Lecture 14Power Flow

Professor Tom OverbyeDepartment of Electrical and

Computer Engineering

ECE 476

POWER SYSTEM ANALYSIS

2

Announcements

Be reading Chapter 6, also Chapter 2.4 (Network Equations). HW 5 is 2.38, 6.9, 6.18, 6.30, 6.34, 6.38; do by October 6 but

does not need to be turned in. First exam is October 11 during class. Closed book, closed

notes, one note sheet and calculators allowed. Exam covers thru the end of lecture 13 (today)

An example previous exam (2008) is posted. Note this is exam was given earlier in the semester in 2008 so it did not include power flow, but the 2011 exam will (at least partially)

3

Modeling Voltage Dependent Load

So far we've assumed that the load is independent of

the bus voltage (i.e., constant power). However, the

power flow can be easily extended to include voltage

depedence with both the real and reactive l

Di Di

1

1

oad. This

is done by making P and Q a function of :

( cos sin ) ( ) 0

( sin cos ) ( ) 0

i

n

i k ik ik ik ik Gi Di ik

n

i k ik ik ik ik Gi Di ik

V

V V G B P P V

V V G B Q Q V

4

Voltage Dependent Load Example

22 2 2 2

2 22 2 2 2 2

2 2 2 2

In previous two bus example now assume the load is

constant impedance, so

P ( ) (10sin ) 2.0 0

( ) ( 10cos ) (10) 1.0 0

Now calculate the power flow Jacobian

10 cos 10sin 4.0( )

10

V V

Q V V V

V VJ

x

x

x2 2 2 2 2sin 10cos 20 2.0V V V

5

Voltage Dependent Load, cont'd

(0)

22 2 2(0)

2 22 2 2 2

(0)

1(1)

0Again set 0, guess

1

Calculate

(10sin ) 2.0 2.0f( )

1.0( 10cos ) (10) 1.0

10 4( )

0 12

0 10 4 2.0 0.1667Solve

1 0 12 1.0 0.9167

v

V V

V V V

x

x

J x

x

6

Voltage Dependent Load, cont'd

Line Z = 0.1j

One Two 1.000 pu 0.894 pu

160 MW 80 MVR

160.0 MW120.0 MVR

-10.304 Deg

160.0 MW 120.0 MVR

-160.0 MW -80.0 MVR

With constant impedance load the MW/Mvar load atbus 2 varies with the square of the bus 2 voltage magnitude. This if the voltage level is less than 1.0,the load is lower than 200/100 MW/Mvar

7

Dishonest Newton-Raphson

Since most of the time in the Newton-Raphson iteration is spend calculating the inverse of the Jacobian, one way to speed up the iterations is to only calculate/inverse the Jacobian occasionally– known as the “Dishonest” Newton-Raphson– an extreme example is to only calculate the Jacobian for

the first iteration( 1) ( ) ( ) -1 ( )

( 1) ( ) (0) -1 ( )

( )

Honest: - ( ) ( )

Dishonest: - ( ) ( )

Both require ( ) for a solution

v v v v

v v v

v

x x J x f x

x x J x f x

f x

8

Dishonest Newton-Raphson Example

2

1(0)( ) ( )

( ) ( ) 2(0)

( 1) ( ) ( ) 2(0)

Use the Dishonest Newton-Raphson to solve

( ) - 2 0

( )( )

1(( ) - 2)

21

(( ) - 2)2

v v

v v

v v v

f x x

df xx f x

dx

x xx

x x xx

9

Dishonest N-R Example, cont’d

( 1) ( ) ( ) 2(0)

(0)

( ) ( )

1(( ) - 2)

2

Guess x 1. Iteratively solving we get

v (honest) (dishonest)

0 1 1

1 1.5 1.5

2 1.41667 1.375

3 1.41422 1.429

4 1.41422 1.408

v v v

v v

x x xx

x x

We pay a pricein increased iterations, butwith decreased computationper iteration

10

Two Bus Dishonest ROC

Slide shows the region of convergence for different initialguesses for the 2 bus case using the Dishonest N-R

Red regionconvergesto the highvoltage solution,while the yellow regionconvergesto the lowvoltage solution

11

Honest N-R Region of Convergence

Maximum of 15

iterations

12

Decoupled Power Flow

The completely Dishonest Newton-Raphson is not used for power flow analysis. However several approximations of the Jacobian matrix are used.

One common method is the decoupled power flow. In this approach approximations are used to decouple the real and reactive power equations.

