ece4762007_lect12

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Lecture 12

Power Flow

Professor Tom OverbyeDepartment of Electrical and

Computer Engineering

ECE 476

POWER SYSTEM ANALYSIS

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Announcements

Homework 5 is due on Oct 4Homework 6 is due on Oct 18First exam is 10/9 in class; closed book, closednotes, one note sheet and calculators allowedPower plant, substation field trip, 10/11 starting at1pm. We’ll meet at corner of Gregory and Oak streets.Be reading Chapter 6, but Chapter 6 will not be onthe example

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In the News

Ground was broken Monday for the new 1600 MWPrairie State Energy Campus coal-fired power plantProject is located near Lively Grove, IL, and will be

connected to grid via 3 345 kV lines (38 new miles)Prairie State, which will cost $ 2.9 billion to build(about $1.82 per watt), should be finished by 2011or 2012.Output of plant will be bought primarily bymunicipal and coop customers throughout region(Missouri to Ohio)

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Prairie State New 345 kV Lines

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Gauss Iteration

There are a number of different iterative methodswe can use. We'll consider two: Gauss and Newton.

With the Gauss method we need to rewrite our equation in an implicit form: x = h(x)

To iterate we fir (0)

( +1) ( )

st make an initial guess of x, x ,

and then iteratively solve x ( ) until we

find a "fixed point", x, such that x (x).ˆ ˆ ˆ

v vh x

h

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Gauss Iteration Example

( 1) ( )

(0)

( ) ( )

Example: Solve - 1 01

Let = 0 and arbitrarily guess x 1 and solve

0 1 5 2.61185

1 2 6 2.61612

2 2.41421 7 2.61744

3 2.55538 8 2.61785

4 2.59805 9 2.61798

v v

v v

x x x x

v

v x v x

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Stopping Criteria

( ) ( ) ( 1) ( )

A key problem to address is when to stop theiteration. With the Guass iteration we stop when

with

If x is a scalar this is clear, but if x is a vector weneed to generalize t

v v v v x x x x

( )

2i2

1

he absolute value by using a norm

Two common norms are the Euclidean & infinity

max x

v

j

n

i ii

x

x

x x

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Gauss Power Flow

** * *

i1 1

* * * *

1 1

*

* 1 1,

*

*

1,

We first need to put the equation in the correct form

S

S

S

S1

i i

i

i

n n

i i i ik k i ik k k k

n ni i i ik k ik k

k k

n ni

ik k ii i ik k k k k i

ni

i ik k

ii k k i

V I V Y V V Y V

V I V Y V V Y V

Y V Y V Y V V

V Y V Y

V

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Gauss Two Bus Power Flow Example

A 100 MW, 50 Mvar load is connected to a generator through a line with z = 0.02 + j0.06 p.u. and linecharging of 5 Mvar on each end (100 MVA base).

Also, there is a 25 Mvar capacitor at bus 2. If thegenerator voltage is 1.0 p.u., what is V 2?

SLoad = 1.0 + j0.5 p.u.

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Gauss Two Bus Example, cont’d

2

2 bus

bus

22

The unknown is the complex load voltage, V .To determine V we need to know the .

15 15

0.02 0.065 14.95 5 15

Hence5 15 5 14.70

( Note - 15 0.05 0.25)

j j

j j

j j

B j j j

Y

Y

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*2

2 *22 1,2

2 *2

(0)2

( ) ( )2 2

1 S

1 -1 0.5( 5 15)(1.0 0)

5 14.70

Guess 1.0 0 (this is known as a flat start)

0 1.000 0.000 3 0.9622 0.0556

1 0.9671 0.0568 4 0.9622 0.0556

2 0

nik k

k k i

v v

V Y V Y V

jV j

j V

V

v V v V

j j

j j

.9624 0.0553 j

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2

* *1 1 11 1 12 2

1

0.9622 0.0556 0.9638 3.3Once the voltages are known all other values can

be determined, such as the generator powers and the

line flows

S ( ) 1.023 0.239

In actual units P 102.3 MW

V j

V Y V Y V j

12

2

, Q 23.9 Mvar

The capacitor is supplying V 25 23.2 Mvar

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Slack Bus

In previous example we specified S 2 and V 1 andthen solved for S 1 and V 2.We can not arbitrarily specify S at all buses because

total generation must equal total load + total lossesWe also need an angle reference bus.To solve these problems we define one bus as the"slack" bus. This bus has a fixed voltage magnitudeand angle, and a varying real/reactive power injection.

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Gauss with Many Bus Systems

*( )( 1)

( )* 1,

( ) ( ) ( )1 2

( 1)

With multiple bus systems we could calculatenew V ' as follows:

S1

( , ,..., )

But after we've determined we have a better estimate of

i

i

nvv i

i ik k vii k k i

v v vi n

vi

s

V Y V Y V

h V V V

V

its voltage , so it makes sense to use this

new value. This approach is known as the

Gauss-Seidel iteration.

