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LINEAR PROGRAMMING CONCEPTS FOR UNIT

COMMITMENT

January 1992

US. DEPARTMENT OF THE INTERIOR Bureau of Reclamation

Denver Office Research and Laboratory Services Division

Electric Power Branch

1. REPORT NO. I R-92-03 ....................................... .,.... .......................

4. TITLE AND SUBTITLE

LINEAR PROGRAMMING CONCEPTS FOR UNIT COMMITMENT

I January 1992 1 6. PERFORMING ORGANIZATION CODE

Bureau of Reclamation Denver Office

7. AUTHOR(S)

Nancy C. Farabee Steve C. Stitt

9. PERFORMING ORGANIZATION NAME AND ADDRESS

11. CONTRACT OR GRANT NO.

D-3722

8. PERFORMING ORGANIZATION REPORT NO.

R-92-03 10. WORK UNIT NO.

Same I

Denver CO 80225 12. SPONSORING AGENCY NAME AND ADDRESS

14. SPONSORING AGENCY CODE I DIBR

13. TYPE OF REPORT AND PERIOD COVERED

I

15. SUPPLEMENTARY NOTES

Microfiche and hard copy available at the Denver OEce, Denver, Colorado. Ed: RNW

The optimization of the Lower Colorado Powerplants, Hoover, Davis, and Parker, was studied using a linear programming model. The goal of the optimization was to minimize the flow required to produce a given amount of power, thereby increasing the average efficiency of the Lower Colorado Basin. The optimization produces an hourly generation schedule termed unit commitment.

17. KEY WORDS AND DOCUMENT ANALYSIS

a. DESCRIPTORS-- *linear optimization1 unit commitment/ operations research1 turbine flow curves

b. IDENTIFIERS- Davis D a d Parker D a d Hoover D a d Lower Colorado River Dams Project

c. COSA TI Fleld/Group 09C COWRR: 0901.1 SRIM: 18. DISTRIBUTION STATEMENT I 19. SECURITY CLASS I 21. NO. OF PAGES

(THIS REPOAT) UNCLASSIFIED 38

20. SECURITY CLASS 1 22. PRICE

LINEAR PROGRAMMING CONCEPTS FOR UNIT COMMITMENT

Nancy C. Farabee and

Steve C. Stttt

Electric Power Branch Research and Laboratory Sewices Division

Denver Office Denver, Colorado

January 1992

UNITED STATES DEPARTMENT OF THE INTERIOR * BUREAU OF RECLAMATION

Mission: As the Nation's principal conservation agency, theDepartment of the Interior has responsibility for most of ournationally owned public lands and natural and culturalresources. This includes fostering wise use of our land andwater resources, protecting our fish and wildlife, preservingthe environmental and cultural values of our national parksand historical places, and providing for the enjoyment of lifethrough outdoor recreation. The Department assesses ourenergy and mineral resources and works to assure that theirdevelopment is in the best interests of all our people. TheDepartment also promotes the goals of the Take Pride inAmerica campaign by encouraging stewardship and citizenresponsibility for the public lands and promoting citizenparticipation in their care. The Department also has a majorresponsibility for American Indian reservation communitiesand for people who live in Island Territories under U.S.Administration.

The research portion of the activities covered by this report wasfunded under the Bureau of Reclamation PRESS (ProgramRelated Engineering and Scientific Studies) allocation No. ROOO1,Resource Optimization, Water and Power Delivery.

The information contained in this report regarding commercialproducts or firms may not be used for advertising or promotionalpurposes and is not to be construed as an endorsement of anyproduct or firm by the Bureau ofReclamation.

ii

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Linearprogrammingdefinition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Model format . . . . . . . . . . . . . . . . . . . . . . .

Modeltypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lindosinglecyclemodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lindo24-hoursinglecyclemodel . . . . . . . . . . . . . . . . . . . . . . . . . . .Lindo24-hourmultiplecyclemodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lindohistoricalmodel. . . . . . . . . . . . . . . . .What'sBest24-hoursinglecyclemodel. . . . . . . . . . . . . . . .Lindomultisegmentflowcurvemodel. . . . . . . . . . . . . . . . . . . . . . . .

Curvefitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Error calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modelcomparisons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Concepttests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Manuallycontrolledreleasemethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . .Automaticallycontrolledreleasemethods. . . . . . . . . . . . . . . . . . . . . . . . . .

Shapingunit commitment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIXESA- DevelopmentofGrandCouleelinearizedturbineflowcurves. . . . . . . . . . . . . . .B - Lower Colorado linearized turbine flow curves . . . . . . . . . .C- Runnerreplacementtests at DavisDam. . . . . . . . . . . . . . . . . .D- Computerfilenamesand contents. . . . . . . . . . . . . . . . . . . . . . . . .

FIGURESFigure

123456789

Lindo historical model and Pascal model. . . . . . . . . . . . . . . . . . . . . . . .Lindo single cyclemodel and what's best single cyclemodels. . . . .Lindo cyclicmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Lindo single cyclemodel, unrestricted flows. . . . . . . . . . . . . . . . . . . . . .Lindo single cyclemodel, Parker Dam, flowrestricted.Lindo multiple cyclemodel, unrestricted flow. . . . . . . . . . . . . . . . . . . . .Lindo historical model. . . . . . . . . . . . . '. . . . . . . . . . . . . . . . . . . . .What's Best single cyclemodels, Davis and Parker Dams, flows limited. . . . . . .What's Best single cyclemodel, Davis and Parker Dams, flows limited

to :t10percentofthe averageremainingflow. . . . . . . . . . . . . . . . . . . .What's Best single cycle model, Davis and Parker Dams, flows limited

to :t20percentofthe averageremainingflow. . . . . . . . . . . . . . . . . . . .What's Best single cyclemodel, Davis and Parker Dams, flowslimited

to :t30 percent ofthe average remaining flow. . . . . . . . . .What's Best single cyclemodel-Davis Dam 15 and 3 percent and

Parker Dam 50 and 24 percent. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

11. . . . . .

