Optimal Operation of a Flood Control Systen a Thesis

The University of Manitoba

Optimal Operation of a Flood Control Systen

John Nairn MacKenzie

A Thesis Submitted to the Faculty of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of It{aster of Science

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Arrmrcf I q7R C)PTiIIAL OPEIIAÏIÛN OT A FLOOD CÛNTRC]L SYSÏËi4

JOHN I.IÄIRÎ''J I.4¡CKEI{Z] E

A clissertation submiÈted to tåte Facr*lty of Cr*ci".¡aÉe Stuse.åies of' tlre University oá' Munitoba ire partLrl f'uifilln'le¡rt ol' t!re requle'ernemts ol'tlre clegrce ot

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S¡TV Ol.' h{.4þ]lTO!3"4 tr¡ lenei q¡r s¿:l! copies oü t!¡is cåÈ*;ertati{!r¡" 8o the N,4TltNAL [.N8¡4,ARY ûFr {lA"l\Ai}A to l¡riar¡:*'ilrsr tl¡ls ciissertafio¡r and tr: lend or sell copies of the felnr, und {,JNl'VUåì,5åTY h{lCfì,OFlLh{S tr: pubiish i¡¡r ahsåraet of this rJissert¿¡f ion.

The ar¡thor re$erve$ otå:cr publ!c¿¡Èir¡¡r rights, ¿¡¡rJ ¡"¡either the dissertation ncr exÉensive cxtri¡cfs fr¡:nr it mluy h*: gx'int*d eir otl¡er- wise repi:oeíLlced evithout t !¡c uuf l¡o¡''s writËc¡r 3:cnrr ieslr.xr" ACI$IOWLEDffiIvE\TS

I would like to e4press my appTeciation to my advisor, Professor G. Booy for his guidance and patience throughout the somewhat long gestation period of this thesis.

I would also like to express my appreciation to Mr. P. Flatt of charles Howard and Associates, consulting Engineers, for making available the linear progranrning solution algorithm used in this thesis and his comments regarding the solution algorithm in particular and the linear programning technique in general-

I would like to express my most sincere appreciation to my wife,

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onbruol^l TABLE OF CONTENTS

Page

Acl,noivledgements i

I i cf nf Fi or rrpq Vi

1.0 INTRODUCTION 2

1.1 A Flood Control System La

7.2 Sinulation Analysis 3

| ( f lnlrmf 7aT1î\'**-rr furdaj/5_L> 4

7.4 Optimization-Símulation Comparison 5

r.J1 q T'harlrv r-ÌnfimirationvIJur¡ttr¿ l4ode1 Selected in this Tþesis 6

2.0 TTIE ASSINIBOINE RIVER SYSTEM l-U

2.I History 10

2.2 Flood Control Tnvestigations 10

2.3 Shellmouth Reservoir 72

2.4 Assiniboine River Diversion 13

3.O THE GENERAL LINEAR PROGRA}O4ING Ì\4ODEL FOR A FLOOD CONTROL SYSTEM 22

3. 1 The Linear Progranming Tecirnique 22

3.2 The }{odel Descríption 25 3.3 Linear Progrannning }lodel Coirstraints

3.3.7 Physical Constraints 27

3 .3.7.7 Storage Constraints 27

3.3.7.2 Reservoir Mass Balance Constraint 28

111 Table of Contents cont. Ðc'_*Þ: ¡a 3.3.I.3 Reservoir Release Constraint 28 3.3.7.4 Cha:r¡rel Continuity of Storage Constraint 29

3 .3 .1.5 RiParian Constraint 29 3.3.7.6 Diversion Capacity Constraint JU 3.3.1.7 Upper Limit on Diversion Flow Values 30 3.3.2 Coniputational- Constraints 3L 3.3.2.I Peak Discharge Constraint 3L 3.3.2.2 Non-Negativity Constraint 32 3.4 Objective Functíon )L 3.5 Time Dependency of Objective Fr-urctíon at Danage Centres 33 3.6 NonlinearitY 35 3.7 Size of the Linear Progranu.ning Problem 3B

3.7.7 Elinination of Variable Qi^t 38 3.7.2 Elinination of Fixed Valued Variables 39

4.0 ftIE LINEAR PROGRA}MING MODEL APPLIED TO 'IT]E ASSINIBOINE RIVER SYSTM,Í 46 4.7 TLe Assiniboine River lt{odel 46 4.2 Study Operational Rules 47 4.3 Flood Damage Fulctions 4B 4.4 Specific Linear Prograruning }lode1 for the Assjniboine River SYstem 53 4.4.I Physical Constraints 53 4.4.7.I Storage Constraints 53 4.4.7.2 Reservoir Mass Balance Equation 53 4.4.7.3 Reservoir Release Constraint trA Table of Contents cont.