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Gauss-Seidel Iteration

( 1) ( ) ( ) ( )2 12 2 3

( 1) ( 1) ( ) ( )2 13 2 3

( 1) ( 1) ( 1) ( ) ( )2 14 2 3 4

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

2 1 2 3 4

Immediately use the new voltage estimates:( , , , , )

( , , , , )

( , , , , )

( , , , ,

v v v vn

v v v vn

v v v v vn

v v vv v

n n

V h V V V V

V h V V V V

V h V V V V V

V h V V V V V

)

The Gauss-Seidel works better than the Gauss, and

is actually easier to implement. It is used instead

of Gauss.

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Three Types of Power Flow Buses

There are three main types of power flow buses – Load (PQ) at which P/Q are fixed; iteration solves for

voltage magnitude and angle. – Slack at which the voltage magnitude and angle are fixed;

iteration solves for P/Q injections – Generator (PV) at which P and |V| are fixed; iteration

solves for voltage angle and Q injection

special coding is needed to include PV buses in theGauss-Seidel iteration

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Inclusion of PV Buses in G-S

i

i

* *

1

( )( ) ( )*

1

( ) ( )

To solve for V at a PV bus we must first make aguess of Q :

Hence Im

In the iteration we use

k

n

i i ik k i ik

nvv v

i i ik k

v vi i i

S V Y V P jQ

Q V Y V

S P jQ

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Inclusion of PV Buses, cont'd

( 1)

( )*( )( 1)

( )*

1,( 1)

i i

Tentatively solve for

S1

But since V is specified, replace by Vi

vi

v nvv i

i ik k v

ii k k iv

i

V

V Y V Y V

V

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Two Bus PV Example

Bus 1(slack bus)

Bus 2 V1 = 1.0 V2 = 1.05

P2 = 0 MW

z = 0.02 + j 0.06

Consider the same two bus system from the previousexample, except the load is replaced by a generator

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Two Bus PV Example, cont'd

*2

2 21 1*22 2

* *2 21 1 2 22 2 2

2( ) ( 1) ( 1)2 2 2

1

Im[ ]

Guess V 1.05 0

0 0 0.457 1.045 0.83 1.050 0.83

1 0 0.535 1.049 0.93 1.050 0.93

2 0 0.545 1.050 0.96 1.050 0.96

v v v

S V Y V Y V

Q Y V V Y V V

v S V V

j

j

j

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Generator Reactive Power Limits

The reactive power output of generators varies tomaintain the terminal voltage; on a real generator this is done by the exciter

To maintain higher voltages requires more reactive power Generators have reactive power limits, which aredependent upon the generator's MW outputThese limits must be considered during the power flow solution.

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Generator Reactive Limits, cont'd

During power flow once a solution is obtainedcheck to make generator reactive power output iswithin its limits

If the reactive power is outside of the limits, fix Q atthe max or min value, and resolve treating thegenerator as a PQ bus – this is know as "type-switching" – also need to check if a PQ generator can again regulateRule of thumb: to raise system voltage we need tosupply more vars

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Accelerated G-S Convergence

( 1) ( )

( 1) ( ) ( ) ( )

(

Previously in the Gauss-Seidel method we werecalculating each value x as

( )

To accelerate convergence we can rewrite this as

( )

Now introduce acceleration parameter

v v

v v v v

v

x h x

x x h x x

x

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

With = 1 this is identical to standard gauss-seidel.

Larger values of may result in faster convergence.

v v v x h x x

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Accelerated Convergence, cont’d

( 1) ( ) ( ) ( )Consider the previous example: - 1 0

(1 )

Comparison of results with different values of

1 1.2 1.5 20 1 1 1 1

1 2 2.20 2.5 3

2 2.4142 2.5399 2.6217 2.4643 2.5554 2.6045 2.6179 2.675

4 2.59

v v v v x x

x x x x

k

81 2.6157 2.6180 2.596

5 2.6118 2.6176 2.6180 2.626

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Gauss-Seidel Advantages

Each iteration is relatively fast (computational order is proportional to number of branches + number of

buses in the system

Relatively easy to program

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Gauss-Seidel Disadvantages

Tends to converge relatively slowly, although thiscan be improved with accelerationHas tendency to miss solutions, particularly on large

systemsTends to diverge on cases with negative branchreactances (common with compensated lines)

Need to program using complex numbers

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Newton-Raphson Algorithm

The second major power flow solution method isthe Newton-Raphson algorithmKey idea behind Newton-Raphson is to use

sequential linearization

General form of problem: Find an x such that

( ) 0ˆ f x

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Newton-Raphson Method (scalar)

( )

( ) ( )

( )( ) ( )

2 ( ) 2( )2

1. For each guess of , , defineˆ

-ˆ

2. Represent ( ) by a Taylor series about ( )ˆ

( )( ) ( )

ˆ

( )higher order terms

v

v v

vv v

vv

x x

x x x

f x f x

df x f x f x x

dx

d f x xdx

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Newton- Raphson Method, cont’d

( )( ) ( )

( )

1( )( ) ( )

3. Approximate ( ) by neglecting all termsˆ

except the first two

( )( ) 0 ( )ˆ

4. Use this linear approximation to solve for

( )( )

