1
2 Steady State Calculation by the Using of Newton Iteration Method
Before applying this method to the steady state calculation, in next will be shown
principle of this method for solution of the common system of non-linear algebraic
equations.
At first, let’s assume one-dimensional non-linear equation .
0
)
(
f
x
Finding of the solution of this equation:
1) choosing of the initial approximation
2) determination of the tangential equation
in this point
(from definition of the function derivation)
:
3) calculation of the first approximation
(0)
x
)
0
(
)
0
(
'
)
0
(
f
f
f
x
x
x
x
x
(1)
x
)
0
(
)
1
(
)
0
(
'
)
0
(
f
f
0
x
x
x
x
)
0
(
'
)
0
(
)
0
(
)
1
(
f
f
x
x
x
x
4) verification of the computing accuracy
)
0
(
)
1
(
x
x
root of the equation
tangent
line
5) if accuracy condition isn’t fulfilled, the computation
is repeated from event 3) according to:
)
(
'
)
(
)
(
)
1
(
f
f
k
k
k
k
x
x
x
x
2
Definition of the function derivation:
x
y
x
y
x
0
x
'
lim
d
d
f
x
x
y
y
d
d
Common expression for Newton iteration method:
3
A system of non-linear equations can by written as:
0
x
f
y
)
(
x
- vector of unknowns (with dimension
n
)
-
n
dimensional vector function of
x
)
(x
f
y
Common expression for Newton iteration method in this case:
x
J
x
x
x
f
y
n
n
n
n
n
n
n
n
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
y
y
y
2
1
2
1
2
2
2
1
2
1
2
1
1
1
2
1
)
(
f
)
(
f
)
(
f
)
(
f
)
(
f
)
(
f
)
(
f
)
(
f
)
(
f
This equation can be decomposed as:
J
- Jacobian matrix
4
Steady State Calculation by the Using of Newton Iteration Method for Given
(
P
,
Q
) nodes
Let's assume, the network containing
n
nodes is given. In each node, except the
balance one, is given the power:
i
i
i
i
i
I
U
Q
P
S
*
.
j
With regard to MNV we can write:
n
j
j
ij
i
i
i
i
U
Y
U
Q
P
I
1
*
.
j
n
i
,
,
3
,
2
n
i
,
,
3
,
2
n
j
j
ij
i
i
i
U
Y
U
Q
P
1
*
.
j
Voltage and admittance phasors can be written by the component or polar form.
When applying the polar form let’s assume:
i
i
i
U
U
i
i
i
U
U
*
j
i
j
i
ij
Y
Y
n
i
,
,
3
,
2
5
We divide the right hand-side of this equation into the real and imaginary parts:
j
i
j
i
j
i
j
n
j
i
i
Y
U
U
P
cos
1
j
i
j
i
j
i
j
n
j
i
i
Y
U
U
Q
sin
1
n
i
,
,
3
,
2
n
i
,
,
3
,
2
We obtain 2.(
n
-1) non - linear equations with (
n
-1) unknown voltages and their (
n
-1)
angles.
Notation: We know absolute value of voltage and its angle (usually set as zero) in balance node.
Then
n
j
ij
j
i
n
j
j
ij
i
n
j
j
ij
i
i
i
j
i
j
i
j
j
i
i
Y
U
U
U
Y
U
U
Y
U
Q
P
1
j
-
1
j
j
j
-
1
*
e
e
e
e
.
j
6
U
Q
U
Q
P
U
P
Q
P
Expression of Newton iteration method for calculation of this equation system:
Short expression:
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
U
U
U
Q
Q
Q
U
Q
U
Q
U
Q
Q
Q
Q
U
Q
U
Q
U
Q
Q
Q
Q
U
Q
U
Q
U
Q
P
P
P
U
P
U
P
U
P
P
P
P
U
P
U
P
U
P
P
P
P
U
P
U
P
U
P
Q
Q
Q
P
P
P
3
2
3
2
3
2
3
2
3
3
3
2
3
3
3
3
2
3
2
3
2
2
2
2
3
2
2
2
3
2
3
2
3
3
3
2
3
3
3
3
2
3
2
3
2
2
2
2
3
2
2
2
3
2
3
2
7
We obtain individual elements of Jacobian matrix by differentiation of previous
equations with respect to appropriate variables as follows:
U
P
)
cos(
cos
.
