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Reactive Power Considerations in Linear Load Flow with Applications to Reactive Power Considerations in Linear Load Flow with Applications to Available Transfer CapabilityAvailable Transfer Capability

Pete SauerPete Sauer(With a lot of help from Santiago Grijalva)(With a lot of help from Santiago Grijalva)

University of Illinois at Urbana-ChampaignUniversity of Illinois at Urbana-Champaign

PSPSERCERC Internet Seminar Internet SeminarDecember 3, 2002December 3, 2002

© 2002 University of Illinois© 2002 University of Illinois

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OverviewOverview

• Linear load flow methods Linear load flow methods

• Linear transfer capability calculationsLinear transfer capability calculations

• Reactive linear ATC calculationsReactive linear ATC calculations

• ExamplesExamples

• ConclusionsConclusions

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Linear Load FlowLinear Load Flow

• An approximation used to estimate An approximation used to estimate the result of a change in operating the result of a change in operating conditions from some base case conditions from some base case

• Small change sensitivitiesSmall change sensitivities

• Large-change sensitivitiesLarge-change sensitivities

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PTDFsPTDFs

• Power Transfer Distribution Factors Power Transfer Distribution Factors (PTDFs) indicate how “injection (PTDFs) indicate how “injection power” flows in the linespower” flows in the lines

• A 5% PTDF for a given injection set A 5% PTDF for a given injection set and line means that 5% of the and line means that 5% of the injection flows in that lineinjection flows in that line

• Principle is simple current divisionPrinciple is simple current division

Base CaseBase Case

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P12 = 10 MWP12 = 10 MW

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Numerical PTDFsNumerical PTDFs Real power distribution factors for real power Real power distribution factors for real power

transfer from area 1 to 2 (area 2 reduced by 10 MW) transfer from area 1 to 2 (area 2 reduced by 10 MW)

jk,srjk,sr = = PPjkjk / / PPsrsr

12,12 12,12 == 0.4, 0.4, 13,12 13,12 = 0.6, = 0.6, 32,1232,12 = 0.6 = 0.6

(40% of the transfer goes down line 12, 60% goes (40% of the transfer goes down line 12, 60% goes down line 13, and 60% goes down line 32)down line 13, and 60% goes down line 32)

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Use these to predict the result of a Use these to predict the result of a 100 MW transaction from 1 to 2100 MW transaction from 1 to 2

PP12 12 == 0+0.4*100 = 40 MW0+0.4*100 = 40 MW

PP13 13 == 0+0.6*100 = 60 MW0+0.6*100 = 60 MW

PP32 32 == 0+0.6*100 = 60 MW0+0.6*100 = 60 MW

P12 = 100 MWP12 = 100 MW

OK, but look at the VARS

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How about 200 MW?How about 200 MW?

PP12 12 == 0+0.4*200 = 80 MW0+0.4*200 = 80 MW

PP13 13 == 0+0.6*200 = 120 MW0+0.6*200 = 120 MW

PP32 32 == 0+0.6*200 = 120 MW0+0.6*200 = 120 MW

P12 = 200 MWP12 = 200 MW

OK, but look at the VARS

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Behavior of Distribution Factors Close to CollapseBehavior of Distribution Factors Close to Collapse

p p* p0

1.0

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Analytical Distribution FactorsAnalytical Distribution Factors

• These PTDFs can be analytically These PTDFs can be analytically constructed from line admittances only, constructed from line admittances only, or line admittances plus base case or line admittances plus base case operating point values.operating point values.

• There are also “Line Outage Distribution There are also “Line Outage Distribution Factors” that estimate the change in Factors” that estimate the change in flows due to a line outage. flows due to a line outage.

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Total Transfer Capability (TTC)Total Transfer Capability (TTC)

• Specify sending point or pointsSpecify sending point or points

• Specify receiving point or pointsSpecify receiving point or points

• Increase sending power injectionIncrease sending power injection

• Decrease receiving power injectionDecrease receiving power injection

• Monitor security limitsMonitor security limits

• Stop when limit reachedStop when limit reached

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ComputationComputation

• Linear algebraicLinear algebraic

• Non-linear algebraicNon-linear algebraic

• Time domain simulationTime domain simulation

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1996 NERC definitions 1996 NERC definitions

• Transmission Reliability Margin Transmission Reliability Margin (TRM) is supposed to account for (TRM) is supposed to account for uncertainty in conditions and modeluncertainty in conditions and model

• Capacity Benefit Margin (CBM) is Capacity Benefit Margin (CBM) is supposed to account for reliability supposed to account for reliability criteria (neighboring reserve etc.)criteria (neighboring reserve etc.)

