linear analysis techniques in - powerworld · [email protected] 2001 south first street...

[email protected]://www.powerworld.com

2001 South First StreetChampaign, Illinois 61820+1 (217) 384.6330

2001 South First StreetChampaign, Illinois 61820+1 (217) 384.6330

Linear Analysis Techniques in PowerWorld Simulator

An overview of the underlying mathematics of the power flow and

linearized analysis techniques

2© 2015 PowerWorld CorporationLinear Analysis Techniques

AC Power Flow Equations• Full AC Power Flow Equations

• Solution requires iteration of equations

• Note: the large matrix (J) is called the Jacobian

LkGk

N

kmm

mkkmmkkmmkkkkk

LkGk

N

kmm

mkkmmkkmmkkkkk

QQbgVVbVQ

PPbgVVgVP

1

2

1

2

cossin0

sincos0

QP

JQP

VQ

δQ

VP

δP

Vδ 1

1


Full AC Derivatives• Real Power derivative equations are

• Reactive Power derivative equations are

N

kmm

mkkmmkkmmkkkk

k bgVgVVP

1

sincos2

N

kmm

mkkmmkkmmkk

k bgVVP1

cossin

N

kmm

mkkmmkkmmkkkk

k bgVbVVQ

1

cossin2

N

kmm

mkkmmkkmmkk

k bgVVQ1

sincos

mkkmmkkmkm

k bgVVP

cossin mkkmmkkmmk

m

k bgVVP

sincos

mkkmmkkmkm

k bgVVQ

cossin mkkmmkkmmkm

k bgVVQ

sincos


Decoupled Power Flow Equations• Make the following assumptions

• Derivatives simplify to

• Note: If assumption is insteadthen replace all with

• Option in Simulator Options under Power Flow Solution, DC Options sub‐tab

1kV 0kmg0 mk 0sin mk

1cos mk

0

k

k

VP

N

kmm

kmk

k bP1

0

m

k

VP

kmm

k bP

N

kmm

kmkkk

k bbVQ

1

)(2

0

k

kQkm

m

k bVQ

0

m

kQ

0kmrkmb kmx1


Other Typical Assumptions about Transformers

• Ignore Transformer Impedance Correction Tables– If we do not ignore, then as tap or phase changes the series

branch impedances change• Ignore Phase Shift Angle Effects

– If we do not ignore, then as the phase angle changes the series impedance seen between the buses will vary

• Both of these are normally done because if they are not ignored, then the solution matrices become a function of the system state (i.e. impedance will vary with tap or phase)– This defeats the purposed of using the Decoupled equations.

Recommendation:Leave these checked


B’ and B’’ Matrices• Define

• Now Iterate the “decoupled” equations

• What are B’ and B’’?– B’ is the imaginary part of the Y-Bus with all the “shunt

terms” removed– B’’ is the imaginary part of the Y-Bus with all the “shunt

terms” double counted

'BδP

''B

VQ

QBV

PBδ

1''

1'


“DC Power Flow”• The “DC Power Flow” equations are simply the real part of the decoupled power flow equations– Voltages and reactive power are ignored– Only angles and real power are solved for by iterating

PBδ 1'


Bus Voltage and Angle Sensitivities to a Transfer

• Power flow was solved by iterating

• Model the transfer as a change in the injections P– Buyer:– Seller:

• Then solve for the voltage and angle sensitivities by solving

• These are the sensitivities of the Buyer and Seller “sending power to the slack bus”

QP

JVδ 1

TSySxS PFPF 0000T 1

1

N

zSzPF

TBgBfB PFPF 0000T 1

1

N

hBhPF

0T

JVδ S

S

S 1

0T

JVδ B

B

B 1


What about Losses?• If we assume the total sensitivity to the transfer is the seller minus the buyer sensitivity, then and

• This makes assumption that ALL the change in losses shows up at the slack bus.

• Simulator assigns the change to the BUYER by defining

• Then

BS δδδ BS VVV

slack power to sendingbuyer for generation busslack in Changeslack power to sendingseller for generation busslack in Change

B

S

SlackSlackk

B

B

S

S kVδ

Vδ

Vδ


Why does “k” assign the losses to the Buyer?


