WO2003032131A2

WO2003032131A2 - Energy market maintenance planning

Info

Publication number: WO2003032131A2
Application number: PCT/US2002/032636
Authority: WO
Inventors: Adolfo M. Fonseca; John D. Finney; Haiwu Ma
Original assignee: Abb Inc.
Priority date: 2001-10-12
Filing date: 2002-10-11
Publication date: 2003-04-17
Also published as: AU2002357652A1; WO2003032131A3

Abstract

Maintenance planning is performed by combining the techniques of random process modeling, clustering, and dividing the maintenance planning problem into a master problem and sub-problems and iteratively solving the master problem (220) and the sub-problems (230). The master problem (220) may include maintenance variables and an objective function for minimizing maintenance cost and the sub-problems (230) may include operation variables and an objective function for maximizing operational profit.

Description

ENERGY MARKET MAINTENANCE PLANNING

Cross Reference to Related Applications

This application claims priority under 35 U.S.C. § 119(e) of U.S. Provisional Patent

Application Serial No. 60/329,230, entitled "System and Method for Utility Resource Portfolio

Maintenance Planning," filed October 12, 2001, and incorporated by reference herein in its

entirety.

Field of the Invention

The invention relates generally to maintenance and energy transaction planning in

energy markets, and more particularly to scheduling unit maintenance and operation and

transaction exercise loading in an effort to increase profitability based on an analysis of

deterministic and stochastic variables.

Background of the Invention

Energy companies provide power to consumers via power generation units. A power

generation unit may be a coal-fired power plant, a hydro-electric power plant, a gas turbine and

a generator, a diesel engine and a generator, a nuclear power plant, and the like. Each of these

generation units is taken out of service occasionally to perform scheduled maintenance. For

example, a gas turbine may be taken out of service once a year for regularly scheduled

maintenance and a nuclear power plant may be taken out of service every ten years for regularly

scheduled maintenance and between eighteen to twenty-four months for refueling. Performing maintenance on a generation unit costs money, not only in terms of labor

expenses associated with the maintenance, but also in terms of lost revenues. Delaying

maintenance increases the chance of breakdowns (which usually results in shutdowns that are

longer than shutdowns for schedule maintenance) and hence increases the chance of large

revenues losses. Scheduling the maintenance at an optimum time can reduce the total cost

associated with the maintenance; however, determining an optimum maintenance time can be

extremely difficult. For one thing, there is a very large number of variables that effect the total

cost of maintenance, such as the demand for power, the cost of fuel, and the like. Moreover,

many of these variables vary over time (i.e., are stochastic) and cannot be predicted with

absolute accuracy. The imperfect knowledge of future energy and fuel prices, customer

demand, hydro energy inflows, and unit availability expose the system to uncertain events that

can invalidate any maintenance schedule or transaction decision that was based on deterministic

assumptions, leading also to unexpected loss of revenues (or increased costs). Existing

techniques for determining an optimal maintenance schedule that take into consideration this

imperfect knowledge of the future are very computationally intensive, generally to the point of

being impractical.

Therefore, there is a need for a system and method for scheduling maintenance and

determining transaction loading levels that can process stochastic and deterministic variables

and provide a maintenance schedule that approximates a near-optimal maintenance schedule

within a predefined confidence level.

Summary of the Invention

Maintenance planning for generation units of an energy market includes: (a) generating a plurality of scenarios based on a plurality of stochastic variables representative

of maintenance of the generation units, a plurality of stochastic variables representative of operation of the generation units, and a plurality of stochastic variables representative of

transactions of the energy market; (b) clustering the plurality of scenarios into groups of

similar scenarios; (c) generating, for each group of scenarios, a scenario that is

representative of the scenarios of the group; (d) iteratively determining a maintenance

solution based on a first objective function of maintenance variables and a set of operation

solutions based on a second objective function of operation and transaction variables, based on the representative scenarios until convergence between the maintenance solution and the

set of operation solutions; and (e) iteratively increasing the number of the plurality of

scenarios and repeating steps (a) through (d) until convergence among the set of operation

solutions.

The scenarios may be generated based on a random process model, such as, a normal

model, a lognormal model, a mean reversion model, a seasonal mean reversion model, a

generalized Wiener process model, and a two-state Poisson model for a binary variable.

Each scenario may include a value for each stochastic variable for each time period

of a maintenance planning period. Each scenario may include a plurality of consecutive

time periods. Each time period may include a plurality of mutually exclusive time sub-

intervals that may be of different duration and each time sub-interval may correspond to an

operation variable and an operation constraint.

The representative scenarios may be generated by averaging, for each time period of

a maintenance plarining period, each value for each stochastic variable of each scenario of a

group of similar scenarios.

The maintenance solution may¹ be and maintenance schedule that may be determined

based on maintenance variables. The set of operation solutions may be a set of operation

schedules and may be determined based on operation variables. The maintenance solution

may be based on an objective function that may be modified by iteratively adding a

coupling variable and a coupling constraint to the first objective function, for example,

using Benders decomposition.