13

Decoupled Power Flow Formulation

( ) ( )

( ) ( )( )

( )( ) ( ) ( )

( )2 2 2

( )

( )

General form of the power flow problem

( )( )

( )

where

( )

( )

( )

v v

v vv

vv v v

vD G

v

vn Dn Gn

P P P

P P P

P Pθθ V P x

f xQ xVQ Q

θ V

x

P x

x

14

Decoupling Approximation

( ) ( )

( )

( ) ( )( )

( ) ( ) ( )

Usually the off-diagonal matrices, and

are small. Therefore we approximate them as zero:

( )( )

( )

Then the problem

v v

v

v vv

v v v

P QV θ

P0

θ P xθf x

Q Q xV0V

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

can be decoupled

( ) ( )v v

vv v v

P Qθ P x V Q x

θ V

15

Off-diagonal Jacobian Terms

Justification for Jacobian approximations:

1. Usually r x, therefore

2. Usually is small so sin 0

Therefore

cos sin 0

cos sin 0

ij ij

ij ij

ii ij ij ij ij

j

ii j ij ij ij ij

j

G B

V G B

V V G B

P

V

Qθ

16

Decoupled N-R Region of Convergence

17

Fast Decoupled Power Flow

By continuing with our Jacobian approximations we can actually obtain a reasonable approximation that is independent of the voltage magnitudes/angles.

This means the Jacobian need only be built/inverted once.

This approach is known as the fast decoupled power flow (FDPF)

FDPF uses the same mismatch equations as standard power flow so it should have same solution

The FDPF is widely used, particularly when we only need an approximate solution

18

FDPF Approximations

ij

( ) ( )( )( ) 1 1

( ) ( )

bus

The FDPF makes the following approximations:

1. G 0

2. 1

3. sin 0 cos 1

Then

( ) ( )

Where is just the imaginary part of the ,

except the slack bus row/co

i

ij ij

v vvv

v v

V

j

P x Q xθ B V B

V VB Y G B

lumn are omitted

19

FDPF Three Bus Example

Line Z = j0.07

Line Z = j0.05 Line Z = j0.1

One Two

200 MW 100 MVR

Three 1.000 pu

200 MW 100 MVR

Use the FDPF to solve the following three bus system

34.3 14.3 20

14.3 24.3 10

20 10 30bus j

Y

20

FDPF Three Bus Example, cont’d

1

(0)(0)2 2

3 3

34.3 14.3 2024.3 10

14.3 24.3 1010 30

20 10 30

0.0477 0.0159

0.0159 0.0389

Iteratively solve, starting with an initial voltage guess

0 1

0 1

bus j

V

V

Y B

B

(1)2

3

0 0.0477 0.0159 2 0.1272

0 0.0159 0.0389 2 0.1091

21

FDPF Three Bus Example, cont’d

(1)2

3

i

i i1

(2)2

3

1 0.0477 0.0159 1 0.9364

1 0.0159 0.0389 1 0.9455

P ( )( cos sin )

V V

0.1272 0.0477 0.0159

0.1091 0.0159 0.0389

nDi Gi

k ik ik ik ikk

V

V

P PV G B

x

(2)2

3

0.151 0.1361

0.107 0.1156

0.924

0.936

0.1384 0.9224Actual solution:

0.1171 0.9338

V

V

θ V

22

FDPF Region of Convergence

23

“DC” Power Flow

The “DC” power flow makes the most severe approximations:

– completely ignore reactive power, assume all the voltages are always 1.0 per unit, ignore line conductance

This makes the power flow a linear set of equations, which can be solved directly

1θ B P

24

Power System Control

A major problem with power system operation is the limited capacity of the transmission system

– lines/transformers have limits (usually thermal)– no direct way of controlling flow down a transmission line

(e.g., there are no valves to close to limit flow)– open transmission system access associated with industry

restructuring is stressing the system in new ways

We need to indirectly control transmission line flow by changing the generator outputs

25

DC Power Flow Example

26

DC Power Flow 5 Bus Example

slack

One

Two

ThreeFourFiveA

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.000 pu 1.000 pu

1.000 pu

1.000 pu

1.000 pu 0.000 Deg -4.125 Deg

-18.695 Deg

-1.997 Deg

0.524 Deg

360 MW

0 Mvar

520 MW

0 Mvar

800 MW 0 Mvar

80 MW 0 Mvar

Notice with the dc power flow all of the voltage magnitudes are 1 per unit.

27

Indirect Transmission Line Control

What we would like to determine is how a change in generation at bus k affects the power flow on a line from bus i to bus j.

The assumption isthat the changein generation isabsorbed by theslack bus

28

Power Flow Simulation - Before

One way to determine the impact of a generator change is to compare a before/after power flow.

For example below is a three bus case with an overload

Z for all lines = j0.1

One Two

200 MW 100 MVR

200.0 MW 71.0 MVR

Three 1.000 pu

0 MW 64 MVR

131.9 MW

68.1 MW 68.1 MW

124%

29

Power Flow Simulation - After

Z for all lines = j0.1Limit for all lines = 150 MVA

One Two

200 MW 100 MVR

105.0 MW 64.3 MVR

Three1.000 pu

95 MW 64 MVR

101.6 MW

3.4 MW 98.4 MW

92%

100%

Increasing the generation at bus 3 by 95 MW (and hence decreasing it at bus 1 by a corresponding amount), resultsin a 31.3 drop in the MW flow on the line from bus 1 to 2.

