Download Optimal Power Flow

ECE 476
POWER SYSTEM ANALYSIS
Lecture 17
Optimal Power Flow, LMPs
Professor Tom Overbye
Department of Electrical and
Computer Engineering
Announcements
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Homework 7 is due now.
Homework 8 is 7.1, 7.17, 7.20, 7.24, 7.27
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Should be done before second exam; not turned in
Be reading Chapter 7
Design Project is assigned today (see website for
details). Due date is Nov 20.
Exam 2 is Thursday Nov 13 in class.
Grainger Power Engineering Award Applications
Due Nov 1. See below for details:
http://energy.ece.uiuc.edu/Grainger.html
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Back of Envelope Values

Often times incremental costs can be approximated
by a constant value:
–
–
–
–
$/MWhr = fuelcost * heatrate + variable O&M
Typical heatrate for a coal plant is 10, modern
combustion turbine is 10, combined cycle plant is 7 to 8,
older combustion turbine 15.
Fuel costs ($/MBtu) are quite variable, with current
values around 2 for coal, 7 for natural gas, 0.5 for
nuclear, probably 10 for fuel oil.
Hydro costs tend to be quite low, but are fuel (water)
constrained
2
Aside: Levelized Cost of Generation
Technology
$/MWh (2007 Dollars) (IOU)
Advanced Nuclear
104
Wind – Class 5
67
Solar – Photovoltaic
686
Solar – Concentrating
434
Solar – Parabolic Trough
281
Ocean Wave (Pilot)
838
Small Scale Hydro
118
Geothermal
63
Keep in mind these numbers involve LOTs of assumptions
that can drastically affect the value, and that many
technology costs are site dependent.
Source: California Energy Commission:
http://energyalmanac.ca.gov/electricity/levelized_costs.html
3
Area Supply Curve
The area supply curve shows the cost to produce the
next MW of electricity, assuming area is economically
dispatched
10.00
7.50
Supply
curve for
thirty bus
system
5.00
2.50
0.00
0
100
200
Total Area Generation (MW)
300
400
4
Economic Dispatch - Summary


Economic dispatch determines the best way to
minimize the current generator operating costs
The lambda-iteration method is a good approach for
solving the economic dispatch problem
–
–
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generator limits are easily handled
penalty factors are used to consider the impact of losses
Economic dispatch is not concerned with
determining which units to turn on/off (this is the
unit commitment problem)
Economic dispatch ignores the transmission system
limitations
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Optimal Power Flow
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The goal of an optimal power flow (OPF) is to
determine the “best” way to instantaneously operate
a power system.
Usually “best” = minimizing operating cost.
OPF considers the impact of the transmission system
OPF is used as basis for real-time pricing in major
US electricity markets such as MISO and PJM.
ECE 476 introduces the OPF problem and provides
some demonstrations.
6
Electricity Markets
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Over last ten years electricity markets have moved
from bilateral contracts between utilities to also
include spot markets (day ahead and real-time).
Electricity (MWh) is now being treated as a
commodity (like corn, coffee, natural gas) with the
size of the market transmission system dependent.
Tools of commodity trading are being widely
adopted (options, forwards, hedges, swaps).
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Electricity Futures Example
Source: Wall Street Journal Online, 10/30/08
8
“Ideal” Power Market

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Ideal power market is analogous to a lake.
Generators supply energy to lake and loads remove
energy.
Ideal power market has no transmission constraints
Single marginal cost associated with enforcing
constraint that supply = demand
–
–