The convergence of the maintenance solution and the set of operation solutions may

be determined by estimating a first estimated profit based on the maintenance solution,

estimating a second estimated profit based on the maintenance solution and the set of

operation solutions, and comparing the difference of the first estimated profit and the second

estimated profit to a predefined threshold profit.

The convergence of the set of operation solutions may be determined by estimating a

first estimated profit based on the maintenance solution, estimating a set of second

estimated profits based on the maintenance solution and the set of operation solutions,

calculating a set of third estimated profits by adding the first estimated profit to each profit of the set of second estimated profits, determining the mean and standard deviation of the

set of third estimated profits, and determining convergence based on the mean and standard

deviation.

Other features of the disclosed maintenance planning are described below.

Brief Description of the Drawings

Systems and methods for maintenance planning are further described with reference to the accompanying drawings in which:

Figure la is a system diagram of an exemplary computing environment and an

illustrative system for maintenance planning, in accordance with an embodiment of the

invention;

Figure lb is a system diagram of an exemplary computing network environment and an

invention;

Figure lc is a system diagram showing the interaction between exemplary computing

components and an illustrative system for maintenance planning, in accordance with an

embodiment of the invention; and

Figure 2 is a flow diagram of an illustrative method for maintenance planning, in accordance with an embodiment of the invention.

Detailed Description of Illustrative Embodiments

Overview

The maintenance planning systems and methods described herein determine a near- optimal maintenance schedule for each power generation unit. Further, the systems and

methods may also determine an operation schedule for each unit and a transaction schedule

(i.e., a fuel and power trading schedule). The systems and methods may be used by an energy

company to determine a maintenance schedule that is close to the optimum maintenance

schedule in terms of maximizing profit. Further, the user may select a confidence level for how

close the determined maintenance schedule is to the optimum maintenance schedule.

To attain the selected confidence level and process a large number of stochastic

variables that are associated with power generation, the systems and methods combine the

techniques of random process modeling, clustering, and dividing the problem into a master

problem and sub-problems. With such a combination of techniques, the systems and methods

provide a maintenance planning solution that is close to the optimum maintenance solution.

Moreover, the systems and methods are computationally practical. Therefore, the systems and

methods may assist an energy company to maximize profits realized from the generation and

sale of power. The systems and methods may be implemented in one or more of the exemplary

computing environments described in more detail below, or in other computing environments.

Exemplary Computing Environment and Illustrative Computing Application

Figure la shows computing system 100 that comprises computer 20a. Computer 20a

comprises display device 20a' and interface and processing unit 20a' ' . Computer 20a executes

computing application 180. As shown, computing application 180 comprises a computing

application processing and storage area 180a and a computing application display 180b.

Computing application processing and storage area 180a may contain maintenance planning

engine 180a(l), maintenance portfolio data store 180a(2), and maintenance planning parameters data store 180a(3). Computing application display 180b may comprise display content 180b'.

In operation, a user (not shown) may interface with computing application 180 through

computer 20a. The user may navigate through computing application 180 to input, display, and

generate data for maintenance planning. Computing application 180 may generate

maintenance, operation, and transaction schedules using maintenance planning engine 180a(l),

maintenance portfolio data store 180a(2), and maintenance planning parameters data store

180a(3). The generated schedules may be displayed to the user as display content 180b' on

computing application display 180b.

Exemplary Network Environment and Illustrative Computing Application

Computer 20a, described above, can be deployed as part of a computer network. In

general, the above description for computers may apply to both server computers and client

computers deployed in a network environment. Figure lb illustrates an exemplary network

environment having server computers in communication with client computers, in which

systems and methods for maintenance planning may be implemented. As shown in Figure lb, a

number of server computers 10a, 10b, etc., are interconnected via a communications network

160 with a number of client computers 20a, 20b, 20c, etc., or other computing devices, such as,

a mobile phone 15 , and a personal digital assistant 17. Communication network 160 may be a

wireless network, a fixed-wire network, a local area network (LAN), a wide area network

(WAN), an intranet, an extranet, the Internet, or the like. In a network environment in which

the communications network 160 is the Internet, for example, server computers 10 can be Web

servers with which client computers 20 communicate via any of a number of known

communication protocols, such as, hypertext transfer protocol (HTTP), wireless application protocol (WAP), and the like. Each client computer 20 can be equipped with a browser 180a to

communicate with server computers 10. Similarly, personal digital assistant 17 can be

equipped with a browser 180b and mobile phone 15 can be equipped with a browser 180c to

display and communicate various data.

In operation, the user may interact with computing application 180 to generate

maintenance, operation, and transaction schedules. The generated schedules may be stored on

server computers 10, client computers 20, or other client computing devices. The generated

schedules may be communicated to users via client computing devices or client computers 20.

Thus, the systems and methods for maintenance planning can be implemented and used

in a computer network environment having client computing devices for accessing and

interacting with the network and a server computer for interacting with client computers. The

systems and methods can be implemented with a variety of network-based architectures, and

thus should not be limited to the examples shown.

