PSERC
A Security Constrained OPF That Produces
Correct Market-Based Pricing
Robert J. Thomas
Cornell University
PSerc Summer Workshop
Lake Tahoe, CA
August 6, 2008
PSERC
Contributors to the “SuperOPF”
• Alberto Lamadrid
• Carlos Murillo-Sánchez
• Robert Thomas
• Zhifang Wang
• Hongye Wang
• Ray Zimmerman
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PSERC
Why a SuperOPF?
• State of the art for planning & real-time
operations is primitive w.r.t. ancillary
services & pricing needed to support market
design.
• Tools are needed to capture true value of
resources, e.g. demand response and
uncertain energy sources.
• Proper allocation and valuation of resources
requires true simultaneous co-optimization.
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PSERC
What is the SuperOPF?
Traditional Approach
SuperOPF
Break into
manageable
sub-problems.
Combine into
single mathematical
programming
framework.
sequential
optimization
true co-optimization
incorrect prices
correct prices
Computational Complexity is an Issue!
4
Simple Example of Sequential OPF
System Parameters
•Generators
• 1: [0,400] MW, $25/MWh, 10MW/min
• 2: [0,300] MW, $30/MWh, 10MW/min
• 3: [0,200] MW, $80/MWh, 15MW/min (Unavailable)
•Lines: All X=0.01
• 1-3: [-240,240] MW
• 1-2: [-300,300] MW
• 2-3: [-300,300] MW
•Load: 450MW, 175 MW is shed-able. VOLL=$5,000/MWh
•Contingencies
• Loss of line 1-2
• Loss of largest generating unit - Gen 1
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PSERC
PSERC
Simple Example
[0,400] MW
$25/MWh,
10MW/min
[0,300] MW,
$30/MWh,
10MW/min
Dispatch Cost - $12,150
Unavailable
Unconstrained
Dispatch Cost =
$11,250
This is the OPF dispatch that would occur if line limits were included but
contingencies are not. Call this the no-contingency dispatch (NCD).
This configuration is unreliable with respect to contingencies.
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A feasible dispatch arrived at through sequential optimization
PSERC
Dispatch Cost - $12,300
In this configuration, an outage of line 1-2 would produce a flow on line 1-3 of 240MW’s. An outage
of Gen 1 would require a load shed of 150MW’s and a ramp of Gen 2 to 300 MW which it could do
in 9 minutes.. But this dispatch does not take advantage of Gen 1’s ramping capability.
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PSERC
The least-cost reliable intact system dispatch
produced by the SuperOPF
Dispatch Cost - $12,250
In this configuration if the line contingency occurs Gen 1 would have to back off 10 MW and Gen
2 would have to ramp up 10 MW to avoid overloading line 1-3. If Gen 1 is outaged then Gen 2
would have to ramp up 100 MW to its 300MW limit and 150 MW’s of load would have to be shed.
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Problems included in the
formulation
• standard OPF with full AC non-linear
network model and constraints
• n–1 contingency security with static and
dynamic constraints
• procurement of adequate supply of
active and reactive energy and
geographically distributed reserves
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PSERC
• uncertainty of demand, wind,
contingencies
• stochastic cost, including cost of postcontingency states
• correct prices for day-ahead contracts
for energy, reactive supply and reserve
• consistent mechanism for subsequent
re-dispatching and pricing, given
specific realization of uncertain
quantities
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PSERC
PSERC
Problems not yet included
• thermal / hydro unit commitment
•We have a formulation that could be included at a
later stage.
• transient upset following contingencies
•Only included in the the form of angle difference
limits.
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PSERC
•
Conceptual
Framework
Replicate network for multiple
scenarios.
• Treat as single large network with
islands.
• All standard OPF variables for all
islands are available to impose costs &
constraints.
• Additional variables can be defined and
included in costs and linear constraints.
• Additional linear constraints on all
variables
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PSERC
•
Day-ahead Market
Scenarios include projected base case
plus set of credible post-contingency
cases.
• All standard limits (voltage, flow, etc.)
apply to base case and postcontingency cases.
• Standard OPF cost function on active
and reactive output included for base
case and post-contingency cases,
weighted by probabilities.
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•
•
PSERC
Additional variables include:
‣
day-ahead contracted energy allocation
‣
positive & negative post-contingency
deviations from day-ahead contract
‣
reserve quantities (max of these
deviations)
Additional costs include:
‣
probability weighted cost of deviations
from contract
‣
cost of reserves
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• Additional constraints include:
‣
limits on reserve quantities
‣
physical ramp rate limits on deviations
of post-contingency output from base
case output
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Real-time Re-dispatch
•Same as day-ahead, except:
•Updated demand forecasts, outages,
credible contingency list, probabilities, etc.
•Day-ahead contracted quantity is now
fixed.
•Reserve quantities are now fixed limits.
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Determining the Economic Benefits
of Avoiding Loss-of-Load during
Contingencies: A Case Study
Area 1
- Urban
- High Load
- High Cost
- VOLL =
$10,000/MWh
Area 2
- Rural
- Low Load
- Low Cost
- VOLL =
$5,000/MWh
Area 3
- Rural
- Low Load
- Low Cost
- VOLL =
$5,000/MWh
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PSERC
The Underlying Rationale for Doing
this Case Study
PSERC
•Region 1 represents an urban load pocket with high load, high cost
generation, and high VOLL.
•Regions 2 and 3 represent rural areas with low load, low cost
generation, and low VOLL.
•Transmission capacity into Region 1 from Regions 2 and 3 is
relatively limited.
•An economic dispatch would use generation in Regions 2 and 3 as
much as possible and use generating capacity in Region 1 for
reserves to maintain operating reliability (e.g. guard against line
outages).
•In this Case Study, operating costs remain constant, offers equal
the true marginal costs, and all loads in Region 1 are increased in
increments until things start to go wrong (i.e. load outages occur).
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
PSERC
The Basic Objective Function
K
min
G, R

