Lecture 17 Economic Dispatch, OPF, Markets Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS
Dec 22, 2015
Lecture 17Economic Dispatch, OPF, Markets
Professor Tom OverbyeDepartment of Electrical and
Computer Engineering
ECE 476
POWER SYSTEM ANALYSIS
2
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.
3
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 when doing the economic dispatch
Losses can be included by slightly rewriting the Lagrangian:
G1 1
L( , ) ( ) ( ( ) ) m m
i Gi D L G Gii i
C P P P P P
P
4
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 PP d
P
P
1
( )(1 ) 0
( ) 0
L G
Gi Gi
m
D L G Gii
P PP P
P P P P
5
Impact of Transmission Losses
thi
i
Solving each equation for we get
( ) ( )(1 0
( )1
( )1
Define the penalty factor L for the i generator
1L
( )1
i Gi L G
Gi Gi
i Gi
GiL G
Gi
L G
Gi
dC P P PdP P
dC PdPP P
P
P PP
The penalty factorat the slack bus isalways unity!
6
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 PP
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.
7
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 PP
8
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 MWP P MW
L L
9
Thirty Bus ED Example
Because of the penalty factors the generator incremental costs are no longer identical.
10
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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Thirty Bus ED Example
Case is economically dispatched without considering the 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.
14
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).
15
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
17
“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
23
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
24
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
25
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
28
10
119
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 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 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/disadvantages of both?
Talk about supply and demand curves, scarcity, withholding, market power
39
Market Experiments