EE 369 POWER SYSTEM ANALYSIS Lecture 14 Power Flow Tom Overbye and Ross Baldick 1.

EE 369POWER SYSTEM ANALYSIS

Lecture 14Power Flow

Tom Overbye and Ross Baldick

1

AnnouncementsRead Chapter 12, concentrating on sections

12.4 and 12.5. Homework 12 is 6.43, 6.48, 6.59, 6.61,

12.19, 12.22, 12.20, 12.24, 12.26, 12.28, 12.29; due Tuesday Nov. 25.

2

400 MVA15 kV

400 MVA15/345 kV

T1

T2800 MVA345/15 kV

800 MVA15 kV

520 MVA

80 MW40 Mvar

280 MVAr 800 MW

Line 3 345 kV

Line

2

Line

1345 kV 100 mi

345 kV 200 mi

50 mi

1 4 3

2

5

Single-line diagram

The N-R Power Flow: 5-bus Example

3

Bus Type

|V|

per unit

θ

degrees

PG

per

unit

QG

per

unit

PL

per

unit

QL

per

unit

QGmax

per

unit

QGmin

per

unit

1 Slack 1.0 0 0 0

2 Load 0 0 8.0 2.8

3 Constant voltage

1.05 5.2 0.8 0.4 4.0 -2.8

4 Load 0 0 0 0

5 Load 0 0 0 0

Table 1. Bus input data

Bus-to-Bus

R

per unit

X

per unit

G

per unit

B

per unit

Maximum

MVA

per unit

2-4 0.0090 0.100 0 1.72 12.0

2-5 0.0045 0.050 0 0.88 12.0

4-5 0.00225 0.025 0 0.44 12.0

Table 2. Line input data


4

Bus-to-Bus

R

per

unit

X

per

unit

Gc

per

unit

Bm

per

unit

Maximum

MVA

per unit

Maximum

TAP

Setting

per unit

1-5 0.00150 0.02 0 0 6.0 —

3-4 0.00075 0.01 0 0 10.0 —

Table 3. Transformer input data

Bus Input Data Unknowns

1 |V1 |= 1.0, θ1 = 0 P1, Q1

2 P2 = PG2-PL2 = -8

Q2 = QG2-QL2 = -2.8

|V2|, θ2

3 |V3 |= 1.05

P3 = PG3-PL3 = 4.4

Q3, θ3

4 P4 = 0, Q4 = 0 |V4|, θ4

5 P5 = 0, Q5 = 0 |V5|, θ5

Table 4. Input data and unknowns


5

Let the Computer Do the Calculations! (Ybus Shown)