T)a¡a

4.4.7.4 Riparian Constraint 54 4.4.L.5 Diversion Capacity Constraint 55 4.4.7.6 Upper Limit on Divcrsion Florv Values 55

4 .4.2 Contputational Constraints 56 4.4.2.1 Peak Discharge Constraint 56

4.5 Sumrmry of Problem Constraints 57

5. 0 COIPUTATIONAL TECFil{IQUES 77 5.1 Ïte ComPuter lt{odel 77 5.2 lt{atrix Generator 77 5.3 Thc Liucar Plogramning Solution Algorithm 79

5 .4 'Ihe RePort Writer 79

6.0 RESULTS B2 6.1 The Optirriization }4ode1 BZ 6.2 Discr-rssion of Results 84 6.3 Model Run Cost B6

6 .4 Surnurary Conclus ions B7

APPENDIX A 100

APPENDIX B L46

BIBLIORAPFry 1s8 LIS'I OF FIüMES

Þa oa

II-1 Assiniboine River Drainage Basín 77

II-2 Shellnouth Reservoir, Location 1B II-'3 Assiniboine River Díversion, Location 19 II=4 Assiniboine River Diversion, Layout Details 20 III.1 Sanple Linear Progranirning Problent 47 III=2 Hypothetical River Basin System 42 III-5 Definítion of Tine periods used in Aria11'sis A7 III-4 Scparable Programning Example Figure 44 IV-1 Assiniboine River Drainage Basin IV-2 l\ratcr Sur-vey of Canada Streamflow Gauging Locations in the Study Area 64 IV-3 Discharge Damage Relationship - Assiniboine Rir¡er at

Russell 65

IV-4 Discharge Damagc Relationship - Assiniboine River al-

Ir{iniota 66 n^*^ n^1^+: TV- 5 Di sclrarse UcLllldgv^^ L l-uj^*^llio l>.--r - Assiniboine River AL Þ- ^f'-Ld Brandon 67

a1- IV-6 Discharge Damage Relationship - Assiniboine River dL

Pnrt:oe le Prairie 68

IV-7 Discharge Damage Relationship - Assiniboine River Diversion ov rr--^-^ Tr^1^+;'^-^hin TV-B Discharp-e----- ò- lJdllidgE r\Uf d Ll-\-rlI>rrrlr - /Ass'inìboine LJJ !rrruvri River

!{oodino1 er¡ 70

V1 List of Figures coqt_.

T)c ra

IV-9 Discharge Damage fte1atíonsìlip - Assiniboine Ríver at

l\Tinnipeg /I ry-10 Elevation.Ðisc]-iarge Relationsliip Shellmouth Reservoir 72 W-11 Elevation=storage Rclationship Shcllmouth Reservoir 73

IV-72 Outflow-storage Relationship Shellmouth Resen'oir 74

IV=13 Schematic Repr-esentation of Assiniboiire River Basin 75

W-1 Natural and Optimal Flow Values, Russell BB VÍ-2 Natural and Optimal Flow Val.ues, tr{iniota 89 VI-3 Natural and Optimal Flor.^¡ Values, Blandon 90 VI-4 Natural and Optimal Flow Values, Portage 1a Prairie 91

VI-S Natural arrd Optimal Florv Values, I"leadingley ^a 'l{imipeg Vi - 6 Natural and Optinal Flow Values , VI-7 Optimal Shellmouth Reservoir Operating Schedule 94 VI-8 Optinal Assiniboine River Divelsion Operating Schedule 95 \iI-9 Output fr'om Report lfriter 96 VI-10 Output from RePort l\rriter 97

VI-11 Output from Rcport j\'riter 98

vii 1.0

INIRODUCTION OPTIMAL OPERATION OF A FLOOD CONTROL SYSTEM

1.0 INTRODUCTION

1.1 A Flood Control SYstem

Two kinds of flood control works may be actively operated in order to alleviate flood danage, reservoirs and diversiors. Reservoirs reduce flood damage by storing flood waters and releasing them at a later date so as to reduce the peak of the flood wave and lengthen its duration. Diversions sinply divert flood waters away to an area less prone to flood damage. An operable flood control system as discussed in this thesis is composed of reservoirs and diversions in addition to passive flood control rvorks such as dykes. The object of operation of any flood control system is to minimlze flood damage. This requirement defines the optimal operating schedule for each of the system components. The degree of complexity of the optimal operation of a flood control systeilì is deternined by the nr¡rber of flood control reseryoirs and/or diversions which conrprise the system for each flood control component requires its oirrn operating schedule. Since the operating schedules of the components of the system may be hi-ghly inter-related it may be appreciated that the difficulty in arriving at optinal operating schedules increases very rapidly as the ru.mber of components of the flood control systen increases. The opti-mal operating procedure is based on anticipated florvs. The r:ncertainty in forecasting flows therefore adds a degree of r.rncertaintY to the optimal operating procedure. It need not add to the complexity of the analysis if the rincertainty itself is not a factor in the objective of the system oPeration. 2 In this thesis all flow values have been treated as being deterministic and no allowance for stochastic variation has been included.

