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Least Cost System Operation: Economic Dispatch 1

Smith College, EGR 325February 6, 2018

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Overview

• Complex system time scale separation• Microeconomics: Supply & Demand

– Market clearing price– Marginal cost

• Least cost system operation– Generator cost characteristics– Heat Rate (efficiency measure)

• Constrained Optimization

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Complex System Analysis• Divide the full system into sub-systems• In power systems we can analyze...

– By analysis question• Cost, policy objectives, environmental impacts...

– By system sector: generation, transmission, distribution, customer...

– By geographic region– By time scale

• Time scale separation of events

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Time Scale Separation1. Given the plants that are generating, decide

how to maintain the supply-demand balance cycle to cycle à power flow analysis

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

3. Given the plants that are built, decide which plants to start for use tomorrow, next week, next month… (not covering this semester)

4. Decide what to build (system planning)

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Generator Cost Characteristics

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To minimize total system generating costs we must first develop equations to represent the cost of generating power

Body of water

Stack

GGenerator

CoolingTower

Condenser

Pump

Boiler

Coal feeder

Burner

Thermal Turbine

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Generator Cost Curves

• Generator costs are determined by fuel costs and generator efficiency– We typically fit a quadratic equation to

empirical data from the generator• These costs are represented by four

graphs defining unit performance1) input/output (I/O) curve2) fuel-cost curve3) heat-rate curve4) unit generating cost curve, and incremental

cost curve

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Input/Output Curve• The I/O curve plots fuel input (in MBtu/hr)

versus net MW output.

0 50 100 150 200 250 300 350 4000

500

1000

1500

2000

2500

3000

3500

4000

Pg (MW output)

Fuel

Rat

e (m

mBt

u/hr

)

Input Output Curve

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Fuel-cost Curve• The fuel-cost curve is the I/O curve scaled

by fuel cost

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

Pg (MW output)

Fuel

Cos

t ($/

hr)

Fuel Cost Curve

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An Efficiency Curve• Efficiency = Output vs. Input• Interpret this curve...

** Efficiency changes with output level **

Most Efficient generation level

MWh/mmBtu

Pgen

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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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Heat Rates• What is a heat rate?

– Is a large or a small value preferable?– What are the units for a heat rate?

• Typical heat rate values – Coal plant is 10 mmBtu/MWh– Modern combustion turbine is 10 mmBtu/MWh– Combined cycle plant is 7 to 8 mmBtu/MWh– Older combustion turbine 15 mmBtu/MWh

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Generator Quadratic Cost Curve

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• … and the derivative of the cost curve, which is the marginal, or incremental, cost curve

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

MCi (PGi ) =dCi (PGi )dPGi

= βi + 2γ iPGi $/MWh

Generator Cost Curve

140 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?

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

– What are the units of each point on the graph?

0 50 100 150 200 250 300 350 4000

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Pg (MW output)

Mar

gina

l (In

crem

enta

l) C

ost (

$/M

Wh)

Marginal (Incremental) Cost Curve

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Mathematical Formulation of Costs

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

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

$/hr )( 2GiiGiiiGii PPPC gba ++=

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System Operations

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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Power System Economic Operation• The total capacity of generators operating is greater than

the load at any specific moment• This allows for much flexibility in deciding which generators

to use to meet the load at any moment

Aug 25-31, 2000 California ISO Load

050100150200250300350400450

1 15 29 43 57 71 85 99 113 127 141 155Hour of week

Dem

and

(GW

)

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Economic Dispatch Discussion

• Formulating the objective– What are our goals in operating the power

system to serve our customers?

• What does solving our (to be developed) set of equations help us to decide?

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Economic Dispatch Formulation

• Formulating the objective– How do we represent our objective

mathematically?

– What mathematical tool do we use to obtain this objective?

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Economic Dispatch Formulation

• We need to understand– How to represent system generating costs

mathematically• Costs of operating (dispatching) generators

– How to find the minimum system cost given• Generator costs and• System constraints

– Such as: total generation must equal total demand

– Constrained optimization via linear programming

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Example Supply Curve – Costs of Different Generating Technologies

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Diurnal Load Shape

050100150200250300350400450

1 3 5 7 9 11 13 15 17 19 21 23Hour of Day

Dem

and

(GW

)

Categories for Generators

Intermediate

Baseload

Peak Load

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Constrained Optimization

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Linear Programming Definition• Optimization is used to find the “best” value

– “Best” defined by us, the analysts and designers

• Constrained optimization– Minimize/maximize an objective, subject to

certain constraints• Linear programming

– Linear constraints– Some binding, some non-binding

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Formulating the Linear Programming Problem

• Objective function– Decision variables, what you need to decide

such as how much pizza to buy• Constraints

– Bounds (limits) on the variables; • Pizza parlor capacity

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

xmin <= x <= xmax

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Formulating the Linear Programming Problem

• For power systems:

• Our “decision variables” are ______?

minCT = ΣCi (PGi )s.t. Σ(PGi ) = Pdemand PGi_min ≤ PGi ≤ PGi_max

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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

Then L = ?

Formulate the ED Problem Using the Lagrangian

L = CT -λ ΣPGi - PL( )

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Summary• Introduce ‘time scale separation’• Examine the mathematical origin for

generator costs– Define heat rate

• Formulate the economic dispatch problem conceptually

• Develop mathematical formulation of the economic dispatch problem

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Energy Conversions• For reference

- 1 Btu (British thermal unit) = 1054 J- 1 MBtu = 1x106 Btu- 1 MBtu = 0.29 MWh- Conversion factor of 0.2928MWh/MBtu