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

11

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

12

Thirty Bus ED Example

Case is economically dispatched without considering the incremental impact of the system losses

13

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

16

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

18

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

20

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

21

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

22

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)

26

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.

27

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.

32

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.

33

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?

34

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.

36

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.”

37

MISO and PJM Joint LMP Contour

http://www.miso-pjm.com/markets/contour-map.html

38

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