5. Solve for a new estim

vv v

v

vv v

f x

df x f x f x x

dx x

df x x f x

dx

( 1) ( ) ( )

ate of xˆ

v v v x x x

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Newton-Raphson Example

2

1( )( ) ( )

( ) ( ) 2( )

( 1) ( ) ( )

( 1) ( ) ( ) 2( )

Use Newton-Raphson to solve ( ) - 2 0The equation we must iteratively solve is

( )( )

1(( ) - 2)

2

1(( ) - 2)

2

vv v

v vv

v v v

v v vv

f x x

df x x f xdx

x x x

x x x

x x x x

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Newton- Raphson Example, cont’d

( 1) ( ) ( ) 2( )

(0)

( ) ( ) ( )

3 3

6

1 (( ) - 2)2

Guess x 1. Iteratively solving we get

v ( )

0 1 1 0.5

1 1.5 0.25 0.08333

2 1.41667 6.953 10 2.454 10

3 1.41422 6.024 10

v v vv

v v v

x x x x

x f x x

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Sequential Linear Approximations

Function is f(x) = x 2 - 2 = 0.Solutions are points wheref(x) intersects f(x) = 0 axis

At eachiteration the

N-R methoduses a linear approximation

to determinethe next valuefor x

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Newton-Raphson Comments

When close to the solution the error decreases quitequickly -- method has quadratic convergencef(x (v)) is known as the mismatch, which we would

like to drive to zeroStopping criteria is when f(x (v)) < Results are dependent upon the initial guess. Whatif we had guessed x (0) = 0, or x (0) = -1?A solution’s region of attraction (ROA) is the set of initial guesses that converge to the particular solution. The ROA is often hard to determine

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Multi-Variable Newton-Raphson

1 1

2 2

Next we generalize to the case where is an n-dimension vector, and ( ) is an n-dimension function

( )

( )( )

( )

Again define the solution so ( ) 0 andˆ ˆ

n n

x f

x f

x f

x

f x

x

xx f x

x

x f x

x ˆx x

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Multi- Variable Case, cont’d

i

1 11 1 1 2

1 2

1

n nn n 1 2

1 2

n

The Taylor series expansion is written for each f ( )f ( ) f ( )

f ( ) f ( )ˆ

f ( ) higher order terms

f ( ) f ( )f ( ) f ( )ˆ

f ( )higher order terms

nn

nn

x x x x

x x

x x x x

x

x

x

x xx x

x

x xx x

x

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Multi- Variable Case, cont’d

1 1 1

1 2

1 12 2 22 2

1 2

1 2

This can be written more compactly in matrix form( ) ( ) ( )

( )( ) ( ) ( )( )

( )ˆ

( )( ) ( ) ( )

n

n

n n n n

n

f f f x x x

f x f f f f x

x x x

f f f f

x x x

x x x

xx x x

xf x

xx x x

higher order terms

n x

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Jacobian Matrix

1 1 1

1 2

2 2 2

1 2

1 2

The n by n matrix of partial derivatives is knownas the Jacobian matrix, ( )

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

n

n

n n n

n

f f f

x x x f f f

x x x

f f f x x x

J x

x x x

x x x

J x

x x x

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Multi-Variable N-R Procedure

1

( 1) ( ) ( )

( 1) ( ) ( ) 1 ( )

( )

Derivation of N-R method is similar to the scalar case( ) ( ) ( ) higher order termsˆ

( ) 0 ( ) ( )ˆ

( ) ( )

( ) ( )

Iterate until ( )

v v v

v v v v

v

f x f x J x x

f x f x J x x

x J x f x

x x x

x x J x f x

f x

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Multi-Variable Example

1

2

2 21 1 2

2 22 1 2 1 2

1 1

1 2

2 2

1 2

xSolve for = such that ( ) 0 wherex

f ( ) 2 8 0

f ( ) 4 0First symbolically determine the Jacobian

f ( ) f ( )

( ) =f ( ) f ( )

x x

x x x x

x x

x x

x f x

x

x

x x

J xx x

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Multi- variable Example, cont’d

1 2

1 2 1 2

11 1 2 1

2 1 2 1 2 2

(0)

1(1)

4 2( ) =2 2

Then

4 2 ( )

2 2 ( )

1Arbitrarily guess 1

1 4 2 5 2.1

1 3 1 3 1.3

x x x x x x

x x x f

x x x x x f

J x

x

x

x

x

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Multi- variable Example, cont’d

1(2)

(2)

2.1 8.40 2.60 2.51 1.8284

1.3 5.50 0.50 1.45 1.2122

Each iteration we check ( ) to see if it is below our

specified tolerance

0.1556( )

0.0900

If = 0.2 then we wou

x

f x

f x

ld be done. Otherwise we'd

continue iterating.

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NR Application to Power Flow

** * *

i1 1

We first need to rewrite complex power equationsas equations with real coefficients

S

These can be derived by defining

Recal

i

n n

i i i ik k i ik k k k

ik ik ik

ji i i i

ik i k

V I V Y V V Y V

Y G jB

V V e V

jl e cos sin j

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