k 0

 I
pk  CGi (Gik )  CR i (Rik ) +

i1


VOLLj LNS(Gk,Rk) jk


j =1
J
Subject to meeting LOAD and all of the nonlinear AC CONSTRAINTS of the network
Where
k is a CONTINGENCY
i is a GENERATOR
j is a LOAD
CG(Gi) is the COST of generating G MWh
CR(Ri) is the COST of providing R MW of RESERVES
VOLLj is the VALUE OF LOST LOAD
LNS(G, R)j is the LOAD NOT SERVED
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Contingencies Considered
•Number
PSERC
Probability
•0 = base case
•1 = line 1 : 1-2 (between gens 1 and 2, within area 1)
•2 = line 2 : 1-3 (from gen 1, within area 1)
•3 = line 3 : 2-4 (from gen 2, within area 1)
•4 = line 5 : 2-5 (from gen 2, within area 1)
•5 = line 6 : 2-6 (from gen 2, within area 1)
•6 = line 36 : 27-28 (main tie from area 3 to area 1)
•7 = line 15 : 4-12 (main tie from area 2 to area 1)
•8 = line 12 : 6-10 (other tie from area 3 to area 1)
•9 = line 14 : 9-10 (other tie from area 3 to area 1)
•10 = gen 1
0.2%
•11 = gen 2
0.2%
•12 = gen 3
0.2%
•13 = gen 4
0.2%
•14 = gen 5
0.2%
•15 = gen 6
0.2%
•16 = 10% increase in load
•17 = 10% decrease in load
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95%
0.2%
0.2%
0.2%
0.2%
0.2%
0.2%
0.2%
0.2%
0.2%
1.0%
1.0%
Expected Nodal Prices for Generators
PSERC
Price Differences are Caused by Congestion
Expected Nodal Prices (Generators)
$90/MWh
90
80
70
G1
G2
G3
G4
G5
G6
$/MWhr
60
50
40
30
$20/MWh
20
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
Scale Factor for Load
Higher Load in the Load Pocket (Region 1) ->
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1.60
Expected Nodal Prices for Loads
PSERC
The Large Price Differences Within Area 1 (blue) are Caused by
Load Shedding --- A Very Localized Effect
Expected Nodal Prices (Loads -- log scale)
$10,000/MWh
10000
Area1
Area1
Area1
Area1
1000
Area1
Area2
Area2
Area2
$/MWhr
Area2
Area2
Area2
Area2
Area2
Area2
Area3
100
Area3
Area3
Area3
Area3
Area3
$10/MWh
10
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60
Scale Factor for Load
Higher Load in the Load Pocket (Region 1) ->
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The Expected Cost of Lost Load
(Weighted by the Probability of Each Contingency)
PSERC
Contingency
$4,500/MWh
Higher Load in the Load Pocket (Area 1) ->
Problems show up first in contingencies as system load increases
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PSERC
The Shadow Price of the Flow on Tie Line 15
Linking Areas 2 and 3 by Contingency
Contingency
$80/MWh
Higher Load in the Load Pocket (Area 1) ->
Typical Example of Binding Flow Limits on Power Transfers
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The Shadow Price of the Flow on Line 10
in Area 1 by Contingency
PSERC
$10,000/MWh
Contingency
Choose Scale
Factor = 1.1278
for next part
of the analysis
(System Load
= 200MW)
Higher Load in the Load Pocket (Area 1) ->
Caused by a Binding Flow Limit WITHIN Area 1 --- Like NYC?
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The Question for Planners I:
PSERC
Will it Pay to Increase the Capacity of Line 10?
STEP 1
SCALED LOAD DURATION CURVES
120
Percentage of the Peak Load
100
80
60
40
20
Area 1
