lecture 17 optimal power flow, lmps professor tom overbye department of electrical and computer...
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Lecture 17Optimal Power Flow, LMPs
Professor Tom OverbyeDepartment of Electrical and
Computer Engineering
ECE 476
POWER SYSTEM ANALYSIS
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Announcements
Homework 7 is due now. Homework 8 is 7.1, 7.17, 7.20, 7.24, 7.27
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
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Aside: Levelized Cost of Generation
Source: California Energy Commission: http://energyalmanac.ca.gov/electricity/levelized_costs.html
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 assumptionsthat can drastically affect the value, and that many technology costs are site dependent.
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Area Supply Curve
0 100 200 300 400Total Area Generation (MW)
0.00
2.50
5.00
7.50
10.00
The area supply curve shows the cost to produce thenext MW of electricity, assuming area is economicallydispatched
Supplycurve forthirty bussystem
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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– 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
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.
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Electricity Markets
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
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“Ideal” Power Market
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 :
Bus A Bus B
300.0 MWMW
199.6 MWMW 400.4 MWMW300.0 MWMW
8459 $/hr Area Lambda : 13.02
AGC ON AGC ON
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Market Marginal (Incremental) Cost
0 175 350 525 700Generator Power (MW)
12.00
13.00
14.00
15.00
16.00
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.
Current generator operating point
0 350 700 1050 1400Total Area Generation (MW)
12.00
13.00
14.00
15.00
16.00
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Real Power Markets
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)
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
Inequality constraints– 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
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OPF Solution Methods
Non-linear approach using Newton’s method– 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
Solution iterates between– solving a full ac power flow solution
enforces real/reactive power balance at each busenforces generator reactive limitssystem controls are assumed fixed takes into account non-linearities
– solving a primal LPchanges system controls to enforce linearized
constraints while minimizing cost
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Two Bus with Unconstrained Line
Total Hourly Cost :
Bus A Bus B
300.0 MWMW
197.0 MWMW 403.0 MWMW300.0 MWMW
8459 $/hr Area Lambda : 13.01
AGC ON AGC ON
13.01 $/MWh 13.01 $/MWh
Transmission line is not overloaded
With no overloads theOPF matchesthe economicdispatch
Marginal cost of supplyingpower to each bus (locational marginal costs)
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Two Bus with Constrained Line
Total Hourly Cost :
Bus A Bus B
380.0 MWMW
260.9 MWMW 419.1 MWMW300.0 MWMW
9513 $/hr Area Lambda : 13.26
AGC ON AGC ON
13.43 $/MWh 13.08 $/MWh
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
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 – 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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Bus 2 Bus 1
Bus 3
Total Cost
0.0 MW
0 MW
180 MW
10.00 $/MWh
60 MW 60 MW
60 MW
60 MW120 MW
120 MW
10.00 $/MWh
10.00 $/MWh
180.0 MW
0 MW
1800 $/hr
120%
120%
B3 with Line Limits NOT Enforced
Line from Bus 1to Bus 3 is over-loaded; all buseshave same marginal cost
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B3 with Line Limits Enforced
Bus 2 Bus 1
Bus 3
Total Cost
60.0 MW
0 MW
180 MW
12.00 $/MWh
20 MW 20 MW
80 MW
80 MW100 MW
100 MW
10.00 $/MWh
14.00 $/MWh
120.0 MW
0 MW
1920 $/hr
100%
100%
LP OPF redispatchesto remove violation.Bus marginalcosts are nowdifferent.
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Bus 2 Bus 1
Bus 3
Total Cost
62.0 MW
0 MW
181 MW
12.00 $/MWh
19 MW 19 MW
81 MW
81 MW100 MW
100 MW
10.00 $/MWh
14.00 $/MWh
119.0 MW
0 MW
1934 $/hr
81%
81%
100%
100%
Verify Bus 3 Marginal Cost
One additional MWof load at bus 3 raised total cost by14 $/hr, as G2 wentup by 2 MW and G1went down by 1MW
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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
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 – 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 Bus 1
Bus 3
Total Cost
100.0 MW
4 MW
204 MW
12.00 $/MWh
0 MW 0 MW
100 MW
100 MW100 MW
100 MW
10.00 $/MWh
20.00 $/MWh
100.0 MW
0 MW
2280 $/hr
100% 100%
100% 100%For bus 3 loadsabove 200 MW,the load must besupplied locally.Then what if thebus 3 generator opens?
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Profit Maximization: 30 Bus Example
1.000
slack
Gen 13 LMP3
1
4
2
576
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10
119
8
22 2125
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27
24
15
14
16
12
17
18
19
13
20
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29 30
16 MW
11 MW
21 MW
2 MW
11 MW
19 MW
10 MW
A
MVA
A
MVA
A
MVA
A
MVA
66%A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
68%A
MVA
67%A
MVA
52%A
MVA
A
MVA
A
MVA
A
MVA
52%A
MVA
73%A
MVA
A
MVA
A
MVAA
MVA
A
MVA
A
MVA
A
MVA
A
MVA
56%A
MVA
62%A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
A
MVA
33.46 $/MWh
52.45 MW 69.58 MW
16.00 MW
35.00 MW
40.00 MW
24.00 MW
82%A
MVA
84%A
MVA
87%A
MVA
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Typical Electricity Markets
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
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– 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
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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.