30

Analytic Calculation of Sensitivities

Calculating control sensitivities by repeat power flow solutions is tedious and would require many power flow solutions. An alternative approach is to analytically calculate these values

The power flow from bus i to bus j is

sin( )

So We just need to get

i j i jij i j

ij ij

i j ijij

ij Gk

V VP

X X

PX P

31

Analytic Sensitivities

1

From the fast decoupled power flow we know

( )

So to get the change in due to a change of

generation at bus k, just set ( ) equal to

all zeros except a minus one at position k.

0

1

0

θ B P x

θ

P x

P

Bus k

32

Three Bus Sensitivity Example

line

bus

12

3

For the previous three bus case with Z 0.1

20 10 1020 10

10 20 1010 20

10 10 20

Hence for a change of generation at bus 3

20 10 0 0.0333

10 20 1 0.0667

j

j

Y B

3 to 1

3 to 2 2 to 1

0.0667 0Then P 0.667 pu

0.1P 0.333 pu P 0.333 pu

33

Balancing Authority Areas

An balancing authority area (use to be called operating areas) has traditionally represented the portion of the interconnected electric grid operated by a single utility

Transmission lines that join two areas are known as tie-lines.

The net power out of an area is the sum of the flow on its tie-lines.

The flow out of an area is equal to

total gen - total load - total losses = tie-flow

34

Area Control Error (ACE)

The area control error (ace) is the difference between the actual flow out of an area and the scheduled flow, plus a frequency component

Ideally the ACE should always be zero.Because the load is constantly changing, each utility

must constantly change its generation to “chase” the ACE.

int schedace 10P P f

35

Automatic Generation Control

Most utilities use automatic generation control (AGC) to automatically change their generation to keep their ACE close to zero.

Usually the utility control center calculates ACE based upon tie-line flows; then the AGC module sends control signals out to the generators every couple seconds.

36

Power Transactions

Power transactions are contracts between generators and loads to do power transactions.

Contracts can be for any amount of time at any price for any amount of power.

Scheduled power transactions are implemented by modifying the value of Psched used in the ACE calculation

37

PTDFs

Power transfer distribution factors (PTDFs) show the linear impact of a transfer of power.

PTDFs calculated using the fast decoupled power flow B matrix

1 ( )

Once we know we can derive the change in

the transmission line flows

Except now we modify several elements in ( ),

in portion to how the specified generators would

participate in the pow

θ B P x

θ

P x

er transfer

38

Nine Bus PTDF Example

10%

60%

55%

64%

57%

11%

74%

24%

32%

A

G

B

C

D

E

I

F

H

300.0 MW 400.0 MW 300.0 MW

250.0 MW

250.0 MW

200.0 MW

250.0 MW

150.0 MW

150.0 MW

44%

71%

0.00 deg

71.1 MW

92%

Figure shows initial flows for a nine bus power system

39

Nine Bus PTDF Example, cont'd

43%

57% 13%

35%

20%

10%

2%

34%

34%

32%

A

G

B

C

D

E

I

F

H

300.0 MW 400.0 MW 300.0 MW

250.0 MW

250.0 MW

200.0 MW

250.0 MW

150.0 MW

150.0 MW

34%

30%

0.00 deg

71.1 MW

Figure now shows percentage PTDF flows from A to I

40

Nine Bus PTDF Example, cont'd

6%

6% 12%

61%

12%

6%

19%

21%

21%

A

G

B

C

D

E

I

F

H

300.0 MW 400.0 MW 300.0 MW

250.0 MW

250.0 MW

200.0 MW

250.0 MW

150.0 MW

150.0 MW

20%

18%

0.00 deg

71.1 MW

Figure now shows percentage PTDF flows from G to F

41

WE to TVA PTDFs

42

Line Outage Distribution Factors (LODFS)

LODFs are used to approximate the change in the flow on one line caused by the outage of a second line– typically they are only used to determine the change in

the MW flow– LODFs are used extensively in real-time operations– LODFs are state-independent but do dependent on the

assumed network topology

,l l k kP LODF P

43

Flowgates

The real-time loading of the power grid is accessed via “flowgates”

A flowgate “flow” is the real power flow on one or more transmission element for either base case conditions or a single contingency

– contingent flows are determined using LODFs

Flowgates are used as proxies for other types of limits, such as voltage or stability limits

Flowgates are calculated using a spreadsheet

44

NERC Regional Reliability Councils

NERCis theNorthAmericanElectricReliabilityCouncil

45

NERC Reliability Coordinators

Source: http://www.nerc.com/page.php?cid=5%7C67%7C206