12

111

Page

1

1

2

2

2233333

4444

5

666

7

17313537

889910101111

12

12

13

13

Figure

CONTENTS - Continued

FIGURES - Continued

13 Lindo multiple cycle model, shaping unit commitment-comparison of averageefficiencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lindo multiple-hour model - Davis Dam restricted to 1 to 5 units, Parker damrestricted to 1 to 4 units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lindo multiple-hour model - Davis and Parker Dams - units restricted to totalmegawattsgenerated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lindo multiple cycle model - Davis Dam restricted to 10 to 20 percent andParker Dam restricted to 4 to 7 percent of the total megawatts and flow. . . . .

14

15

16

IV

Page

14

14

15

15

INTRODUCTION

The Bureau of Reclamation is committed to maximizing power production through increasingthe efficiency of generator operations on the Colorado River. This effort started with theconsolidation of generator operations at Hoover Dam and Powerplant in i987, resulting inpowerplant efficiency increases up to 5 percent. In continuing this effort, a proposal was madeto study the benefits of consolidating all generator resources on the Colorado River.1Consolidating generator resources means that the generators are pooled and that any generatorin the pool can be used to provide the Basin (Colorado River Basin) generation and capacityrequirement. Capacity is defmed as the available megawatt output of a generator.

This study consisted of a linear programming optimization of the consolidated Basin whichincluded Hoover, Davis, and Parker Dams. Glen Canyon Dam will be added to the study at alater date. The goal of optimization was to increase efficiency by increasing power productionwith the available water. The optimization produces an hourly generation schedule that istermed unit commitment. The study involved optimizing the above powerplants actualoperations from April 17, 1988. The daily water releases from Davis and Parker powerplantswere restricted to be identical to the historical water releases; Hoover powerplant's daily waterrelease was not restricted.

CONCLUSIONS

The results of the study indicated that the Basin average efficiency can be increased by3 percent using the unit commitment schedule from the linear programming optimization. A3-percent increase in Basin average efficiency will result in an estimated increase in energy of176000 kilowatt hours per year. The efficiency increase was achieved simply by generating withthe most efficient generators at their maximum efficiency levels and by motoring the lessefficient generators to provide capacity as spinning reserve. Daily water releases from Davisand Parker powerplants can be shaped as desired without affecting this increase in efficiency.The shape can vary from a flat water release to a water release having multiple peaks.

The Electric Power Branch is designing the generation control function for the proposed LowerColorado SCADA (Supervisory Control and Data Acquisition) system. One element of thecontrol function will be to produce an op¥llzed unit commitment schedule. In this paper,linear programming was evaluated as an optimization method for the unit commitmentschedule. The advantages of linear programmiIig are: (1) ease of modeling, and (2) calculatingsolution in reasonable time. In addition, linear programming consistently produced the optimalsolution. The main drawback of linear programming was that it required linearization of thenonlinear turbine flow curves. The turbine flow curves used in this study were not completefrom 0 megawatts to capacity. A portion of the curve near 0 megawatts had to be extrapolatedas a linear segment. Linearization of the turbine flow curves introduces error. The errorgenerated from linearizing the extrapolated turbine flow curves is unknown. The HydraulicMachinery Section of the Electrical and Mechanical Engineering Division is developingcomplete flow curves for the Basin turbines. If linearizing the new turbine flow curves does not

1 S. C. Stitt, Colorado River Systems Operations Consolidation Proposed Concepts and Benefits, USBR Project Notes 90-24,September 1990.

introduce significant error, then implementing the optimization with linear programming willproceed.

LINEAR PROGRAMMING DEFINITION

Linear programming is a mathematical procedure for optimizing resources. It can be used onlyfor problems that are linear. The restrictions for variables in linear programming problems arethat they be proportional, additive and continuous. The solution of the linear programmingmodel is based on the Simplex method which proceeds from an initial feasible solution, throughbetter solutions, until the optimal solution is obtained. The linear programming softwarepackages (commercially available) used in this study were Lindo and What's Best.

MODEL FORMAT

The Linear Programming Model is optimized by minimizing a quantity called water energy inmegawatts.

n

Water energy = 8.46 x 10-5 L Hi x Qii=l

(1)

where: .

Hi = constant head of the ith turbine in feetQi = flow through the ith turbine in cubic feet per second

Each generator is modeled by a turbine flow curve. The turbine flow curve is produced from theturbine flow equation which contains head and megawatts as variables. Flow, Qi , used inequation 1 is calculated with the ith turbine flow equation using the head of the ith turbine andthe megawatts of the ith generator.

The historical data for all models consists of a 24-hour generation and capacity schedule andtotal 24-hour water releases for Davis and Parker powerplants from April 17, 1988. The basis ofall models is a I-hour linear programming model. whose function is to produce the hourlygeneration and capacity schedule and release a portion of the total Davis and Parker waterreleases. In order to generate the complete 24-hour schedule, the I-hour model must be run24 times.

MODEL TYPES

Lindo Single Cycle Model

The Lindo Single Cycle Model is the basis for all model development in this study; all othermodels are variations of this model. It runs a single-hour generation and capacity schedule andconstrains Davis and Parker Dam water releases.

2

The Lindo Single Cycle Model was used both with integer and noninteger variables. Integervariables can only assume a value of 0 or 1, while noninteger variables in our model werecontinuous from 0 to 1. The model contains variables for each generator which indicategenerator functions, i.e., on, off, motoring, or not motoring. In real operations, these functionsalways would be represented by integer variables. Initially, noninteger variables were used todecrease the problem's solution time. Later, it was discovered that using the IPTOLcommand-with integer variable problems-would decrease the solution time withoutsacrificing optimality in the solution.

The IPTOL command establishes a tolerance of a fractional number between 0 and 1. Once afeasible integer solution is determined, other solutions will only be considered if they canimprove on the current best solution by the tolerance. For example, if the IPTOL is 0.02 and thefIrst feasible solution is 100, then the program will only search for solutions better than 102. Allsubsequent models presented here use integer variables.

