Lecture 16: Sensitivity Analysis

ECEN 615Methods of Electric Power Systems Analysis

Prof. Tom Overbye

Dept. of Electrical and Computer Engineering

Texas A&M University

overbye@tamu.edu

Announcements

• Exam average was 85.7, with a high of 100

• Read Chapter 7 (the term reliability is now used instead

of security)

• Homework 4 is assigned today, due on Thursday Nov 1

2

Linearized Sensitivity Analysis

• By using the approximations from the fast

decoupled power flow we can get sensitivity values

that are independent of the current state. That is,

by using the B’ and B’’ matrices

• For line flow we can approximate

3

2

( ) ( ) ( ), ,

By using the FDPF appxomations

( )( ) ( ) , ,

i i j i j i j i j

i ji j

h s g V V V cos b V V sin i j

h s b i jX

Linearized Sensitivity Analysis

• Also, for each line

and so,

0

4

0V

h

a

θ

hb

1 1

h

a a A Bθh

x 0h

V

L

TL

b b

0

Sensitivity Analysis: Recall the Matrix Notation

• The series admittance of line is g +jb and we

define

• We define the LN incidence matrix

5

1 2B , , ,

Ldiag b b b

1

2

a

aA

aL

T

T

T

where the component j of ai isnonzero whenever line i is

coincident with node j. Hence

A is quite sparse, with at most

two nonzeros per row

Linearized Active Power Flow Model

• Under these assumptions the change in the real power

line flows are given as

• The constant matrix

is called the injection shift factor matrix (ISF)

1

1B 0 I

f B A 0 p B A B p Ψ p0 B 0

1Ψ BA B

6

Injection Shift Factors (ISFs)

• The element in row and column n of is called

the injection shift factor (ISF) of line with respect to

the injection at node n

– Absorbed at the slack bus, so it is slack bus dependent

• Terms generation shift factor (GSF) and load shift

factor (LSF) are also used (such as by NERC)

– Same concept, just a variation in the sign whether it is a

generator or a load

– Sometimes the associated element is not a single line, but

rather a combination of lines (an interface)

• Terms used in North America are defined in the NERC

glossary (http://www.nerc.com/files/glossary_of_terms.pdf)

n

7

line

np

n nf

i j

slackbusp

ISF Interpretation

is the fraction of the additional 1 MW injection atnode n that goes though line

+ 1

n

slack

node

1

8

ISF Properties

• By definition, depends on the location of the

slack bus

• By definition, for since the

injection and withdrawal buses are identical in this case and, consequently, no flow arises on any line

• The magnitude of is at most 1 since

n

1 1n

slackbus0 L

n

Note, this is strictly true only for the linear (lossless)

case. In the nonlinear case, it is possible that a

transaction decreases losses. Hence a 1 MW injection

could change a line flow by more than 1 MW.9

Five Bus Example Reference

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

42 MW

67 MW

100 MW

118 MW

1.040 pu

1.042 pu

A

MVA

A

MVA

A

MVA

1.042 pu

A

MVA

1.044 pu

33 MW

MW200

258 MW

MW118

260 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

10

Five Bus ISF, Line 4, Bus 2 (to Slack)

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

52 MW

63 MW

100 MW

128 MW

1.040 pu

1.042 pu

A

MVA

A

MVA

1.042 pu

A

MVA

1.044 pu

37 MW

MW200

238 MW

MW118

280 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

86%A

MVA

4

2 128 118

20

0.5

11

Five Bus Example

-1 0 0 0

0 -1 0 0

0 0 -1 0A

1 -1 0 0

0 1 -1 0

0 0 1 -1

=

The row of A correspond to

the lines and transformers,

the columns correspond to

the non-slack buses (buses 2

to 5); for each line there

is a 1 at one end, a -1 at the

other end (hence an

assumed sign convention!).

Here we put a 1 for the

lower numbered bus, so

positive flow is assumed

from the lower numbered

bus to the higher number

B 6.25, 12.5, 12.5, 12.5, 12.5, 10= diag

12

Five Bus Example

18.75 12.5

12.5 37.5 12.5B A B A

12.5 35 10

10 10

T

0 0

0= =

0

0 0

1

-0.4545 -0.1818 -0.0909 -0.0909

-0.3636 -0.5455 -0.2727 -0.2727

-0.1818 -0.2727 -0.6364 -0.6364

0.5455 -0.1818 -0.0909 -0.0909

0.1818 0.2727 -0.3636 -0.3636

-1.0000

B A B

0 0 0

13With bus 1 as the slack, the buses (columns) go for 2 to 5

Five Bus Example Comments

• At first glance the numerically determined value of

(128-118)/20=0.5 does not match closely with the

analytic value of 0.5455; however, in doing the

subtraction we are losing numeric accuracy

– Adding more digits helps (128.40 – 117.55)/20 = 0.5425

• The previous matrix derivation isn’t intended for actual computation; is a full matrix so we would

seldom compute all of its values

• Sparse vector methods can be used if we are only

interested in the ISFs for certain lines and certain

buses

14

Distribution Factors

• Various additional distribution factors may be

defined – power transfer distribution factor (PTDF)

