Production Cost Models

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PRODUCTION COST

MODELS

Production cost models are computational models designed to calculate information

for long-range system planning:

generation system production costs

energy import requirements

availability of energy for sale to other systems

fuel consumption

employs models of expected load patterns and simulated

operation of the systems generation uncertainty of load forecasts

reliability of generating units

expected need for emergency energy and capacity supplies

uses statistical computational methods for solving problems

Stochastic production cost models

used for long range studies

the risk of sudden, random, generating unit failures andrandom deviations for the mean forecasted load aretreated as probability distributions

load modeling considers the behaviors of the expectedload patterns that cover periods of weeks, months, and/oryears the load duration cover expresses the time period that the

loading is expected to equal or exceed a given power value

generating unit modeling includes fuel costs usuallyexpressed over a monthly basis generating unit scheduled maintenance outages may involve time

periods from days to years

Introduction

Types of production cost studies


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Load Duration Curves Representation of future loads in which the impact

of capacity limitations will be studied

Building the load-duration curve

consider the expected load pattern

Load Duration Curves build a histogram of

load for a given timeperiod and find theload density function,p(x)

integrate the loaddensity function toobtain the loaddistribution function,

Load Duration Curves

Building the loadduration curve

multiply the probabilityby the period length to

show the number of hoursthat load equals orexceeds a given powerlevel, L

common convention hasthe load on the verticalaxis

Block Loading

simulates the economic dispatch

procedure with this type of load model

generating units are ordered by cost

units are assumed to be fully loaded or

loaded up to the limitation of the load-

duration curve


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Example: Niagara-Mohawk System

Internal Peak Load is 1700 MW

Example 8A

consider a two generating unit system

that will serve the following expected

load pattern:

construct a load-duration curve in tabular

and graphic form

load-duration curve as a form of probability

distribution.

The load density and distribution functions, p(x) and

Pn(x), respectively, are probabilities. Thus, p(20) = 0, p(40) = 20/100 = 0.2, p(60) = 0,

Pn(20) = Pn(40) = 1.0, and Pn(60) = 0.8, and so

forth.

the two generating units have the following characteristics

the fuel cost rate for each unit is a linear function of the

power output i.e.

F(P) = fuel cost at zero output + incremental cost rate x P.

block-load the two units onto the load-duration curve

unit #1 is used first because of its lower average cost per

MWh


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unit #1 is on-line for 100 h

80 MW output for 80 h


unit #2 is on-line for 20 h


Summary:

the production costs for unit 1

= hours on line x no load fuel cost rate + energy generated x

incremental fuel cost rate

= 100 h x 160 $/h + (6400 + 800) MWh x 8 $/MWh

= 16,000 $ + 57,600 $ = 73,600

Forced Outages

the time that the unit is not available due to a failure ofsome sort

represents a random event

taken out of the total time that the unit should be available

for service the forced outage rate is the ratio of forced outage

time over the total time available

schedule outage times for maintenance are excluded inboth the total time available and the forced outage time

Forced outage rates for all generating units must beaccounted for in the expected production costs

Example 8b

reconsider the previous example, but now including the effects of forced

outages

evaluate by load levels

Load = 40 MW; duration 20 h

Unit 1: on-line for 20 h, operates for 0.95 20 = 19 houtput: 40 MW, energy delivered: 19 40 = 760 MWh

Unit 2: on-line for 1 h, operates for 0.90 1 = 0.9 h

output: 40 MW, energy delivered: 0.9 40 = 36 MWh

load energy = 800 MWh

generation = 796 MWh

unserved energy = 4 MWh

shortage = 40 MW for 0.1 h


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Comments

it was necessary to make an arbitrary assumption that thesecond unit will be on-line for any load level that equals orexceeds the capacity of the first unit

the enumeration of the possible states is not complete

need to separate the periods when there is excess capability,exact matching of generation and load, and shortages

when there is an exact matching of generation and load, it isreferred to as a zero-MW shortage

there are two such periods in the example

40 MW loading, 20 h duration, unit 2 on: 0.05 0.9 20 = 0.9 h

80 MW loading, 60 h duration, unit 1 on: 0.95 0.1 60 = 5.7 h

total zero-reserve expected duration: 6.6 h

Summary of all possible states


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Summary of generation cost results unserved load

