Transcript Pricing of Ancillary Services - Power Systems Engineering

Reactive Power Considerations in
Linear Load Flow with Applications to
Available Transfer Capability
Pete Sauer
(With a lot of help from Santiago Grijalva)
University of Illinois at Urbana-Champaign
PSERC Internet Seminar
December 3, 2002
© 2002 University of Illinois
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Overview
• Linear load flow methods
• Linear transfer capability calculations
• Reactive linear ATC calculations
• Examples
• Conclusions
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Linear Load Flow
• An approximation used to estimate
the result of a change in operating
conditions from some base case
• Small change sensitivities
• Large-change sensitivities
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PTDFs
• Power Transfer Distribution Factors
(PTDFs) indicate how “injection
power” flows in the lines
• A 5% PTDF for a given injection set
and line means that 5% of the
injection flows in that line
• Principle is simple current division
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Base Case
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P12 = 10 MW
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Numerical PTDFs
Real power distribution factors for real power
transfer from area 1 to 2 (area 2 reduced by 10 MW)
jk,sr = Pjk / Psr
12,12 = 0.4, 13,12 = 0.6, 32,12 = 0.6
(40% of the transfer goes down line 12, 60% goes
down line 13, and 60% goes down line 32)
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Use these to predict the result of a
100 MW transaction from 1 to 2
P12 = 0+0.4*100 = 40 MW
P13 = 0+0.6*100 = 60 MW
P32 = 0+0.6*100 = 60 MW
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P12 = 100 MW
OK, but look
at the VARS
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How about 200 MW?
P12 = 0+0.4*200 = 80 MW
P13 = 0+0.6*200 = 120 MW
P32 = 0+0.6*200 = 120 MW
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P12 = 200 MW
OK, but look
at the VARS
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Behavior of Distribution Factors Close to Collapse
1.0
p0
12
p*
p
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Analytical Distribution Factors
• These PTDFs can be analytically
constructed from line admittances only,
or line admittances plus base case
operating point values.
• There are also “Line Outage Distribution
Factors” that estimate the change in
flows due to a line outage.
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Total Transfer Capability (TTC)
• Specify sending point or points
• Specify receiving point or points
• Increase sending power injection
• Decrease receiving power injection
• Monitor security limits
• Stop when limit reached
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Computation
• Linear algebraic
• Non-linear algebraic
• Time domain simulation
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1996 NERC definitions
• Transmission Reliability Margin
(TRM) is supposed to account for
uncertainty in conditions and model
• Capacity Benefit Margin (CBM) is
supposed to account for reliability
criteria (neighboring reserve etc.)
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TRM Components
• Forecasting error
• Data uncertainty
– Impedances
– Ratings
– Measurements
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TRM Alternatives
• Fixed MW amount
• Fixed %
• Resolve with limits reduced by
some amount (I.e. 4%)
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CBM Issues
• Loss of the biggest unit will result in
import from neighbors. There must
be capability to allow this import.
• Some companies use CBM = 0 and
include the loss of units in the
contingency list.