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TRM ComponentsTRM Components

• Forecasting errorForecasting error

• Data uncertaintyData uncertainty– ImpedancesImpedances

– RatingsRatings

– MeasurementsMeasurements

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TRM AlternativesTRM Alternatives

• Fixed MW amountFixed MW amount

• Fixed %Fixed %

• Resolve with limits reduced by Resolve with limits reduced by

some amount (I.e. 4%)some amount (I.e. 4%)

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CBM IssuesCBM Issues

• Loss of the biggest unit will result in Loss of the biggest unit will result in import from neighbors. There must import from neighbors. There must be capability to allow this import.be capability to allow this import.

• Some companies use CBM = 0 and Some companies use CBM = 0 and include the loss of units in the include the loss of units in the contingency list.contingency list.

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Available Transfer CapabilityAvailable Transfer Capability

ATC = TTC - TRM - CBM - ETCATC = TTC - TRM - CBM - ETC

Available = Total - Margins - ExistingAvailable = Total - Margins - Existing

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TTC Computation ErrorsTTC Computation Errors

• Linear vs nonlinear flow calculationsLinear vs nonlinear flow calculations

• MW vs MVA limitsMW vs MVA limits

• Neglecting voltage constraintsNeglecting voltage constraints

• Neglecting stability constraintsNeglecting stability constraints

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Estimating Maximum Power TransfersEstimating Maximum Power Transfers

Recall the 3-bus example:Recall the 3-bus example:

Real power distribution factors for real Real power distribution factors for real power transfer from area 1 to 2 power transfer from area 1 to 2

12,12 12,12 == 0.4, 0.4, 13,12 13,12 = 0.6, = 0.6, 32,1232,12 = 0.6 = 0.6

Maximum transfer 1-2 is minimum of:Maximum transfer 1-2 is minimum of:

100/.4 = 250, 130/.6 = 217, 140/.6 = 233100/.4 = 250, 130/.6 = 217, 140/.6 = 233

P12 = 217 MWP12 = 217 MW

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P12 = 203 MWP12 = 203 MW

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P12 = 212 MWP12 = 212 MW

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P12 errorP12 error

• Linear vs nonlinear error plus MW Linear vs nonlinear error plus MW vs MVA error = 217 MW vs 203 MW vs MVA error = 217 MW vs 203 MW (7%)(7%)

• Linear vs nonlinear error only = 217 Linear vs nonlinear error only = 217 MW vs 212 MW (2%)MW vs 212 MW (2%)

LIMITINGCIRCLE

Qjk

Pjk

(Pjk0, Qjk

0)

Pjk*

Linear ATC

Psr* = Pjk

*/ jk,sr

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Line power flow relationsLine power flow relations

j kj X jk

Pjk + j Q jk Pkj + j Q kj

PPjkjk = = VVjj V Vkk B Bjkjk sin ( sin (jj - -kk) )

QQjk jk = V = Vjj22 B Bjkjk - V - Vjj V Vkk B Bjkjk cos ( cos (jj - -kk))

PPjkjk22 + (V + (Vjj

22 B Bjkjk - Q - Qjkjk))22 = (V= (VjjVVkkBBjkjk))22

kVk jVj

jkBus j j X

Pjk + j Q jk Pkj+ j Q kjjVj

kVk

Bus kRjk +

-j(1/Bjj) -j(1/Bkk)

Pjk = + Vj2 Gjk - Vj Vk Yjk cos (j -k+jk)

Qjk = - Vj2 Bjj - Vj

2 Bjk - Vj Vk Yjk sin (j -k+jk)

(Pjk -Vj2 Gjk)2 +(Qjk +Vj

2 Bjj +Vj2 Bjk)2 = (Vj Vk Yjk)2

Line power flow relations

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Limiting Circle:

Pjk2 + Qjk

2 = (Sjkmax)2

Pjk

Operating and Limiting Circles

Operating Circle:

(Pjk -Vj2 Gjk)2 +(Qjk +Vj

2 Bjj +Vj2 Bjk)2 = (Vj Vk Yjk)2

r = Sjkmax

(Pjk0, Qjk0)

Q jkr =Sjk0

(Pjk*, Qjk*)

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SolutionsSolutions

Solve: (Pjk -Vj

2 Gjk)2 +(Qjk +Vj2 Bjj +Vj

2 Bjk)2 = (Vj Vk Yjk)2

Pjk2 + Qjk

2 = (Sjkmax

)2

Define: - M02 = Pjk0

2 +Qjk02 -Sjk0

2

A = (Pjk02 + Qjk0

2)