Lindo 24-Hour Single Cycle Model

The Lindo 24-Hour Single Cycle Model uses the Lindo Single Cycle Model to run the 24-hourschedule by running the model 24 times.

Lindo 24-Hour Multiple Cycle Model

The Lindo 24-Hour Multiple Cycle Model uses a multiple hour version of the Lindo Single CycleModel to run the 24-hour schedule. Because of limitations in the number of variables, thelargest multiple-hour model we could produce was 3 consecutive hours. The 3-hour model hadthe same hourly generation and capacity schedule as the single cycle model. The water releasesfor the 3-hour model were 3 times the I-hour model water releases and were allowed to bereleased in any hour during the 3-hour period.

Lindo Historical Model

The Lindo Historical Model was formulated to force the Lindo 24-Hour Single Cycle Model torun the generators exactly as they ran on April 17, 1988.

What's Best 24-Hour Single Cycle Model

The What's Best 24-Hour Single Cycle Model formulated the model in What's Best. What's Bestruns Lindo in a Lotus 1-2-3 environment.

Lindo 24-Hour Multisegment Flow Curve Model

The Lindo 24-Hour Multisegment Flow Curve Model was a variation on the Lindo Single CycleModel that accounted for the nonlinear turbine flow curves of the Grand Coulee generators.Unlike the Basin turbine flow curves, the Grand Coulee turbine flow curves could not be fit

3

with a single linear curve. To fit the curves, multiple linear segments were used. In the model,each linear segment was treated as if it were a separate generator.

CURVE FITTING

General

In using linear programming to model generators, linear approximations for the nonlinearturbine flow curves must be made. The PC-Matlab software program was used to fit linearcurves to the nonlinear turbine flow curves. The Lotus 1-2-3 software program was used with theturbine flow curve formulas to calculate the flows at various megawatt and head values.

Error Calculations

The error between the fitted curves and the turbine flow curves was calculated by subtractingthe fitted curve from the turbine flow curve and dividing by the full-scale turbine flow. Themaximum acceptable error between the two curves was 2 percent.

Algorithm

An algorithm was developed to linearize the turbine flow curves that consists of the followingsteps:

1. Select a head value of interest for each powerplant.

2. Use the turbine flow curve equations to find the following data for each turbine at itshead value.

a Flow at speed-no-Ioadb. Maximum efficiency loadc. Flow at maximum efficiency load

*d. 130 percent maximum efficiency load*e. Flow at 130 percent maximum efficiency load*f. 160 percent maximum efficiency load*g. Flow at 160 percent maximum efficiency loadh. Flow at capacity

.May not exist at higher values of head.

3. The linear segments are located between the following points:a Speed-no-Ioad and maximum efficiency loadb. Maximum efficiency load and 130 percent maximum efficiency load

*c. 130 percent maximum efficiency load and 160 percent maximum efficiency load*d. 160 percent maximum efficiency load and capacity.May not be required at higher values of head.

4. The equations for the linear segments between the upper and lower points given aboveare in the formy = mx + b.

4

m = flow at upper point - flow at lower pointMW at upper point - MW at lower point

(2)

b = flow at lower point (3)

The Lower Colorado Basin and Grand Coulee Dam turbine flow curves were fit with thealgorithm given above. The single segment linearized turbine flow curves developed for theLower Colorado Basin Powerplants are a good approximation for all generators except those atDavis Dam. The linearized turbine flow curves for Grand Coulee Dam must be multisegmented.The difference is that the Grand Coulee turbines are smaller relative to the size of theirgenerators than the turbines and generators of the Basin.

MODEL COMPARISONS

The data for all Basin models used in comparisons was from April 17, 1988. The generation andcapacity schedule came directly from the operation of the generators on that day. The waterreleases for Davis and Parker powerplants used in model comparisons were releases calculatedin the Lindo Historical Model. All models were required to produce:

. the exact generation schedule,

. capacity at or above the capacity schedule, and

. water releases approximately equal to the required releases.

A comparison was done between the Lindo Historical Model and the Pascal Model to comparethe accuracy of the results generated from the two different types of models. The Pascal Modelis a nonlinear simulation while the Lindo Historical Model is a linear simulation. Figure 1shows comparison of the average efficiency from the two models. The Pascal Model results areconsistently 2 to 2-1/2 percent below the Lindo Historical Model for all comparisons. The PascalModel releases slightly more water at each powerplant which is reflected in the decrease inaverage efficiency noted on figure 1.

The What's Best 24-Hour Single Cycle Model and the Lindo 24-Hour Single Cycle Model werecompared to verify that both these linear programming models were formulated in the sameway. What's Best runs Lindo in a Lotus 1-2-3 environment. With What's Best, the parametersof the model are entered into a Lotus 1-2-3 spreadsheet and What's Best converts theparameters into Lindo program code. The code generated by What's Best is not exactly thesame as the code we generated for the Lindo Model. This can be seen on figure 2 which showsthat while both models have the same Basin average efficiency, the efficiencies of the individualplants differ. Therefore, the models are not formulated in exactly the same way. The macrocapabilities of What's Best offer distinct advantages for performing calculations and alteringthe problem between optimizations. A macro is a program that executes commands in What'sBest.

5

CONCEPT TESTS

Manually Controlled Release Methods

To compare the Lindo 24-Hour Single Cycle or Multiple Cycle Models with the Lindo HistoricalModel, the Davis Dam and Parker Dam powerplant releases must be the same in all models.With the cyclic models, this was done by choosing the upper and lower limits for Davis andParker and running the models until the limits that produced the correct releases were found.

The models used in the comparison were a Lindo 24-Hour Single Cycle Model with anunrestricted flow and with the flow at Parker restricted to release between 9,000 to 9,270 cubicfeet per second per hour. The Lindo 24-Hour Multiple Cycle Model was a 3-consecutive hourmodel having an unrestricted flow.