6

Ybus Details

02321 YY

2424 24

1 10.89276 9.91964

0.009 0.1Y j per unit

R jX j

2525 25

1 11.78552 19.83932

0.0045 0.05Y j per unit

R jX j

24 2522

24 24 25 25

1 1

2 2

B BY j j

R jX R jX

2

88.0

2

72.1)83932.1978552.1()91964.989276.0( jjjj

unitperj 624.845847.284590.2867828.2

Elements of Ybus connected to bus 2

7

Here are the Initial Bus Mismatches

8

And the Initial Power Flow Jacobian

9

Five Bus Power System Solved

slack

One

Two

ThreeFourFiveA

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.000 pu 0.974 pu

0.834 pu

1.019 pu

1.050 pu 0.000 Deg -4.548 Deg

-22.406 Deg

-2.834 Deg

-0.597 Deg

395 MW

114 Mvar

520 MW

337 Mvar

800 MW 280 Mvar

80 MW 40 Mvar

10

37 Bus Example Design Case

slack

Metropolis Light and Power Electric Design Case 2SLACK345

SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.03 pu

1.02 pu

1.03 pu

1.03 pu

1.01 pu

1.00 pu1.01 pu

1.00 pu

1.02 pu

1.01 pu

1.00 pu

1.01 pu

1.01 pu

1.01 pu

1.01 pu

1.02 pu

1.00 pu

1.00 pu

1.02 pu

0.99 pu

0.99 pu

1.00 pu

1.02 pu

1.00 pu1.01 pu

1.01 pu

1.00 pu 1.00 pu

1.01 pu

1.02 pu 1.02 pu

1.02 pu 1.03 pu

A

MVA

1.02 pu

A

MVA

A

MVA

LYNN138

A

MVA

1.02 pu

A

MVA

1.00 pu

A

MVA

System Losses: 10.70 MW

220 MW 52 Mvar

12 MW 3 Mvar

20 MW 12 Mvar

124 MW 45 Mvar

37 MW

13 Mvar

12 MW 5 Mvar

150 MW 0 Mvar

56 MW

13 Mvar

15 MW 5 Mvar

14 MW

2 Mvar

38 MW 3 Mvar

45 MW 0 Mvar

25 MW 36 Mvar

36 MW 10 Mvar

10 MW 5 Mvar

22 MW 15 Mvar

60 MW 12 Mvar

20 MW 28 Mvar

23 MW 7 Mvar

33 MW 13 Mvar

15.9 Mvar 18 MW 5 Mvar

58 MW 40 Mvar

60 MW 19 Mvar

14.2 Mvar

25 MW 10 Mvar

20 MW 3 Mvar

23 MW 6 Mvar 14 MW

3 Mvar

4.9 Mvar

7.3 Mvar

12.8 Mvar

28.9 Mvar

7.4 Mvar

0.0 Mvar

55 MW 25 Mvar

39 MW 13 Mvar

150 MW 0 Mvar

17 MW 3 Mvar

16 MW -14 Mvar

14 MW 4 Mvar

KYLE69A

MVA

11

Good Power System Operation• Good power system operation requires that there be

no “reliability” violations (needing to shed load, have cascading outages, or other unacceptable conditions) for either the current condition or in the event of statistically likely contingencies:• Reliability requires as a minimum that there be no

transmission line/transformer limit violations and that bus voltages be within acceptable limits (perhaps 0.95 to 1.08)

• Example contingencies are the loss of any single device. This is known as n-1 reliability.

12

Good Power System Operation

• North American Electric Reliability Corporation now has legal authority to enforce reliability standards (and there are now lots of them).

• See http://www.nerc.com for details (click on Standards)

13

http://www.nerc.com/

Looking at the Impact of Line Outages

slack


SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.03 pu

1.02 pu

1.03 pu

1.03 pu

1.01 pu

1.00 pu1.01 pu

1.00 pu

1.02 pu

1.01 pu

1.00 pu

1.01 pu

1.01 pu

1.01 pu

1.01 pu

1.02 pu

1.01 pu

1.00 pu

1.02 pu

0.90 pu

0.90 pu

0.94 pu

1.01 pu

0.99 pu1.00 pu

1.00 pu

1.00 pu 1.00 pu

1.01 pu

1.01 pu 1.02 pu

1.02 pu 1.03 pu

A

MVA

1.02 pu

A

MVA

A

MVA

LYNN138

A

MVA

1.02 pu

A

MVA

1.00 pu

A

MVA


227 MW 43 Mvar

12 MW 3 Mvar

20 MW 12 Mvar

124 MW 45 Mvar

37 MW

13 Mvar

12 MW 5 Mvar

150 MW 4 Mvar

56 MW

13 Mvar

15 MW 5 Mvar

14 MW

2 Mvar

38 MW 9 Mvar

45 MW 0 Mvar

25 MW 36 Mvar

36 MW 10 Mvar

10 MW 5 Mvar

22 MW 15 Mvar

60 MW 12 Mvar

20 MW 40 Mvar

23 MW 7 Mvar

33 MW 13 Mvar


58 MW 40 Mvar

60 MW 19 Mvar

11.6 Mvar

25 MW 10 Mvar

20 MW 3 Mvar

23 MW 6 Mvar 14 MW

3 Mvar

4.9 Mvar

7.2 Mvar

12.8 Mvar

28.9 Mvar

7.3 Mvar

0.0 Mvar

55 MW 32 Mvar

39 MW 13 Mvar

150 MW 4 Mvar

17 MW 3 Mvar

16 MW -14 Mvar

14 MW 4 Mvar

KYLE69A

MVA

80%A

MVA

135%A

MVA

110%A

MVA

Opening one line (Tim69-Hannah69) causes overloads. This would not be Allowed.