buy from the least cost unit that is not at a limit
this price is the marginal cost
This solution is identical to the economic dispatch
problem solution
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Two Bus ED Example
Total Hourly Cost : 8459 $/hr
Area Lambda : 13.02
Bus A
Bus B
300.0 MW
199.6 MW
AGC ON
300.0 MW
400.4 MW
AGC ON
10
Market Marginal (Incremental) Cost
Below are some graphs associated with this two bus
system. The graph on left shows the marginal cost for each
of the generators. The graph on the right shows the
system supply curve, assuming the system is optimally
dispatched.
16.00
16.00
15.00
15.00
14.00
14.00
13.00
13.00
12.00
12.00
0
175
350
525
Generator Power (MW)
700
0
350
700
1050
Total Area Generation (MW)
1400
Current generator operating point
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Real Power Markets
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Different operating regions impose constraints -total demand in region must equal total supply
Transmission system imposes constraints on the
market
Marginal costs become localized
Requires solution by an optimal power flow
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Optimal Power Flow (OPF)
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OPF functionally combines the power flow with
economic dispatch
Minimize cost function, such as operating cost,
taking into account realistic equality and inequality
constraints
Equality constraints
–
–
–
bus real and reactive power balance
generator voltage setpoints
area MW interchange
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OPF, cont’d
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Inequality constraints
–
–
–
–
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transmission line/transformer/interface flow limits
generator MW limits
generator reactive power capability curves
bus voltage magnitudes (not yet implemented in
Simulator OPF)
Available Controls
–
–
generator MW outputs
transformer taps and phase angles
14
OPF Solution Methods
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Non-linear approach using Newton’s method
–
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handles marginal losses well, but is relatively slow and
has problems determining binding constraints
Linear Programming
–
–
fast and efficient in determining binding constraints, but
can have difficulty with marginal losses.
used in PowerWorld Simulator
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LP OPF Solution Method
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Solution iterates between
–
–
solving a full ac power flow solution
 enforces real/reactive power balance at each bus
 enforces generator reactive limits
 system controls are assumed fixed
 takes into account non-linearities
solving a primal LP
 changes system controls to enforce linearized
constraints while minimizing cost
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Two Bus with Unconstrained Line
With no
overloads the
OPF matches
the economic
dispatch
Bus A
Total Hourly Cost : 8459 $/hr
Area Lambda : 13.01
13.01 $/MWh
Bus B
300.0 MW
197.0 MW
AGC ON
Transmission
line is not
overloaded
13.01 $/MWh
300.0 MW
403.0 MW
AGC ON
Marginal cost of supplying
power to each bus
(locational marginal costs)
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Two Bus with Constrained Line
Total Hourly Cost : 9513 $/hr
Area Lambda : 13.26
Bus A
13.43 $/MWh
Bus B
380.0 MW
260.9 MW
AGC ON
13.08 $/MWh
300.0 MW
419.1 MW
AGC ON
With the line loaded to its limit, additional load at Bus A
must be supplied locally, causing the marginal costs to
diverge.
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Three Bus (B3) Example
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Consider a three bus case (bus 1 is system slack),
with all buses connected through 0.1 pu reactance
lines, each with a 100 MVA limit
Let the generator marginal costs be
–
–
–
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Bus 1: 10 $ / MWhr; Range = 0 to 400 MW
Bus 2: 12 $ / MWhr; Range = 0 to 400 MW
Bus 3: 20 $ / MWhr; Range = 0 to 400 MW
Assume a single 180 MW load at bus 2
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B3 with Line Limits NOT Enforced
Bus 2
60 MW
60 MW
Bus 1
10.00 $/MWh
0.0 MW
10.00 $/MWh
120 MW
120%
180.0 MW
0 MW
60 MW
120%
Total Cost 60 MW
1800 $/hr
120 MW
10.00 $/MWh
Bus 3
180 MW
0 MW
Line from Bus 1
to Bus 3 is overloaded; all buses
have same
marginal cost
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B3 with Line Limits Enforced
Bus 2
20 MW
20 MW
Bus 1
10.00 $/MWh
60.0 MW 12.00 $/MWh
100 MW
100%
120.0 MW
0 MW
80 MW
100%
Total Cost 80 MW
1920 $/hr
100 MW
14.00
Bus 3
180 MW
0 MW
LP OPF redispatches
$/MWh
to remove violation.
Bus marginal
costs are now
different.
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Verify Bus 3 Marginal Cost
Bus 2
19 MW
19 MW
Bus 1
10.00 $/MWh
62.0 MW 12.00 $/MWh
100 MW
81%
100%
119.0 MW
0 MW
81 MW
Total Cost 81 MW
1934 $/hr
81%
100%
14.00
Bus 3
181 MW
0 MW
One additional MW
of load at bus 3
$/MWh
raised total cost by
14 $/hr, as G2 went
up by 2 MW and G1
went down by 1MW
100 MW
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Why is bus 3 LMP = $14 /MWh