Figure 3 shows the comparison of the average efficiencies of the Cyclic Models and theHistorical Model. The Cyclic Models differ in the way they run the individual powerplants, butall show the same increase in Basin average efficiency of 3 percent over the Historical Model.An interesting aspect in this comparison can be seen on figures 4, 5, and 6 is that the hourlyflows for Davis and Parker powerplants can be extremely different without effecting the Basinaverage efficiency. Figure 7 shows the Lindo Historical Model hourly flows. There appears to beno advantage in Basin average efficiency in using a multihour model versus the single-hourmodel, and the multihour model tends to produce unit commitment schedules that are notpractical.

Automatically Controlled Release Methods

The What's Best 24-Hour Single Cycle Model includes a macro that controls Davis Dam andParker Dam releases by adjusting the high and low limits of the water releases before eachoptimization. The limits were calculated using plus and minus percentages of the averageremaining flow. The average remaining flow is the total flow required minus the total flowaccumulated divided by the remaining hours and is recalculated after each optimization. Thelimits are calculated either with a fixed or variable percentage. A fixed percentage uses thesame percentage for every hour and produces a flat shaped limit around the average remainingflow. A variable percentage is a percentage that decreases by the same amount each hour andproduces a funnel shaped limit around the average remaining flow.

The limits used in the average efficiency comparisons on figure 8 were fixed percentages ofj:10 percent, X20 percent, ffiO percent and a variable percentage in which the percentage atDavis Dam decreased by 0.5 percent per hour from 15 to 3 percent and the percentage atParker Dam decreased by 1.5 percent per hour from 50 to 24 percent. Figures 9 through 12show the hourly flows of each run for the 24-hour schedule.

Several runs using variable percentages flow limits were made and it was found that the usablepercentage range at Parker Dam could be quite large, while the range at Davis had to berelatively small. If the initial percentage at Davis was large (>20%), the model solution did notrelease enough water in the beginning and a feasible solution could not be reached.

6

Although figure 8 indicates that the Basin average efficiency, with the:1:30 percent limit run isincreased, this is not the case. The increase in efficiency is due to the model solution notreleasing as much water through Davis Dam. Also, it should be noted on figure 11 that thehourly flows for Davis Dam double at the end of the 24-hour schedule. While the :1:30percentlimit run produces the same Basin average efficiency as the other runs, it's water releasepattern is the least desirable.

SHAPING UNIT COMMITMENT

An attempt was made at shaping unit commitment according to the hourly generation andcapacity schedules. A Lindo 24-Hour Multiple Cycle Model was used to deliver the exact waterreleases at Davis and Parker Dams. Shaping unit commitment was attempted by:

. Forcing at least 1 unit to remain on at all times.Varying the number of units running depending upon the generation required.Restricting the powerplants to provide the same percentage of generation they did in thehistorical data.

In the first run, Davis was allowed to run 1 to 5 units and Parker 1 to 4 units. Each 3-hourwater release was equal to an eighth of the total water release. In the second run, the unitcommitment was scheduled to correspond to the generation required per hour. If the requiredgeneration was 500 MW, then Davis could run 1 to 2 units and Parker could run 1 to 2 units. Ifthe required generation was 1,000 MW, then Davis could run 3 to 5 units and Parker could run2 to 4 units. The water releases of the second run were a percentage that varied according tothe generation schedule. In the third run, the unit commitment is shaped so that Davisprovided 10 to 20 percent and Parker provided 4 to 7 percent of the generation required eachhour. These numbers came from the average operation of Davis Dam and Parker Dam in thehistorical data. The water releases were proportional to the generation that each plant wasproviding during the 3-hour period.

A comparison of the average efficiencies for the above three runs and the Historical Model isshown on figure 13. Again, it is seen that the Lindo 24-Hour Multiple Cycle Models provide anincrease in Lower Colorado Basin average efficiency of approximately 3 percent over the LindoHistorical Model. As observed in the hourly flows for each run, on figures 14, 15, and 16,shaping the unit commitment according to the megawatts required for that hour produces thebest efficiency and hourly flow results.

7

90

a LINDO HISTORICAL MODEL ~ PASCAL MODEL

88

t) 86

~tJfi:~tlJ~ 84

82

80BASIN HOOVER DAVIS PARKER

Figure 1. - Lindo Historical. Model and Pascal Model. Comparison of average efficiencies.(4/17/88)

90 mLINDO SINGLE CYCLE MODEL ~ WHATS BEST SINGLE CYCLE MODEL

88

>- 86UZtlJtJfi:~tlJ~ 84

82


Figure 2. - Lindo Single Cycle and What's Best Single Cycle Models. Comparison ofaverage efficiencies. (4117/88)

8

90 mSING I ,E CYCLE, UNRESTRICTED FLOW SING LE CYCLE, PARKER FLOW RESTRICTEDMUI.TIPI.E CYCLE, UNRESTRICTED FLOW

~ IIISTORICAL MODEL88

t) 86Z~0&::~~

t~ 84


Figure 3. - Lindo Cyclic Models. Comparison of average efficiencies. (4117/88)

25

"b...x.!!1:: 15

§""'

~ 10

~ ~/

/

20

5

-- HOOVER~-+--

DAVIS--~-

PARKER

00 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 4.- Lindo Single Cycle Model, unrestricted flows. (4117/88)

9

'b...x-!!1: 15

ii

~

~ 10

~

'b...x.!!1: 15

§Iioo

~ 10

~

25

20,"

-,,"~ /

',----r/Y

/!~'i//

. . . . ..'~

/

--..,

/--",/.

"

, ').,

'~~/

~\',

///.