14

Contingency Analysis

Contingencyanalysis providesan automaticway of lookingat all the statisticallylikely contingencies. Inthis example thecontingency setis all the single line/transformeroutages

15

Power Flow And Design• One common usage of the power flow is to determine

how the system should be modified to remove contingencies problems or serve new load• In an operational context this requires working with the

existing electric grid, typically involving re-dispatch of generation.

• In a planning context additions to the grid can be considered as well as re-dispatch.

• In the next example we look at how to remove the existing contingency violations while serving new load.

16

An Unreliable Solution:some line outages result in overloads

slack


SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.02 pu

1.01 pu

1.02 pu

1.03 pu

1.01 pu

1.00 pu1.01 pu

1.00 pu

1.02 pu

1.01 pu

1.00 pu

1.01 pu

1.01 pu

1.01 pu

1.01 pu

1.02 pu

0.99 pu

1.00 pu

1.02 pu

0.97 pu

0.97 pu

0.99 pu

1.02 pu

1.00 pu1.01 pu

1.01 pu

1.00 pu 1.00 pu

1.01 pu

1.02 pu 1.02 pu

1.02 pu 1.03 pu

A

MVA

1.02 pu

A

MVA

A

MVA

LYNN138

A

MVA

1.02 pu

A

MVA

1.00 pu

A

MVA


269 MW 67 Mvar

12 MW 3 Mvar

20 MW 12 Mvar

124 MW 45 Mvar

37 MW

13 Mvar

12 MW 5 Mvar

150 MW 1 Mvar

56 MW

13 Mvar

15 MW 5 Mvar

14 MW

2 Mvar

38 MW 4 Mvar

45 MW 0 Mvar

25 MW 36 Mvar

36 MW 10 Mvar

10 MW 5 Mvar

22 MW 15 Mvar

60 MW 12 Mvar

20 MW 40 Mvar

23 MW 7 Mvar

33 MW 13 Mvar


58 MW 40 Mvar

60 MW 19 Mvar

13.6 Mvar

25 MW 10 Mvar

20 MW 3 Mvar

23 MW 6 Mvar 14 MW

3 Mvar

4.9 Mvar

7.3 Mvar

12.8 Mvar

28.9 Mvar

7.4 Mvar

0.0 Mvar

55 MW 28 Mvar

39 MW 13 Mvar

150 MW 1 Mvar

17 MW 3 Mvar

16 MW -14 Mvar

14 MW 4 Mvar

KYLE69A

MVA

96%A

MVA

Case now has nine separate contingencies having reliability violations(overloads in post-contingency system).