– line outage distribution factor (LODF)

– line addition distribution factor (LADF)

– outage transfer distribution factor (OTDF)

• These factors may be derived from the ISFs making

judicious use of the superposition principle

15

Definition: Basic Transaction

• A basic transaction involves the transfer of a

specified amount of power t from an injection node

m to a withdrawal node n

t

n

t

m

16

Definition: Basic Transaction

• We use the notation

to denote a basic transaction

injection

node withdrawal

node quantity

, ,w m n t

17

Definition: PTDF

• NERC defines a PTDF as

– “In the pre-contingency configuration of a system under

study, a measure of the responsiveness or change in

electrical loadings on transmission system Facilities due

to a change in electric power transfer from one area to

another, expressed in percent (up to 100%) of the change

in power transfer”

– Transaction dependent

• We’ll use the notation to indicate the PTDF

on line with respect to basic transaction w

• In the lossless formulation presented here (and

commonly used) it is slack bus independent

( )w

18

PTDFs

line

nf

i j

t

m

t

f

t+t+

( )w f

t

Note, the PTDF is

independent of the

amount t; which is often

expressed as a percent19

PTDF Evaluation in Two Parts

line

( )w

i j

nm

1

=

m

i j

0+

m

1

1

1

i j

0

n

1

n

1 20

line

nf

i j

t

m

t

0

PTDF Evaluation

+ 1 + 1

m n

1 1

( )w m n

21

Calculating PTDFs in PowerWorld

• PowerWorld provides a number of options for

calculating and visualizing PTDFs

– Select Tools, Sensitivities, Power Transfer Distribution

Factors (PTDFs)

Results are

shown for the

five bus case

for the Bus 2

to Bus 3

transaction

22

Five Bus PTDF Visualization

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

52 MW

63 MW

100 MW

128 MW

1.040 pu

1.042 pu

9%PTDF

27%PTDF

73%PTDF

1.042 pu

9%PTDF

1.044 pu

37 MW

MW200

238 MW

MW118

280 MW

100 MW

MW100

18%PTDF

One Two

Three

Four

Five

PowerWorld Case: B5_DistFact_PTDF 23

Nine Bus PTDF Example

slack

43%PTDF

57%PTDF

13%PTDF

30%PTDF

35%PTDF

20%PTDF

10%PTDF

2%PTDF

34%PTDF

34%PTDF

32%PTDF

A

G

B

C

D

E

I

F

H

34%PTDF

MW 400.0 MW 400.0 MW 300.0

MW 250.0

MW 250.0

MW 200.0

MW 250.0

MW 150.0

50.0 MW

PowerWorld Case: B9_PTDF

Display shows

the PTDFs

for a basic

transaction

from Bus A

to Bus I. Note

that 100% of

the transaction

leaves Bus A

and 100%

arrives at

Bus I24

Eastern Interconnect Example: Wisconsin Utility to TVA PTDFs

In this

example

multiple

generators

contribute

for both

the seller

and the

buyer

Contours show lines that would carry at least 2% of a

power transfer from Wisconsin to TVA 25

Line Outage Distribution Factors (LODFs)

• Power system operation is practically always limited

by contingencies, with line outages comprising a large

number of the contingencies

• Desire is to determine the impact of a line outage

(either a transmission line or a transformer) on other

system real power flows without having to explicitly

solve the power flow for the contingency

• These values are provided by the LODFs

• The LODF is the portion of the pre-outage real power line flow on line k that is redistributed to line as a result of the outage of line k

