03-Economic Dispatch 1 - drhenrylouie · 03-Economic Dispatch 1 ... heat-rate curve, the generator ... Operating the generator at the same incremental cost is the optimal solution

03-Economic Dispatch 1EE570

Energy Utilization & Conservation

Professor Henry Louie

1

2

Topics

Generator Curves

Economic Dispatch (ED) Formulation

ED (No Generator Limits, No Losses)

ED (No Losses)

ED Example

Dr. Henry Louie

Introduction

How should the real power output of a fleet of generators be determined?

What factors are important in making the decision?

4

Economic Dispatch

Generic problem: for an m generator system, what should P1, …, Pm be set to?

Several choices: minimize losses, minimize costs, etc.

We focus on cost minimization• known as Economic Dispatch (ED)

This is an optimization problem

Dr. Henry Louie

Note: we will use $ as a unit of cost, but the

results are generalizable to Kwacha

Optimization Problems

Minimize or maximize an objective function• Cost

• Profit

• CO2 emissions

Subject to constraints• Generator output limits

• Power line thermal or stability limits

• CO2 limits

• KCL, KVL, Ohm’s Law, conservation of energy, etc

Dr. Henry Louie5

6

Economic Dispatch

What factors influence the total cost of operation?• fuel

• labor

• maintenance

• etc

For simplicity, we will only analyze fuel costs

We also assume that we have access to the fuel-cost curves for each generator

Note: we are concerned with three-phase power, but the superscript will be suppressed

Dr. Henry Louie

7

Fuel-Cost & Heat-Rate Curves Fuel cost curve

Heat-rate curve• determined by field

testing

• heat energy supplied by burning fuel

• supplies efficiency information

• 34-39%

• 8.6 – 10 MBtu/MWh

PG (MW)

Ci(P

G)

($/h

r)

PG (MW)

Hi(P

G)

(MB

tu/M

Wh

)Dr. Henry Louie

Heat-Rate Curve Point x: 1200 MBtu to produce

100 MWh of electrical energy

Point y: 1600 MBtu to produce 200 MWh of electrical energy

At which point does the generator have the highest efficiency?

At the minimum point of the heat-rate curve, the generator is most efficient

Dr. Henry Louie8

PG (MW)

Hi(P

G)

(MB

tu/M

Wh

)

100

128

200

xy

Heat-Rate Curve Modern fossil-fuel

power plants: minimum heat-rate is approx. 8.6 - 10 MBtu/MWh

What efficiency does this correspond to?

Approx. 1,055 joules/Btu

• 39%

Dr. Henry Louie9

PG (MW)

Hi(P

G)

(MB

tu/M

Wh

)

100

128

200

xy

10

Input-Output Curve

Heat input energy rate:

• Hi : MBtu/MWh

• PGi: three-phase power, MW

• Fi: heat input energy rate, MBtu/hr

Plot of Fi(PGi) is the input-output curve

If the cost of fuel is K dollars/MBtu then:

• fuel cost to supply per hour to supply PGi MW of electricity

i Gi Gi i GiF(P ) P H(P )

i Gi i GiC(P ) KF(P )

Dr. Henry Louie

Example

Compute the heat input energy rate (Mbtu/hr) and cost ($/hour) of a generator producing 100 MW of power given:• Heat rate = 10 MBtu/MWh

• Fuel: $3/MBtu

Dr. Henry Louie11

Example

Compute the heat input energy rate (Mbtu/hr) and cost ($/hour) of a generator producing 100 MW of power given:• Heat rate = 10 MBtu/MWh

• Fuel: $3/MBtu

F = 100 x 10 = 1,000 MBtu/hr

C = 3 x 1,000 = $3,000/hr

Dr. Henry Louie12

i Gi Gi i GiF(P ) P H(P )

i Gi i GiC(P ) KF(P )

0 20 40 60 80 1000

1000

2000

3000

4000

5000

6000

7000

power (MW)

cost

($ /

hr)

13

Fuel Cost CurvesFuel cost curves can be approximated by

2

G G GC P P P dollars/hr

Dr. Henry Louie

Nearly linear

(quadratic form)

14

Example

Assume a 50-MW gas-fired generator has the following properties:• 25 % of rating: 14.26 MBtu/MWh

• 40 % of rating: 12.94 MBtu/MWh


Assume cost of gas is $5/MBtu

Find C(PG) in the form of

Dr. Henry Louie

2

G G GC P P P

Example

We are given the heat rate, H, at three points




We need to relate C(PG) and H(PG)