17

A Reliable Solution:no line outages result in overloads

slack


SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.03 pu

1.01 pu

1.02 pu

1.03 pu

1.01 pu

1.00 pu1.01 pu

1.00 pu

1.02 pu

1.01 pu

1.00 pu

1.01 pu

1.01 pu

1.01 pu

1.01 pu

1.02 pu

1.00 pu

0.99 pu

1.02 pu

0.99 pu

0.99 pu

1.00 pu

1.02 pu

1.00 pu1.01 pu

1.01 pu

1.00 pu 1.00 pu

1.01 pu

1.02 pu 1.02 pu

1.02 pu 1.03 pu

A

MVA

1.02 pu

A

MVA

A

MVA

LYNN138

A

MVA

1.02 pu

A

MVA

A

MVA


266 MW 59 Mvar

12 MW 3 Mvar

20 MW 12 Mvar

124 MW 45 Mvar

37 MW

13 Mvar

12 MW 5 Mvar

150 MW 1 Mvar

56 MW

13 Mvar

15 MW 5 Mvar

14 MW

2 Mvar

38 MW 4 Mvar

45 MW 0 Mvar

25 MW 36 Mvar

36 MW 10 Mvar

10 MW 5 Mvar

22 MW 15 Mvar

60 MW 12 Mvar

20 MW 38 Mvar

23 MW 7 Mvar

33 MW 13 Mvar


58 MW 40 Mvar

60 MW 19 Mvar

14.1 Mvar

25 MW 10 Mvar

20 MW 3 Mvar

23 MW 6 Mvar 14 MW

3 Mvar

4.9 Mvar

7.3 Mvar

12.8 Mvar

28.9 Mvar

7.4 Mvar

0.0 Mvar

55 MW 29 Mvar

39 MW 13 Mvar

150 MW 1 Mvar

17 MW 3 Mvar

16 MW -14 Mvar

14 MW 4 Mvar

KYLE69A

MVA

Kyle138A

MVA

Previous case was augmented with the addition of a 138 kV Transmission Line

18

Generation Changes and The Slack Bus

• The power flow is a steady-state analysis tool, so the assumption is total load plus losses is always equal to total generation• Generation mismatch is made up at the slack bus

• When doing generation change power flow studies one always needs to be cognizant of where the generation is being made up• Common options include “distributed slack,” where the

mismatch is distributed across multiple generators by participation factors or by economics.

19

Generation Change Example 1

slack

SLACK345

SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69

HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

0.00 pu

-0.01 pu

0.00 pu

0.00 pu

0.00 pu

-0.03 pu-0.01 pu

0.00 pu

0.00 pu

0.00 pu

-0.03 pu

-0.01 pu0.00 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu

-0.002 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu0.00 pu

0.00 pu

0.00 pu 0.00 pu

0.00 pu

0.00 pu0.00 pu

0.00 pu0.00 pu

A

MVA

-0.01 pu

A

MVA

A

MVA

LYNN138

A

MVA

0.00 pu

A

MVA

0.00 pu

A

MVA

162 MW 35 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

-157 MW -45 Mvar

0 MW

0 Mvar

0 MW 0 Mvar

0 MW 2 Mvar

0 MW

0 Mvar

0 MW 0 Mvar

0 MW

0 Mvar

0 MW 3 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 4 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

-0.1 Mvar 0 MW 0 Mvar

0 MW 0 Mvar 0 MW

0 Mvar

-0.1 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar 0 MW

0 Mvar

-0.1 Mvar

0.0 Mvar

-0.1 Mvar

-0.2 Mvar

0.0 Mvar

0.0 Mvar

0 MW 51 Mvar

0 MW 0 Mvar

0 MW 2 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

Display shows “Difference Flows” between original 37 bus case, and case with a BLT138 generation outage; note all the power change is picked up at the slack

Slack bus

20

Generation Change Example 2

slack

SLACK345

SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69

HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

0.00 pu

-0.01 pu

0.00 pu

0.00 pu

0.00 pu

-0.03 pu0.00 pu

0.00 pu

0.00 pu

0.00 pu

-0.03 pu

-0.01 pu-0.01 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu

-0.003 pu

0.00 pu

0.00 pu

0.00 pu

0.00 pu0.00 pu

0.00 pu

0.00 pu 0.00 pu

0.00 pu

0.00 pu0.00 pu

0.00 pu0.00 pu

A

MVA

0.00 pu

A

MVA

A

MVA

LYNN138

A

MVA

0.00 pu

A

MVA

0.00 pu

A

MVA

0 MW 37 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

-157 MW -45 Mvar

0 MW

0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW

0 Mvar

0 MW 0 Mvar

0 MW

0 Mvar

42 MW -14 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

99 MW -20 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

-0.1 Mvar 0 MW 0 Mvar

0 MW 0 Mvar 0 MW

0 Mvar

-0.1 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar 0 MW

0 Mvar

0.0 Mvar

0.0 Mvar

-0.1 Mvar

-0.2 Mvar

-0.1 Mvar

0.0 Mvar

19 MW 51 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

0 MW 0 Mvar

Display repeats previous case except now the change in generation is picked up by other generators using a “participation factor” (change is shared amongst generators) approach.

