Electrical Power Systems

170 Electrical Power Systems
Given a set of non linear equations,
y 1 = f 1(x 1, x 2, ...x n)
y 2 = f 2 (x 1 , x 2 , ...x n ) ...(7.43)
... ........................
y n = f n(x 1, x 2, ...x n)
and the initial estimate for the solution vector
( 0)
( 0)
( 0)
x1 , x2 , ................. xn
( 0)
( 0)
( 0) Assuming Dx1 , Dx2 , .........., Dxn are the corrections required for x1 , x2 , ... xn respectively,
so that the equations (7.43) are solved i.e.,
e j
( 0)
e + D
( 0)
( 0)
1,,........., 2 + D 2
n + D nj
( 0)
( 0)
( 0)
1 1 1 2 2
n n
y1 = f x + Dx ,,.........,
x + Dx x + Dx
y 2 = f x x x x x x
2 1
..... ............................................................................ ...(7.44)
e j
( 0)
( 0)
( 0)
n 1 1 2 2
n n
yn = f x + Dx ,,.........,
x + Dx x + Dx
Each equation of the set (7.44) can be expanded by Taylor’s series for a function of two or
more variables. or example, the following is obtained for the first equation.
e j
( 0)
( 0)
( 0)
1 1 1 2 2
n n
y1 = f x + Dx ,,.........,
x + Dx x + Dx
( 0)
( 0) ( 0)
f1
f1
f1
= f1ex1,,........., x2 xn j+ Dx1+ Dx2+ ... Dxn
+ y
x x
x
Where y 1 is a function of higher powers of Dx 1 , Dx 2 , .... , Dx n and second, third ..., derivatives
of the function f 1. Neglecting y 1, the linear set of equations resulting is as follows:
( 0)
( 0) ( 0)
f
f
y1 = 1 1
fex1,,........., x2 xj+ Dx1+ Dx2+ ... Dx
x x
1 n n
1 0
2 0
( 0)
( 0) ( 0)
f2
f2
y2 = f x1,,........., x2 x Dx1Dx2... Dx
x x
2 n n
1 0
2 0
1 0
2 0
e j + + +
..........................................................................................................
( 0)
( 0) ( 0)
f f
yn = n n
fnex1,,........., x2 xn j+ Dx1+ Dx2+ ... Dx
x x
1
2
n
f
x
f
x
2
n
n
1
n
0
0
f
2
x
0
n
0
1
...(7.45)