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Fast Load Flowand

Contingency Analysis

Dr. Swapan Kr. GoswamiProfessor

Department of Electrical EngineeringJadavpur University

Applications of load flow As an independent tool

As a support tool

Speed requirements

Accuracy requirements

Fast and accurate : Fast decoupled load flow

Fast but approximate : DC load flow

Fast Decoupled Load Flow:Newton – Raphson method in polar co-ordinate

V

VLJ

NH

Q

P P-V & Q - are weakly coupled

V1X

V2 /

P+jQ

111

112

1

V

Pxjx

V

QV

V

jQPjxVV

V

jQPI

V

V0

0

L

H

Q

P

Assuming complete decoupling

V1

(Q/V1.X)(PX/V1)

V2

kmkmkmkmm

mkSpecifiedkk

kmkmkmkmm

mkSpecified

kk

BGVVQQ

BGVVPP

cossin

sincos

kmkmkm

kkkkkkkkkkkk

kmkmkmkmmkkmkm

jBGY

QVBLQVBH

BGVVLH

22 ;

;cossin

Normally,Vk, Vm 1.0 p.u.

R

xratio of lines are high

km are small

Thus, cos km 1, Gkm sin km << Bkm;

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kmkmkmkmm

mkk BGVVQ cossin

ko

kmm

kmko

kmm

km

kmm

kmkkm

km

YyYy

BBB

and

2

Thus,

kkkkkkk

kmmkkmkm

kmkm

ikokikk

VBLH

BVVLH

yY

yyY

The decoupled load flow equation is thus,

V

VVBVQ

VBVP

..

..

The elements of the matrices [B′ ] and [B″] are strictly elementsof [-B].Further simplification:(a)Taking the left hand V terms on to the left hand sides of the

equations and by setting all the right hand V terms to 1 p.u.(b) Omitting from [B′ ] the representation of those network

elements that predominantly affect MVAR flows i.e., shuntreactances and off-nominal in phase transformer taps.

(c) Omitting from [B″] the angle shifting effects of phase shifters.(d) Neglecting series resistances in calculating the elements of

[B′ ].

The final fast decoupled load flow equations

VBV

Q

BV

P

kmkmkk

kmkm

km kmkk

kmkm

bB

kmbB

xB

kmx

B

for,

1

for,1Where,

22kmkm

kmkm

kmkmkm

xr

xb

jbgy

The matrics B’ and B” are real, sparse and have the structuresof [H] and [L] respectively. Order of B’ and B” are (N – 1) and(N – M) respectively, where N is the number of bus bars and Mis the number of PV bus bars. B” is symmetric in value and so isB’ if phase shifters are ignored. The elements of the matricesare constant and need to be evaluated and triangulated onlyonce for a network.

The speed of iterations of the fast decoupled method is aboutfive times that of formal Newton – Raphson and about two thirdsthat of Gauss – Seidel method. Storage requirements are about60 percent of the formal Newton – Raphson method.

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KP = KQ = 1

Calculate [P/V]

Converged? KP=0

Solve for and update

KQ=1

KQ=0?

NO

YES

Out

put

Calculate [Q/V]

Converged ?

Solve forV andupdate V KQ=0

KP=1KP=0 ?

YES

NO

YES

NO

Y

NO

Flow Chart:

P-, Q- V equations to be converged simultaneously

Branch Outage Calculation:Equation to be solved in fast decoupled load flow are of the form

[R] = [Bo] [Eo]

For which a solution, [Eo] = [Bo]-1 [R] , can be obtained using thefactors of [Bo].For outage of element k-m

MbMBbBB boo

010100

0

1

0

1

0

0

1

k mk

m

b = line or transformer series admittance.

Using matrix inversion lema,

111

oo BMXCBB

to MBX

XMb

C

1

11

Where,

The solution vector [E1] to the outage problem is

oo

oo

EMXCE

RBMXCB

RBE11

111

quantityScalar11

;1

1

1

1

XMX

bb

XMb

CMB

Xt

o

M=

k

k

m

m

1

1

EE

ECE

EMXCEE

o

oo

oo

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The DC Power flow : Simplified further by dropping the Q-Vequation i.e., assuming V = 1 p.u.

BP

Useful for calculating MW flows through lines and transformers.

ik

kiik x

P

Extensively used in contingency analysis.

System Security:Involves practices designed to keep the system operating when componentsfail.

Mechanism: Maintaining proper spinning reserve Maintaining proper network flow margin.

Functions: System monitoring: Measuring critical power system quantities Estimation of system states.

Contingency analysis:To determine which contingencies cause limit violations and alsoseverity of such violation. Results of this study allow system to beoperated defensively.