,",,~~'5

~-- HOOVER--+- DAVIS ~~ PARKER

0

0 1 2 3 4 5 6 7 8 9 10 1I 12 13 14 15 16 17 18 19 20 21 22 23HOUR

Figure 5.- Lindo Single Cycle Mode~arker Dam, flow restricted. (4117/88)

25

20

5

---HOOVER --+- DAVIS -- PARKER

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 6. - Lindo Mu~iple Cycle Model, unrestricted flow. (4117/88)

10

25

~...x-!!1:

§""'

~

~

15

20

10

5

-- HOOVER --+- DAVIS --+- PARKER

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 7. - Lindo Historical Model. (4117/88)

90

88

~ +/- 20% LIMIT

~ DAVIS /5%-3%, PARKER 50%-24%

m+/- 10% LIMIT

~ +/- 30% LIMIT

t) 86Z~0R::IJ..~~ 84

82


Figure 8. - Whafs Best Single Cycle Models, Davis and Parker Dams, flowslimited-comparison of average efficiencies. (4117/88)

11

25

~'"~ "\

,-- .. -'e- -8 \./8/ ~"'./'

- -~,

r~

20

'7:...x..!!t:ii9 15

""'~...=~

'--\/~

/~"\II,

.~ \ \\

10

./"'f>/--\ ~ /\.

/ \v. \ /+'

~/

'~\

\\

/r . ..10

.//../ -- HOOVER -- DAVIS -- PARKER

/

/"

52 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR0

Figure 9. - What's Best Single Cycle Madel-flows at Davis and Parker Dams are limitedto:t10 percent of the average remaining flow. (4/17/88)

25

20/

'7:...x..!!t: 15ii9""'

1.0Eo

/~.5

-8- HOOVER-+- DAVIS -.- PARKER

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 10. - What's Best Single Cycle Model-flows at Davis and Parker Dams are limitedto:t20 percent of the average remaining flow. (4/17/88)

12

~...x

~1: 15

~~ 10

~

25

20

/. . . /

5

0

I~.~/V

-- HOOVER -+-- DAVIS -- PARKER

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23HOUR

Figure 11. - What's Best Single Cycle Model-Flows at Davis and Parker Dams arelimitedto :i:30percent of the average remainingflow.(4/17/88)

~...x~1:: 15

~

~ 10

~

25

20

+--

5

/--

"/..

0

-- HOOVER-+-- DAVIS -- PARKER

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23HOUR

Figure 12. - What's Best Single Cycle Model-Flows at Davis Dam limitedto a variablepercentage from 15 to 3 percent and flows at Parker Dam limited to a variablepercentage from50 to 24 percent. (4/17/88)

13

90 e DAVIS 1-5 UNITS, PARKER 1-4lJNITS ~ UNIT COMMITMENT SHAPED TO MW

~ UNIT COMMITMENT SHAPED TO % OF MW

~ HISTORICAL MODEL88

~ 86ZJ,1J0~~J,1J

~ 84

82

80BASIN IIOOVER DAVIS PARKER

Figure 13. - Lindo Multiple Cycle Model, shaping unit commitment-comparison ofaverage efficiencies. (4117/88)

25

20

..0...x

~1:: 15

~~ 10

~

5

HOOVER -+- DAVIS PARKER

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 14. - LindoMultiple-Hour Model-Davis Dam restricted to 1 to 5 units, Parker Damrestricted to 1 to 4 units. (4117/88)

14

20

'b...x-!!!1: 15

§1100

~10...=

'b...x.

1: 15

§1100

~ 10

~

25

5

-- HOOVER --+- DAVIS --- PARKER

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 15. - Lindo Multiple-Hour Model- The number of Davis and Parker Dam un~s thatcan operate are restricted according to the generation required. (4117/88)

25

20

+--.

5

-- HOOVER --+- DAVIS--- PARKER

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

HOUR

Figure 16. - Lindo Multiple-HourModel-Davis Dam provided 10 to 20 percent of thegeneration required and Parker Dam provided 4 to 7 percent of the generation required.(4/17/88)

15

APPENDIX A

DEVELOPMENT OF GRAND COULEE LINEARIZED FLOW CURVES-Units G1 through G6

Appendix A contains the Matlab plot program generated during the development of the linearcurve fitting algorithm.

17

~x.

.;;-

*=~0ii:

.c:e::I...

l

sr

Max1mum Error - 53.181I

Average Error - 12.881.

1

Fined curve

2

0

Original curve

-10 40 60

Power 1n Megawatts

Figure 1 A.-

Using the first degree polynomial function of Matlab to fita single linear curve

to (units G1 through G6) turbine flow curves for a head of 220 feet (sheet 1 of 3).

20 80 100 120

MaxillUlI Error - 7.637S Average Error - 2.42416000

1000

Fined curve5000

4000

..;;-

*=

~ii: 3000.c:e::I...

Original curve

2000

00 40 60

Power 1n Megawatts

Figure 1 A.-

Using the first degree polynomial function of Matlab to fita single linear curve

to (units G1 through G6) turbine flow curves for a head of 300 feet (sheet 2 of 3).

20 80 100 120

18

5000MaX18U. Error - 1.577S

.Averege Error - 0.B299I

I

4500

1000

Original curve

4000

3500

.!!

'= 3000

~

~ 2500c:e::I

to- 2000

1500

Fined curve

500

00

~ L--------20 40 60

Power ln Megewetts80 100 120

Figure 1A. - Using the first degree polynomial function of Matlab to fit a single linear curveto (units G1 through G6) turbine flow curves for a head of 360 feet (sheet 3 of 3).

3500 II3000

ri

2500 -

M8x1- Error - O.4182S-r- ,-- Avera... Error - 0.08629S

i

1500 I

Fined curve

~cS~ 2000.c:e::I

to-

~

Original curve

1000

0 I t ~- --1-- I

0 5 ~ ~ 20 25 30Power 1n Megawatts

- I

35 40 45 50

Figure 2A. - Fitting turbine flow curves (units G1 through G6) from speed-no-Ioad tomaximum efficiency load for a head of 220 feet (sheet1 of5).

19

Fitted curve

500

0 -, --t-

0 10 20 30 40 50 60 70 80Power in Megawatte

Max1- Error - 0.- AverageError - 0.12081

1 r'---~---~' :..",,,,,

-'-r-- -

i

~~ ~

! II

I

aoor1

~

-; ~oo~,I

Originalcurve 1~ i Ii ~~

l

-t1

-~- ~-- - ~ J!SOO

0 L- - --4-- -- - -1--0 10 20 30

Power in Magawette40 !SO 60

Figure 2A. - Fitting turbine flow curves (units G1 through G6) from speed-no-Ioad tomaximumefficiencyload for a head of 240 feet (sheet 2 of 5).