Figure lc shows the cooperation of various computing elements during maintenance

planning in a network computing environment. The user may employ client computer 20a to

send a request for maintenance planning to computing application 180 executing on server

computer 10a. In response, computing application 180 may process the request by executing

maintenance planning engine 180a(l) to generate a maintenance, operation, and transaction

schedule. The schedules can then be communicated to client computer 20a via communications

network 160. At client computer 20a, the generated schedules are displayed and may be viewed

and manipulated by the user.

Illustrative Maintenance Planning The overall function of the maintenance planning system is to detemiine a near-optimal

maintenance, operation, and transaction schedule that yields near-maximum profit for an energy

company. The complete problem to be solved is to maximize the expected profit based on

maintenance variables, operation variables, and transaction variables. An illusfrative objective

function corresponding to the complete problem is given by:

Maximize Expected {

+ Energy sales revenue

+ Revenue of final pond energy with reference value + Reserve sale revenue - Maintenance costs

- Energy purchase cost

- Advance maintenance penalties

- Optional maintenance financial penalties

- Optional group maintenance financial penalties - Cost of nuclear fuel

- Cost of fossil fuels

- Minimum system down reserve penalties

- Minimum consumption of time bucket fossil fuel penalties

- Minimum consumption of time period fossil fuel penalties - Minimum consumption of time interval fossil fuel penalties}

Some maintenance, operation, and transaction schedules are impossible to implement,

therefore, constraints may be used to model such cases in which schedules are impossible to

implement. Constraints may be either linear or non-linear. Moreover, there are some situations

in which a limitation should not be violated, but may be violated for a price. These situations

may be modeled as "soft" constraints or penalties. That is, the overall objective function should be solved subject to such constraints and penalties. Illustrative constraints and penalties are

given as follows:

Maintenance time for generating units, linear model

Mode continuity constraints

Maintenance startup constraints

Maintenance end consfraints

Constraints on the ability to perform maintenance

End maintenance requests

Optional maintenance requests • Maximum requests per crew

Unit group simultaneous maintenances

Maintenance request groups

Energy balance equations

Coordination of maximum loading with modes of generating units • Coordination of maximum pumping with modes of hydro pumping units

Equations for heat and loading balance by operating modes of fossil generating units

Equations for fuel and heat balance of fossil generating units

Equations for unit fuel and fuel balance of fossil generating units • Capacity vs. energy since last refueling (stretching out nuclear units)

Outage limit of unit groups

Reserve contributions of generating units

System regulation reserve requirements

Availability of generating units • Operating capacity of generating units

System capacity requirements

Penalty for violation of minimum fuel burn by time period and time interval Balance equation for fuel limitations in time interval

Equation for fuel limitations in time period

Balance equations for water stored in reservoirs

Hydro arc consfraints

Area capacity requirements

Area global import/export limits

Area transfer paths

Unit outage models

Unit groups

Forced outage groups

Fuel contracts for each time bucket, period, and interval

Penalties for violation of contracts for each time bucket, period, and interval

Fuel unit group for each time bucket, period, and interval

Penalties for violation of fuel unit groups for each time bucket, period, and interval

Fuel unit for each time bucket, period, and interval

Penalties for violations of fuel units for each time bucket, period, and interval

Start restriction of maintenances

Fixed transactions

Dispatchable transactions

Transactions by unit group

Information representative of such constraints and penalties are stored in maintenance

portfolio data store 180a(2). In addition to storing constraint information and penalty

information, data representative of the portfolio of assets of an energy company are stored in

maintenance portfolio data store 180a(2). The following illustrative portfolio asset data (which

includes maintenance variables, operation variables, and transaction variables) may be stored in

maintenance portfolio data store 180a(2). Portfolio assets: Fossil fuel units; Nuclear units; Hydro-electric units; Energy transactions; Primary and secondary reserves.

Fossil fuel units: Name; Power station; Default fossil mode; Nominal capacity; Active flag; Multiple fossil modes.

Fossil modes: Active flag; Quick start flag; Heat rate; Primary and secondary reserve capability; Determimstic maximum generation; Stochastic maximum generation (typically Poisson); Stochastic parameters for normal state, failure state, failure rate, and repair rate (typically Poisson); Fuel model with price and fossil fuel; Measurement unit, heat content, and volumetric constraint for each fuel; Minimum and maximum fuel level constraints for various time spans.

Nuclear units: Name; Power station; Default nuclear mode; Nominal capacity; Active flag; Multiple nuclear modes.

Nuclear modes: Name; Active flag; Maximum generation; Date of last refueling; Energy generated since last refueling; Stretch out function including energy at stretch-out point, capacity reduction rate, and capacity; Maximum number of rods; Price per rod; Refueling rod fraction.

Hydro-electric units: Name; Power station; Default hydro mode; Nominal capacity; Active flag; Multiple hydro modes; Type; Upstream pond.

Hydro-electric modes: Name; Active flag; Generating efficiency; Pumping efficiency;

Multiple reserve capabilities in generating and pumping modes; Deterministic maximum generation and pumping; Stochastic maximum generation and pumping (typically two-state

Poisson); Stochastic parameters for normal state, failure state, failure rate, and repair rate (typically Poisson) Reservoirs: Name; Active flag; Maximum energy storage; Initial energy storage; Volume constraints, minimum and maximum; Deterministic inflow; Stochastic inflow (typically normal or lognormal).