Security constrained optimal power flow:Contingency analysis incorporated into power flow optimization

Why contingency analysis is needed:

250MW500MW

Unit 1 250 MW1200MW

700MW

Unit 2

Optimal dispatch

500MW

Unit 1 500MW

700MW

Unit 2

1200 MWPost contingency dispatch

200MW400MW

Unit 1 200 MW1200MW

800MW

Unit 2

Secure dispatch

400MW

Unit 1 400MW

800MW

Unit 2

1200 MW

Secure post contingency dispatch

Contingency Analysis Procedure:Start

Set system model to initialconditions

i = 1

Simulate outage of Generator i

Line flowor

Bus voltage limitviolated ?

last generatorconsidered

?

Display alarmmessage

i = i + 1

Yes

No

l = 1

Simulate outage of line l

line flowor

bus voltage limitviolated

?

Display alarmmessage

l = l + 1

last lineconsidered

?

No

No

end

Problem : Time Consuming

No

Yes

Yes

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Fast Solution: Two approaches1. Linear sensitivity factors : Uses simple multiplying

factors very fast andapproximate solutions.

2. Contingency ranking : Contingencies are rankedaccording to their severity, fullAC load flow performed for theselected cases.

Linear sensitivity factors:1. Generation shift factors.

2. Line outage distribution factors.

;i

lli P

fa

oii PP

fl = Change is flow through line lPi = Change is generation at bus i.

Pre-calculated set of a factors can be used to determine theChange in power flow on each line.

For outage of the generator at bus i,

ij

l

Generation shift factor of line ‘l’ due to a shift of generation atBus i,

DC Load flow equation:n m

lXl

i

minil

i

m

i

n

l

l

mn

ii

lli

XXx

dP

d

dP

d

x

xdP

d

dP

dfa

PX

BP

1

1

Generation Shift Factor:

i

mi

i

m

n

P

X 0

0

0

...............

.....X.......... ni

1

ilio

ll Paff ˆ

jljilio

ll PaPaff ˆ

If the generation outage is made up at bus j,

If all generations pick up in proportion to their rating

ikk

jji

ijijiljili

oll

P

PPaPaff

max

max

;ˆ

Post contingency flow through line l

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Line outage distribution factors:

oklk

oll

ok

llk

fdff

f

fd

.ˆ

k

l

i j

Outage of line k is simulatedas

Pi = + PijPj = - Pij

kp q

+Ppq

Ppq

- Ppql

r sPrs + drs - pq

Ppq

pqpqqqpprs

sqsprqrppq

pq

rspqrs

xXXXx

XXXXx

P

Pd

2

Contingency Selection:Severe contingencies are identified and ranked.Full AC load flow run for the selected cases.

ContingencySelection and

ranking

List

of p

ossi

ble

outa

ges

Shor

t lis

t of m

ost

seve

re c

ases

i = 1

Pick outage i from short list andremove the component from the power

flow model

Run AC power flow

Test for over load / voltage limitviolation

all cases done?

i = i + 1

end

Alar

m li

st

No

Yes

Contingency Ranking : According to Performance index (PI)Most severe contingency ranked first followed by lesssevere ones.

Active power contingency :n

element l

ll P

PWPI

2

Wl = weightage factor for element ‘l’Pl = power flow through element ‘l’_Pl = power rating of element ‘I’

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Reactive power / voltage contingencies:

2ll PxPI

xl = reactance of element l

This index can identity contingencies creating wide spreadvoltage problem. However, this index show little sensitivityto smaller local problems, where the voltage on one or a fewBuses has dropped only slightly.

Alternative approaches for the identification oflocal problems:

Local solution method:Voltage problems generally occur in the neighbourhoodof the dropped element. A modified Gauss – Seidelsolution algorithm is used by dividing the network into

layers. Voltage at buses near the outage are updated keepingthe voltage on the distant buses constant.

layer - 3

layer - 3layer - 2

layer - 1

outage line

layer =1end buses of the

outage line

Consider voltage at thebuses connected tolayer as constant

Determine voltage ofthe buses in the layer

bus voltages closeto the pre-contingencies

values?

Expand the layer byincluding the buses

directly connected tothe previous layer

Yesend

No

1P1Q method:One iteration of fast decoupled power flow.

B', B" matrices

model outage case

solve P - eqn.for , update

solve Q-v eqn. forv, update V

Calculate line flows and PI

Pick next outage case

Full o

utag

eca

se lis

t

PI lis

t(o

ne e

ntry

for

each

out

age

case

)

Questions ?

THANK YOU