Stochastic variation of flow values could be addressed by mearìs of new estinates of flow values which would then be subject to another analysis.

In an operational flood control context this might take the form of a high, best and low estimate of systen in flows with each estimate requiring a separate analysis. In this manner any stochastic variation in flow estimates can be partially addressed within the deterministic frame- work of the model constructed in a nanner acceptable for operational flood

control ÐurDoses,

7.2 Sinrulation Analysis Sinrulation analysis is a conmonly accepted method of arrivi¡rg at an operating schedule for a flood control systen. In a simulation analysis, the river system is modelled so that movement in time and space of river

flows throughout the river basin may be simulated. This is generally done by mathenatical nodels of the river basin rather than by analog models and is carried out on an electronic computer. Forecast inflows to the river basin are innut to the simulation

^-c +L^ ^--^fem -^1^-1,-1 ^ i- f,^'- -^J^1llluL-tgl_ uI Lllg 5y>LCilr drruencl d1An 'JpErdLIrrBl-lne.ì2*:-- >LlluuL]l.c 15 -^^+,.'1PU5Lur-d"LUL.L -+^J luI eaCh system component. The output from the simulation model consists of discharges and water 1eve1s at various locations of interest in the river basin. Each nrn results in a calculated value of flood damage. Operating schedules for each system component are changed between runs so as to jm- prove the perfonnance and the sj-mulation analysi-s is repeated. The re= sults of individual sjmulation runs are compared and revised operating

schedules are postulated until a best operating schedule for the system 3 components is arrived at. No systematic search for optirnal operating schedules of system components is necessarily iniplied in a simulation analysis although it can be incorporated therein' The nain difficulty with this technique lies in the revision of the operating schedules for individuaL system components so as to improve the performance. If the variables are changed one by one by sma1l amouttts' the amormt of work required to achieve the optimr.un becomes prohibitive. If larger steps are taken or Several variables changed at once, the analysis may well miss the optimr-m altogether. Therefore, simulation analysis works well to refine initial postulates of system corrponent operating schedules. If the initial operation postulates are far from optimal simulation nay give problems.

When the flood control system is not overly complex the limitations of sinrulation a¡ralysis are not severe. However, as the complexity of the system increases it may prove very difficult, if not impossible, to find the optimal operating schedule through simulation analysis-

1.3 Optinization AnalYsís Optini.zatj-on techniques differ from simulation in that a systematic search for the optinir-mr operating schedule is incorporated in the technique. Theoretica1Jy, this would solve the problem- However, in practice it is often found that the large nulber of variables involved makes an optimi zation technique much too laborior'rs even with the aid of large electronic computers. It is possible in nany cases to set the target in optimization somewhat lower and to aim for a solution that is not too far away from the optimrln' The so ca11ed fine tuning can then be acconplished by means of sintulation analysis'

¿. In optimization, all the components of the flood control system must be arnlyzed simultaneously over their entire range of possible operating schedules. In addition, the technique mrst realistically reflect any operating schedule inter=relationships between individual system components. The time interval, however, caÌr be increased substantially to reduce the nurber of variables.

An optini zation technique l

I.4 Optimization-Slmulation Conrparison

LIUII allu Jlllul@Lavl¡ nrocesses fL^ LUllqJdI^^--^*;^"rn -L>U-^ of-- the---* ont'imization-l- --..-'@ and simulation y¡vuvJJvJ is in order. Optimization tends to look at the overall picture with respect rn nner:fino s,chedules of individual flood control systeni components. In an optin,ization analysis the entire physically feasible range of operating schedules for operating system components is ana1yzed. Generally, the flood control system is not modelled in great detail with respect to either the time step employed in the analysis or the degree of exact modelling of the systen under consideration. Simulation analysis, on the other hand, tends to look at a very 1in- ited picture with respect to operating schedules of the individual flood

5 Control System components. fn sjmulation analysis a nalro\^I range of operating schedules for individuaL system coniponents is analysed. Gen- era11y, the system is modelled in considerable detail with respect to the tìme step employed and the degree of exact modelling of the system. It may be appreciated that a trade off in tenns of computational burden a¡d exactness of results is involved in sirnulation and optì:r'iization analysis. A point may be made that the optiinization and sjmulation analysis processes are always carried out together in the determíning of optímal operating schedule for flood control components. For the case of less coniplex flood control systems the optimization analysis takes the form of intuitive reasoning by the person responsible for the analysis ' For more conplex flood control systems, a more formal optimi-zation process, together with the all important intuitive reasoning of the person responsible for the analysis, is felt to be more appropriate.