Transactions: Name; Active flag; Type (e.g., sale or purchase); Hours of day filter (e.g., off-peak, on-peak, weekends); Type (e.g., forced, optimizable, switchable); Fixed dispatch or minimum and maximum limit; Deterministic price model; Stochastic price model (e.g., mean reversion, seasonal mean reversion, normal, lognormal, or geometric Brownian walk)

Reserves: Name; Active flag; Type (e.g., spinning, non-spinning); System obligation, requirement.

Areas: Name; Active flag; Deterministic demand; Stochastic demand (typically normal or lognormal).

Maintenance request: Name; Active flag; Request date; Unit; Duration; Start and end date of maintenance window; Fixed schedule flag and start date; Crew; Mode before maintenance done; Mode if maintenance not done; Mode after maintenance done; Maintenance optional flag; Financial penalty for delaying maintenance; Maintenance cost; Financial penalty for advancing maintenance.

Crews: Name; Active flag; Maximum number of maintenances at one time.

Simultaneous outage groups: Name; Active flag; Units; Maximum number of units in

maintenance; Maximum capacity of units in maintenance.

As can be seen, some of the data may be modeled as a stochastic variable, for example, transaction prices may be modeled with a normal distribution function. Modeling a price as a

stochastic variable is more realistic than modeling a price with a deterministic variable because

prices typically vary somewhat randomly over time. While conventional systems and methods

have difficulty processing the large amount of data associated with stochastic variables, the

systems and methods disclosed herein process stochastic variables efficiently and within a

predefined confidence level.

Figure 2 shows a flow chart of an illusfrative method for maintenance planning that can

efficiently handle stochastic variables. The method combines the techniques of random process

modeling, clustering, and dividing the problem into a master problem and sub-problems, as

described in more detail below.

Prior to requesting a maintenance schedule, the user may enter and computing

application 180 may receive the following parameters; a start date and an end date for a

planning horizon (that can be, for example, a few weeks, several months, a few years, or the

like), a time period and interval resolution for the planning horizon (e.g., daily, weekly,

monthly, or any other duration or combination), buckets for each time period (e.g., high peak,

peak, low peak, mid-load, low peak), an interest rate, a sampling convergence threshold, a

Benders convergence threshold, a maximum number of sampling iterations, a maximum

number of Benders iterations, a number of clusters, a seed value, an integration step for sample

generation, a number of points in cumulative distribution functions, a study mode (i.e.,

simulation or optimization mode described in more detail below), and the like. The parameters

may be stored in maintenance planning parameters data store 180a(3). Alternatively, the user

may choose to use default parameters already stored in maintenance planning parameters data

store 180a(3). To describe time periods and intervals in more detail, given an illustrative planning

horizon of one year, the planning horizon may be divided into fifty-two week long time periods

and a value, for each variable, may be associated with each of the fifty-two weeks. Time

periods refer to periods of time that are chronologically consecutive, such as, for example, a

week, a day, a combination of thereof, or the like. Time periods of different lengths may also

be used in the same planning horizon. For example, given the illustrative one year planning

horizon, the first six months may be divided into daily time periods and the second six months

may be divided into weekly time periods. Time periods may also be further divided into

mutually exclusive and collectively exhaustive time buckets or sub-intervals that are not

necessarily chronologically consecutive or of the same length. For example, a time bucket may

be associated with load conditions, such as, for example, a peak operating time (e.g., 8 am to 8

pm weekdays) or the like. A time interval is a set of consecutive time periods. With division of

the planning horizon into time periods, intervals, and buckets, computing application 180 may

analyze each time period, interval, and bucket to provide a more realistic and reliable solution.

The user may configure the time periods, intervals and buckets for a planning horizon. With

such parameters, the method illustrated in Figure 2 may begin maintenance planning.

At step 200, computing application 180 generates a plurality of scenarios based on the

stochastic and deterministic variables (e.g., maintenance, operation, and fransaction variables)

of portfolio data store 180a(2). For determimstic variables, computing application 180 uses the

respective deterministic value stored in portfolio data store 180a(2). For stochastic variables,

computing application 180 generates a value based on the respective random process model

stored in portfolio data store 180a(2). The random process model may include any model, such

as, for example, normal, lognormal, mean reversion, seasonal mean reversion, generalized Wiener process, two-state Poisson for binary variables, and the like.

Each scenario includes a value for each variable and for each time period (and possibly

for each time bucket. For example, computing application 180 may generate one-hundred

scenarios wherein each scenario includes a value for each variable and for each time period. A

first scenario may include a fuel price of $100 for a first day and a fuel price of $101 for a

second day, and so on. The first scenario may include a load of 2000 megawatt for the first day

and a load of 2050 megawatt for the second day, and so on. A second scenario may include a

fuel price of $99 for the first day and a fuel price of $102 for the second day, and so on. The

second scenario may include a load of 2005 megawatt for the first day and a load of 2010

megawatt for the second day, and so on.