3500M8x1- Error - 0.9842S

IAverage Error - 0.32991

1000

Original curve

3000

2500

...--= 2000

~u::.c:e 1!S00:::II-

Figure 2A. - Fitting turbine flow curves (units G1 through G6) from speed-no-Ioad tomaximumefficiencyload for a head of 300 feet (sheet3 of 5).

20

M8x18U8 Error - 1.72~14000 ,--T--r--- -t--- 'T-'

3500 rI!

3000

r

.;e 2100~

-= I

~I

i

-[

1500III

1000~

500l

\\

Average Error - 0.62781I I r r---

Fitted curve

~i

1

1

J

I

J

I

i;

1- ---- .-1--- ~

Original curve

OL-- --- .1--- 1- 1 l L 1 I.----

0 W ~ ~ ~ 50 ~ ~Power in Megawatts

80 90 100

Figure 2A. - Fitting turbine flow curves (un~s G1 through G6) from speed-no-Ioad tomaximum efficiency load for a head of 240 feet (sheet 2 of 5).

41000M8X18U8 Error - 2.!S44I. , Aver818 Error - 0.91321

. . .

1500

Original curve

~

3!IOO

3000

.!!! 2500..

-=

~~ 2000c:e::II-

1000 Fitted curve

!!GO

00

--I10 ~ 30

-L .-L L--~o 50 60 70

Power ln Megawatts80 90 100

Figure 2A. - Fitting turbine flow curves (un~s G1 through G6) from speed-no-Ioad tomaximum efficiency load for a head of 360 feet (sheet 5 of 5).

21

4.5

4

0 3.5:-...x i..;;-

3r-=

~ii:

2':~

.c:D!:iI-

,.:~

0.5

0 ~-- -~---1---0 20

5

4.5

4

I3.5~

0 i...

3~x

~-=

~2.5ii:.c:e

~2I-

1.51

1r

0.51-

00

5

Maxi..,. Error - 1.4991-r--- I

Average Error - 0.88511I r--

Maximum error = 1.5%


Fitted curve


Original curve

---~-~---1--~ L---~-- L--_~ I--~ _1 120 40 60 SO 100 120

Power in Megawatts

Figure3A.- Fitting turbine flow curves (unitsG1 through G6) past maximum efficiency

load for 50 percent of maximum efficiency load for a head of 220 feet (sheet 1 of 3).

MIld..,. Error - 17.41y------

Average Error - 5.7991I I


Fitted curve

...;~ '='17'~


Original curve

40 60Power in MII,awatts

SO 100 120

Figure 4A.- Fitting turbine flow curves (units G1 through G6) past maximum efficiency


22

MeX1.u8 Error - 1.5851 Aver.ge Error - 0.7B231

-[, ~ -- ---~ ,------

: I

I

""'mom.~ - '-73"

]

1

Fitted curve

"

1L

=

-:~oO

ol \~

ll

'

D Maximum error=1.6%

iI

'"~ 4000r '"

~

-~Ori.~~

~

2OO0t --~

~~---~ ~--~ , J0 40 60

Power 1n Megewatt.

Figure 4A. - Fitting turbine flow curves (un~s G1 through G6) past maximum efficiency


20 80 100 120

Maxl8U8 Error - 1.817S6000 ~_n ,--- - --r-

Aver... Error - 0.78421-, ,-'--

Fitted curve


I

~

I

I

"'gl~ ~N'

~

I

5000

~

~

j ~I~ 3000

II

I

_°,

,...

r

/

--nL

20_L ---- -- L--- L---- -- - J.__._-----40 60 80 100 120

Power 1n Megew.tte

00

Figure 4A. - Fitting turbine flow curves (un~s G1 through G6) past maximum efficiency


23

5

4.5

4

3.5

~x3.

..--=

~2.5ii:CDc:e 2:I....

1.5

1

0.5

00

Maximum Error - 0.3B05~-----.

Average Error - 0.126B~


Fitted curve


Maximum error= 1.8%

Original curve

- ---1 ~ 1-------20 40 60

Power in 'Megawatts

Figure 5A. - Fittingturbine flow curves (units G1 through G6) past maximum efficiencyload for 30 percent of maximumefficiencyload fora head of 220 feet (sheet 1 of 3).

BO---1 ~

100

Maximum Error - 2.1311 Average Error - 0.7E181

:~~-~-r, u . -~~-r r-m._-

I,i

7000 ~,I

~...0

r~ ,~ 5000 iCD

'c:e:I

.... 4000

ri

3000 :-IIII

2000 l-

I

1000

Maximum error = 2.1 %


Original curve

IO~

0--1 --

20--L ~~.-L ~

- --J--- --. J --- -40 60 80 100

Power 1n Magawatts

Figure 5A. - Fittingturbine flow curves (units G1 through G6) past maximum efficiencyload for 30 percent of maximumefficiencyload fora head of 260 feet (sheet 2 of 3).

24

120

iI

1

~III

11

~

- J120

.!!!..4::

~u::

I!! 3000

:e::I...

.!!..4::

~u::.c:e::I...

Maximum E~~o~ - 0.853916000

5000

4000

Ave~8ge E~~o~ - 0.34981I I

Fined curve


Maximum error =~

2000

1000

00 20 40 60

Powe~ in Megawatts

Original curve

80 100 120

Figure 5A. - Fitting turbine flow curves (un~s G1 through G6) past maximum efficiencyload for 30 percent of maximum efficiency load for a head of 300 feet (sheet 3 of 3).

480MIIx &~or Seg 1-1.3681 Max E~~o~ Seg 2-0.71991 Mex E~~or Seg 3-0.623S

-r-'I I

460

440

420

400

380

360

340

320

Original curve

300

280220

L240 260

..L-280

Head in feet

Fined curve

/

/Segment 2

1

.

