POWER SYSTEM ANALYSIS
Lecture 17
Optimal Power Flow, LMPs
Tom Overbye and Ross Baldick
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 Read Chapter 7.
 Homework 12 is 6.59, 6.61, 12.19, 12.22,
12.24, 12.26, 12.28, 7.1, 7.3, 7.4, 7.5, 7.6,
7.9, 7.12, 7.16; due Thursday, 12/3.
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• Over last 20 years electricity markets have moved from bilateral contracts between utilities to also include centralized markets operated by Independent System
Operators/Regional Transmission Operators:
– Day-ahead market that establishes unit commitment and “forward financial positions,”
– Real-time market, run every 5 or 15 minutes that arranges for physical dispatch, the “spot” market.
• Basic “engine” for operating centralized markets is Optimal Power Flow (OPF).
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 OPF is used as basis for day-ahead and realtime dispatch pricing in US ISO/RTO electricity markets:
 MISO, PJM, ISO-NE, NYISO, SPP, CA, and ERCOT.
 Electricity (MWh) is treated as a commodity
(like corn, coffee, natural gas) but with the extent of the market limited by transmission system constraints.
 Tools of commodity trading have been widely adopted (options, forwards, hedges, swaps).
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Source: Wall Street Journal Online, 10/30/08
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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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Total Hourly Cost : 8459 $/hr
Area Lambda : 13.02
Bus A Bus B
AGC ON AGC ON
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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
15.00
14.00
13.00
12.00
0
16.00
15.00
14.00
13.00
175 350 525
G enerator Power (MW)
700
12.00
0
Current generator operating point
350 700 1050
Total Area G eneration (MW)
1400
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 Different operating regions impose constraints
– may limit ability to achieve economic dispatch “globally.”
 Transmission system imposes constraints on the market:
 Marginal costs differ at different buses.
 Optimal dispatch solution requires solution by an optimal power flow
 Charging for energy based on marginal costs at different buses is called “locational marginal pricing” (LMP) or “nodal” pricing.
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 LMP indicates the additional cost to supply an additional amount of electricity to bus.
 All North American ISO/RTO electricicity markets price wholesale energy at LMP.
 If there were no transmission limitations then the
LMPs would be the same at all buses:
 Equal to value of lambda from economic dispatch.
 Transmission constraints result in differing LMPs at buses.
 Determination of LMPs requires the solution of an
“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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 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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 Non-linear approach using Newton’s method:
– handles marginal losses well, but is relatively slow and has problems determining binding constraints
 Linear Programming (LP):
– fast and efficient in determining binding constraints, but can have difficulty with marginal losses.
– used in PowerWorld Simulator
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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 an LP
 changes system controls to enforce linearized constraints while minimizing cost
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With no overloads the
OPF matches the economic dispatch
Total Hourly Cost : 8459 $/hr
Area Lambda : 13.01
Transmission line is not overloaded
13.01 $/MWh
Bus B
Bus A 13.01 $/MWh
AGC ON AGC ON
Marginal cost of supplying power to each bus (locational marginal costs)
This would be price paid by load and paid to the generators.
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Total Hourly Cost : 9513 $/hr
Area Lambda : 13.26
13.43 $/MWh
Bus B
13.08 $/MWh Bus A
AGC ON AGC ON
With the line loaded to its limit, additional load at Bus A must be supplied locally, causing the marginal costs to diverge.
Similarly, prices paid by load and paid to generators will differ bus by bus.
(In practice, some markets such as ERCOT charge zonal averaged price to load.)
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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:
– 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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60 MW
Bus 2
60 MW
Bus 1
10.00 $/MWh
0.0 MW 10.00 $/MWh
120 MW
120%
180.0 MW
0 MW
60 MW
Total Cost
1800 $/hr
Bus 3
0 MW
120%
120 MW
10.00 $/MWh
180 MW
Line from Bus 1 to Bus 3 is overloaded; all buses have same marginal cost
(but not allowed to dispatch to overload
18 line!)

20 MW
Bus 2
20 MW
Bus 1
10.00 $/MWh
60.0 MW 12.00 $/MWh
100 MW
100%
120.0 MW
0 MW
80 MW
Total Cost
1920 $/hr
Bus 3
0 MW
100%
100 MW
14.00 $/MWh
180 MW
LP OPF redispatches to remove violation.
Bus marginal costs are now different.
Prices will be different
19 at each bus.

19 MW
Bus 2
19 MW
Bus 1
10.00 $/MWh
62.0 MW 12.00 $/MWh
81%
0 MW
81 MW
Total Cost
1934 $/hr
Bus 3
81%
100 MW
100%
100%
100 MW
14.00 $/MWh
181 MW
119.0 MW
0 MW
One additional MW of load at bus 3 raised total cost by
14 $/hr, as G2 went up by 2 MW and G1
20 went down by 1MW.

 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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 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:
– Extra production of 1MW: P g1
+ P g2
= 1 MW
– No more flow on line 1 to 3: 2/3 P g1
+ 1/3 P g2
= 0;
 Solving requires we increase P g2 by 2 MW and decrease P g1 by 1 MW – for a net increase of
$14/h for the 1 MW increase.
 That is, the marginal cost of delivering power to bus 3 is $14/MWh.
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0 MW
Bus 2
0 MW
Bus 1
10.00 $/MWh
100.0 MW 12.00 $/MWh
100%
0 MW
100 MW
Total Cost
2280 $/hr
Bus 3
100%
100 MW
100%
100%
100 MW
20.00 $/MWh
204 MW
4 MW
100.0 MW
For bus 3 loads above 200 MW, the load must be supplied locally.
Then what if the bus 3 generator breaker opens?
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 Electricity markets trade various 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 (offer roughly represents marginal costs); OPF is used to determine who gets dispatched based upon forecasted conditions. Results are “financially” binding: either generate or pay for someone else.
– Real-time: Modifies the conditions from the day ahead market based upon real-time conditions.
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 Generators are not paid their offer, rather they are paid the LMP at their bus, while the loads pay the LMP:
 In most systems, loads are charged based on a zonal weighted average of LMPs.
 At the residential/small commercial level the
LMP costs are usually not passed on directly to the end consumer. Rather, these consumers typically pay a fixed rate that reflects time and geographical average of LMPs.
 LMPs differ across the system due to transmission system “congestion.”
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LMPs at 8:55 AM on one day in Midwest.
Source: www.midwestmarket.org
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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 involves setting a price and emitter of CO
2 emitted. pays based upon how much CO
2 is
A cap-and-trade system limits emissions by requiring permits (allowances) to emit CO
2
. The government sets the number of allowances, allocates them initially, and then private markets set their prices and allow trade.
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