9 EE 334 Powfbdzer Flow Solutions

7/29/2019 9 EE 334 Powfbdzer Flow Solutions

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EE 334 Power systems

Power Flow Solutions

Prof. S. V. Kulkarni


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Ketan Badgujar

EE 334 Power systems Prof. S. V. Kulkarni

Topic 9 : Power Flow Solutions

Power flow studies : useful in planning and designing futureexpansion of power systems.

Also useful in determining best operating condition of an

existing system.

Information obtained from load flow study : the magnitude and

phase angle of voltage at each bus and the real and reactive

power flowing in each line.

Lines are represented by their per-phase nominal-equivalent

circuits (medium-length line representation)

Introduction

2


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Ketan Badgujar

EE 334 Power systems Prof. S. V. Kulkarni


Nominal-equivalent circuit

3

VRY/2

IR

Y/2

Z = R + jX

VS

IS+

_

+

_


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Ketan BadgujarEE 334 Power systems Prof. S. V. Kulkarni


Y bus is used for power-flow studies.

Other essential information: transformer ratings and

impedances, tap settings

Voltage at any bus:

Net current injected at bus i,

4

( )iiiiii jVVV sincos +==

cos sinij ij ij ij ij ij ij ij ijY Y Y j Y G jB= = + = +

1 1 2 2

1

n

i i i in n ij j

j

I Y V Y V Y V Y V=

= + + + =


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Let and be the net real and reactive powers entering the

network at bus i,

The power flow problem is handled more easily through the use

of rather than

i = 1, 2, 3, n

5

iP iQ

*

iI* *

1

n

i i i i i ij j

j

P jQ V I V Y V =

= =

*i i i i iS P jQ V I = + =

iI


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Equating real and imaginary parts,

i = 1, 2, 3,..n

i = 1, 2, 3,..n

The above equations represent 2n power flow equations at nbuses of a power system (n real power flow equations and n

reactive power flow equations)

Each bus is characterized by four variables:

resulting in a total of 4n variables

Hence, this system of equations can be solved for 2n variables

if the remaining 2n variables are specified

6

, , ,i i i i

P Q V

1

cos( ) (1)n

i i j ij ij j i

j

P V V Y =

= +

1

sin( ) (2)n

i i j ij ij j i

j

Q V V Y =

= +


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Hence, two variables for each bus are fixed a priori

Still the solution is not straight-forward since these are non-linear

algebraic equationsThe solution can only be obtained by using an iterative numerical

technique

Depending upon which two variables are fixed a priori, the busesare classified into 3 categories:

(i)

7

iP iQ

diQdiP

giP giQ

iV i

PQ bus (load bus)

At this bus, the net powers and are known

and (demand) are known from load forecasting and (generation) are usually zero

Two unknowns are and


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(ii) PV bus / generator bus / voltage controlled bus

At this bus the voltage magnitude is kept constant: voltage

controlled bus.Real power generation is controlled by adjusting the prime

mover, and the voltage magnitude is controlled by adjusting the

generator excitation.

Therefore, at each PV bus, and are specified.

If this bus is also loaded, and are known

Generator reactive power required to support the specified

voltage magnitude is unknown

(net) and are the unknown variables at a PV bus

8

giP iV

diP diQ

giQ

iQ i


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(iii) Slack bus / swing bus / reference bus

Real and reactive powers at this bus are not specified

Voltage magnitude and phase angle (normally set equal to zero)are specified

Usually, there is only one bus of this type in a given network

Need for this bus: real and reactive powers cannot be fixed apriori at all buses as the net complex power flow into the

network is not known

System losses are not known till the power flow analysis is completed

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Hence, it is necessary to have one bus at which complex power

is unspecified so that it supplies the difference,

where loss of lines and transformers

Currents in transmission lines cannot be calculated until and

are known at every bus.

Losses cannot be specified in advance.

The bus connected to the largest generating station is usually

taken as the slack/swing bus.

10

V

( )gi di LP P P +

LP


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By shifting all the variables on one side, equations 1 and 2 can

be written in vector form:

where u = vector of 2n unspecified variables

v = vector of 2n specified variables

For a realizable solution, all the specified and unspecifiedvariables must lie within practical limits (governed by operating

constraints and specifications of various equipment)

11

( , v) 0f u =


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(1) ( 5 or 10% voltage variation)

(2) A few of the variables must satisfy the

jkn inequality constraint.(imposed by stability considerations)

(3)

(imposed by limitations ofP and/or Q generation

sources)

Also, equality constraints must be satisfied,

12

min maxi i iV V V

i

maxi j i j

( ) ( )min max ;gi gi giP P P ( ) ( )min maxgi gi giQ Q Q

gi di L gi di LP P P Q Q Q= + = +


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The power flow problem is now defined as follows:

Specify at all the PQ buses

Specify (i.e ) and at all the PV buses

Specify and (= 0) at the slack bus (1)

Thus 2n variables ofv vector are specified

Now the 2n equations can be solved iteratively to determine thevalues of 2n variables of vector u

At the end of power flow solution, we obtain

- voltage magnitudes and angles at the PQ buses- reactive powers and voltage angles at the PV buses

- active and reactive powers at the slack bus

13

i iP jQ+

iP iV

iV i

giP


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An iterative algorithm for solving a set of non-linear algebraic

equations

PQ busesTo start with, a solution vector is assumed (flat voltage start or

uniform voltage profile)

(Here, it is assumed that buses 2 to n are PQ buses).

