Overview
•  Complex system time scale separation
•  Microeconomics: Supply & Demand
Least Cost System Operation:
Economic Dispatch 1
Smith College, EGR 325
September 22, 2014
–  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
2
Time Scale Separation
•  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
3
1.  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)
4
1
St
ea
m
Ga
se
s
To minimize total system generating costs
we must first develop equations to
represent the cost of generating power
Stack
Generator Cost
Characteristics
Boiler
Thermal Turbine Generator
Cooling
Tower
G
Condenser
Pump
Coal
feeder
Burner
Body of water
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Input/Output Curve
Generator Cost Curves
•  The I/O curve plots fuel input (in MBtu/hr)
versus net MW output.
•  Generator costs are determined by fuel
costs and generator efficiency
Input Output Curve
–  We typically fit a quadratic equation to
empirical data from the generator
4000
3500
•  These costs are represented by four
graphs defining unit performance
input/output (I/O) curve
fuel-cost curve
heat-rate curve
unit generating cost curve, and incremental
cost curve
Fuel Rate (mmBtu/hr)
1) 
2) 
3) 
4) 
3000
2500
2000
1500
1000
500
7
0
0
50
100
150
200
250
Pg (MW output)
300
350
400
8
2
Fuel-cost Curve
An Efficiency Curve
•  The fuel-cost curve is the I/O curve scaled
by fuel cost
•  Efficiency = Output vs. Input
•  Interpret this curve...
** Efficiency changes with output level **
Fuel Cost Curve
MWh/
mmBtu
6000
Fuel Cost ($/hr)
5000
4000
3000
Most Efficient
generation level
2000
1000
0
0
50
100
150
200
250
Pg (MW output)
300
350
400
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The Heat Rate Curve
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Heat Rates
•  Plots the average number of MBtu/MWhr of fuel
input per MW of output
•  What is a heat rate?
–  Is a large or a small value preferable?
–  What are the units for a heat rate?
–  The inverse of the standard efficiency (output/input)
•  Heat-rate curve is the I/O curve scaled by MW
* and is not constant *
mmBtu/
MWh
Pgen
•  Typical heat rate values
Level for most efficient
unit operation
Pgen
–  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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12
3
Generator Quadratic Cost Curve
•  … and the derivative of the cost curve,
which is the marginal, or incremental, cost
curve
Generator Cost Curve
•  Plots $/hr as a function of Pg output
–  What are the units of each point on the graph?
Generator Cost Curve
4
2
x 10
1.8
Ci (PGi ) = α i + βi PGi + γ P
MCi (PGi ) =
1.6
$/hr (fuel cost)
dCi (PGi )
= βi + 2γ i PGi $/MWh
dPGi
1.4
Cost ($/hr)
2
i Gi
1.2
1
0.8
0.6
0.4
0.2
0
0
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Marginal Cost Curve
–  What are the units of each point on the graph?
Marginal (Incremental) Cost Curve
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Marginal (Incremental) Cost ($/MWh)
100
150
200
250
Pg (MW output)
300
350
400
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Mathematical Formulation of Costs
•  Plots the $/MWh as a function of Pgen MW output
•  Typically curves can be approximated
using
–  quadratic or cubic functions
–  piecewise linear functions
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•  Building from the quadratic nature of HR,
we will use a quadratic cost equation
10
Ci ( PGi ) = α i + β i PGi + γ i PGi2
5
0
0
50
50
100
150
200
250
Pg (MW output)
300
350
$/hr
400
15
16
4
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
System Operations
Aug 25-31, 2000 California ISO Load
450
Demand (GW)
400
350
300
250
200
150
100
50
0
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What is “Economic Dispatch?”
•  Economic dispatch (ED) determines the
least cost dispatch of generation for a
system.
1
15
29
43
57 71 85 99 113 127 141 155
Hour of week
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Economic Dispatch Discussion
•  Formulating the objective
–  What are our goals in operating the power
system to serve our customers?
–  To dispatch ≡ To control generators to generate
(more or less) energy
–  To determine the MW output
•  Economic Dispatch (from EPACT 1992)
–  The operation of generation facilities to produce
energy at the lowest cost to reliably serve consumers,
recognizing any operational limits of generation and
transmission facilities.
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•  What does solving our (to be developed)
set of equations help us to decide?
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5
Economic Dispatch Formulation
Economic Dispatch Formulation
•  Formulating the objective
•  We need to understand
–  How do we represent our objective
mathematically?
–  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
–  What mathematical tool do we use to obtain
this objective?
–  Such as: total generation must equal total demand
–  Constrained optimization via linear
programming
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22
Example Supply Curve – Costs of
Different Generating Technologies
Categories for Generators
Demand (GW)
Diurnal Load Shape
450
400
350
300
250
200
150
100
50
0
Peak Load
Intermediate
Baseload
1
23
3
5
7
9
11 13 15
Hour of Day
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19 21
23
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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
•  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
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Formulating the
Linear Programming Problem
•  Objective function
•  For power systems:
–  Decision variables, what you need to decide
such as how much pizza to buy
minCT = ΣCi (PGi )
•  Constraints
s.t. Σ(PGi ) = Pdemand
–  Bounds (limits) on the variables;
PGi _min ≤ PGi ≤ PGi _max
•  Pizza parlor capacity
•  Standard form
•  Our “decision variables” are ______?
–  min f (x)
–  s.t. Ax = b
xmin <= x <= xmax
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Summary
Energy Conversions
•  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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•  For reference
- 
- 
- 
- 
1 Btu (British thermal unit) = 1054 J
1 MBtu = 1x106 Btu
1 MBtu = 0.29 MWh
Conversion factor of 0.2928MWh/MBtu
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