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Least Cost System Operation 3: Interpreting Economic Dispatch

Smith College, EGR 325 February 8, 2018

Puerto Rico Project Options

•  Final deliverable options A.  Long report B.  Poster for Collaborations and a short

handout to go with the poster §  Collaborations applications due February 28

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Overview

•  Least cost system operation – Economic dispatch

•  Constrained optimization – Linear programming – Formulating and solving the Lagrangian

•  Interpretation of economic dispatch results –  Interpreting marginal cost and ‘System λ’

Regional Electricity Prices

•  ISOne (New England) – http://www.iso-ne.com/ – http://www.iso-ne.com/isoexpress/

•  PJM (Pennsylvania-New Jersey-Maryland) – http://www.pjm.com/markets-and-operations/

interregional-map.aspx

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The Heat Rate Curve •  Plots the average number of MBtu/MWhr of fuel

input per MW of output –  The inverse of the standard efficiency (output/input)

•  Heat-rate curve is the I/O curve scaled by MW * and is not constant *

Level for most efficient unit operation

mmBtu/MWh

Pgen

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Mathematical Formulation of Costs

•  Cost curves can be approximated using – quadratic or cubic functions – piecewise linear functions

•  Building from the quadratic nature of HR, we use a quadratic cost equation

$/hr )( 2GiiGiiiGii PPPC γβα ++=

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Generator Cost Curve

7 0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 104

Pg (MW output)

Cos

t ($/

hr)

Generator Cost Curve

•  Plots $/hr as a function of Pg output – What are the units of each point on the graph?

Generator Quadratic Cost Curve

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•  The cost curve and the derivative of the cost curve are both important

•  The derivative of the cost curve is the marginal, or incremental, cost curve

•  Notice the units of each curve

Ci (PGi ) =αi +βiPGi +γ iPGi2 $/hr (fuel cost)

MCi (PGi ) =dCi (PGi )dPGi

= βi + 2γ iPGi $/MWh

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Marginal Cost Curve •  Plots the $/MWh as a function of Pgen MW output

0 50 100 150 200 250 300 350 4000

5

10

15

20

Pg (MW output)

Mar

gina

l (In

crem

enta

l) C

ost (

$/M

Wh)

Marginal (Incremental) Cost Curve

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Significance of Marginal Costs

•  From total cost to marginal cost...

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

of power (the next MW) –  (Marginal cost = incremental cost)

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Economic Dispatch Definition

•  Deciding which generators to ‘dispatch’ at what MW output level

•  The objective is to serve electrical load at minimum cost

•  So … sum the cost curves of all the generators, and minimize this total cost – Take the derivative with respect to __what__?

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Economic Dispatch Equations

•  Constrained optimization – Minimize an objective function, subject to given

constraints •  Linear programming implies…

– Linear constraints – Some binding, some non-binding

•  Economic dispatch is a constrained optimization, linear programming, problem

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Formulating the Linear Programming Problem

•  Objective function – A function of your decision variables

•  Constraints – Bounds (limits) on the decision variables

•  Standard form – min f (x) – s.t. Ax = b

xmin <= x <= xmax

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Formulating the Linear Programming Problem

•  For power systems: min CT = ΣCi(PGi)

s.t. Σ(PGi) = PL PGi min <= PGi <= PGi max

•  Our “decision variables” are ______?

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To Solve: Formulate the “Lagrangian”

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

(unconstrained optimization) to solve – Need to introduce a new variable – the

“Lagrange multiplier” lambda, λ •  The task is to interpret the results correctly

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min CT = ΣCi(PGi)

s.t. Σ(PGi) = PL PGi min <= PGi <= PGi max

Introduce a new variable, λ, and write our problem as:

Formulate the ED Problem Using the Lagrangian

L = CT -λ ΣPGi - PL( )Ci (PGi ) =αi +βiPGi +γ iPGi

2 $/hr

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Economic Dispatch: Formulation

•  What is the economic dispatch for a two generator problem with: (The cost curve units are $/hr)

€

C1(PG1) =1000 + 20PG1 + 0.01PG12

C2(PG2) = 400 +15PG2 + 0.03PG22

PG1 + PG2 = PL = 500MW

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•  Formulate the Lagrangian •  Take derivatives •  Solve

∂L(PG2)∂PG2

=15+0.06PG2 .λ=0

€

500 −PG1 −PG2 = 0

Economic Dispatch: Formulation

∂L(PG1)∂PG1

=20+0.02PG1 ,λ=0

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•  Solve these three linear equations

€

20+ 0.02PG1 −λ = 015+ 0.06PG2 −λ = 0500 −PG1 −PG2 = 0

€

0.02 0 −10 0.06 −1−1 −1 0

#

$

% % %

&

'

( ( (

PG1PG2λ

#

$

% % %

&

'

( ( (

=

−20−15−500

#

$

% % %

&

'

( ( (

€

PG1PG2λ

#

$

% % %

&

'

( ( (

=

312.5MW187.5MW$26.25 /MWh

#

$

% % %

&

'

( ( (

Economic Dispatch: Solution

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Economic Dispatch: Solution

•  For our problem, we find that Ø PG1 = 312.5MW Ø PG2 = 187.5MW

Ø  λ = $26.25/MWh

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Marginal Cost Discussion •  With PL unchanged, find the following:

–  MC equation for each generator –  MC value at operating point (312.5MW, 187.5MW) –  What is the total cost of serving this load?