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Available Transfer Capability
ATC = TTC - TRM - CBM - ETC
Available = Total - Margins - Existing
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TTC Computation Errors
•
•
•
•
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Linear vs nonlinear flow calculations
MW vs MVA limits
Neglecting voltage constraints
Neglecting stability constraints
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Estimating Maximum Power Transfers
Recall the 3-bus example:
Real power distribution factors for real
power transfer from area 1 to 2
12,12 = 0.4, 13,12 = 0.6, 32,12 = 0.6
Maximum transfer 1-2 is minimum of:
100/.4 = 250, 130/.6 = 217, 140/.6 = 233
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P12 = 217 MW
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P12 = 203 MW
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P12 = 212 MW
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P12 error
• Linear vs nonlinear error plus MW
vs MVA error = 217 MW vs 203 MW
(7%)
• Linear vs nonlinear error only = 217
MW vs 212 MW (2%)
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Qjk
Linear ATC
Psr* = Pjk*/ jk,sr
Pjk
Pjk
(Pjk0, Qjk0)
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*
LIMITING
CIRCLE
Line power flow relations
Vj j
Pjk + j Qjk
j
Vk k
Pkj + j Q kj
j Xjk
k
Pjk = Vj Vk Bjk sin (j -k)
Qjk = Vj2 Bjk - Vj Vk Bjk cos (j -k)
Pjk2 + (Vj2 Bjk - Qjk)2
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= (VjVkBjk)2
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Line power flow relations
Vj j
Bus j
-j(1/Bjj)
Pjk + j Q jk
Rjk + j X jk
Pkj + j Q kj
Vk k
Bus k
-j(1/Bkk)
Pjk = + Vj2 Gjk - Vj Vk Yjk cos (j -k+jk)
Qjk = - Vj2 Bjj - Vj2 Bjk - Vj Vk Yjk sin (j -k+jk)
 (Pjk -Vj2 Gjk)2 +(Qjk +Vj2 Bjj +Vj2 Bjk)2 = (Vj Vk Yjk)2
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Operating and Limiting Circles
Operating Circle:
r =Sjk0
r = Sjkmax
Q jk
(Pjk0, Qjk0)
(Pjk -Vj2 Gjk)2 +(Qjk +Vj2 Bjj +Vj2 Bjk)2 = (Vj Vk Yjk)2
(Pjk*, Qjk*)
Pjk
Limiting Circle:
Pjk2 + Qjk2 = (Sjkmax)2
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Solutions
Solve:
(Pjk -Vj2 Gjk)2 +(Qjk +Vj2 Bjj +Vj2 Bjk)2 = (Vj Vk Yjk)2
Pjk2 + Qjk2 = (Sjkmax )2
Define: - M02 = Pjk02 +Qjk02 -Sjk02
A = (Pjk02 + Qjk02)
B = - Pjk0 ((Sjkmax )2 -M02)
C = [(Sjkmax )2 -M02]2 /4 - Qjk02 (Sjkmax )2
Then:
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Pjk* = [ - B  (B2-4AC)1/2]/2A
Qjk*= [(Sjkmax )2 - Pjk* 2 ] 1 / 2
Reactive Linear ATC
r =Sjk0
Qjk
OPERATING
CIRCLE
(Pjk*, Qjk*)#2
r = Skmax
(Pjk0,Qjk0)
(Pjk*, Qjk*)#1
Pjk
*
Pjk
Psr* = Pjk*/ jk,sr
(Pjk0, Qjk0)
LIMITING
CIRCLE
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Feasibility in Reactive Linear ATC Computation
Qjk
OPERATING
CIRCLE
Qjk
D
C
(0, Vj2Yjk)
C
(0, Vj2Yjk)
B
Pjk
Pj jk
A
A
A to B
(Thermal limit)
Limiting circle I
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A to B to C
(Feasibility limit)
Limiting circle II
B
Test for infeasible cases in reactive linear
ATC computation
[(±Vj2Gjk+VjVkYjk)2 + (-Vj2 Bjj -Vj2Bjk)2]1/2 < Sjkmax
 An estimate of voltage collapse limits.
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Estimation of Reactive Power Support
Consider:
Qjk = Qjk0 + [Sjk02 - (Pjk - Pjk0)2]1/2
Valid for line complex flow if voltages ~ constant.
Then, for a variation in the injection at bus i:
Qjk = Qjk0 -Qjk0 +[Sjk02-(jk,sr Psr +Pjk0 -Pjk0)2]1/2
Therefore, the new reactive power at bus j:
Qj = Qj0 + k {Qjk0 -Qjk0 +[Sjk02-(jk,sr Psr +Pjk0 -Pjk0)2]1/2}
All terms are known except Psr which is independent.
 A way to estimate the reactive power support required for
large variations in active power transactions.
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Base Case
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TRANSACTION
S/B LINE
LINE
1-2
1-3
2-3
Actual
LINEAR
REACTIVE
jk,sr
Rating
P*
P*
Error
P*
Error
-
p.u.
p.u.
p.u.