Probabilistic Production Cost

Operating experience indicates that the force outage rateof thermal-generating units tends to increase with unit size the frequent long-duration outages of the more efficient base-

load units require running the less efficient, more expensive plantsat higher than expected output and the importation of

emergency energy Two measures of system unreliability due to random

forced generator failures: time period when the load is greater than available generation

capacity

the expected levels of power and energy that must be importedto satisfy the load demand

these are sensitivity indicators of the need to add generatoror tie-line capacity

Probabilistic Production Cost Computations

mathematical methods that make use of probabilistic models load models

energy and capacity models for generators generator models represent the unavailability of basic energy resources,

random forced outages, and the effects of energy sale/purchase contracts

tie-line models represent the expected cost of emergency energy, also

called the cost of unsupplied energy

techniques for the convolution of the load distributions with thecapacity-probability density functions numerical convolution of discrete functions

Unserved load distribution method

Expected cost method

analytical convolution of continuous functions Cumulants method


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The Unserved Load Distribution Method

The area under the load distributions represent needed

energy

The individual unit probability capacity density functions areconvolved one by one with the load distribution function

each convolution result is an unserved load distribution, which will be

used for the next convolution

the overall result is a series of unserved load distributions

the sequence of convolutions is determined by a fixed economicloading criterion that sets the order of the generator units

A units energy production is found from the difference in the

energy from the unserved load distribution prior to convolution

and the energy from the new unserved load distribution after

the convolution

The Expected Cost Method

The individual unit probability capacity densityfunctions are convolved with one another

the sequence of convolutions is determined by a fixedeconomic criterion that sets the order of the generator units

the convolutions of the generating units produces a set ofavailable capacity distributions, each with an associatedcost curve as a function of total power generated

The resultant expected cost curve is convolved withthe load distribution function this produces the expected value of production cost to serve

the given load forecast

Data Representation


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The Unserved Load Method

Load is modeled as a load-duration curve

a probability distribution that is expressed in hours that the load equals

or exceeds a given power value

monotonically decreasing function with increasing load

can be treated as cumulative probability density function, Pn(x) or can be

expressed in hours, T Pn(x) where T is the duration of time interval

for numerical analysis

treat it in terms of regular discrete steps in a recursive algorithm

procedural steps if there is a segment of capacity with a total of C MW available for

scheduling

let q be the probability that C MW are unavailable

and p = 1 q (that is the availability probability of the segment)


then after the segment is scheduled

the probability of needing x MW or more is now

because the occurrence of loads and unexpected

unit outages are statistically independent events, the

new probability distribution function is a combination

of the two mutually exclusive events is the probability that new capacity C MW is

unavailable and needing x MW or more

is the probability that C is available andneeding (x + C) MW or more


first, the generation requirements for anygenerating segment are determined by theknowledge of the distribution that existsprior to the dispatch of the particular generating

segment the value of determines the required hours of

operation of a new unit, and the area under the distribution between zero and the rating of the unit determinesthe energy production

if a generation segment is not perfectly reliable, there willbe a residual distribution of demands that cannot be servedby this particular segment because of forced outages


Representing the forced outage when the unit is neededfor hours

on average it is only available for hours energy required by the load distribution that the unit could serve

the unit can only generate because of its expected

unavailability

suppose that the unit has a linear input-output cost function the

expected production cost for this period


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there is a residual of unserved demands due to

the forced outages of the unit

the new distribution of the probabilities ofunserved load

the process may be repeated until all units have

been scheduled a residual distribution remains that gives the

final distribution of unserved demand

Example 8C

Example 8C Example 8C


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Documents

Production Cost Models