.67825
30.59639
66
13.90746
51.44682
615
6.53796
24.08149
617
7.4295
27.36532
77
13.81057
46.88945
722
1.76607
4.4249
735
3.74355
13.19844
88
5.66517
19.64528
820
2.48434
8.63067
821
3.18083
11.0146
99
6.94882
12.10935
911
5.14539
8.9666
930
1.80343
3.14275
1010
18.01201
45.70547
1037
9.62425
26.04736
1039
5.31241
11.95284
1040
3.07535
7.70527
1111
11.37208
24.5679
1129
6.22669
15.60129
1212
22.89817
24.3483
1220
3.20594
8.03246
1223
19.69224
34.31584
1313
14.12767
52.03695
1315
4.97453
18.32287
1317
9.15314
33.71408
Bus to Bus
G(P.U.),
conductance
B(P.U.),
susceptance
1414
14.65299
53.97187
1415
6.65199
24.50151
1516
32.89171
110.77638
1515
8.17245
30.10186
1534
3.47942
6.06339
1616
12.98062
47.81029
1717
93.63839
244.98614
1727
5.05963
8.81716
1733
6.96555
12.1384
1818
6.60826
12.71954
1825
0.90372
2.03338
1827
4.23597
7.38181
1831
1.46855
3.30425
1919
34.11817
85.48443
2020
11.00269
28.61598
2121
15.63407
32.7162
2132
3.56803
6.21779
2222
5.26614
12.30004
2230
1.59093
3.57961
2231
1.90912
4.29553
2323
27.24222
53.23229
2326
7.54998
18.91645
2424
9.75381
33.01641
2425
1.07556
2.42001
2525
11.97929
4.4534
2626
9.43165
32.55575
2628
1.88166
4.6393
2727
9.29561
16.19897
2828
7.37256
18.39672
2929
12.82056
32.12222
3030
3.39437
6.72236
3131
3.37768
7.59978
3232
3.56803
6.21779
3333
6.96555
12.1384
3434
3.47942
6.06339
3535
3.74355
13.19844
3636
21.26757
53.28579
3737
9.62425
26.04736
3838
21.02305
62.30049
3840
12.8506
32.19863
3939
5.31241
11.95284
4040
15.92595
39.9039


REFERENCES

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Sasson A. M., "An optimal ordering algorithm for sparse matrix application", IEEE Trans. power App., Vol. PAS97, No. 6, pp. 860867, Nov. 1978.

Kubba H.A.,"A rapid and more reliable load flow solution method for illconditioned power system", Engineering and technology Journal Vol17 , No.5, pp. 550568. 1998.

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