1.5 The Optjmlzation Model Selected in this Thesis A linear programming technique was selected as the optimization model in this thesis. It is intended to optìmize operation of flood control works so as to mÍnimize damage caused by flooding. Flood damage is represented by flooded area in this context. Iniplicit in this statement is the assumption that all flooded areas in the model have the same do11ar value per acre with regard to flood damage for a gíven ti-rne of yeaT. This assr.nrption may be relaxed quite easily should a particular river systen require different dollar peï acle values of flood damage' The linear progranmring model constructed herein produces opti:na1 mean seven day operating schedules for each of a flood control system's control

o works on the basis of ninimizing total flooded area as a function of peak mear seven day discharges at selected damage centres throughout the flood control system. Peak mean seven d.ay d"ischarges at each d.amage centre in the analysis, together with their corresponding damage coefficients, form the objective firnction in the anal-ysis. The total value of this objective function is to be ininimized. No limitation as to the nunber of flood control reservoirs or diver= sions in the flood control system is i-rqplied by the model although in reality the computational capacity of the electronic computer employed in the analysis or the cost factor of the analysis itself would be the limiting factor as to the size of the systen lvhich could be optimized attd/or the detail to which the system could be modelled in the optim:_zat:.on. The optimtzatíon model is formulated to address river basins in which the predominant danages due to floodíng are agricultural in nature. In this respect the phenornenon that agricultural flood danages are tjme dependent is addressed in the model. This feature is an extension rather than a restriction to the use of the model as non=agricultural damages, such as flooding of buildings or loss of bridges are non time dependent and may be addressed directly by the model constructed. The general optimization model constructed ín this thesis was applied to the Assiniboine River in lt4anitoba to assess its applicability. It should be noted that the flood control system in existence on the Assiniboine River system may be treated as one j-n which acceptable postulates for a simulation analysis may be obtained by intuitive reasoning rather than requiring the use of an optimization model. However, it should also be noted that it is this ver/ norr.co:nplexity of the flood control system in terms of the mmber of operable system coilponents that allows a

7 reasonable check of the applicabiltty of. the optimlzation model as com= pared to intuitive reasoning. The linear prograilûlting optimization model constructed in this thesis effectively reflects the inter-relationships of the flood control system on the Assiniboine River in ]t{anitoba and produces acceptable starting postulates for a simulation analysis. 2.0

THE ASS]NIBOINE RT\ER SYSTHVÍ 2.0 THE ASSINIBOINE RI\/ER SYSTE}4

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Since the begirLning of the settl-ement of the west the Assiniboine River has had a record of destructive floods. 0n several occasions during the 1800's floods did occur but nuch of this information is very sketchy. Detailed records have been kept since 1913. In the latter period there were seventeen years in which major flooding occurred-

The Assiniboine Ri-ver flows in a valley from Kamsack to Portage 1a Prairie. Flooding in this reach would generally affect the river valley bottem land on1y. Downstream of Portage 1a Prairie the Assiniboine en- ters the flat prairie which once comprised the bottem of Lake Agassis. In this reach the adjacent land fa11s away from the river banks which have been raised by silt deposits. Once the banks are overtopped the water ca¡mot return to the river chamel when the flood peak has passed- The flood waters eventually drain via old creeks ancl char¡rels to Lake X{a¡itoba to the north and into the La Sa1le River to the south. However, drainage is generally poor and flood waters may remain on the land for several weeks.

2.2 Flood Control Investigatiorx Following the 1950 f1ood, measures were studied for both the Assiniboine River and the Red River. In 1958 the Royal Conmission on Flood Cost Benefit reconrnended specific flood control rvorks. ÌVith respect to the Assiniboine River, the Royal Conmission invest- igated storage as wel-1 as diversion proposals. For every principal tributary investigations were conducted to assess the feasibility of 10 flood control storage; each analysis, however, showed that tríbutary contribution was insufficient to warra¡tt the construction of a major flood control dam. It was finally decided that a substantial storage reservoir on the main stem of the Assiniboine was required in order to achieve the desired level of flood control. Three sites for a major dam were thoroughly in- vestigated; the first at St. Lazare, the second west of Russell and the third below the confluence of the She1l and Assiniboine Rivers- All three sites were found to be econornically feasible. The Shellmouth site was finally chosen because of more satisfactory foundation conditions, a smaller acreage of agricultural land in the reservoir area and a better source of construction materials. Extensive investigations to divert flood waters into Lake Manitoba were conducted. Several different locations and diversion capacities were examined. Several alternative diversion routes were found which is not surprising since areal photographs clearly show that the Assiniboine at one time flowed north to Lake Manitoba. It followed several different routes to the lake before it broke through to its present course eastward to the Red River. The diversion route finally selected has its begin¡ring two miles west of Portage 1a Prairie. Froni there the chalmel runs almost due north to Lake Manitoba. The length of river betrveen Portage la Prairie and l{innipeg has been

