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Least Cost System Operation: Economic Dispatch 2

Smith College, EGR 325March 10, 2006

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Overview

• Complex system time scale separation

• Least cost system operation– Economic dispatch first view– Generator cost characteristics

• System-level cost characterization

• Constrained optimization – Linear programming– Economic dispatch completed

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Time Scale Separation1. Decide what to build

2. Given the plants that are built decide which plants to have warmed up and ready to go this month, week...

3. Given the plants that are ready to generate decide which plants to use to meet the expected load today, the next 5 minutes, next hour...

4. Given the plants that are generating Decide how to maintain the supply and demand balance cycle to cycle

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Economic Dispatch Recap

• Economic dispatch determines the best way to minimize the current generator operating costs

• 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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Constrained Optimization& Economic Dispatch

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Mathematical Formulation of Costs

• Generator cost curves are not actually smooth• Typically curves can be approximated using

– quadratic or cubic functions– piecewise linear functions

2( ) $/hr (fuel-cost)

( )( ) 2 $/MWh

i Gi i Gi Gi

i Gii Gi Gi

Gi

C P P P

dC PIC P P

dP

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Mathematical Formulation of Costs

• The marginal cost is one of the most important quantities in operating a power system

• Marginal cost = incremental cost: the cost of producing the next increment (the next MWh)

• How do we find the marginal cost?

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Economic Dispatch

• An economic dispatch results in all the generator generating at a level where they have equal marginal costs (for a lossless system)

IC1(PG,1) = IC2(PG,2) = … = ICm(PG,m)

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Incremental Cost Example

21 1 1 1

22 2 2 2

1 11 1 1

1

2 22 2 2

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For a two generator system assume

( ) 1000 20 0.01 $ /

( ) 400 15 0.03 $ /

Then

( )( ) 20 0.02 $/MWh

( )( ) 15 0.06 $/MWh

G G G

G G G

GG G

G

GG G

G

C P P P hr

C P P P hr

dC PIC P P

dP

dC PIC P P

dP

10

G1 G2

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22

1

2

If P 250 MW and P 150 MW Then

(250) 1000 20 250 0.01 250 $ 6625/hr

(150) 400 15 150 0.03 150 $6025/hr

Then

(250) 20 0.02 250 $ 25/MWh

(150) 15 0.06 150 $ 24/MWh

C

C

IC

IC

Incremental Cost Example

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Economic Dispatch: Formulation

• The goal of economic dispatch is to – determine the generation dispatch that

minimizes the instantaneous operating cost– subject to the constraint that total generation

= total load + losses

T1

m

i=1

Minimize C ( )

Such that

m

i Gii

Gi D Losses

C P

P P P

Initially we'll ignore generatorlimits and thelosses

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Unconstrained Minimization• This is a minimization problem with a

single inequality constraint

• For an unconstrained minimization a necessary (but not sufficient) condition for a minimum is the gradient of the function must be zero,

• The gradient generalizes the first derivative for multi-variable problems:

1 2

( ) ( ) ( )( ) , , ,

nx x x

f x f x f xf x

( ) f x 0

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Minimization with Equality Constraint

• When the minimization is constrained with an equality constraint we can solve the problem using the method of Lagrange Multipliers

• Key idea is to modify a constrained minimization problem to be an unconstrained problem

That is, for the general problem

minimize ( ) s.t. ( )

We define the Lagrangian L( , ) ( ) ( )

Then a necessary condition for a minimum is the

L ( , ) 0 and L ( , ) 0

T

x λ

f x g x 0

x λ f x λ g x

x λ x λ

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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

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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

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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 0 500

312.5 MW

187.5 MW

26.2 $/MW

G

G

G G

G

G

G

G

P

P

P P

P

P

P

P

h

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Constrained Optimization & Linear Programming

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Linear Programming Definition

• Optimization is used to find the “best” value– “Best” defined by us, the analysts and

designers

• Constrained opt Linear programming– Linear constraints– Complicates the problem

• Some binding, some non-binding

• Visualize via a ‘feasible region’

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Formulating the Problem

• Objective function

• Constraints

• Decision variables

• Variable bounds

• Standard form– min cx– s.t. Ax = b

xmin <= x <= xmax

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Formulating the Problem

• For power systems:

min CT = ΣCi(PGi)

s.t. Σ(PGi) = PL

PGi min <= PGi <= PGi max

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Constrained Optimization& Economic DispatchThe Lagrangean

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Formulating the Lagrangean

• Rewrite the constrained optimization problem as an unconstrained optimization problem !– Then we can use the simple derivative

(unconstrained optimization) to solve

• The task is to interpret the results correctly

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• We are minimizing gradients of both multivariate equations– CT & ΣPGi = PL

• For both equations to be at a minimum these gradients must be linearly dependent vectors

CT – λw = 0• with w ≡ ΣPG – PL = 0

• The “Lagrangean multiplier”– λ is defined to be the scaling variable that

brings CT and w into linear alignment

Formulating the Lagrangean

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max g(x) = 5x12x2

s.t. h(x) = x1 + x2 = 6 or x1 + x2 – 6 = 0

Formulate L =

L = g(x) – λh(x)

Find ?

dL/dx1, dL/dx2, dL/dλ

x1 = 4, x2 = 2, λ = 80

Lagrangean Example

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min CT = ΣCi(PGi)

s.t. Σ(PGi) = PL

PGi min <= PGi <= PGi max

Then L = ?

Economic Dispatch & the Lagrangean

LGiT PPCL

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Economic Dispatch Example

• What is the economic dispatch for the two generator problem withPG1 + PG2 = PD = 500MW

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Economic Dispatch Example

• Formulate the Lagrangean

• Take derivatives

• Solve

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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 0 500

312.5 MW

187.5 MW

26.2 $/MW

G

G

G G

G

G

G

G

P

P

P P

P

P

P

P

h

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Economic Dispatch: Formulation

• We find that – PG1 = 312.5MW;

– PG2 = 187.5MW

= $26.2/MWh

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Discussion

• Key results for Economic Dispatch?– Incremental cost of all generating units is

equal– This incremental cost is the Lagrangean

multiplier, – ‘’ is called the ‘System ’ and is the system-

wide cost of generating electricity• This is the price charged to customers

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Power System Control Center

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Power System Control Center

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New England Power Grid Operator

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Regional Prices and Constraints

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The Hong Kong Trade Development Council

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Summary

• Economic dispatch is used to determine the least cost means of using existing generating plants to meet electric demand

• To calculate the economic dispatch for a power system, the techniques of linear programming + the Lagrangean are used

• Now to a review of the production cost homework results...