RHS: the most recently calculated values of voltages should be used

Gauss-Seidel method

14

1

1 n

i i ij j

jii j i

V I Y V Y =

=

1

1; 2,3,......

ni i

ij j

jii i j i

P jQY V i n

Y V

=

= =

1

n

i ij jj

I Y V==


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Voltages at buses i = 2, 3, n are updated in a sequence during

each iteration

V1 Slack bus voltage is fixed, therefore not updated

Example (4-bus system):

The entire process is repeated till the changes in voltages at all

buses are less than a pre-decided small number

Therefore, generalizing,

15

( )( )

( ) ( )( )1 1 03 3

3 31 1 32 2 34 4033 3

1 P jQV Y V Y V Y V Y V

= + +

( )

( )

( ) ( )1

1

11 1

1 i nm m mi ii ik k ik k m

k k iii i

P jQV Y V Y V

Y V

= = +

=


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Acceleration Factor:

The number of iterations required may be reduced considerably ifthe voltage in an iteration is corrected by using an acceleration

factor and the previous iteration value:

=1.6 usually, and not greater than 2 - otherwise the algorithm

may diverge

16

( ) ( ) ( ) ( )( )1 1

(accelerated)m m m m

i i i iV V V V + += +


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Here, voltage magnitudes are specified and Qis are unknown

Therefore, only the angle of the voltage at ith bus is updated

PV buses

17

( ) ( ) ( )

( )( )

( )

( ) ( )

( )( )

( )

* *

1 1

11 * 1( )

1

11

11 1

Im

Im

1

(corrected)

i

n n

i i ij j i i i i i ij j

j j

i nm m mm

i ij j ij j

j j i

m i nm m mi i

i ij j ij jmj j iii i

mm i

i i m

i

Q V Y V P jQ V I V Y V

Q V Y V Y V

P jQV Y V Y V

Y V

V

V V V

= =

= =

= = +

= = =

= +

=

=


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By making suitable approximations, it is possible to linearize

power flow equations in order to obtain quick solutions.

This approach is adopted for planning studies wherein powerflow solutions are required repeatedly (high degree of accuracy

is not essential)

Approximations

Line resistances are neglected, i.e PL=0

is small (


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19

* *

1

1 1

1

1

2

1

| | | || | cos( ) | | | || | cos[90 ( )]

| | | || | ( ) (3)

| | | || | sin( )

| | | || | cos( ) | | |

n

i i i i i ij j

j

n n

i i j ij ij j i i j ij i j

j j

n

i j ij i jj

n

i i j ij ij j i

j

n

i j ij i j i

jj i

P jQ V I V Y V

P V V Y V V Y

V V Y

Q V V Y

V V Y V

=

= =

=

=

=

= =

= + =

=

= +

= +

| (4)iiY


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Since |Vi| are specified in (3), it represents a set of linear

algebraic equations in (which are (n-1) in number) as at

slack bus is specifiedThe equation corresponding to slack bus (n=1) is redundant as

the real power injected at this bus is fully specified

Equation (3) can be now solved explicitly (i.e non-iteratively)

for which when substituted in (4) give Qis , the

reactive power bus injections.

20

)0with(22

1 == ==L

n

igi

n

idi PPPP

1

2 3, ... n


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Equations (3) and (4) are decoupled, and therefore need not be

solved simultaneouslyDecoupling:- since all Vis are specified

Earlier some Vis were not specified and were function of Qitherefore there was coupling between (3) and (4)

(3) and (4) are solved sequentially

Computational burden is reduced

- 2n to (n1) equations

- non-iterative solution

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We need to solve n non-linear algebraic equations

Let initial values of unknowns beSuppose that are the corrections, to be added to

satisfy the set of equations

Expanding using Taylor series,

The Newton-Raphson Method

22

( )1 2, , ......, ; 1,2,3...,i n if x x x K i n = =

0 0 0

1 2, ,....., nx x x

0 0 0

1 2, ,....., nx x x

( )0 0 0 0 0 0

1 1 2 2, ,......, ; 1,2,3...,i n n if x x x x x x K i n + + + = =

( )

00 0

0 0 00 0 0 1 21 2 1 2

....., ,.....,

higher order terms

i i in

i n in

f f fx x x

f x x x Kx x x + + +

+ =

+


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derivative offi w.r.t x1, evaluated at

Neglecting higher order terms,

23

0

1

if

x

( )0 0 01 2, ,....., nx x x

00 0

1 1 1

0 01 21 1

0 0

2 2

0 000 0

1 2

. . .

. . . . . .

. . . . . .. .

. . . . . .. .

. . . . . .. .

. . .

n

n nn n n

n

f f f

x x xf x

f x

f xf f f

x x x

+

.

.

.

.

i

n

K

K

=


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In matrix notations,

is known as Jacobian matrix

These are linear algebraic equations.

24

0 0 0f J x K+ =

0J

0 0 0

1 1 1 1

0 0 0 0

0 0 0

n n n n

K f K x

K f J x J

K f K x

=

= = =

0 0

1 11

0

0 0

n n

x K

J

x K

=


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The approximate values of corrections can be obtained by

inverting Jacobian or by using LU factorization.

25

( ) ( ) ( )

1 0 0

1

or generalizing,

m m m

x x x

x x x+

= +

= +


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Iterations are continued until

26

( )

( )

( ) ( )

( ) ( )

1

1

cos

sin

n

i i j ij ij j i

j

n

i i j ij ij j i

j

i i ispecified calculated

i i ispecified calculated

P V V Y

Q V V Y

P P P

Q Q Q

=

=

= +

= +

=

=

( )( ) ( ) (a specified value) = 1, 2, ....,m mi i iK f x K i n =

9 EE 334 Powfbdzer Flow Solutions

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