dyked. The drainage basin of the Assiniboine River in lr{anitoba together with the location of the above noted flood control works are shown on Figure II-1. Both the Shellmouth Reservoir and the Portage Diversion will be discussed

in more detail below. 11 2.3 Shellmouth Reservoir The Shellnouth Reservoir, on the Assiniboine River, is located approximately two miles north and two miles east of the Village of Shellnouth in an area where the Assiniboine River Va11ey is wide, with high banks. The earth dam is 75 feet high and 4000 feet 1ong. It is equipped with a concrete conduit to control releases fron the reservoir. An uncontrolled concrete spillway passes flows in excess of the conduit capacity. The storage capacity is used for water conservation as well as flood control. The conservation pool is up to an elevation of 1591.0, representing 165,000 acre=feet of storage, while con'bined conservation a¡d flood control capacity is between elevations 1391.0 and 1402.5 resulting in a storage of 13ór000 acre-feet. A further 87,000 acre-feet of flood storage capacity is available between elevation 1402.5 and 1408.5, the elevation of the uncontrolled sPi11waY. The location of the dam is shown on Figure II-2-

The prime purpose of Shellurouth Reservoir is to reduce the flood damage along the Assiniboine River and in tr{innipeg by storing the majority of the flood n-rnoff that originates l4lstream from the Shellmouth Reservoir in the Assiniboine River Basin. A secondary benefit that can be achieved by operation of the reservoir is that of augmenting 1ow flows on the Assiniboine River during dry periods.

The Water Resources Division of the Department of Mines, Resources and Environmental Management of the Government of lr4alitoba has determined the following principles of operation. In order that the ful1 capacity of the reservoir for flood control purposes be available in the spring the reservoir is lolvered from a sumner leve1 of 1402.5, to 1391.0 over the period Novenùer tr, to l'larch 31.

T2 The lowering is carríed out at as r.niforn rate o{ release as is possible based on an early winter forecast of the inflow to the reservoir during the winter period and recogniztng the anount of water in storage that nust be released. In years with evidence of a high spring runoff the reservoir may be drarnm down below 739L.0 by March 51. In order that storage spece be available in the reservoir subsequent to the spring runoff to reduce flood damage from sumner floods on the Assiniboine River, the water level in the reservoir is lowered to 1402.50 immedíately after the spring mnoff at a rate which does not cause downstream flooding. The reservoir is then mai¡rtained at 1402.50 throughout the sunmer a¡rd early fall until Novenber 1 when releases begin to lower the reservoir for spring flood control as noted earlier. If drought conditions plus high water sr-pply demands prevail, these elevations may be impossible to obtain. This system of allocating the available storage to the various purposes makes it possible to provide sufficient storage to reduce peak flows along the Assiníboine River downstream of the Shellmouth Reservoir and at the same time make it possible to maintain a flow of 250 c.f.s. in the Assiniboine River at Brandon compared to the recorded minimun of 7 c.f .s.

2.4 Assiniboine River Diversion The Assiniboine River Diversion channel begins two miles lvest of Portage 1a Prairie and mns almost due north to Lake Manitoba. It is 18 nriles long and is designed to carry up to 25,000 c.f.s. away from the Assiniboine River. The removal of water through the Diversion gives flood protection to the cities of Portage 1a Prairie and Winnipeg and the areas bet-ween them.

1.5 An earthfill darn across the Assiniboine River with a concrete spi11- r,ray control structure creates a small reservoir with a storage capacity of 14,600 acre-feet. North and west of the dam at the upper end of the diversion cha¡lnel an inlet control structure regulates flow to Lake Manitoba. The diversion cha¡¡rel has three drop structure along its route so as to keep water velocities below those which would cause erosion. For economic reasons approxìmately the last 3 niles of the diversion, which are located in the Delta Marsh, have been designed to carry only 15,000 c.f.s. The excess flow, which at fu1l d,esign discharge could be up to 10,000 c.f.s. is spil-led over into the west Delta Marsh. A section of the rvest dyke of the diversion cha¡ne1 in the vicinity of Cram Creek was designed and constructed at a lower elevation than the remaining portion of the dykes through the marsh. This particular reach of lower designed dyking concentrates the overflol and reduces the probability of an extended faí1ure along the dyke. The location of the diversion and details of its layout are shown on Figure II=3 a¡d Figure II-4. The nrrrnose of the Assiniboine River Diversion is to provide flood protection to i{innipeg and the area from Portage 1a Prairie to l\rirmipeg.