21

Voltage Regulation Example: 37 Buses

Display shows voltage contour of the power system

slack

SLACK345

SLACK138

RAY345

RAY138

RAY69

FERNA69

A

MVA

DEMAR69

BLT69

BLT138

BOB138

BOB69

WOLEN69

SHI MKO69

ROGER69

UI UC69

PETE69

HI SKY69

TI M69

TI M138

TI M345

PAI 69

GROSS69

HANNAH69

AMANDA69

HOMER69

LAUF69

MORO138

LAUF138

HALE69

PATTEN69

WEBER69

BUCKY138

SAVOY69

SAVOY138

J O138 J O345

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVAA

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

A

MVA

1.03 pu

1.01 pu

1.02 pu

1.03 pu

1.01 pu

1.00 pu1.00 pu

0.99 pu

1.02 pu

1.01 pu

1.00 pu

1.01 pu1.01 pu

1.01 pu

1.01 pu

1.02 pu

1.00 pu

1.00 pu

1.02 pu

0.997 pu

0.99 pu

1.00 pu

1.02 pu

1.00 pu1.01 pu

1.00 pu

1.00 pu 1.00 pu

1.01 pu

1.02 pu1.02 pu

1.02 pu 1.03 pu

A

MVA

1.02 pu

A

MVA

A

MVA

LYNN138

A

MVA

1.02 pu

A

MVA

1.00 pu

A

MVA

219 MW 52 Mvar

21 MW 7 Mvar

45 MW 12 Mvar

157 MW 45 Mvar

37 MW

13 Mvar

12 MW 5 Mvar

150 MW 0 Mvar

56 MW

13 Mvar

15 MW 5 Mvar

14 MW

2 Mvar

38 MW 3 Mvar

45 MW 0 Mvar

58 MW 36 Mvar

36 MW 10 Mvar

0 MW 0 Mvar

22 MW 15 Mvar

60 MW 12 Mvar

20 MW 9 Mvar

23 MW 7 Mvar

33 MW 13 Mvar 15.9 Mvar 18 MW

5 Mvar

58 MW 40 Mvar 51 MW

15 Mvar

14.3 Mvar

33 MW

10 Mvar

15 MW 3 Mvar

23 MW 6 Mvar 14 MW

3 Mvar

4.8 Mvar

7.2 Mvar

12.8 Mvar

29.0 Mvar

7.4 Mvar

20.8 Mvar

92 MW 10 Mvar

20 MW 8 Mvar

150 MW 0 Mvar

17 MW 3 Mvar

0 MW 0 Mvar

14 MW 4 Mvar

1.010 pu 0.0 Mvar


22

Automatic voltage regulation system controls voltages.

Real-sized Power Flow Cases

• Real power flow studies are usually done with cases with many thousands of buses• Outside of ERCOT, buses are usually grouped into various

balancing authority areas, with each area doing its own interchange control.

• Cases also model a variety of different automatic control devices, such as generator reactive power limits, load tap changing transformers, phase shifting transformers, switched capacitors, HVDC transmission lines, and (potentially) FACTS devices.

23

Sparse Matrices and Large Systems

• Since for realistic power systems the model sizes are quite large, this means the Ybus and Jacobian matrices are also large.

• However, most elements in these matrices are zero, therefore special techniques, sparse matrix/vector methods, are used to store the values and solve the power flow: • Without these techniques large systems would be

essentially unsolvable.

24

Eastern Interconnect Example

Peoria

Rockford

Nort h Chi cago

Abbot t Labs ParkU. S. N Trai ni ng

O l d El m

Deerf i el d

Nort hbrook

Lakehurst

Waukegan

Zi on

G urnee

Ant i och

Pl easant

Round Lake

Zi on (138 kV)

Lake Zur i ch

Lest hon

Apt aki si c

Buf f al o G roove

Wheel i ng

Prospect Hei ght s

Pal at i ne

Ar l i ngt on

M ount Prospect

Prospect

G ol f M i l l

Des Pl ai nes

El mhurst

I t asca

Garfi eld

Tol l w ay

W407 ( Ferm i )

Wi l son

Barr i ngt on

D undee

Si l ver Lake

Cherry Val l ey

Wempleton

N elson

H -471 (N W Steel )

Paddock

Ponti ac Midpoi nt

Brai dw ood

State Li ne

Shefi eld

Chiave

Munster

St. John

El ect r i c Junct i on

Pl ano

La Sal l e

Lombard

Li sle

Col l i ns

D resden

Lockport

East Frankfort

Goodi ngs Grove

Li ber t yvi l l e345 kV

Li ber t yvi l l e138 kV

Lake George

D unacr

Green Acres

Schahfer

Tower Rd

Babcock

Hei ght s

Prai ri e

Racine

Michigan Ci ty

El wood

D equine

Louisa

East Mol ine

Sub 91

Walcott

D avenport

Sub 92

Rock Crk.