Lossless DC Voltage and Angle Sensitivities

• Use the DC Power Flow Equations

• Then determine angle sensitivities

• The DC Power Flow ignores losses, thus

PBδ 1'

SS TBδ 1' BB TBδ

1'

BS δδδ


Lossless DC Sensitivities with Phase Shifters Included

• DC Power Flow equations• Augmented to include an equation that describes the change in flow on a phase‐shifter controlled branch as being zero.

• Thus instead of DC power flow equations we use

• Otherwise process is the same.

PδB '

0αBδB Change Flow Line

0P

BB0B

αδ

1'


Lossless DC with Phase Shifters

• Phase Shifters are often on lower voltage paths (230 kV or less) with relatively small limits– They are put there in order to

manage/prevent the flow on a path that would commonly see overloads

– Thus, they constantly show up as “overloaded” when using linear analysis if they are not accounted for

• Example: Border of Canada with Northwestern United States– PTDFs between Canada and US without

Phase‐Shifters• 85% on 500 kV Path • 15% on Eastern Path

– PTDF With Phase‐Shifters• 100% goes on 500 kV Path• 0% on Eastern Path• This better reflects real system

230 kV Phase Shifter

115 kV Phase Shifter

115 kV Phase ShifterCalifornia

Canada

BPA

500 kV Path


Power Transfer Distribution Factors (PTDFs)

• PTDF: measures the sensitivity of line MW flows to a MW transfer. – Line flows are simply a function of the voltages and angles at its terminal buses

– Thus, the PTDF is simply a function of these voltage and angle sensitivities.

• This is the “Chain Rule” from Calculus

• Pkm is the flow from bus k to bus m

mm

kmK

k

kmm

m

kmK

k

kmkm

PPVVPV

VPPPTDF

Voltage and Angle Sensitivities that were just determined


Pkm Derivative Calculations• Full AC equations

• Lossless DC Approximations yields

PV

V g V g bkm

kk kk m km k m km k m2 cos sin

PV

V g bkm

mk km k m km k mcos sin

PV V g bkm

kk m km k m km k m

sin cos

PV V g bkm

mk m km k m km k m

sin cos

0

k

km

VP 0

m

km

VP

kmk

km bP

km

m

km bP


What do Flow Sensitivities (PTDFs, GSFs, TLRs, …) give us?

• Give the ability to model a change in power injection without actually doing a new power flow solution– Transfer of power between two places– Generator outage– Load outage

• Still can’t model a line outage or line closure yet, but that is what LODFs will give us


Line Outage Distribution Factors (LODFs)

• LODFl,k: percent of the pre-outage flow on Line K will show up on Line L after the outage of Line K

• Linear impact of an outage is determined by modeling the outage as a “transfer” between the terminals of the line

k

klkl P

PLODF ,

,


AssumeThen the flow on the Switches is ZERO, thusOpening Line K is equivalent to the “transfer”

Modeling an LODF as a Transfer

Line k

n m

Other Line l

The Rest of System

Pn Pm kP~

Switches

mnk PPP ~

mn PP and

Create a transfer defined by


Modeling an LODF as a Transfer• Thus, setting up a transfer of MW from Bus n to Bus m is equivalent to outaging the transmission line

• Let’s assume we know what is equal to, then we can calculate the values relevant to the LODF.

kP~

kP~


Calculation of LODF• Estimate of post‐outage flow on Line L

• Estimate of flow on Line L after transfer

• Thus we can write

• We have a simple function of PTDF values

klkl PPTDFP ~*,

kkkk PPTDFPP ~*~ k

kk PTDF

PP

1

~

k

k

kl

k

kl

k

klkl P

PTDFPPTDF

PPPTDF

PP

LODF

1

*~*,,

k

lkl PTDF

PTDFLODF

1,


Line Closure Distribution Factors (LCDFs)

• LCDFl,k: percent of the post‐closure flow on Line K will show up on Line L after the closure of Line K

• Linear impact of an closure is determined by modeling the closure as a “transfer” between the terminals of the line

k

klkl P

PLCDF ~

,,


Modeling the LCDF as a Transfer

Line k n m

Other Line l

The Rest of System

PnPm

kP~

Net flow from rest of the system

Net flow to rest ofthe system

AssumeThen the net flow to and from the rest of the system are both zero, thus closing line k is equivalent the “transfer”

mnk PPP ~mn PP and

Create a “transfer” defined by


Modeling an LCDF as a Transfer• Thus, setting up a transfer of MW from Bus n to Bus m is linearly equivalent to outaging the transmission line

• Let’s assume we know what is equal to, then we can calculate the values relevant to the LODF.