The diversion will acconrnodate and regulate the Assiniboine River flow up to a maximum of 45,000 c.f.s. At this design flood f1ow, 25,000 c.f.s. is diverted i¡to Lake Manitoba, while the remaining 20,000 c.f.s. passes downstream into the Assiniboine River. At Assiniboine River flows of greater than 45,000 c.f.s. flood damage will occur either along the Diversion or the Assiniboine River depending on whether the flow is di- r¡erted or alloived to flow down the Assiniboine. The following points illustrate the procedures now being followed by the Water Resources Division in the operation of the Diversion. 1¿" l\tt I{hile there is ice on the AssÍniboine Riyer downstream of the Portage Diversion it is desirabLe to maintaln flows less than 5,000 c.f.s.',in the river because of the possibility of ice jams. 7l After the ice has gone from the Assiniboine River downstream of the Portage Diversion it is desirable to maintain flows less than 10,000 c.f.s. in the river. Flows greater than 10,000 c.f.s" are above the natural bank stage of the river and backup of 1ocal streams which outlet into the Assiniboine may occur at this 1eve1. There also rnay be seepage problens through dykes, leakage through gated through-,dyke culverts a¡rd flooding of cultivated land between dykes.

\l The bankfull capacity of the Assiniboíne River downstream of the Assiniboine River Diversion is 20,000 c.f.s. 4l The overflow section of the west dyke of the Portage Diversion rvhich al1ows flows in excess of 15,000 c.f.s. to spi11 out into

the west marsh of Delta Marsh should only be overtopped when dictated by an extreme condition on the main stem of the Assiniboi¡re. The design capacity of the Portage Diversion is 25,000 c.f.s.

6) If possible flows on the Assiniboine River dorn¡nstream of the Diversion while ice is stil1 present should only exceed 5,000 c.f.s. if the I{inlipeg James Avenue stage is below 745.57. The 1evel of Lake Ma¡ritoba should not be taken into account while there is ice on the Assiniboine River, as the period during which there is ice on the river during the sprin runoff is only a few days, and diverted flows for this short period

of time would have a negligible effect on the level of Lake I{anitoba. 15 7) For Assíniboine River inflows to the Portage Reservoir in the

25,000 to 35,000 c.f.s. range, the diverted flow nay i¡ some insta:rces be limited to 15,000 c.f.s. to prevent overtopping of the west dyke overflow section. Thus flows in the 10,000 to 20,000 c.f.s. ra-rìge would occur on the river. rn this instance

the Janes Avenue stage would be the deciding factor as to how

much water should be sent down the river and how nuch diverted.

Flows in excess of 10,000 c.f.s. in the river should only be pennitted if Lake lvfanitoba is high and the James Avenue 1eve1 is 1ow. tf both the lake and the Janes Avenue levels are high,

presr.unably flows would be diverted to the lake rather than down the river.

8) The 20,000 c.f.s. 1ínit to flows down the Assiniboine River

should only be exceeded in the event of inflows greater tha¡r 45,000 c.f.s.

9) The 15,000 c.f.s. 1ímit to flows doi^¡n the Diversion should only

be exceeded when,

a) there is the possibility of exceeding a flow of 20,000 c.f.s. in an ice=free Assiniboine River. b) there is the possibility of exceeding a flow of 5,000 c.f.s. in an ice-borind Assiniboine River. c) the stage of James Avenue is at such a 1evel that flows down the Assiniboine River must be reduced. This condition

may be reached at all inflows as conditions at Janes Avenue nay require that the entire inflow or a very large portion of it be diverted.

ì \ I RESERVOIR

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I N LET CON TROL STR UCTUR E 20 3.0

TFIE ffi{ERAI, L]NEAR PROGRAMMING

}4]DEL FOR A FLOOD CONTROL SYSTH\4

2I 3.0 THE ffiNERAL LINEAR PROGRAilMING X4]DEL FOR A FLOOD CONTROL SYSTEM

3.I The Linear Progranuning Technique In the development of an optimization model to determine optirnal operating schedules for flood control system components it l.vas declded in this thesis to employ a linear progrannning analysis. Linear progrannning has decíded advantages in that if the problem at hand can be adapted to the linear programming technique a-n optimal solution can be obtained very quickly at a reasonable cost. Most non-linear opti:nization techniques on the other hand, such as a pattern or gradient search may arrive at a 1oca1 optimr-mi solution which could be quite different fron thetrueglobal or overall optimal solution to the problem at hand. Many tj¡nes the cost of the non-linear techaique may be prohibitive. This probleni does not exist with the linear progranmring technique.

