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Lecture 17
Economic Dispatch, OPF, Markets
Professor Tom Overbye
Department of Electrical and
Computer Engineering
ECE 476
POWER SYSTEM ANALYSIS
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Announcements
Be reading Chapter 7
HW 7 is 12.26, 12.28, 12.29, 7.1 due October 27 in class.
US citizens and permanent residents should consider applying
for a Grainger Power Engineering Awards. Due Nov 1. See
http://energy.ece.illinois.edu/grainger.html for details.
The Design Project, which is worth three regular homeworks,
is assigned today; it is due on Nov 17 in class. It is Design
Project 2 from Chapter 6 (fifth edition of course).
For tower configuration assume a symmetric conductor spacing, with
the distance in feet given by the following formula:
(Last two digits of your UIN+50)/9. Example student A has an EIN of
xxx65. Then his/her spacing is (65+50)/9 = 12.78 ft.
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Inclusion of Transmission Losses
The losses on the transmission system are a function
of the generation dispatch. In general, using
generators closer to the load results in lower losses
This impact on losses should be included whendoing the economic dispatch
Losses can be included by slightly rewriting the
Lagrangian:
G1 1
L( , ) ( ) ( ( ) )m m
i Gi D L G Gi
i i
C P P P P P
P
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Impact of Transmission Losses
G1 1
G
This small change then impacts the necessary
conditions for an optimal economic dispatch
L( , ) ( ) ( ( ) )
The necessary conditions for a minimum are now
L( , ) ( )
m m
i Gi D L G Gii i
i Gi
Gi
C P P P P P
dC P
P d
P
P
1
( )(1 ) 0
( ) 0
L G
Gi Gi
m
D L G Gi
i
P P
P P
P P P P
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Impact of Transmission Losses
th
i
i
Solving each equation for we get( ) ( )
(1 0
( )1( )
1
Define the penalty factor L for the i generator1
L( )
1
i Gi L G
Gi Gi
i Gi
GiL G
Gi
L G
Gi
dC P P P
dP P
dC PdPP P
P
P P
P
The penalty factor
at the slack bus is
always unity!
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Impact of Transmission Losses
1 1 1 2 2 2
i Gi
The condition for optimal dispatch with losses is then
( ) ( ) ( )
1Since L if increasing P increases
( )1
( )the losses then 0 1.0
This makes generator
G G m m Gm
L G
Gi
L Gi
Gi
L IC P L IC P L IC P
P P
P
P PL
P
i
i appear to be more expensive
(i.e., it is penalized). Likewise L 1.0 makes a generator
appear less expensive.
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Calculation of Penalty Factors
i
Gi
Unfortunately, the analytic calculation of L is
somewhat involved. The problem is a small change
in the generation at P impacts the flows and hence
the losses throughout the entire system. However,
Gi
using a power flow you can approximate this function
by making a small change to P and then seeing how
the losses change:
( ) ( ) 1
( )1
L G L Gi
L GGi Gi
Gi
P P P PL
P PP P
P
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Two Bus Penalty Factor Example
2
2 2
( ) ( ) 0.370.0387 0.037
10
0.9627 0.9643
L G L G
G Gi
P P P P MW
P P MW
L L
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Thirty Bus ED Example
Because of the penalty factors the generator incrementalcosts are no longer identical.
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Area Supply Curve
0 100 200 300 400
Total 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 economically
dispatched
Supply
curve for
thirty bus
system
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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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Thirty Bus ED Example
Case is economically dispatched without consideringthe incremental impact of the system losses
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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 acommodity (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, 10/19/11
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Historical Variation in Oct 11 Price
Source: Wall Street Journal Online, 10/19/11
Price has dropped, following the drop in natural gas prices
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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 MWMW
300.0 MWMW
8459 $/hrArea 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 bussystem. 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 inequalityconstraints
Equality constraints
bus real and reactive power balance
generator voltage setpoints
area MW interchange
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OPF, contd
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 Newtons 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 bus
enforces generator reactive limits system controls are assumed fixed
takes into account non-linearities
solving a primal LP
changes system controls to enforce linearizedconstraints 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 MWMW
300.0 MWMW
8459 $/hr
Area Lambda : 13.01
AGC ON AGC ON
13.01 $/MWh 13.01 $/MWh
Transmission
line is not
overloaded
With nooverloads the
OPF matches
the economic
dispatch
Marginal cost of supplying
power 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 MWMW
300.0 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 1
to Bus 3 is over-
loaded; all buses
have 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 MW
100 MW
100 MW
10.00 $/MWh
14.00 $/MWh
120.0 MW
0 MW
1920 $/hr
100%
100%
LP OPF redispatches
to remove violation.
Bus marginal
costs are now
different.
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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 MW
100 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 MW
of load at bus 3raised total cost by
14 $/hr, as G2 went
up by 2 MW and G1
went 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 takedirect 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, contd
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 MW
100 MW
100 MW
10.00 $/MWh
20.00 $/MWh
100.0 MW
0 MW
2280 $/hr
100% 100%
100% 100%
For bus 3 loads
above 200 MW,
the load must besupplied locally.
Then what if the
bus 3 generator
opens?
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Profit Maximization: 30 Bus Example
1.000
slack
Gen 13 LMP3
1
4
2
576
28
10
11
9
8
22 2125
26
27
24
15
14
16
12
17
18
19
13
20
23
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 differentcommodities, 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 arepaid 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 endconsumer. 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 and PJM Joint LMP Contour
http://www.miso-pjm.com/markets/contour-map.html
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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/disadvantagesof both?
Talk about supply and demand curves, scarcity,
withholding, market power
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Market Experiments