26

kd

line

f

i j

LODFs

f

kline

kf

i j

line

f

i j

outaged

klinei j

,k

k

k

fd d

f

base caseoutage case

Best reference is Chapter 7 of the course book27

LODF Evaluation

We simulate the impact of the outage of line k by

adding the basic transaction , ,k k

w i j t

and selecting tk in such

a way that the flows on

the dashed lines become

exactly zeroline

f f

i j

kline

i j

kt k

t

k kf f

In general this tk is not

equal to the original line

flow

28

LODF Evaluation

• We select tk to be such that

where ƒ k is the active power flow change on the line

k due to the transaction wk

• The line k flow from wk depends on its PTDF

it follows that

kk kf f t 0

( )kw

k kkf t

( )1 1k

k k

k w i j

k k k

f ft

29

LODF Evaluation

• For the rest of the network, the impacts of the

outage of line k are the same as the impacts of the

additional basic transaction wk

• Therefore, by definition the LODF is

( )

( )

( )1

k

k

k

w

w

k w

k

kf t f

( )

( )1

k

k

w

k

w

k k

fd

f

30

Five Bus Example

• Assume we wish to calculate the values for the outage

of line 4 (between buses 2 and 3); this is line k

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

52 MW

63 MW

100 MW

128 MW

1.040 pu

1.042 pu

A

MVA

A

MVA

1.042 pu

A

MVA

1.044 pu

37 MW

MW200

238 MW

MW118

280 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

86%A

MVA

Say we wish

to know

the change

in flow

on the line

3 (Buses 3

to 4).

PTDFs for

a transaction

from 2 to 3

are 0.7273

on line 4

and 0.0909

on line 3 31

Five Bus Example

• Hence we get

4

4

( )

3 34

3 ( )

4 4

3 4

0.09090.333

1 1 0.7273

0.333 0.333 128 42.66MW

w

w

fd

f

f f

( )

128469.4

1 1 0.7273k

k

k w

k

ft

32

Five Bus Example Compensated

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

184 MW

108 MW

100 MW

465 MW

1.040 pu

1.016 pu

A

MVA

1.027 pu

A

MVA

1.029 pu

8 MW

MW200

238 MW

MW587

749 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

124%A

MVA

312%A

MVA

Here is the

system with the

compensation

added to bus

2 and removed

at bus 3; we

are canceling

the impact of

the line 4 flow

for the reset

of the network.

33

Five Bus Example

• Below we see the network with the line actually

outaged

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

180 MW

106 MW

100 MW

0 MW

1.040 pu

1.044 pu

A

MVA

1.042 pu

A

MVA

1.044 pu

6 MW

MW200

238 MW

MW118

280 MW

100 MW

MW100

A

MVA

One Two

Three

Four

Five

121%A

MVA

The line 3

flow changed

from 63 MW

to 106 MW,

an increase

of 43 MW,

matching the

LODF

value

34

Developing a Critical Eye

• In looking at the below formula you need to be

thinking about what conditions will cause the formula

to fail

Here the obvious situation is when the denominator is

zero

• That corresponds to a situation in which the

contingency causes system islanding

– An example is line 6 (between buses 4 and 5)

– Impact modeled by injections at the buses within each viable

island

( )

( )

( )1

k

k

k

w

w

k w

k

kf t f

35

Calculating LODFs in PowerWorld

• Select Tools, Sensitivities, Line Outage Distribution

Factors

– Select the Line using dialogs on right, and click Calculate

LODFS; below example shows values for line 4

36

Blackout Case LODFs

• One of the issues associated with the 8/14/03 blackout

was the LODF associated with the loss of the Hanna-

Juniper 345 kV line (21350-22163) that was being

used in a flow gate calculation was not correct because

the Chamberlin-Harding 345 kV line outage was

missed

– With the Chamberlin-Harding line assumed in-service the

value was 0.362

– With this line assumed out-of-service (which indeed it was)