In general, generating and analyzing more scenarios results in a more reliable

maintenance planning. However, generating more scenarios creates a higher processing

requirement, which may slow the generation of a maintenance plan. Therefore, at step 210,

computing application 180 clusters the plurality of scenarios into clusters of similar scenarios

and creates a representative scenario for each cluster of similar scenarios. The representative

scenario may be generated, for example, by averaging the values of each variable for each time

period in each scenario in the cluster or by some other technique. For example, the exemplary

one-hundred scenarios generated in step 200 maybe clustered into ten representative scenarios.

Any known clustering technique may be implemented.

At step 220, computing application 180 generates a proposed maintenance schedule by

solving a master problem that is a subset of the overall maintenance planning problem obj ective

function. To explain in more detail, rather than solving for a maintenance schedule, an operation schedule, and a fransaction schedule at the same time, the overall maintenance problem objective function is divided into a master problem and a plurality of sub-problems.

The master problem may include maintenance variables and may be used to solve for the

maintenance schedule. The sub-problems may include operation and transaction variables and

may be used to solve for the operation schedule and the transaction schedule. For example, the

objective function for the master problem may be given by

Maximize Expected {

- Maintenance costs

- Advance maintenance penalties - Optional maintenance financial penalties

- Optional group maintenance financial penalties

- Cost of nuclear fuel

+ Estimate of sub-problems' profit (on second and later solution iterations)}

As can be seen, the master problem includes maintenance variables and constraints and

the objective of the master problem is to maximize expects profits based on maintenance costs,

maintenance variables and constraints, nuclear fuel costs, and a profit estimation that is based

on sub-problems. The master problem may include a set of Benders variables that represent the

estimate of sub-problems' profit, and a set of constraints (called Benders cuts) that represent the

relationship between the profit estimate variables and the maintenance variables and nuclear

unit generation. The sub-problem profit estimation may be based on maintenance variables and

consfraints and Benders cuts. As can be seen, the master problem includes information of the

overall maintenance problem; however, the master problem is solved iteratively with the sub-

problems, thereby providing a new (and hopefully better) overall maintenance planning solution with each iteration. The objective function for the sub-problems may be given by:

Maximize Expected {

+ Energy sales revenue + Revenue of final pond energy with reference value

+ Reserve sale revenue

- Energy purchase cost

- Cost of fossil fuels

- Minimum system down reserve penalties - Minimum consumption of time bucket fossil fuel penalties

- Minimum consumption of time period fossil fuel penalties

- Minimum consumption of time interval fossil fuel penalties}

As can be seen, the sub-problems include operation and transaction variables and

constraints (e.g., transaction volumes, unit generation levels, and the like) and the objective of

each sub-problem is to maximize profits based on operation and fransaction variables and

constraints. There is typically one sub-problem for each representative scenario. In step 220,

computing application 180 solves the master problem using mixed-integer programming, for

example, or other techniques. Computing application 180 generates a proposed maintenance

schedule and an estimated profit corresponding to the proposed maintenance schedule.

At step 230, computing application 180 independently solves each sub-problem based

on the proposed maintenance schedule. Computing application 180 generates a proposed

operation schedule, a proposed transaction schedule, and an estimated profit for each

representative scenario. It should be noted that the master problem is solved for the minimum

maintenance cost (and only the minimum maintenance cost on the first iteration), which may not correspond to the maximum profit for the overall maintenance planning problem.

Therefore, the master problem and the sub-problems are solved with an iterative technique until

converging when a threshold condition is met, as described in more detail below.

The iterative technique includes coupling variables that represent the maintenance

schedule and nuclear units' generation level. Coupling variables exist in both the master

problem and the sub-problems but they are determined by the master problem and are used by

the sub-problems. Coupling consfraints exist in sub-problems and combine coupling variables

with operation and transaction variables. Benders cuts exist in the master problem and include

the coupling variables. Benders variables exist in the master problem and represent an estimate

of the objective function of the sub-problems as seen from the master problem. Benders cuts

together with Benders variables are a condensed and linearized representation of sub-problems

as seen from the master problem. From this perspective, the master problem is always a

simplified representation of the entire optimization problem. The representation is a rough

approximation at the beginning and becomes a near-optimal representation at the end of the

iterative solution process. Upon the first iteration, the master problem only includes the

maintenance cost but after the first iteration, the master problem also includes an estimate of the

revenues and the costs based on the sub-problems.

At step 240, computing application 180 determines whether the proposed maintenance,

operation, and fransaction schedules have converged. A Benders convergence test may be used

or any other decomposition convergence test. For example, computing application 180 may

calculate a first estimated profit (actually only a maintenance cost on the first iteration, but an

estimated profit on later iterations) based on the master problem. Computing application 180

may calculate a second estimated profit based on the combined master problem and the sub- problems. The second estimated profit may be determined using probability weightings for

each of the representative scenarios. The first and second estimated profits are compared and if

the difference between the first and second estimated profits is less than a threshold profit

value, there is convergence and the method proceeds to step 250. If, however, the difference

between the first and second estimated profits is greater than a threshold profit value, the

method proceeds to step 245.

