Transcript EE 369 POWER SYSTEM ANALYSIS Lecture 17 Optimal Power Flow, LMPs Tom Overbye and Ross Baldick.

EE 369
POWER SYSTEM ANALYSIS
Lecture 17
Optimal Power Flow, LMPs
Tom Overbye and Ross Baldick
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Announcements
Read Chapter 7.
Homework 12 is 6.43, 6.48, 6.59, 6.61,
12.19, 12.22, 12.20, 12.24, 12.26, 12.28,
12.29; due Tuesday Nov. 25.
Homework 13 is 12.21, 12.25, 12.27, 7.1,
7.3, 7.4, 7.5, 7.6, 7.9, 7.12, 7.16; due
Thursday, December 4.
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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).
• OPF is used as basis for real-time pricing in
major US electricity markets such as MISO,
PJM, CA, and ERCOT (from December 2010).
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Electricity Markets
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 have been 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 : 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
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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
Current generator operating point
0
350
700
1050
Total Area Generation (MW)
1400
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Real Power Markets
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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Pricing Electricity
 LMP indicates the additional cost to supply an
additional amount of electricity to bus.
 Some electric markets price wholesale energy at
LMP:
– ERCOT began this in December 2010.
 In 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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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 (LP):
– 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 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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Two Bus with Unconstrained Line
With no
overloads the
OPF matches
the economic
dispatch
Total Hourly Cost : 8459 $/hr
Area Lambda : 13.01
Bus A
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)
This would be price paid by load
and paid to the generators.
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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.
Similarly, prices paid by load and paid to generators will differ bus by bus.
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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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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
(but not allowed to
dispatch to overload
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line!)
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 $/MWh
Bus 3
180 MW
0 MW
LP OPF redispatches
to remove violation.
Bus marginal
costs are now
different.
Prices will be different
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at each bus.
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%
100 MW
14.00 $/MWh
Bus 3
181 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
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went down by 1MW.
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:
– Extra production of 1MW: Pg1 + Pg2 = 1 MW
– No more flow on line 1 to 3: 2/3 Pg1 + 1/3 Pg2 = 0;
 Solving requires we increase Pg2 by 2 MW and
decrease Pg1 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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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%
100%
100 MW
20.00 $/MWh
Bus 3
204 MW
4 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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Typical Electricity Markets
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. 24
Payment
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 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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LMPs at 9:30 AM on same day
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MISO LMP Contours – 10/30/08
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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 involves setting a price and
•
emitter of CO2 pays based upon how much CO2 is
emitted.
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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