25

Segment 3

- --L__-320 340 360300

Figure 6A. - Fitting head versus turbine flow curves (un~s G1 through G6) in three linearsegments for 0 MW (sheet 1 of 2).

5

4.5

4

'b3.5

...x.

;;- 3..,:

~ii:

2.5CDc:e::J

~2

1.5

1

0.5

0220

4000

3900

3800

~3700..,:

~0

ii: 3600CDc:e::J

~3500

3400

3300

3200

3100220

Max Error Seg 1-31.671 Mex Error Seg 2-1.2231 Max Error Seg 3-0.!1331

Seglllent 1

Fitted curve

Segllent 2

Seglll8nt 3

Original curve

---J.-240

..t

340 360320280Heed 1n feet

Figure 6A. - Fitting head versus turbine flow curves (units G1 through G6) in three linearsegments for 120 MW (sheet 2 of 2).

260 300

Max111UIIIError -3.8091 Average Error - 2.1621

Fitted curve

Original curve

--1-..--240 260 300280

Heed 1n ft.320 340 360

Figure 7A. - The curve is head versus flow (units G1 through G6) at maximum efficiencyload for that value of head. The curves are fit in one or two segments (sheet 1 of 2).

26

Maximum Error in first segment - 0.6981S Maximum Error in second segment - 1.89214000

3900

3800Fitted curve

~ 3700c:

tiL 3600.ci!::I

I- 3500

Original curve

3400

3300

3200

3100 1--- 4 ~220 240 260 280

Head in ft.

Figure lA. - The curve is head versus flow (units G1 through G6) at maximum efficiency

load for that value of head. The curves are fitin one or two segments (sheet 2 of 2).

300 320 340 360

4900, -r- .-Maximum Error - 0.51121

I I ,--,

4500

Original curve4800

4700

~c:

! 4600.ci!::II-

Fined curve

4400

4300 1 J 1--1 I-220 230 240 250 260 270

Head in ft.

Figure 8A. - The curve is head versus the flow (units G1 through G6) at 130 percent

maximum efficiency load for that value of head. The curve isfitin one linear segment.

._~-~280 290 300 310 320

27

=l--~Maximum E~~o~ - 1.7641

- T --.-_d_-

t -.. r---

Ave~age ~ror - 0.68141--r

Fittedcurve

2000

Original curve

~4::

~ii: 1500. ,

C i:e I:I I

I-I

1000 ~I

i

I

I500

I0 L- .. ---L ---I J J ~--I '

0 5 W ffi ~ ~ ~ ~Power 1n Megawatts

~--!----40 45 50

Figure 9A. - Fitting turbine flow curves (pump-generator units PIG 7 and 8) to maximumefficiency load for a head of 260 feet (sheet 1 of 3).

~4::

~ii:.C:e:II-

=l--~I

I

I

1500 ...

I

II

Maximum Error - 1.0841.._--, n-r ,Average ~ror - 0.47781

I , I

1000

\\

Original curve

Fitted curve

I

I

0L I J ~--- -~.-0 5 10 15 20

Power

-~--~--~-- ~- ---~_J~ 30

1n Megawatts35 40 45 50


28

,

BOOl

!I

600r

400 t-!i

2000 5

Mex18U8 Error - 3.8641 Average Error -:l-~---r~-'---'-~'

,

1600rI

1.9691, ,--

i;

{ 1400

r~~ 1200 r-c:e:I...

Original curve

/.

\~I,I

1000 ~I

Fined curveBOO!..

600

400

200 -1. 1 1 1. L-- - 1-- l--. - - 1--- ---1---0 5 W ~ 20 ~ ~ ~ ~ ~

Power in Megawatts


2000,--

.

1800

t

l

1600

I

i~ 1400 rc::: I

! !~ 1200~C I:e:I...

M8X18Um Error - 3.8951--r_n_-T"-_- -.-"T-__n- r I .---

Average Error - 1.8861--y---~ I

I

1000t!

..

Original curve

L 1 L ~- L--- ---1 1------~W ~ 20 ~ ~ ~ ~ ~

Power in Megawatta

Figure 10A. - Fitting turbine flow curves (pump-generator units PIG 9-12) to maximumefficiency load for a head of 370 feet.

29

jI

50

l

1

50

2500 Selllll8nt 1

2400

Fined curve

.!-

I4:: 2300

~ii: SeDlI8nt 2.cI:e 2200

~...

Original curve Sell88nt 32100

~l1900 -- ~----J-

260 280 300 320 340 360 380Head 1n feet

2600Max Error Beg 1-0.25921 Max Error Beg 2-0.6543S Max Error Sell 3-0.2892S

~j

.

-

Figure 11A. - Fitting head versus turbine flow curves (pump-generator un~s PIG 7 and 8)in three linear segments at 50 MW.

Max Error seD 1-0.8871 Max Error Sell 2-1.6711 Max Error seD 3-O.!HI6713200 , r r -'-r-- r--

3000

~Jof

2600

~

'

~ii:. I

I 2400

I

2200~

~OO~

1800 L__-260

Fined curve

Sellll8nt 1

Sellll8nt 2

Original curve Sell-nt 3

1--

300 320Head 1n feet

Figure 12A. - Fitting head versus turbine flow curves (pump-generator un~s PIG 9-12) inthree linear segments at 50 MW.

280 340 360 380

30

APPENDIX B

LOWER COLORADO RIVER BASIN LINEAR TURBINE FLOW CURVES

Appendix B contains the linearized Lower Colorado Basin turbine flow curves that producederrors of greater than 2 percent using the curve fitting algorithm.

31

3000

2500

2000

~c:~u: 1!500CDc:e::sI-

1000

3000

2500

..

Vi'c:

~u:CDc

:e::sI-

2000

1500

1000

Max18ull Error - 2.2OtS Average Error - O.!5691S

1500

Original curve

Fitted curve

00 20 040 80 eo

Power in Mell8watts100 120 1040

Figure 1B. - Fitting turbine flow curve (unit AS) in a single segment at a head of 552.47 feet.