At step 245, computing application 180 adds one coupling variable and one coupling

constraint to the master problem (e.g., generates and adds a Benders "cut" to the master

problem). The coupling variable and the coupling consfraint may be selected using a

decomposition approach, such as, for example, Benders decomposition or the like. Also, the

coupling variable may be scaled using the probability of each representative scenario. The

coupling consfraint may be formulated to limit the combined estimated cost of the maintenance

variables with the estimated profits of the operations variables to the value from the current

solution of the sub-problems. In effect, the coupling variable and the coupling consfraint

modify the master problem to reflect operations profits and constraints implied by the

maintenance variables in the representative scenarios. Alternatively, rather than adding one

coupling variable and one coupling consfraint to the master problem, one aggregate coupling

variable and one aggregate coupling constraint may be added to the master problem.

In addition to adding coupling variables and coupling consfraints to the master problem

(wherein each addition of a coupling variable and a coupling constraint is referred to as a

"cut"), the number of cuts can be limited by replacing a cut having a high slack variable with a

newly generated cut.

After adding one coupling variable and one coupling constraint to the master problem at step 245, the method returns to step 220 and computing application 180 solves the modified

master problem (i.e., solves the master problem including the added coupling variable and

coupling constraint). The method then proceeds through steps 222, 230, and 240 until there is

convergence or until a maximum number of iterations has been reached (not shown). If there is

convergence at step 240, the method proceeds to step 250.

At step 250, computing application 180 determines if the proposed maintenance

schedule, operation schedule, and transaction schedule have converged with respect to the

number of scenarios used. To determine convergence, computing application 180 determines

an estimated cost based on the proposed maintenance schedule. Computing application 180

determines an estimated profit based on the operation and transaction schedule for each

representative scenario. Computing application 180 then determines an estimated total profit for

each representative scenario by adding the estimated cost to each estimated total profit.

Computing application 180 calculates a mean and standard deviation for the estimated total

profits. Computing application 180 divides the standard deviation by the mean times the

square root of the quantity given by the number of representative scenarios minus one. The

result of the division is compared to a threshold value. If the result is less than or equal to the

threshold value, convergence is determined and the method stops. If the result is greater than

the threshold value, the method proceeds to step 255.

At step 255, computing application 180 increases the number of scenarios and proceeds

to step 210 where computing application 180 generates new scenarios. This is repeated until

convergence is reached.

Once convergence is reached, computing application 180 may generate several outputs.

For example, computing application 180 may generate a near-optimal maintenance schedule for each unit, a Gantt chart presentation of the schedule, a capacity outage chart, a statistical

summary of fuel burn, unit production, or market activity, an estimated revenue and standard

deviation, an estimated cost and profit for each time period, a firm capacity, an overhaul

confirmation, an estimated unit generation and pumping and standard deviation, an estimated

transaction loading, cost, and revenue and standard deviation, an estimated fuel consumption

and cost and standard deviation, an estimated pond level and standard deviation, an estimated

reserve allocation and standard deviation, a cumulative distribution function for total cost,

revenue, and profit, and the like. Computing application 180 may also generate a near-optimal

operation schedule and transaction schedule based on a probability weighted average of the sub-

problem solutions .

The above discussion illustrates how computing application 180 can determine a

maintenance, operation, and fransaction schedule. This is referred to as an optimization mode

(i.e., computing application 180 determines an optimum maintenance schedule for given

planning horizon). In addition to optimization mode, computing application 180 can operate in

a simulation mode. In the simulation mode, the user specifies a maintenance schedule for each

unit via a user interface of computer 20. Computing application 180 receives the specified

maintenance schedule and determines an optimal operation schedule and an optimal transaction

schedule (e.g., a schedule of resources such as unit dispatch, reserve allocation, fransaction

loading, and the like). In simulation mode, Benders cuts and convergence may be performed if

the nuclear load is unknown. If nuclear load is known, Benders cuts and convergence may be

implemented in a single iteration. In a hybrid mode, the user specifies a maintenance schedule

for some of the units and computing application 180 receives the specified maintenance

schedules and determines a near-optimal maintenance schedule for those units without a specified maintenance schedule, as well as an operation and fransaction schedule for each unit.

In an illustrative embodiment, the systems and methods described herein employ the

commercially available CPLEX ® mathematical software program to solve the master problems

and the sub-problems.

Program code (i.e., instructions) for performing the above-described methods may be

stored on a computer-readable medium, such as a magnetic, electrical, or optical storage

medium, including without limitation a floppy diskette, CD-ROM, CD-RW, DVD-ROM,

DVD-RAM, magnetic tape, flash memory, hard disk drive, or any other machine-readable

storage medium, wherein, when the program code is loaded into and executed by a machine,

such as a computer, the machine becomes an apparatus for practicing the invention. The

invention may also be embodied in the form of program code that is transmitted over some

transmission medium, such as over electrical wiring or cabling, through fiber optics, over a

network, including the Internet or an intranet, or via any other form of transmission, wherein,

when the program code is received and loaded into and executed by a machine, such as a

computer, the machine becomes an apparatus for practicing the above-described processes.