•  With a different dispatch for each generator –  What is the total cost if PG1 generates 325MW? –  What is the MC of each generator at these new

dispatch points? (325MW, 175MW)

€

C1(PG1) =1000+ 20PG1 + 0.01PG12

C2 (PG2 ) = 400+15PG2 + 0.03PG22

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Marginal Cost Discussion •  Cost values from previous slide: •  MC value at operating point? (312.5MW, 187.5MW)

–  $26.25/MWh

•  What is the total cost of serving this load? –  $12,493.75

•  What is the total cost if PG1 generates 325MW? –  $14,750.00

•  What is the MC of each generator at these new dispatch points? (325MW, 175MW) –  $26.5/MWh and $25.5/MWh –  So, decrease PG1 and increase PG2 in order to be at least cost

dispatch.

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Economic Dispatch: Interpretation •  Question: So what happens (to the value of

the objective function) if we dispatch the generators at output levels different from the constrained optimization solution?

CT = (1000+20PG1+0.01PG12 )+(400+15PG2 +0.03PG2

2 )

∂L(PG2)∂PG2

=15+0.06PG2 .λ=0

∂L(PG1)∂PG1

=20+0.02PG1 ,λ=0

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Super-Important Results •  The traditional objective in operating a power system is to

_________________? •  The operating point found is the ____<in terms of cost>____

operating point. •  Operating anywhere else, thus ________________ •  At the least cost operating point the incremental cost of all

generating units ______________ •  This incremental cost is the Lagrange multiplier, λ

–  For power systems, this is the system-wide cost of generating electricity

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Now Include Inequality Constraints

min CT = ΣCi(PGi)

s.t. Σ(PGi) = PL PGi min <= PGi <= PGi max

•  For Homework: Iterate •  The field of operations research includes

theory to solve as one unified problem.

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ED With Inequality Constraints

•  What is the economic dispatch for a two generator problem with

€

C1(PG1) =1000 + 20PG1 + 0.01PG12

C2(PG2) = 400 +15PG2 + 0.03PG22

PG1 + PG2 = PL = 500MWPG1 ≤ 300MW

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ED With Inequality Constraints

•  For our problem, we previously found that – PG1 = 312.5MW; – PG2 = 187.5MW – λ = $26.2/MWh

•  But now PG1 output must be <= 300 MW, so what should we do? – With only 2 generators we can set PG1 = 300

MW and observe that we then require PG2 = 200 MW

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ED With Inequality Constraints

•  Note, for the HW –  If we have three generators (or more), and

you find that one of them must be set equal to a limiting value,

– Then we have two generators remaining, and we cannot simply ‘observe’ the economic dispatch solution…

–  In this case, re-solve the Lagrangian problem with the remaining 2 generators, and the value PL adjusted as is appropriate

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ED With Inequality Constraints

•  Interpret our results – What is system-lambda now, and what does it

represent? – What inefficiencies have been introduced into

our solution as a result of the binding generator limit?

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Four Generator Example

•  A power system has four generators with the following cost characteristics – C1 = 1000 + 15P1 + 0.05P1

2 $/MWh – C2 = 1200 + 25P2 + 0.12P2

2 $/MWh – C3 = 2060 + 20P3 + 0.01P3

2 $/MWh – C4 = 2500 + 12P4 + 0.03P4

2 $/MWh

•  Typical demand, PL, levels are: à 750MW, 1000MW, 1500MW, and 2500MW

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Problem Formulation

•  Write the equations you would use to find the least cost dispatch for this system

•  How might you go about finding the least cost dispatch?

•  What form will your results take?

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Problem Formulation

•  Equations…

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Linear Algebra Solution Setup

Matrix A: Vector B: 0.1 0 0 0 -1 -15 0 0.24 0 0 -1 -25 0 0 0.02 0 -1 -20 0 0 0 0.06 -1 -12 1 1 1 1 0 2500 •  Solve in Matlab using x = (A)-1 ·(B) •  Where x = ?

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Matlab Results

•  For changing load levels:

(the units are ___?)

PL PG1 PG2 PG3 PG4 Lambda750 123 9 363 254 271000 154 22 518 306 301500 216 48 827 409 372000 277 74 1137 512 432500 339 100 1446 615 49

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Matlab Results à Interpret …

•  The solution is: Ø  with PL = 2500 MW Ø  PG1 = 339 MW Ø  PG2 = 100 MW Ø  PG3 = 1446 MW Ø  PG4 = 615 MW Ø  λ = $49/MWh

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Results Plotted Dispatch for Increasing Load

0

200

400

600

800

1000

1200

1400

1600

750 1000 1500 2000 2500

Load Level (MW)

Gen

erat

or O

utpu

t (M

W)

0

5

10

15

20

25

30

35

40

45

50

Syst

em

Lam

bda(

$/M

Wh)

PG1 PG2 PG3 PG4 Lambda

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Visualizing Equal Marginal Cost; Quadratic Cost Curves Graphed

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Visualizing Equal Marginal Cost

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Including Inequality Constraints

•  Include Pmin and Pmax values – Each generator must generate more than

50MW – P1 and P2 must generate less than 500 MW – P3 and P4 must generate less than 1400 MW

•  How would you formulate the problem with these inequality constraints?

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Discussion •  Key results for Economic Dispatch?

– The ‘marginal’ or ‘incremental’ cost of all generating units is equal

– This incremental cost is the Lagrangian multiplier, λ

–  ‘λ’ is called the ‘System λ’ and is the system-wide cost of generating electricity

•  This is the price charged to customers •  On ISO websites, this is called either ‘System λ’ or

LMP (locational marginal price)

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Economic Dispatch Ignores…

•  Economic dispatch determines the best way to minimize the generator operating costs –  It is not concerned with determining which

units to turn on/off (this is the unit commitment problem)

–  It ignores the transmission system limitations

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Summary

•  Rev iewthe mathematical origin for generator costs – Define heat rate

•  Formulate the economic dispatch problem conceptually

•  Develop mathematical formulation for solving the economic dispatch problem

•  Interpret the results, including the Lagrangian multiplier and marginal cost