the value increased to 0.464

37

2000 Bus LODF Example

LODF is for line

between 3048 and

5120; values will

be proportional to

the PTDF values

38

2000 Bus LODF Example

0%PTDF

0%

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WINTERS

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ROWENA

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CHICO

DUBL IN 2

ITAL Y

FRANKSTON

HAMIL TON

BEN WHEEL ER

MEXIA

KIL L EEN 1

MABANK 1

PAL ESTINE 1

L INDAL E

ZEPHYR

TENNESSEE COL ONY

MINEOL A

MIL L SAP

FORRESTON

MARQUEZ

CHANDL ER

KOPPERL

L OVEL ADY

GL EN ROSE 2

THROCKMORTON

NEMO

MINGUS

CORSICANA 1

HARKER HEIGHTS

ABBOTT

GORMAN

GREENVIL L E 2

AL VORD

WOODWAY

TEMPL E 3

BL OOMING GROVE

CROSS PL AINS

KIL L EEN 2

JACKSBORO 2

HIL L SBORO

CL YDE

CISCO

WEATHERFORD 3

HEWITT

SANTO

EL M MOTT

CL IFTON 2

CHINA SPRING

KEMP

DUBL IN 1

TYL ER 9

MERTENS

BRECKENRIDGE

GATESVIL L E 1

JEWETT 2

EASTL AND

COOPER

DIKE

GOL DTHWAITE 2

TRINIDAD 2

GATESVIL L E 2

MAL ONE

BEL TON

TOL AR

PAL ESTINE 2

GRAFORD

EMORY

OAKWOOD

BRYSON 2

WORTHAM

EDGEWOOD

RIO VISTA

PARADISE

MAY

STRAWN

MORGAN

WACO 6

AXTEL L

MOODY

NOL ANVIL L EROSEBUD

EL KHART

TYL ER 8

PERRIN

RICE

BAIRD

POINT

MARL IN

COOL IDGE

MINERAL WEL L S

ATHENS 1

SUL PHUR SPRINGS

GRAND SAL INE

KAUFMAN

HUBBARD

WAL NUT SPRINGS

VAN

MURCHISON

WIL L S POINT

GORDON

OL NEY 2

FAIRF IEL D 3JONESBORO

MERIDIAN

ATHENS 2

BUFFAL O

WA C O 1

A RLINGTO N 1

MC KINNEY 3

MA NSF IELD

GREENV ILLE 1

SILV ER

RO SC O E 2

TRINIDA D 1

C O PPERA S C O V E

SHERMA N 1

A BILENE 2

GRA ND PRA IRIE 3

MIDLO THIA N 1

FA IRF IELD 1

PO O LV ILLEA LLEN 1

SNYDER 2

BRYSO N 1

PA RIS 2

DENTO N 1

MERKEL 1

WA C O 2

BREMO ND

FLUV A NNA 2

STEPHENV ILLE

WINGA TE

GRA NBURY 2

GRA NBURY 1

KELLER 2

A LEDO 1

GA RLA ND 1

JA C KSBO RO 1

ODESSA 1

SAVOY

O DONNELL 1

MCCAMEY 1

FAIRFIELD 2

GLEN ROSE 1

OLNEY 1

GOLDTHWAITE 1

CORSICANA 2

DALLAS 1

GRAHAM

RICHARDSON 2

BROWNWOOD

JEWETT 1

FRANKLIN

CHRISTOVAL

PALO PINTO 1

STERLING CITY 1

ROSCOE 5

PARIS 1RALLS 1

MCKINNEY 1

HERMLEIGHABILENE 1

ALBANY 1

ENNIS

RIESEL 1

BRIDGEPORT

39

Multiple Line LODFs

• LODFs can also be used to represent multiple device

contingencies, but it is usually more involved than just

adding the effects of the single device LODFs

• Assume a simultaneous outage of lines k1 and k2

• Now setup two transactions, wk1 (with value tk1)and

wk2 (with value tk2) so

1

2

1 1 2

2 1 2

k

k

k k k

k k k

f f f t 0

f f f t 0

1 2

1 2

( ) ( )

1 1 1 2 1

( ) ( )

2 1 2 1 2

1

2

k k

k k

w w

k k k k k

w w

k k k k k

k

k

f t t t 0

f t t t 0

40

Multiple Line LODFs

• Hence we can calculate the simultaneous impact of

multiple outages; details for the derivation are given in

C.Davis, T.J. Overbye, "Linear Analysis of Multiple

Outage Interaction," Proc. 42nd HICSS, 2009

• Equation for the change in flow on line for the

outage of lines k1 and k2 is

12

11 2 1

1

22

1

1

k

kk k k

k

kk

fdf d d

fd

41

Multiple Line LODFs

• Example: Five bus case, outage of lines 2 and 5 to

flow on line 4. 1

2

11 2 1

1

22

1

1

k

kk k k

k

kk

fdf d d

fd

1

1 0.75 0.3360.4 0.25 0.005

0.6 1 0.331f

42

Multiple Line LODFs

Line 1

Line 2

Line 3

Line 6

Line 5

Line 4slack

1.050 pu

42 MW

100 MW

100 MW

118 MW

1.040 pu

1.036 pu

A

MVA

A

MVA

A

MVA

1.040 pu

1.042 pu

0 MW

MW200

258 MW

MW118

260 MW

100 MW

MW100

One Two

Three

Four

Five

Flow goes

from 117.5

to 118.0

43