When implemented on a general-purpose processor, the program code combines with the

processor to provide an apparatus that operates analogously to specific logic circuits.

It is noted that the foregoing description has been provided merely for the purpose of

explanation and is not to be construed as limiting of the invention. While the invention has

been described with reference to illusfrative embodiments, it is understood that the words which

have been used herein are words of description and illustration, rather than words of limitation.

Further, although the invention has been described herein with reference to particular structure,

methods, and embodiments, the invention is not intended to be limited to the particulars disclosed herein; rather, the invention extends to all structures, methods and uses that are within

the scope of the appended claims. Those skilled in the art, having the benefit of the teachings

of this specification, may effect numerous modifications thereto and changes may be made

without departing from the scope and spirit of the invention, as defined by the appended claims.

Claims

What is claimed is:

1. A method for maintenance planning for generation units of an energy market, the

method comprising:

(a) generating a plurality of scenarios based on a plurality of stochastic variables

representative of maintenance of the generation units, a plurality of stochastic variables

representative of operation of the generation units, and a plurality of stochastic variables

representative of transactions of the energy market;

(b) clustering the plurality of scenarios into groups of similar scenarios;

(c) generating, for each group of scenarios, a scenario that is representative of the

scenarios of the group;

(d) iteratively determining a maintenance solution based on a first objective function

of maintenance variables and a set of operation solutions based on a second objective

function of operation and transaction variables, based on the representative scenarios until

convergence between the maintenance solution and the set of operation solutions; and

(e) iteratively increasing the number of the plurality of scenarios and repeating steps

(a) through (d) until convergence among the set of operation solutions.

2. The method as recited in claim 1 , wherein generating a plurality of scenarios

comprises generating a plurality of scenarios based on a random process model.

3. The method as recited in claim 2, wherein the random process model comprises one

of a normal model, a lognormal model, a mean reversion model, a seasonal mean reversion

model, a generalized Wiener process model, and a two-state Poisson model for a binary variable.

4. The method as recited in claim 1, wherein generating a scenario comprises

generating a value for each stochastic variable for time periods of a maintenance planning

period.

5. The method as recited in claim 4, wherein generating a representative scenario

comprises averaging, for each time period, each value for each stochastic variable of each

scenario of a group of similar scenarios.

6. The method as recited in claim 1 , wherein each scenario comprises a plurality of

consecutive time periods.

7. The method as recited in claim 6, wherein each of the plurality of time periods

comprises a plurality of mutually exclusive time sub-intervals.

8. The method as recited in claim 6, wherein each of the plurality of consecutive time

periods is not the same duration.

9. The method as recited in claim 9, further comprising receiving an indication of each

of the plurality of consecutive time periods via a user interface.

10. The method as recited in claim 9, wherein each of the plurality of mutually exclusive time sub-intervals corresponds to a value of an operation variable and value of an operation

consfraint.

11. The method as recited in claim 1 , wherein iteratively determining a maintenance

solution comprises determining a maintenance schedule using mixed integer linear

programming.

12. The method as recited in claim 1, wherein iteratively determining a maintenance

solution and a set of operation solutions comprises determining a maintenance schedule

based on maintenance variables and determining a set of operation schedules based on

operation variables.

13. The method as recited in claim 1, wherein iteratively determimng a maintenance

solution based on a first objective function and a set of operation solutions based on a

second objective function comprises adding a coupling variable and a coupling consfraint to

the first objective function.

14. The method as recited in claim 13, wherein iteratively determining a maintenance

second objective function comprises determining the coupling variable and the coupling

consfraint via a Benders decomposition.

15. The method as recited in claim 1 , wherein iteratively determining a maintenance solution and a set of operation solutions until convergence comprises determining

convergence by:

estimating a first estimated profit based on the maintenance solution;

estimating a second estimated profit based on the maintenance solution and the set

of operation solutions; and

comparing the difference of the first estimated profit and the second estimated profit

to a predefined threshold profit.

16. The method as recited in claim 1 , wherein iteratively increasing the number of the

plurality of scenarios until convergence comprises determining convergence by:

estimating a first estimated profit based on the maintenance solution;

estimating a set of second estimated profits based on the maintenance solution and

the set of operation solutions;

calculating a set of third estimated profits by adding the first estimated profit to each

profit of the set of second estimated profits;

determining the mean and standard deviation of the set of third estimated profits;

and

determining convergence based on the mean and standard deviation.

17. A method for solving an overall maintenance planning problem for generation units

of an energy market, the method comprising:

(a) generating a plurality of scenarios based on a plurality of stochastic variables;

(b) clustering the plurality of scenarios into a plurality of representative scenarios; (c) iteratively solving a master problem and a plurality of sub-problems that are

representative of the overall maintenance planning problem and modifying the master

problem with Benders cuts based on the plurality of representative scenarios until

convergence between the solution of the master problem and the solutions of the plurality of

sub-problems; and

(d) iteratively increasing the number of the plurality of scenarios and repeating steps

(a) through (c) until convergence among the solutions of the sub-problems.