Dr. Henry Louie15

2

G G G G G GC(P ) KF(P ) KP H(P ) P P

G G

G

G G

G

P KH(P )P

H(P ) PKP K K

K: dollars/MBtuH: Mbtu/MWhF: MBtu/hrP: MW

Example

Three unknowns, and three points. we can solve• 12.5 MW: 14.26 MBtu/MWh

• 20 MW: 12.94 MBtu/MWh

• 50 MW: 11.70 MBtu/MWh

Dr. Henry Louie16

G G

G

H P PKP K K

2.5 14.2662.5 5

4 12.94100 5

10 11.70250 5

3 equations,3 unknowns

Example

Solution:

Find the fuel cost when fully loaded (50MW)

Dr. Henry Louie17

224.5

53.1

0.0180

2

G G G

2

G G G

C P P P

C P 224.5 53.1P 0.018P

2C 50 224.5 53.1 50 0.018 50

$2924.5 / hr for 50 MW

5.85 cents/hr for 1 kW

Example

Find the fuel cost at 40% loaded, 25% loaded

Dr. Henry Louie18

224.5

53.1

0.0180

Example

Find the fuel cost at 40% loaded, 25% loaded

Dr. Henry Louie19

2

2

C 20 224.5 53.1 20 0.018 20

$1293.7 / hr for 20 MW

6.47 cents/kWh

C 12.5 224.5 53.1 12.5 0.018 12.5

$891.1 / hr for 12.5 MW

7.13 cents/kWh

224.5

53.1

0.0180

20

General Problem Formulation

Now that we have discussed how costs are computed we can formulate the general economic dispatch problem

We want to minimize the total cost while observing all the power flow equations, constraints on generators, line flow and voltage magnitude by selecting generator real power and voltage magnitudes

Dr. Henry Louie

General Problem Formulation Minimize total cost CT ($/hr)

m generators committed (on-line)

n buses

SDi given

Dr. Henry Louie21

m

T i Gii 1

min C C (P )P

min max

Gi Gi Gi

min max

i i i

max

ij Gi

P P P i 1,2, ,m

i 1,2, ,m, ,n

P P

V V V

all lines

power flow equations satisfied

subject to:

Optimization Problem

General Problem Formulation

See text page 404 for further comments (to be provided)

Economic dispatch problem is a nonlinear optimization problem

Nonlinear programming is beyond the scope of this class

We will make some problem alterations

Dr. Henry Louie22

Assumptions

Losses ignored

Weak dependence on voltage magnitude and reactive power demand

Therefore, we expect the voltage magnitudes and reactive power demand to have little effect on the flow of real power

We then make the approximation that all voltages are equal to 1.0 p.u. and formulate the problem entirely based on real power generation and flows

Dr. Henry Louie23

24

Classical Economic Dispatch

Tmin C ( )P

P

m n

Gi D Dii 1 i 1

P P P

min max

Gi Gi GiP P P i 1,2, ,m

subject to

conservation of power

if we ignore generator limits the problem becomes very simple

Dr. Henry Louie

Simplified ED Problem

Classical Economic Dispatch

Incremental costs (IC)

Derivative of fuel cost curve

What are the units?• dollars/MWh

Linear, monotonically increasing

Dr. Henry Louie25

i Gi

i

Gi

i i i Gi

dC PIC

dP

IC 2 P

Optimal Dispatch Rule Consider two generators

operating at different incremental costs to serve a 150 MW load

If we decrease PG1 by 10 MW, we decrease the cost by a significant amount (large slope)

PG2 increases by 10, but the cost only mildly increases (small slope)

Dr. Henry Louie26

PG1 (MW)

C1

(PG

1)

PG2 (MW)

C2

(PG

2)

100

50

slope = IC1

slope = IC2

27

Optimal Dispatch Rule

Using the problem as formulated on the previous slide, if we have no losses and no generator limits then:

Operating the generator at the same incremental cost is the optimal solution

• note 1: this rule only gives an optimality condition, not a solution (but the problem has been greatly reduced)

• note 2: it is implicitly assumed that there are no constraints preventing the generators from operating at the same incremental cost

Dr. Henry Louie

28

Example

Consider two generators with cost curves:

Let the total demand be 700 MW

Find PG1, PG2 that serves the load and minimizes the cost

2

1 G1 G1 G1

2

2 G2 G2 G2

C (P ) 900 45P 0.01P

C (P ) 2500 43P 0.003P

Dr. Henry Louie

29

Example

Find the incremental cost curves:

2

1 G1 G1 G1

2

2 G2 G2 G2

C P 900 45P 0.01P

C P 2500 43P 0.003P

Dr. Henry Louie

30

Example

Find the incremental cost curves:

add the power balance constraint

2

1 G1 G1 G1

2

2 G2 G2 G2

C P 900 45P 0.01P

C P 2500 43P 0.003P

11 G1

G1

22 G2

G2

dCIC 45 0.02P

dP

dCIC 43 0.006P

dP

G1 G2

P P 700

Dr. Henry Louie

31

Example

We have three equations, and three unknowns; solve

11 G1

G1

22 G2

G2

G1 G2

dCIC 45 0.02P

dP

dCIC 43 0.006P

dP

P P 700

G1 G2

G1 G2

G1

G2

45 0.02P 43 0.006P

P 700 P

P 84.6MW

P 615.4MW

Dr. Henry Louie

32

Example

Incremental costs are

Total costs are:

Average cost is: $34,877/700 MW = $49.82/MWh

11

G1

22

G2

dCIC 45 0.02 84.6 $46.69 / MWh

dP

dCIC 43 0.006 615.4 $46.69 / MWh

dP

2 2

TC 900 45 84.6 0.01 84.6 2500 43 615.4 0.003 615.4

$34,877

Dr. Henry Louie

33

Example

Another way to view it

PG (MW)

Incr

emen

tal C

ost

s

200

IC1

400 600

43

45

IC2

Dr. Henry Louie

34

Optimal Dispatch Rule Proof

Using the simplified formulation we need to find the values of PGi that minimize

We have m-1 independent variables

The problem is now unconstrained

T 1 G1 2 G2 m Gm

C C (P ) C (P ) C (P )

G1 G2 Gm D

Gm D G1 G2 Gm 1

P P P P

P P P P P

T 1 G1 2 G2 m D G1 G2 Gm 1C C P C P C P P P P

Dr. Henry Louie

35


Remember how to solve an unconstrained minimization problem?

We set the partial derivative wrt all the other variables equal to zero

T 1 G1 2 G2 m D G1 G2 Gm 1C C (P ) C (P ) C (P P P P )

GmT i m

Gi Gi Gm Gi

i m

Gi Gm

PC dC dC

P P dP P

dC dC0

P dPfor i = 1, 2, … , m-1

Dr. Henry Louie

36


Note that we have assumed convexity

Assuming that the cost curves are monotonically increasing, we expect only one solution to this problem

Therefore, equal ICs is a necessary and sufficient condition to determine for optimality

Dr. Henry Louie

37

Solution Via Lagrange Multipliers

It is often easier to use Lagrange Multipliers to minimize or maximize a constrained function

See Appendix 3 for more details

We rewrite the cost function with an augmented cost function

l is the Lagrange Multiplier

l

m

T T Gi Di 1

C C P P

Dr. Henry Louie

38


Optimal point: stationary points of CT wrt l and all the PGi

variables

note: this point satisfies the constraintl

T

Gi

T

C0 i 1,2, ,m

P

C0

l

m

T T Gi Di 1

C C P P

Dr. Henry Louie

39


applied to our problem

l is the system incremental cost

l

T

Gi

m

Gi Di 1

dC i 1,2, ,m

dP

P P 0

l

m

T T G Gi Di 1

C C P PP

Dr. Henry Louie

40


Incr

emen

tal C

ost

s

IC1 IC2

l

PG1 PG2 PG3

IC3

Dr. Henry Louie

41


If the cost curves are quadratic then the incremental cost curves are linear and the problem can be readily solved

If the cost curves are nonlinear, then an iterative process is used

1) pick an initial value of l

2) find the corresponding PG1(l), PG2(l), PG3(l)

3) if the generation is less than load, then increase l and repeat step 2. if the generation is greater than the load, decrease l and repeat step 2. If generation and load balance then stop.

Dr. Henry Louie

42

Generator Limits Included

Formulation so far as ignored generator constraints, we now will add them

Tmin CP

P

m n

Gi D Dii 1 i 1

min max

Gi Gi Gi

P P P

P P P i 1,2, ,m

subject to

conservation of power

Dr. Henry Louie

43


Now we add limits to our generator output

At the current operating point, there is no problem

Incr

emen

tal C

ost

s IC1 IC2

l1

PG1 PG2 PG3

IC3

Dr. Henry Louie


What if the system load increases?

Dr. Henry Louie44

Incr

emen

tal C

ost

s IC1 IC2

PG1 PG2 PG3

IC3

l1

l2

45

Generator Limits Included What if the system load increases further?