All lines have equal impedance. Power flow in a
simple network distributes inversely to impedance
of path.
–
–
For bus 1 to supply 1 MW to bus 3, 2/3 MW would take
direct path from 1 to 3, while 1/3 MW would “loop
around” from 1 to 2 to 3.
Likewise, for bus 2 to supply 1 MW to bus 3, 2/3MW
would go from 2 to 3, while 1/3 MW would go from 2 to
1to 3.
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Why is bus 3 LMP $ 14 / MWh, cont’d
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With the line from 1 to 3 limited, no additional
power flows are allowed on it.
To supply 1 more MW to bus 3 we need
–
–
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Pg1 + Pg2 = 1 MW
2/3 Pg1 + 1/3 Pg2 = 0; (no more flow on 1-3)
Solving requires we up Pg2 by 2 MW and drop Pg1
by 1 MW -- a net increase of $14.
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Both lines into Bus 3 Congested
Bus 2
0 MW
0 MW
Bus 1
10.00 $/MWh
100.0 MW12.00 $/MWh
100 MW
100%
100%
100.0 MW
0 MW
100 MW
Total Cost100 MW
2280 $/hr
100%
Bus 3
4 MW
For bus 3 loads
100% 100 MW
above 200 MW,
the load must be
20.00 $/MWh
supplied locally.
204 MW
Then what if the
bus 3 generator
opens?
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Profit Maximization: 30 Bus Example
52.45 MW
69.58 MW
slack
A
1
84%
2
MVA
18
15
A
68%
1.000
MVA
A
A
A
A
MVA
MVA
MVA
A
A
62%
MVA
MVA
19
MVA
28
A
14
67%
A
35.00 MW
MVA
MVA
3
A
A
MVA
16 MW
A
MVA
MVA
A
56%
82%
MVA
7
8
Gen 13 LMP
A
4
A
12
MVA
5
MVA
A
13
MVA
6
33.46 $/MWh
A
A
A
A
MVA
MVA
MVA
MVA
9
11 MW
11
A
16
17
19 MW
MVA
A
11 MW
A
MVA
MVA
10 MW
A
20
66%
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A
21 MW
10
24.00 MW
MVA
A
23
MVA
MVA
A
73%
MVA
25
22
A
21
24
52%
2 MW
A
MVA
16.00 MW
MVA
40.00 MW
A
A
52%
MVA
87%
MVA
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27
30
A
MVA
A
A
MVA
MVA
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Typical Electricity Markets
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Electricity markets trade a number of different
commodities, with MWh being the most important
A typical market has two settlement periods: day
ahead and real-time
–
–
Day Ahead: Generators (and possibly loads) submit
offers for the next day; OPF is used to determine who
gets dispatched based upon forecasted conditions.
Results are financially binding
Real-time: Modifies the day ahead market based upon
real-time conditions.
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Payment
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Generators are not paid their offer, rather they are
paid the LMP at their bus, the loads pay the LMP.
At the residential/commercial level the LMP costs
are usually not passed on directly to the end
consumer. Rather, they these consumers typically
pay a fixed rate.
LMPs may differ across a system due to
transmission system “congestion.”
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MISO LMP Contours – 10/30/08
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Why not pay as bid?

Two options for paying market participants
–
–
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Pay as bid
Pay last accepted offer
What would be potential advantages/disadvantages
of both?
Talk about supply and demand curves, scarcity,
withholding, market power
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Market Experiments
31
Limiting Carbon Dioxide Emissions
• There is growing concern about the need to limit
•
carbon dioxide emissions.
The two main approaches are 1) a carbon tax, or 2)
a cap-and-trade system (emissions trading)
• The tax approach is straightforward – pay a fixed rate
•
based upon how the amount of CO2 is emitted. But there
is a need to differentiate between carbon and CO2
(related by 12/44).
A cap-and-trade system limits emissions by requiring
permits (allowances) to emit CO2. The government sets
the number of allowances, allocates them initially, and
then private markets set their prices and allow trade.
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