18. A computer-readable medium having computer-executable instructions stored

thereon for maintenance planning for generation units of an energy market, the instructions

when executed on a processor performing the following:

representative of transactions of the energy market;

(b) clustering the plurality of scenarios into groups of similar scenarios;

scenarios of the group;

function of operation and fransaction variables, based on the representative scenarios until

(e) iteratively increasing the number of the plurality of scenarios and repeating steps (a) through (d) until convergence among the set of operation solutions.

19. The computer-readable medium as recited in claim 18, wherein generating a scenario

comprises generating a value for each stochastic variable for time periods of a maintenance

planning period and wherein generating a representative scenario comprises, for each time

period, averaging each value for each stochastic variable of each scenario of a group of

similar scenarios.

20. The computer-readable medium as recited in claim 18, wherein iteratively

determining a maintenance solution and a set of operation solutions comprises determining

a maintenance schedule based on maintenance variables and determining a set of operation

schedules based on operation variables.

21. The computer-readable medium as recited in claim 18, wherein iteratively

determining a maintenance solution based on a first objective function and a set of operation

solutions based on a second objective function comprises adding a coupling variable and a

coupling consfraint to the first objective function.

22. The computer-readable medium as recited in claim 18, wherein iteratively

determining a maintenance solution and a set of operation solutions until convergence

comprises determimng convergence by:

estimating a first estimated profit based on the maintenance solution;

estimating a second estimated profit based on the maintenance solution and the set of operation solutions; and

to a predefined threshold profit.

23. The computer-readable medium as recited in claim 18, wherein iteratively increasing

the number of the plurality of scenarios until convergence comprises determining

convergence by:

estimating a first estimated profit based on the maintenance solution;

the set of operation solutions;

profit of the set of second estimated profits;

and

determining convergence based on the mean and standard deviation.

24. A method for solving an overall maintenance planning problem for generation units

of an energy market, the method comprising:

(b) clustering the plurality of scenarios into a plurality of representative scenarios;

(c) decomposing the overall maintenance plan into a master problem and a plurality

of sub-problems with Benders decomposition;

(d) iteratively solving the master problem and the plurality of sub-problems and modifying the master problem with Benders cuts based on the plurality of representative

scenarios until convergence between the solution of the master problem and the solutions of

the plurality of sub-problems; and

(a) through (d) until convergence among the solutions of the sub-problems.

25. A system for maintenance planning for generation units of an energy market, the

system comprising:

a first data store comprising a plurality of stochastic variables representative of

maintenance of the generation units, a plurality of stochastic variables representative of

operation of the generation units, and a plurality of stochastic variables representative of

transactions of the energy market;

a computing application cooperating with the first data store and (a) retrieving the

plurality of stochastic variables representative of maintenance of the generation units, the

plurality of stochastic variables representative of operation of the generation units, and the

plurality of stochastic variables representative of fransactions of the energy market, (b)

generating a plurality of scenarios based on the plurality of stochastic variables

representative of maintenance of the generation units, the plurality of stochastic variables

representative of operation of the generation units, and the plurality of stochastic variables

representative of transactions of the energy market, (c) clustering the plurality of scenarios

into groups of similar scenarios, (d) generating, for each group of scenarios, a scenario that

is representative of the scenarios of the group, (e) iteratively determining a maintenance

solution based on a first objective function of maintenance variables and a set of operation solutions based on a second objective function of operation and fransaction variables, based

on the representative scenarios until convergence between the maintenance solution and the

set of operation solutions, and (f) iteratively increasing the number of the plurality of

scenarios and repeating steps (b) through (e) until convergence among the set of operation

solutions.

26. The system as recited in claim 25, wherein generating a scenario comprises

period and wherein generating a representative scenario comprises averaging, for each time

period, each value for each stochastic variable of each scenario of a group of similar

scenarios.

27. The system as recited in claim 25, wherein iteratively determining a maintenance

operation variables.

28. The system as recited in claim 25, wherein iteratively determining a maintenance

second objective function comprises adding a coupling variable and a coupling constraint to

the first objective function.

29. The system as recited in claim 25, wherein iteratively determining a maintenance solution and a set of operation solutions until convergence comprises determining

convergence by:

estimating a first estimated profit based on the maintenance solution;

of operation solutions; and

to a predefined threshold profit.

30. The system as recited in claim 25, wherein iteratively increasing the number of the

plurality of scenarios until convergence comprises determining convergence by:

estimating a first estimated profit based on the maintenance solution;

the set of operation solutions;

profit of the set of second estimated profits;

and

determining convergence based on the mean and standard deviation.

31. A method for operation planning for generation units of an energy market, the

method comprising:

(a) receiving a predefined maintenance schedule;

(b) generating a plurality of scenarios based on a plurality of stochastic variables; (c) clustering the plurality of scenarios into a plurality of representative scenarios;

(d) iteratively determining an operation solution based on the predefined

maintenance schedule and an objective function of operation and transaction variables; and

(b) through (e) until convergence among the operation solutions.