ECE 476 Power System Analysis Lecture 16: Economic Dispatch, Optimal Power Flow Prof. Tom Overbye Dept. of Electrical and Computer Engineering University.

ECE 476 Power System Analysis

Lecture 16: Economic Dispatch, Optimal Power Flow

Prof. Tom Overbye

Dept. of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

[email protected]

Announcements

• Chapter 12.4 and 12.5, Chapter 7• HW 7 is due now• HW 8 is 12.19, 12.20, 12.26, 12.28; due October 29 in

class (no quiz)

2

Economic Dispatch Lagrangian

G1 1

G

For the economic dispatch we have a minimization

constrained with a single equality constraint

L( , ) ( ) ( ) (no losses)

The necessary conditions for a minimum are

L( , )

m m

i Gi D Gii i

Gi

C P P P

dCP

P

P

1

( )0 (for i 1 to m)

0

i Gi

Gi

m

D Gii

PdP

P P

3

Economic Dispatch Example

D 1 2

21 1 1 1

22 2 2 2

1 1

1

What is economic dispatch for a two generator

system P 500 MW and

( ) 1000 20 0.01 $/

( ) 400 15 0.03 $/

Using the Largrange multiplier method we know

( )20 0

G G

G G G

G G G

G

G

P P

C P P P hr

C P P P hr

dC PdP

1

2 22

2

1 2

.02 0

( )15 0.06 0

500 0

G

GG

G

G G

P

dC PP

dP

P P

4

Economic Dispatch Example, cont’d

1

2

1 2

1

2

1

2

We therefore need to solve three linear equations

20 0.02 0

15 0.06 0

500 0

0.02 0 1 20

0 0.06 1 15

1 1 500

312.5 MW

187.5 MW

26.2 $/MWh

G

G

G G

G

G

G

G

P

P

P P

P

P

P

P

5

Lambda-Iteration Solution Method

• The direct solution only works well if the incremental cost curves are linear and no generators are at their limits

• A more general method is known as the lambda-iteration– the method requires that there be a unique mapping

between a value of lambda and each generator’s MW output

– the method then starts with values of lambda below and above the optimal value, and then iteratively brackets the optimal value

6

Lambda-Iteration Algorithm

L H

m mL H

Gi Gii=1 i=1

H L

M H L

mM H M

Gii=1

L M

Pick and such that

P ( ) 0 P ( ) 0

While Do

( ) / 2

If P ( ) 0 Then

Else

End While

D D

D

P P

P

7

Lambda-Iteration: Graphical View

In the graph shown below for each value of lambda

there is a unique PGi for each generator. This

relationship is the PGi() function.

8

Lambda-Iteration Example

1 1 1

2 2 2

3 3 3

1 2 3

Gi

Consider a three generator system with

( ) 15 0.02 $/MWh

( ) 20 0.01 $/MWh

( ) 18 0.025 $/MWh

and with constraint 1000MW

Rewriting as a function of , P ( ), we have

G G

G G

G G

G G G

IC P P

IC P P

IC P P

P P P

G1 G2

G3

15 20P ( ) P ( )

0.02 0.0118

P ( )0.025

9

Lambda-Iteration Example, cont’d

m

Gii=1

m

Gii=1

1

H

1

Pick so P ( ) 1000 0 and

P ( ) 1000 0

Try 20 then (20) 1000

15 20 181000 670 MW

0.02 0.01 0.025

Try 30 then (30) 1000 1230 MW

L L

H

mL

Gii

m

Gii

P

P

10

Lambda-Iteration Example, cont’d

1

1

Pick convergence tolerance 0.05 $/MWh

Then iterate since 0.05

( ) / 2 25

Then since (25) 1000 280 we set 25

Since 25 20 0.05

(25 20) / 2 22.5

(22.5) 1000 195 we set 2

H L

M H L

mH

Gii

M

mL

Gii

P

P

2.5

11

Lambda-Iteration Example, cont’d

H

*

*Gi

G1

G2

G3

Continue iterating until 0.05

The solution value of , , is 23.53 $/MWh

Once is known we can calculate the P

23.53 15P (23.5) 426 MW

0.0223.53 20

P (23.5) 353 MW0.01

23.53 18P (23.5)

0.025

L

221 MW

12

Generator MW Limits

• Generators have limits on the minimum and maximum amount of power they can produce

• Often times the minimum limit is not zero. This represents a limit on the generator’s operation with the desired fuel type

• Because of varying system economics usually many generators in a system are operated at their maximum MW limits.

13

Lambda-Iteration with Gen Limits

Gi

Gi ,max Gi ,max

Gi ,min Gi ,min

In the lambda-iteration method the limits are taken

into account when calculating P ( ) :

if P ( ) then P ( )

if P ( ) then P ( )

Gi Gi

Gi Gi

P P

P P

14

Lambda-Iteration Gen Limit Example

G1 G2

G3

1 2 31

In the previous three generator example assume

the same cost characteristics but also with limits

0 P 300 MW 100 P 500 MW

200 P 600 MW

With limits we get

(20) 1000 (20) (20) (20) 100m

Gi G G Gi

P P P P

1

0

250 100 200 450 MW (compared to -670MW)

(30) 1000 300 500 480 1000 280 MWm

Gii

P

15

Lambda-Iteration Limit Example,cont’d

Again we continue iterating until the convergence

condition is satisfied. With limits the final solution

of , is 24.43 $/MWh (compared to 23.53 $/MWh

without limits). The presence of limits will alwa

G1

G2

G3

ys

cause to either increase or remain the same.

Final solution is

P (24.43) 300 MW

P (24.43) 443 MW

P (24.43) 257 MW

16

Back of Envelope Values

• Often times incremental costs can be approximated by a constant value:– $/MWhr = fuelcost * heatrate + variable O&M– Typical heatrate for a coal plant is 10, modern

combustion turbine is 10, combined cycle plant is 7 to 8, older combustion turbine 15.

– Fuel costs ($/MBtu) are quite variable, with current values around 1.5 for coal, 4 for natural gas, 0.5 for nuclear, probably 10 for fuel oil.

– Hydro, solar and wind costs tend to be quite low, but for this sources the fuel is free but limited

17

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

18

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

19

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!

20

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.

21

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

22

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

23

Thirty Bus ED Example

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

24

Thirty Bus ED Example, cont

• Because of the penalty factors the generator incremental costs are no longer identical.

25

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 the next MW of electricity, assuming area is economically dispatched

Supplycurve forthirty bussystem

26

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

27

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.

28

Electricity Markets

• Over last fifteen 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).

29

Electricity Futures Example

Source: Wall Street Journal Online, 10/21/2015 30

Historical Variation in Oct 2015 Price

Source: Wall Street Journal Online, 10/21/2015

Price has dropped, following the drop in natural gas prices

“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

32

Two Bus Economic Dispatch 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

33

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

34

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

35

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

36

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

37