Salem

GILMAN

WATSEKA 17GO D LN D

ELPASO T

MIN ON K T

O GLESBY

1556A TPO TTAWA T

O GLSBY M

O GLES; T

H EN N EPIN

ESK TAP

LTV TP NLTV TP E

H EN N E; T

LTV STL

PRIN C TP

PRIN CTN

RICH LAN D

KEWAN IP

S ST TAP

GALESBRG

N ORMA; BN ORMA; R

R FAL; R

MON MOUTH

GALESBR5

KEWAN ;

H ALLOCK

CAT MO SSFARGO

SPN G BAY

E PEO RIA

RSW EAST

PION EERC

RAD N O R

CAT TAP

CAT SUB1

SB 18 5

E MO LIN E

SB 43 5

SB 112 5

KPECKTP5

SO .SUB 5

SB 85 5

SB 31T 5

SB 28 5

SB 17 5

SB 49 5

SB 53 5

SB 47 5SB 48 5

SB A 5

SB 70 5

SB 79 5

SB 88 5

SB 71 5

BVR CH 65 BVR CH 5 ALBAN Y 5

YO RK 5

SAVAN N A5

GALEN A 5

8TH ST.5

LO RE 5

SO .GVW.5

SALEM N 5

ALBAN Y 6

GARD E;

H 71 ;BT

H 71 ; B

H 71 ; R

R FAL; B

N ELSO; R

N ELSO;RT

STERL; B

D IXO N ;BT

MECCORD 3

CO RD O ;

Q uad Ci ti es

LEECO ;BP

Byron

MARYL; B

MEN D O ; T

STILL;RT

B427 ;1T

LAN CA; R

PECAT; B

FREEP;