B = - Pjk0 ((Sjkmax

)2 -M02)

C = [(Sjkmax

)2 -M02]2 /4 - Qjk0

2 (Sjkmax

)2

Then: Pjk* = [ - B (B2-4AC)1/2]/2A

Qjk*= [(Sjkmax

)2 - Pjk* 2 ] 1 / 2

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LIMITINGCIRCLE

OPERATINGCIRCLE

(Pjk*, Qjk*)#1

r = Skmax

Qjk

Pjk

r =Sjk0

(Pjk0,Qjk0)

(Pjk0, Qjk

0)

(Pjk*, Qjk*)#2

Pjk*

Psr* = Pjk

*/ jk,sr

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Feasibility in Reactive Linear ATC ComputationFeasibility in Reactive Linear ATC Computation

Qjk

Pjk

OPERATINGCIRCLE

(0, Vj2Yjk) B

C

A

j

B

C

D

A

Pjk

Qjk

Limiting circle ILimiting circle II

(0, Vj2Yjk)

A to B (Thermal limit) A to B to C

(Feasibility limit)

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Test for infeasible cases in reactive linear Test for infeasible cases in reactive linear ATC computationATC computation

[(±V[(±Vjj22GGjkjk+V+VjjVVkkYYjkjk))22 + (-V + (-Vjj

22 B Bjjjj -V -Vjj22BBjkjk))22]]1/21/2 < S < Sjkjk

maxmax

An estimate of voltage collapse limits.An estimate of voltage collapse limits.

Estimation of Reactive Power SupportEstimation of Reactive Power Support

Consider:Consider: QQjkjk = Q = Qjk0jk0 + [S + [Sjk0jk022 - (P - (Pjk jk - P- Pjk0jk0))22]]1/21/2

Valid for line complex flow if voltages ~ constant.Valid for line complex flow if voltages ~ constant.

Then, for a variation in the injection at bus i:Then, for a variation in the injection at bus i:

QQjkjk = Q = Qjk0jk0 -Q -Qjkjk00 +[S +[Sjk0jk0

22-(-(jk,sr jk,sr PPsrsr +P +Pjkjk00

-P-Pjk0jk0))22]]1/21/2

Therefore, the new reactive power at bus j:Therefore, the new reactive power at bus j:

QQjj = Q = Qjj00 + + kk {Q {Qjk0jk0 -Q -Qjkjk

00 +[S +[Sjk0jk022-(-(jk,sr jk,sr PPsrsr + +PPjkjk

00 -P-Pjk0jk0))22]]1/21/2}}

All terms are known except All terms are known except PPsrsr which is independent. which is independent.

A way to estimate the reactive power support required for A way to estimate the reactive power support required for large variations in active power transactions.large variations in active power transactions.

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Base CaseBase Case

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TRANSACTION Actual LINEAR REACTIVE