Linear progranming is a nunerical technique that generates a solution to the optirnizatì-on problem at hand by means of a iterative procedure. The linear progranrníng technique involved in optimization nanipulates all the variables in the analysis simultaneously on each iteration subject to the constraints of the problem in the quest for arr optimum solution. A linear programning problen is one in which a linear frinction is the criterion to be minirnized or naximized. This linear criterion to be minimized or maximized is subject to constraints that are also linear functions. A conbination of variables denoted in general by X is said to be linear if the variables can be assenbled in the form:

X C"X.+C^X^+T-L¿¿NN C where the C's are constants. For example the function: ¿¿ 4X, + 3Xr+ 5X, + Z is linear in the vatiable XI , XZ , XS , whereas the fr-mction:

2Å-\+X.X^+3exp[X-')- t- | L is non-linear i-n the same variables. The linear progranrning technique solves equations of the following form:

U = CrX, * CZXZ + . ... * CrrXr, = minimum (or naximum)

Ç.rrlrioef fn. AttXt *AtzX| +... +Ah\ >b1 'Z .L A^"X" + A^^X^ + ... + A^ X > b^ ¿I I ¿¿ L LnnT ¿

A"X-+A^X^+...+Am-L-]. m¿L mn?mX >b

The above equations may be written in a much more concise nathematical fonn. n Minimize u =F, c.x. r'^* rrr---i-;z-r J J lor lvraÀxlrl-zej J =1

Qrrl-rìoef fn' ¿vu J vv tl ¡b* m \-/11 A.*x.,l-=1 ; for i = 1, j''t X.>0 j=7,n J- ;for

The ats, b's and cts in the above equations are constants and the xrs are variables whose values are sought. A simple linear programníng problem will serve to illustrate the teclinique.

23 Let us consider a graphícal solutíon of a s.imple linear progranrning

nrnhlvurvrr!. cm The nrohl em mâv heuv fn-ry¡ufated aS follows: l/r ^vr

Obiective Function: U=4Xr+X,

Çr rhi o¿-f f n' -,.17X X">2

4X 3xz >-3 1 2X, +SXr< 2I LL- 4x.,-x.,<16 I L-

x,, > 0 f- Xrt0 L-

It is desired to naximize tlne objective function in this problem. This problen is shown graphically on Figure III-1.

It is to be noted that all lines on Figure III-1 are straight and ^"-r^^^ :^ ^ dcdrrr-ed theL]1ç \-rPnnfìmizafior LLilLLLA Lrvrl )(]I LduÇ l-5 d -laneP**^- Tt mav readilr¡- -áIJLL/ heLJç uUuLaLçu that the limit equations are satisfied by any solution inside the shaded area. If it proves difficult to determine the direction of feasibility

graphically for a limit equation this may be determined quickly by taking any point, i.rsua1ly the origin, and evaluating the function at hand. This inrnediately tells where the origin is on the feasible or infeasible side of the 1ine. Let us rewrite the obiective fi-nction U in the form: x^=U-4x, ¿I The dashed lines on Figure III-1 represent this equation for various values of the objective fi-rnction U. It is apparent that point D is the opti:num solution with objective function U having a value of z4 3.43. Variable X, having the value 4.93 and variable X, having the value of 3.7L. Essentially the procedure in linear progranmring is to start in any one corner of the shaded polygon shown and jump to a¡l adjacent corner having a higher value of the objective function U. This is continued until there is no adjacent corner having a higher value. The general probleni has more than two variables and cannot be thought of in geome- trical terms. However, the mathematical method is essentially similar. Various computational algorithnis have been developed to solve the linear prograruning problen. The linear programning solution algorithn employed in this thesis r>; ^ L^udrcu -^Å .,-^-uyurr +Lur).e mutual primal dual siniplex method of Michael L. Polinsþ and Ralph E. Gomery. The algorithm in question was supplied through the courtesy of Charles Howard and Associates, Professional Engineers, ltlinnipeg, Manitoba. It is recognized that different algorithms are available for the solutiorLs of linear progranrning problems. However, the algorithm used in this thesis proved extremely efficient. Moreso, valuable assistance was available from Charles Howard and Associates regarding the use of the algorithm in particular and the linear prograruning

'i fLvurur¡Yuv er-hn i nr re r¡rn 6v¡.oeq6¡¿] .

3.2 The Model Description

The linear programning model is conrprised of two cornponents. The first of these is a linear objective function and the second is a linear system of constraints. These two components are as described in tlro preceding section.

Some general conrnents regarding the linear prograrmning model developed

25 in this thesis are in order. It is assrmed in the formulation of the model that all reservoirs and/or diversions have gated or controllable outflohrs and that decisions regarding the size and location of individual flood control system components have already been made. It is not intended, in its present form, to size reservoirs and/or diversions in a planning study. In other words, the linear progranrning nodel constructed herein is an operative model rather than a planning model. The optinal operating schedules for a flood control system proposed in this thesis are based trpon the object of minj¡rizing the total flooded area, with respect to the system as a whole, based on the peak mean seven day discharges at selected damage centres withín the systen. The use of a seven day tíme period in the fornulation of the model a11ows adequate starting postulates for a sjmulation analysis and reduces the conputational burden of the linear prograruning solution.