ELERO ;BT ELERO ;RT

LEN A ; RLEN A ; B

H 440 ;RT

H 440 ; R

STEWA; B

H 445 ;3B

Roscoe

Pi erpont

S PEC; R

FO RD A; R

H arl em

Sand Park

N WT 138

BLK 138

RO R 138

JAN 138

ALB 138

N OM 138

D AR 138

H LM 138

PO T 138 MRE 138

CO R 138 D IK 138

BCH 138

Sabrooke

Bl awkhawk

Al pine

E. Rockford

Charl es

Belvi dere

B465

Marengo

WIB 138

WBT 138ELK 138

N LG 138

N LK GV T

SGR CK5

BRLGTN 1

BRLGTN 2

SGR CK4

UN IVRSTY

UN IV N EU

WH TWTR5

WH TWTR4

WH TWTR3

SUN 138

VIK 138

LBT 138

TICH IGNPARIS WE

ALBERS-2

C434

El mw ood

Ni l es

Evanst on

Devon

Rose Hi l l

Skoki e

Nort hw est

Dr i ver

Ford Ci ty

H ayford

Sawyer

Nort hr i dge

Hi ggi nsDes Pl ai nes

Frankl i n Park

O ak Park

Ri dgel and

D799

G al ew ood

Y450

Congress

Rockw el lCl ybourn

Q uarry

Lasal l e

State

Crosby

Ki ngsbury

Jeff erson

O hio

Taylor

Cl int

D ekov

Fi sk

Crawford

Universi ty

Ri ver

Z-494

Washington Park

H arbor

Calumet

H egewi sch

Z-715

South H ol l and

Evergreen

D amen

Wal l ace

Beverly

G3851

Z-524

G3852

Wi ldwood

H arvey

Green Lake

Sand Ridge

Chicago H ei ghts

Burnham

Lansi ng

F-575

F-503

Gl enwood

Bl oom

Park ForestMatteson

Country Club H i l l s

Al t G E

Nat om a

Woodhi l lU. Park

Moken

M cHenry

Cryst al Lake

Al gonqui n

Hunt l ey

P Val

Woodstock

Bl ue Isl and

G394

Al sip

Crestwood

K-319 # 1

K-319 # 2

Bradl ey

Kankakee

D avi s Creek

Wi lmington

Wi l ton Center

Frankfort

N Len

Brigg

O akbrook

D owners Groove

Woodridge

W604

W603

Bol ingbrook

Sugar Grove

W. De Kal b G l i dden

N Aurora

El gi n

Hanover

Spaul di ngBart l et t

Hof f m an Est at es

S. Schaumberg

Tonne

LandmBusse

Schaumberg

How ard

Berkel ey

Bel l w ood

La G range

Church

Addi son

NordiG l endal e

G l en El l yn

But t e

York Cent er

D775

Bedford Park

Cl earning

Sayre

Bridgevi ew

Ti nley Park

Roberts

Palos

Romeo

Wi l l ow

Burr Ri dge

Jo456

J322

Sout h El gi n Wayne

West Chi cago

Aurora

Warrenvi l l e

W507

Montgomery

O swego

Wol f Creek

Frontenac

W600 ( Napervi l l e)

W602

W601J307

Sandwich

Wat erm an

J323

Mason

J-371

J-375

J-339

Streator

Marsei l l esLasal l e

N LASAL

Mendota

J370

Shore

Goose Lake

J-305

J-390

J-326

Pl ainfi eld

J -332

Archer

Bel l Road

Wi l l Co.

H i l l crest Rockdale

Jol i et

Kendra

Crete

Upnor

LAKEVIEW

BAIN 4

Kenosha

SO MERS

ST RITA

BIG BEN D

MUKWO N GO

N ED 138

N ED 161

LAN 138

EEN 138

CASVILL5

TRK RIV5

LIBERTY5

ASBURY 5

CN TRGRV5

JULIAN 5

MQO KETA5

E CALMS5

GR MN D 5

D EWITT 5

SBH YC5

SUB 77 5

SB 74 5SB 90 5

SB 78 5

D AVN PRT5

SB 76 5

SB 58 5

SB 52 5

SB 89 5

IPSCO 5

IPSCO 3

N EWPO RT5

H WY61 5

WEST 5

9 SUB 5

TRIPP

Z-100

O rlan

Kenda

MPWSPLIT

WYO MIN G5

MT VERN 5

BERTRAM5

PCI 5

SB J IC 5

SB UIC 5

-0.40 deg

2.35 deg

-13.3 deg -13.4 deg

McCook

-1.1 deg

1.9 deg

0.6 deg

93%B

MVA

105%B

MVA

Example, which models the Eastern Interconnectcontains about 43,000 buses. 25

Solution Log for 1200 MW OutageIn this example thelosss of a 1200 MWgenerator in NorthernIllinois was simulated. This caused a generation imbalancein the associated balancing authorityarea, which wascorrected by a redispatch of localgeneration.

26

Interconnected OperationPower systems are interconnected across

large distances. For example most of North America east of

the Rockies is one system, most of North America west of the Rockies is another.

Most of Texas and Quebec are each interconnected systems.

27

Balancing Authority AreasA “balancing authority area” (previously called a

“control area”) has traditionally represented the portion of the interconnected electric grid operated by a single utility or transmission entity.

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-line flow28

Area Control Error (ACE)The area control error is a combination of:

the deviation of frequency from nominal, and the difference between the actual flow out of an area and

the scheduled (agreed) flow.That is, the area control error (ACE) is the difference

between the actual flow out of an area minus the scheduled flow, plus a frequency deviation component:

ACE provides a measure of whether an area is producing more or less than it should to satisfy schedules and to contribute to controlling frequency.

29

actual tie-line flow schedACE 10P P f

Area Control Error (ACE)The ideal is for ACE to be zero.Because the load is constantly changing, each

area must constantly change its generation to drive the ACE towards zero.

For ERCOT, the historical ten control areas were amalgamated into one in 2001, so the actual and scheduled interchange are essentially the same (both small compared to total demand in ERCOT).