S/B LINE jk,sr Rating P* P* Error P* Error

LINE - p.u. p.u. p.u. % p.u. %

1-2 1-2 0.395 1.00 2.330 2.53 8.65 2.261 -2.97

2-1 -0.395 1.00 2.330 2.53 8.65 2.261 -2.97

1-3 0.605 1.30 2.110 2.15 1.84 2.123 0.60

3-1 -0.605 1.30 2.030 2.15 5.85 2.045 0.73

2-3 -0.605 1.40 2.270 2.31 1.94 2.304 1.51

3-2 0.605 1.40 2.190 2.31 5.66 2.218 1.27

1-3 1-2 0.242 1.00 3.490 4.13 18.40 3.690 5.74

2-1 -0.242 1.00 3.490 4.13 18.40 3.690 5.74

1-3 0.758 1.30 1.690 1.72 1.48 1.694 0.24

3-1 -0.758 1.30 1.630 1.72 5.22 1.632 0.13

2-3 0.242 1.40 Unst 5.79 N/A Unst. -

3-2 -0.242 1.40 Unst 5.79 N/A Unst. -

2-3 1-2 -0.190 1.00 4.430 5.26 18.81 4.700 6.10

2-1 0.190 1.00 4.430 5.26 18.81 4.700 6.10

1-3 0.190 1.30 Unst 6.84 N/A Unst. -

3-1 -0.190 1.30 Unst 6.84 N/a Unst. -

2-3 0.810 1.40 1.700 1.73 1.67 1.721 1.24

3-2 -0.810 1.40 1.630 1.73 6.04 1.657 1.63

7-bus system7-bus system

Area A

Area B Area C

1

2

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5

6 7

200.00 MW

100.00 MW

370.00 MW

300 MW 50 MVR

80 MW 30 MVR

130 MW 40 MVR

40 MW 20 MVR

1.02 PU

0.96 PU

0.98 PU1.05 PU

1.00 PU

0.98 PU

1.01 PU

104.05 MW

-101.86 MW

95.95 MW -90.52 MW -120.92 MW 123.24 MW

91.95 MW

-88.56 MW

78.83 MW -76.31 MW

-67.44 MW

-2.75 MVR

10.829 MW

-64.52 MW

65.43 MW

-42.22 MW43.83 MW

-79.88 MW

82.34 MW

125.67 MW

200 MW 40 MVR

200 MW 80 MVR

43.83 MW

-42.22 MW

OFF AGC

OFF AGC

OFF AGC

OFF AGC

8.15 MVR

116.74 MVR

187.24 MVR

-2.04 MVR

-24.98 MVR

-20.00 MVR

-6.35 Deg

-11.37 Deg

0.56 Deg

-1.27 Deg

-3.19 Deg

181.00 MW

21.80 MVR

-12.95 Deg-17.96 MVR

OFF AGC

-30.50 MVR 71.0 MW

-43.41 MVR

46.21 MVR

23.53 MVR

16.01 MVR

-87.82 MVR 93.76 MVR

38.02 MVR

-2.52 MVR

-8.24 MVR -8.62 Deg

-29.239 MVR

-10.12 MW25.46 MVR

-14.54 MVR

9.09 MVR

15.26 MVR

15.26 MVR

-14.54 MVR

96.0 MVA 93.1 MVA

105.6 MVA

104.5 MVA 92.0 MVA 77.2 MVA79.2 MVA 76.35 MVA

65.0 MVA90.9 MVA

94.4 MVA46.4 MVA

46.4 MVA

44.7 MVA

66.1 MVA

44.7 MVA

31.2 MVA

27.4 MVA77.4 MVA

154.8 MVA149.4 MVA

90.0 MVA

60.00 MVR

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7- bus system: P7- bus system: P6 - 46 - 4

LIMITATION Actual LINEAR REACTIVELine Rating P* P* Error P* Error

- - p.u. p.u. p.u. % p.u. %4-2 -0.294 1.00 0.70 0.77 9.82 0.690 1.382-4 0.326 1.00 0.72 0.70 -2.86 0.716 0.546-2 0.740 1.60 1.04 0.89 -14.76 1.043 -0.252-3 0.253 1.20 1.12 1.11 -1.19 1.118 0.172-6 -0.713 1.60 1.13 0.97 -14.24 1.128 0.203-2 -0.235 1.20 1.24 1.28 2.95 1.217 1.896-7 0.130 0.80 2.73 2.58 -5.33 2.694 1.327-5 0.252 1.20 2.31 2.14 -7.41 2.267 1.88

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WSCC Summer CaseWSCC Summer Case

• Forty-two transfers across the BC Hydro, BPA, and PG&E control areas were simulated.

• The simulation did not include contingency sets.

• The model had 7,119 buses, and 9,630 lines and transformers. Total generation was 120GW.

• Simulations run by PowerWorld Corp.

-15

-10

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45

Transfer #

Error % Linear Linear with Reactive

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NYISO Summer caseNYISO Summer case

• Fifty transfers across different control areas in the NYISO

• The simulation did not include contingency sets.

• The model had about 40,000 buses and included more than 6,000 generating units and 139 control areas.

• Simulations run by PowerWorld Corp.

-15

-10

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

Transfer #

Error % Linear Linear with Reactive

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Transmission Loading Relief (TLR)Transmission Loading Relief (TLR)

• Based on the PTDF concept Based on the PTDF concept

• Could benefit from consideration of reactive Could benefit from consideration of reactive power in loadingpower in loading

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The future of ATCThe future of ATC

• The ATC concept has other problemsThe ATC concept has other problems

– chaining does not workchaining does not work

– updates are difficult updates are difficult

• What will the new Standard Market Design What will the new Standard Market Design rules use?rules use?

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ConclusionsConclusions• The inclusion of reactive power considerations in a The inclusion of reactive power considerations in a

linear ATC calculation can reduce error in ATC. linear ATC calculation can reduce error in ATC.

• It may provide a way to estimate the proximity to It may provide a way to estimate the proximity to voltage collapse limits due to a transaction.voltage collapse limits due to a transaction.

• The inclusion of reactive power considerations in a The inclusion of reactive power considerations in a linear ATC calculation is easy. linear ATC calculation is easy.