Operate Gen 3 at PG3, gen 1 and gen 2 should have equal l

Incr

emen

tal C

ost

s IC1 IC2

l1

PG1 PG2 PG3

IC3l2

l3

Dr. Henry Louie

46

Optimal Dispatch Rule (No losses)

Operate all generators that are not at their limits at equal l

Procedure: pick a l such that all generators operate at the same incremental costs and within their constraints

If the generation is not equal to the load at this l, then adjust las in the unconstrained case

Repeat this process until the load is met, or a generator reaches is limit

If a generator reaches its limit, then fix its output to the limit and continue to adjust the other values

Dr. Henry Louie

47

Optimal Dispatch Rule (No losses)

Note 1: it usually helpful to draw graphs of the incremental curves

Note 2: this rule applies to committed generators only (already on-line)

Note 3: this rule can be interpreted as trying to operate the system as closely as possible to equal incremental costs

Dr. Henry Louie

48

Example

Assume there are two generators with cost functions:

Let the generator limits be:

if PD = 700, find the optimal dispatch

2

1 G1 G1 G1

2

2 G2 G2 G2

C P 900 45P 0.01P

C P 2500 43P 0.003P

G1

G2

50MW P 200MW

50MW P 600MW

Dr. Henry Louie

49

Example

Start with an initial l guess• l = 45.5

Is this feasible? • No (PG1 <50MW)

Is the demand met?• No, so increase l• We don’t fix PG1 at its minimum since we need to increase l

1 G1 G1

2 G2 G2

IC 45.5 45 0.02P P 25

IC 45.5 43 0.006P P 416.7

2

1 G1 G1 G1

2

2 G2 G2 G2

C P 900 45P 0.01P

C P 2500 43P 0.003P

Dr. Henry Louie

50

Example

Increase l to 46.69

Is the demand met?• yes

Are the generators within their limits? • No (PG2 > 600)

• fix PG2 at 600, PG1 = 700 – 600 = 100

1 G1 G1

2 G2 G2

IC 46.69 45 0.02P P 84.6

IC 46.69 43 0.006P P 615.4

Dr. Henry Louie

51

Example

PG (MW)

Incr

emen

tal C

ost

s

200

IC1

400 600

46.69IC2

Dr. Henry Louie

52

Line Losses Considered

Line losses can reasonably be neglected if all the generators are located geographically close to one another

However, this may not be the case. we then need to account for transmission loses

Generators closer to the loads will tend to have lower losses than those far away

Dr. Henry Louie

53


Let PL be the total line losses

We then have

If we assume that each PDi is fixed, then:

• assuming bus 1 is the slack bus

m m

L Gi Dii 1 i 1

P P P

L L G L G2 Gm

P P (P ) P P , ,P

Dr. Henry Louie

54


G

T Gmin CP

P

m n

Gi L G2 Gm Dii 1 i 1

min max

Gi Gi Gi

P P P , ,P P 0

P P P i 1,2, ,m

subject to

Dr. Henry Louie

55


Once again, we will use Lagrange multipliers (for now, ignore generator limits)

The stationary points are:

l

m m n

T i Gi Gi L G2 Gm Dii 1 i 1 i 1

C C P P P P , ,P P 0

l

l

l

m

TGi L D

i 1

T 1

G1 G1

T i L

Gi Gi Gi

dCP P P 0

d

dC dC0 (slack bus)

dP dP

dC dC P1 0 i 2,...,m

dP dP P

Dr. Henry Louie

56


rewrite

as

now define:

T i L

Gi Gi Gi

dC dC P1 0 i 2,...,m

dP dP Pl

i

L Gi

Gi

dC1 i 2,...,m

P dP1

P

l

1

iL

Gi

L 1

1L i 2,...,m

P1

P

Dr. Henry Louie

57


Li is called the penalty factor of generator i

We can rewrite the necessary conditions for the optimal solution as

Recall that the incremental costs are:

We can now formulate the optimal dispatch rule with line losses considered and generator limits ignored

1 2 m1 2 m

G1 G2 Gm

dC dC dCL L L

dP dP dPl

i

Gi

dC

dP

Dr. Henry Louie

58

Optimal Dispatch Rule (Line Losses Considered, No Generator Limit)

Operate all generators so that Li x ICi = l for every generator

Note: we no longer need to operate each generator at the same incremental cost

Dr. Henry Louie

59

Optimal Dispatch Rule (Line Losses Considered)

Operate all generators not at their limits so that Li x ICi = l

Dr. Henry Louie

Documents

03-Economic Dispatch 1 - drhenrylouie · 03-Economic Dispatch 1 ... heat-rate curve, the generator ... Operating the generator at the same incremental cost is the optimal solution