%
p.u.
%
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
1-2
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.
-
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 system
1.01 PU
0.56 Deg
95.95 MW
-2.04 MVR
96.0 MVA
-90.52 MW
21.80 MVR
93.1 MVA
123.24 MW
93.76 MVR
154.8 MVA
3
1
104.05 MW
-17.96 MVR
105.6 MVA
-120.92 MW
-87.82 MVR
149.4 MVA
OFF AGC
200.00 MW
-20.00 MVR
-88.56 MW
16.01 MVR
90.0 MVA
4
0.98 PU
-12.95 Deg
1.02 PU
-11.37 Deg
-67.44 MW
38.02 MVR
77.4 MVA
-10.12 MW
25.46 MVR
27.4 MVA
300 MW
50 MVR
Area A
-101.86 MW
23.53 MVR
104.5 MVA
2
40 MW
20 MVR
82.34 MW
46.21 MVR
94.4 MVA
6
1.05 PU
-1.27 Deg
OFF AGC
370.00 MW
116.74 MVR
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71.0 MW
-30.50 MVR
77.2 MVA
91.95 MW
-2.75 MVR
92.0 MVA
1.00 PU
-79.88 MW
-43.41 MVR -3.19 Deg
90.9 MVA
43.83 MW
15.26 MVR
46.4 MVA
78.83 MW
8.15 MVR
79.2 MVA
-76.31 MW
-2.52 MVR
76.35 MVA
-64.52 MW
-8.24 MVR
65.0 MVA
OFF AGC
100.00 MW
-24.98 MVR
Area B
OFF AGC
125.67 MW
187.24 MVR
10.829 MW
-29.239 MVR
31.2 MVA
5
0.96 PU
-8.62 Deg
130 MW
40 MVR
-42.22 MW
-14.54 MVR
44.7 MVA
43.83 MW
15.26 MVR
46.4 MVA
200 MW
40 MVR
80 MW
30 MVR
65.43 MW
9.09 MVR
66.1 MVA
200 MW
80 MVR
-42.22 MW
-14.54 MVR
44.7 MVA
Area C
0.98 PU
-6.35 Deg
7
OFF AGC
181.00 MW
60.00 MVR
7- bus system: P6 - 4
LIMITATION
Line
Rating

4-2
2-4
6-2
2-3
2-6
3-2
6-7
7-5
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-0.294
0.326
0.740
0.253
-0.713
-0.235
0.130
0.252
p.u.
1.00
1.00
1.60
1.20
1.60
1.20
0.80
1.20
Actual
P*
LINEAR
Error
P*
REACTIVE
P* Error
p.u.
0.70
0.72
1.04
1.12
1.13
1.24
2.73
2.31
p.u.
0.77
0.70
0.89
1.11
0.97
1.28
2.58
2.14
p.u.
0.690
0.716
1.043
1.118
1.128
1.217
2.694
2.267
%
9.82
-2.86
-14.76
-1.19
-14.24
2.95
-5.33
-7.41
%
1.38
0.54
-0.25
0.17
0.20
1.89
1.32
1.88
WSCC 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.
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Linear
Error %
Linear with Reactive
25
20
15
10
5
0
-5
-10
T ransfer #
-15
0
41
5
10
15
20
25
30
35
40
45
NYISO 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.
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Linear
E r ro r %
Linear with Reac tive
25
20
15
10
5
0
-5
-1 0
T ra n sf e r #
-1 5
0
43
5
10
15
20
25
30
35
40
45
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Transmission Loading Relief (TLR)
• Based on the PTDF concept
• Could benefit from consideration of reactive
power in loading
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The future of ATC
• The ATC concept has other problems
–
–
chaining does not work
updates are difficult
• What will the new Standard Market Design
rules use?
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Conclusions
• The inclusion of reactive power considerations
in a linear ATC calculation can reduce error in
ATC.
• It may provide a way to estimate the proximity
to voltage collapse limits due to a transaction.
• The inclusion of reactive power considerations
in a linear ATC calculation is easy.
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