.
2
,
1
j
i
j
i
j
i
n
i
j
j
j
i
i
i
i
i
i
i
Y
U
Y
U
U
P
)
cos(
j
i
j
i
j
i
i
j
i
Y
U
U
P
P
)
sin(
.
.
.
,
1
j
i
j
i
j
i
j
n
i
j
j
i
i
i
Y
U
U
P
)
sin(
.
.
.
j
i
j
i
j
i
j
i
j
i
Y
U
U
P
U
Q
Q
)
sin(
sin
.
.
2
,
1
j
i
j
i
j
i
n
i
j
j
j
i
i
i
i
i
i
i
Y
U
Y
U
U
Q
)
sin(
j
i
j
i
j
i
i
j
i
Y
U
U
Q
)
cos(
.
.
.
,
1
j
i
j
i
j
i
j
n
i
j
j
i
i
i
Y
U
U
Q
)
cos(
.
.
.
j
i
j
i
j
i
j
i
j
i
Y
U
U
Q
8
The solution process is evident from the following algorithm:
- network configuration
- parameters of the network elements
)
0
(
)
0
(
2
...,
,
n
U
U
giv
giv
P
i giv
– given
power in i-th node
Calculation of P, Q distribution
Determination of
P in network
9
Steady State Calculation by the Using of Newton Iteration Method for Given
(
P
,
Q
) and (
P
,
U
) Nodes
It is required to keep voltage on the specified value in chosen nodes (
P
,
U
) of the
power system. We could guarantee this condition if we have adequate reactive power
in the power system (let’s assume, that a compensator is available in these nodes).
We can evaluate the magnitude of the compensation reactive power
Q
in individual
nodes by steady state solutions, e. g. with the aid of Newton method.
Let’s see the solution of the given problem in the power system, which contains
n
nodes.
Let’s assume, that in power systems is:
• the 1
st
node - the balance node, i.e.
U
1
and
1
(usually
1
= 0) is given,
• 2
nd
till
m -
th node is of type
P
,
Q
,
• (
m
+1) till
n
– th node is regulating one, i.e. of type
P
,
U
.
We know:
• (
n
– 1) equations with given active power
P
2
, ...,
P
n
and
• (
m
– 1) equations with given reactive power
Q
2
, ...,
Q
m
,
hence we calculate:
• the absolute values of voltage in nodes 2 till
m
, (
U
2
, ...,
U
m
) and
• angles in all nodes except the balance node (
2
, ...,
n
).
10
With application of Newton method we can write:
n
m
m
m
n
n
m
n
n
m
m
m
m
n
n
m
m
m
m
n
n
m
m
m
n
m
m
m
n
m
m
m
m
m
m
m
m
m
n
m
m
m
n
m
m
m
m
m
m
m
m
m
n
m
m
m
U
U
P
P
P
P
P
P
P
P
U
P
U
P
U
P
U
P
Q
Q
Q
Q
Q
Q
Q
Q
U
Q
U
Q
U
Q
U
Q
P
P
P
P
P
P
P
P
U
P
U
P
U
P
U
P
P
P
Q
Q
P
P
1
2
2
1
1
1
1
2
1
2
1
2
1
2
1
1
2
1
2
2
2
2
2
2
2
2
2
1
2
1
2
2
2
2
2
2
2
2
2
1
2
2
For short we can write this relation as:
U
P
P
U
P
Q
Q
U
Q
P
P
U
P
P
Q
P
11
By iterative process solution with desired accuracy we obtain values of:
• voltage
U
2
, ...,
U
m
and
• angles
2
, ...,
n
.
Then the required compensation reactive power for the keeping of specified voltage in
i -
th node is:
j
i
j
i
j
i
j
n
j
i
i
Y
U
U
Q
sin
1
n
m
i
,
,
1
If the calculated reactive power in some regulating node oversteps the allowable
value (i.e. the installed power output of compensation devices), then the type of this
node is changed from type (
P
,
U
) to type (
P
,
Q
), where
Q
is appropriate reactive
power limit. The specified voltage is cancelled and we repeat the calculation.