In ERCOT, ACE is predominantly due to frequency deviations from nominal since there is very little scheduled flow to or from other areas.

30

Automatic Generation Control

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

Usually the control center (either ISO or utility) calculates ACE based upon tie-line flows and frequency; then the AGC module sends control signals out to the generators every four seconds or so.

31

Power TransactionsPower transactions are contracts between

generators and (representatives of) loads.Contracts can be for any amount of time at any

price for any amount of power. Scheduled power transactions between balancing

areas are called “interchange” and implemented by setting the value of Psched used in the ACE calculation:ACE = Pactual tie-line flow – Psched + 10β Δf…and then controlling the generation to bring ACE

towards zero.32

“Physical” power Transactions

• For ERCOT, interchange is only relevant over asynchronous connections between ERCOT and Eastern Interconnection or Mexico.

• In Eastern and Western Interconnection, interchange occurs between areas connected by AC lines.

33

Three Bus Case on AGC:no interchange.Bus 2 Bus 1

Bus 3Home Area

266 MW

133 MVR

150 MW

250 MW 34 MVR

166 MVR

133 MW 67 MVR

1.00 PU

-40 MW 8 MVR

40 MW -8 MVR

-77 MW 25 MVR

78 MW-21 MVR

39 MW-11 MVR

-39 MW

12 MVR

1.00 PU

1.00 PU

101 MW 5 MVR

100 MWAGC ONAVR ON

AGC ONAVR ON

Net tie-line flow is close to zero

Generationis automaticallychanged to matchchange in load

34

100 MW Transaction between areas in Eastern or Western

Bus 2 Bus 1

Bus 3Home Area

Scheduled Transactions

225 MW

113 MVR

150 MW

291 MW 8 MVR

138 MVR

113 MW 56 MVR

1.00 PU

8 MW -2 MVR

-8 MW 2 MVR

-84 MW 27 MVR

85 MW-23 MVR

93 MW-25 MVR

-92 MW

30 MVR

1.00 PU

1.00 PU

0 MW 32 MVR

100 MWAGC ONAVR ON

AGC ONAVR ON

100.0 MW

Scheduled100 MWTransaction from Left to Right

Net tie-lineflow is now100 MW

35

PTDFsPower transfer distribution factors (PTDFs) show

the linearized 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 to evaluate PTDFs.

Note that we can modify several elements in ,

in proportion to how the specified generators would

par

θ B P

θ

P

ticipate in the power transfer. 36

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

37

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 for a change in transaction from A to I

38

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 for a change in transaction from G to F

39

WE to TVA PTDFs

40

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 compared to the pre-contingency flow if a contingency were to occur,

– LODFs are used extensively in real-time operations,– LODFs are approximately independent of flows but

do depend on the assumed network topology.

41


42

,

change in flow on line ,

due to outage of line .

pre-contingency flow on line

,

Estimates change in flow on line

if outage on line were to occur.

l

k

l l k k

P l

k

P k

P LODF P

l

k


43

,

If line initially had 100 MW of flow on it,

and line initially had 50 MW flow on it,

and then there was an outage of line ,

if =0.1 then the increase in flow

on line after a continge

k

l

l k

k P

l P

k

LODF

l

,

ncy of line would be:

0.1 100 10 MW

from 50 MW to 60 MW.

l l k k

k

P LODF P

Flowgates

• The real-time loading of the power grid can be assessed via “flowgates.”

• A flowgate “flow” is the real power flow on one or more transmission elements for either base case conditions or a single contingency– Flows in the event of a contingency are approximated

in terms of pre-contingency flows using LODFs.

• Elements are chosen so that total flow has a relation to an underlying physical limit.

44

Flowgates

• Limits due to voltage or stability limits are often represented by effective flowgate limits, which are acting as “proxies” for these other types of limits.

• Flowgate limits are also often used to represent thermal constraints on corridors of multiple lines between zones or areas.

• The inter-zonal constraints that were used in ERCOT until December 2010 are flowgates that represent inter-zonal corridors of lines.

45

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EE 369 POWER SYSTEM ANALYSIS Lecture 14 Power Flow Tom Overbye and Ross Baldick 1.

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