Area 2 & 3
0
0
10
20
30
40
50
60
70
80
Percentage of the Year
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90
100
STEPS
• Specify the Hourly
Loads for a Year.
• Specify a Peak Load
of 200MW (when
load shedding first
occurs in some
contingencies).
1. Determine the Nodal
Prices, Revenues
and Costs
2. Compute the Annual
Totals
STEPS 2 and 3
Low Load - base system
PSERC
High Load - base system
1.
2.
Low Load - Transmission improvement
3.
High Load - Transmission improvement
Transmission Improvement => line 6-8 capacity doubled
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Load Shedding
AND Congestion
(i.e. High Prices)
only occur for a
Few Hours.
Most of the time
Loads pay the
True Marginal
Cost of Energy.
Net Revenues for
Baseload Units
are Large --Where do the $$$
go?
PSERC
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PSERC
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The Question for Planners III:
PSERC
Will it Pay to Increase the Capacity of Line 10?
Expected Annual Costs, Revenues and Payments ($/Year)
STEP 4
LOW LOAD CASE
DEMAND SIDE
1. Load Payment + Cost of LNS*
2. Load Payment
Initial
Upgrade**
Change
26062479
26058061
25988879
25988879
-73600
-69182
24792467
24506897
9020838
24819019
24533566
9016970
26552
26669
-3868
4418
0
1265594
1169860
285570
285453
15486059
15516596
TOTAL SAVINGS
* LNS is Load-Not-Served and all loads pay real-time nodal prices
** Upgrade to transmission line L10 linking nodes 6 and 8
-4418
-95734
-117
30537
69732
SUPPLY SIDE
3. Energy + Reserves Revenue
4. Energy Revenue
5. Energy Cost
DERIVED VALUES
Reliability Cost of LNS* (1 - 2)
Congestion Cost (2 - 3)
Reliability Cost of Reserves (3 - 4)
Net Revenue for Generation (4 -5)
HIGH LOAD CASE
DEMAND SIDE
1. Load Payment + Cost of LNS*
2. Load Payment
Initial
Upgrade**
Change
97187081
93849516
36810006
36810006
-60377075
-57039510
32768733
32422941
12352036
32849335
32505015
12362575
80602
82074
10539
3337565
0
61080783
3960671
345792
344320
20416697
20486760
TOTAL SAVINGS
* LNS is Load-Not-Served and all loads pay real-time nodal prices
** Upgrade to transmission line L10 linking nodes 6 and 8
-3337565
-57120112
-1472
70063
60389086
SUPPLY SIDE
3. Energy + Reserves Revenue
4. Energy Revenue
5. Energy Cost
DERIVED VALUES
Reliability Cost of LNS* (1 - 2)
Congestion Cost (2 - 3)
Reliability Cost of Reserves (3 - 4)
Net Revenue for Generation (4 -5)
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Conclusions
•A large part of the economic benefit of adding equipment to the
network may be avoiding LOSS OF LOAD when credible
contingencies occur (e.g. N-1 contingencies).
•By using the SuperOPF, it is possible to identify the LOCATION of
weaknesses on the network and the net economic benefit of
upgrading the network.
•The same analytical framework can be used for PLANNING
purposes, and problems tend to occur first when contingencies
occur in a simulation (e.g. when load outages occur and violate
standards of operating reliability).
•GET THE PRICES RIGHT BY MAKING CONTINGENCIES
EXPLICIT AND USING AN AC NETWORK FOR PLANNING.
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