Max111U8 Error - 4.4!56S Aver.g. Error - 2.4491

Fitted curve

Original curve

!50010 50 80 70

Power in Megawatts

Figure 2B. - Fitting turbine flow curve (unit N8) to maximum efficiency load at a head of552.47 feet (sheet 1 of 2).

20 30 40 9080 100 110

32

2.wo

2200

2000

. 1800...c::

~u:: 1600.c:e:a... 1400

1200

1000

800

60010 20

6000

!5OO0

4000.

...c::

~~ 3000c:e:a...

2000

1000

Mlxl8U8 Error - 0.8421 Average Error - O.5457S

Fined curve

\\Original curve

40 !50 60Power 1n Mlgawatte

Figure 28. - Fitting turbine flow curve (untt N8) to maximum efficiency load at a head of

552.47 feet (sheet 2 of 2).

30 70 10080 90

Maxl- Error - 8.288S Aver... Error - 4.097S

Original curve

Fined curve

00 3520 2!5 30

Power 1n "gawetta

Figure 38. - Fittingturbine flow curves (units D1 through D5) in a single segment at a head

of 139.57 feet.

!5 10 15 40 4!5 !50

33

APPENDIX C

RUNNER REPLACEMENT TEST AT DAVIS DAM

A test using the What's Best 24-Hour Single Cycle Model was done to show the increase inefficiency that could be achieved by replacing the turbine runners at Davis Dam. The slope ofDavis turbine flow curves was changed to represent this improvement. Figure IC shows acomparison of the average efficiency between the original runners and the upgraded runners.This test shows that the Basin average efficiency can be increased by 1 percent if the suggestedupgrading the Davis Dam turbine runners occurs.

90

~ Original Runners

~ Upgraded Runners

88

t 86Z~t:J~~~~ 84

82

80TOTAL HOOVER DAVIS PARKER

Figure 1C. - Comparison of average efficiencies between original and replacement turbinerunners (4/17/88).

35

APPENDIX D

COMPUTER FILENAMES AND CONTENTS

There are five directories on the three Linear Programming Concepts for Unit CommitmentDisks. The directories are related to the types offiles they contain.

Disk No.1 of 3

\WP51

LPUC.DOC - this is a Word Perfect 5.1 file that contains the report.

\ 123R31

IHLP -.RES.WK3 - A worksheet that contains the result summary of the Lindo 24-Hour SingleCycle Model with Parker flow restricted.

IHLP _UN.WK3 - A worksheet that contains the result summary of the Lindo 24-Hour SingleCycle Model with no flow restrictions.

3HLP_UN.WK3 - A worksheet that contains the result summary of the Lindo 24-Hour MultipleCycle Model for a three consecutive-hour model with no flow restrictions.

AVE_EFF.WK3 - A worksheet that contains the results of the I-Hour Model, restricted andunrestricted flows, the 3-Hour Model, and the Historical Model.

GC.WK3 - A worksheet that contains Grand Coulee turbine flow curves.

GCLIN.WK3 - A worksheet that contains a summary of the points used to generate the multi-segment linear curves for all Grand Coulee generators.

GCOLD.WK3 - A worksheet that contains Grand Coulee turbine flow curves for the old units.

H3ATEST.WK3 - A worksheet with the result summary from the first run of the 3-hour LPmodel from the unit commitment shaping test.

H3CTEST. WK3 - A worksheet with the result summary from the second run of the 3-hour LPmodel from the unit Commitment shaping test.

H3DTEST.WK3 - A worksheet with the result summary from the third run of the 3-hour LPmodel from the unit commitment shaping test.

HIST.WK3 - A worksheet the contains the result summary from the Lindo Historical Model.

HISTDAYl.WK3 - A worksheet that calculates the releases from Parker and Davis from theoperation of the generator in the historical data of the day.

Disk No.2 of 3

\LINDO

IHRLP.DAT - the Lindo 24-Hour Single Cycle Model.

37

3HLP.DAT - The Lindo 24-Hour Multipl~ Cycle Model for 3 hours.

GCMODEL.DAT - The Grand Coulee I-hour Multisegment curve Model.

HIST.DAT - The Lindo Historical Model.

\MATLAB

GCFITI.M and GCFIT2.M - Are M-files that can produce multisegment linear curves from thenonlinear flow curves.

Disk No.3 of 3

/WB

EFFCOMP.WK3 - A worksheet that contains a result summary of the flow limiting runs inWhat's Best.

LCDATA.WK3 - A worksheet where data for the 24-hour generation and capacity scheduleand water releases are entered and from which a 24-hourWhat's Best Modelis run with amacro.

LCOPT.WK3 - A worksheet that contains the What's Best Model for the Lower Colorado UnitCommitment optimization problem.

LCOPTI.WK3 - A worksheet that contains the What's Best Model with a macro that controlsthe releases at Davis and Parker with a funnel shaped limit.

LCOPT2.WK3 - A worksheet that contains the What's Best model with manual control of thereleases at Davis and Parker.

LCOPT3.WK3 - A worksheet that contains the What's Best model with a macro that controlsthe water releases at Davis and Parker with a fixed percentage.

RESULTI.WK3 - A worksheet that contains the result summary of the :f:20percent run.

RESULT2.WK3 - A worksheet that contains the result summary of the :i:30percent run

RESULT3.WK3 - A workSheet that contains the result summary ofthe What's Best 24-HourModel with Parker restricted.

RESULT4.WK3 - RESULT9.WK3 - Worksheets that contain the result summaries from theDavis runner replacement tests.

38.u.s. GOVERNMENT PRINTING OFFICE 1992-0-673-185/40013

Mission

The mission of the Bureau of Reclamation is to manage, develop, and protect water and related resources in an environmentally and economically sound manner in the interest of the American public.

A free pamphlet is available from the Bureau entitled, "Publications for Sale." It describes some of the technical publications currently available, their cost, and how to order them. The pamphlet can be obtained upon request from the Bureau of Reclamation, Attn D-7921, PO Box 25007, Denver Federal Center, Denver CO 80225-0007.

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