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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.

http://energy.ece.illinois.edu/grainger.htmlhttp://energy.ece.illinois.edu/grainger.html

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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