• Note: The negative sign is used so that the notation is consistent with LODF “transfer”

kP~

kP~


Calculation of LCDF• Estimate of post‐closure flow on Line L

• Thus we can write

• Thus the LCDF, is exactly equal to the PTDF for a transfer between the terminals of the line

klkl PPTDFP ~*,

lkl PTDFLCDF ,

lk

kl

k

klkl PTDF

PPPTDF

PP

LCDF

~~*

~,

,


LODF and LCDF• LODF (LCDF) – Gives the ability to model a single line

outage (closure) event– OTDF – incremental impact on a particular branch while also

considering a single line outage– OMW – estimate of the flow on a particular branch after a single

line outage

• Still can’t model multiple line outages simultaneously– Can not just sum because the lines that are being outaged also

interact with each other

11,1, * PTDFLODFPTDFOTDF MMM

11,1, * MWLODFMWOMW MMM

slineoutagek

kkMMKM PTDFLODFPTDFOTDF *,,

slineoutagek

kkMMKM MWLODFMWOMW *,,


Linear Impact of a Contingency with Multiple Outages

• Outage Transfer Distribution Factors (OTDFs)– The percent of a transfer that will flow on a branch M after the contingency occurs

• Outage Flows (OMWs)– The estimated flow on a branch M after the contingency occurs

1 2 nc. . . . . . M

Contingent Lines 1 through nc Monitored Line M


OTDFs and OMWs• Single Contingency

• Multiple Contingencies

• What are and ?

11,1, * PTDFLODFPTDFOTDF MMM

11,1, * MWLODFMWOMW MMM

Cn

KKMKMCM NetPTDFLODFPTDFOTDF

1, *

Cn

KKMKMCM NetMWLODFMWOMW

1, *

KNetPTDF KNetMW


Determining NetPTDFKand NetMWK

• Each NetPTDFK is a function of all the other NetPTDFs because the change in status of a line affects all other lines.

• Assume we know all NetPTDFs except for NetPTDF1. Then we can write:

• In general for each Contingent Line N, write

C

CC

n

KKK

nn

NetPTDFLODFPTDF

NetPTDFLODFNetPTDFLODFPTDFNetPTDF

211

121211 ...

N

n

NKK

KNKN PTDFNetPTDFLODFNetPTDFC

1


• Thus we have a set of nc equations and ncunknowns (nc= number of contingent lines)

• Thus• Same type of derivation shows

Determining NetPTDFKand NetMWK

CCCCC

C

C

C

nnnnn

n

n

n

PTDF

PTDFPTDFPTDF

NetPTDF

NetPTDFNetPTDFNetPTDF

LODFLODFLODF

LODFLODFLODFLODFLODFLODFLODFLODFLODF

3

2

1

3

2

1

321

33231

22321

11312

1

11

1

CCCC PTDFLODFNetPTDF 1

CCCC MWLODFNetMW 1

Known Values


Operating and Limiting Circles• Operating circle defines a circle of valid MW and Mvar

values for a transmission line as a transfer takes place across the system

• Limiting circle has a radius equal to the MVA limit of the line

• The operating circle is utilized when using one of the DC methods and modeling reactive power by assuming constant voltage magnitudes

• Contingency analysis looks up the Mvar value corresponding to the calculated MW value

• ATC analysis and DC power flow find the intersection of the operating circle and the limiting circle to assign adjusted limits to lines


Operating and Limiting Circles

r =Sjkmax

Limiting Circle

Qjk

P jk

Operating Circle

r =Sjk,0

(Pjk,0,Qjk,0) (Pjk*,Qjk

* )

linear analysis techniques in - powerworld · [email protected] 2001 south first street...

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