For the purpose of illustrating the linear progranmíng model developed in this thesis a hypothetical river system is sholvn on Figure III-2" lthile this system, in general, is much sirpler than an actual flood control system, ít serves to illustrate the linear progranming model for the optimal operation of a flood control system as developed in this thesis.

3.3 Linear Progranuning N4ode1 Constraints

The constraints, which together with the objectíve function comprise the linear progratrming nodel , fray be classified in two t)?es. These are physical constraints and computational constraints. Physical constraints force the linear prograilrning model to fo11ow the physical realities of the flood control system being mode11ed. These would include upper storage limits for reservoirs in the flood control system, diversion capacity of diversions within the flood control system, and minimr¡n flow requirements within the system. Computational constraints adapt the nathematical linear progralming technique to the determining of optirnal operating schedules for individual components of the flood control system. An example of this tlpe of constraint is that which is required to determine peak discharges at individual damage centers in a form suitable for the linear progrannning techlique.

3.3.I Physical Constraints 3.3.7.7 Storage Constraints Each reseryoir in the flood control system is limited to its maximun storage capacj-ty. That is, at the start of any tíme period in the analysi-s the quaatity of water stored in a given reservoir nust be equal to or less than the maximun available flood storage voli¡ne of that reservoir. This nay be represented mathematically as

t s_.V. II]-1 1f where S. is the volume of water stored in reservoir i at the start of a given time period t, and V is the maximun available flood storage volune at the reservoir in question. It should be noted that the maximu'n available storage for a given reservoir may be set in the model at any preselected magnitude less than or equal to the absolute physical reseryoir capacity in order to address, for example, Tecreation use constraints on the reservoir in question.

27 3.3.I.2 Reservoir lt4ass Bá1aúce Constraint This constraint is based on the principle of continuity and states that during any time period the inflow voh¡ne minus the outflow voh-une for a given reservoir mu.st equal the change in reservoir storage. Eva- poration and seepage losses from reservoirs during flood periods are generally an insignificant portion of the total flow and are therefore not included in this mode1. Because of the seven day tin,e period used j-n this study all storage terms are in r-¡nits of c.f.s.-weeks. This constraint may be represented mathematically as

+L1L L-l +1L-I L1L-.1 S -S +I -0 III-Z iiii

+1 L- -L where I is the inflow to a given reservoir over a given ti:ne period :+1

L- r r 0 is the corresponding outflow froni the reservoir in i t t-1 question over the same tjme period and S and S are storage volunes for a given reservoir at the beginning of the tine period in question and the beginning of the preceding ti:r,e period. All terms in this constraint are expressed in a consistent set of units, in this case c.f.s.-weeks. Figure III-3 shows the defi¡rition of time periods.

3.3.7.3 Reservoir Release Constraint Reservoir releases are limited by the conduit discharge capacity which is a function of the reservoir storage volune. In general this is a non-1ínear function. The reservoir release constraint may be erpressed inathenaticalTy by the equation

0 < f(sj) III-3 it z8 where i denotes Reseryoir i and t denotes time period t. f(Si) is the outflow lirniting function for reservoir i"

3.3.L"4 Chânnè1 Continúiti óf Storage Constraint

Inflow must equal outflow for any river cha¡nel reach over each

week in the model as changes in channel storage aïe neglected. Channel

inflows are comprised of the outflow from the i¡mediately upstream

channel or reservoir, loca1 inflow to the reach, expressed as end of

reach inflow, and as a negative term, any flow lost from the reach by

diversion in the reach in question. This may be e4pressed nathematt-ca1;¡y as follows:

LLL 1' T I T^^ ^v - l- T Ll.- - ÐIVR III-4 iii i

with reference to Figure rrr-z this would be expressed as

t ttt _O +LM -DIVR III-5 i 111

L where Q:¡ is the mean seven day discharge at the end of reach i over time -LL period t; 0 is the mean seven day outflow from reservoir i over time i perì-od t, that is, the mean seven day inflow to reach i over tiile períod +u_ t; LOC is the mean seven day local inflow to reach i over time period 11 L' ljlvR is the mean seven ðay : diversion flow diverted from reach i over tjme period t.

3.5.1. 5 Riparian Constraint It{inimun releases are usually specified for any ïeservoir ín order zg to rneet doL\rnstream riparian rights. This may be e>çressed mathematicaTTy

AS follows:

t 0 I]I-6 ii Í where 0 is the mean seven day outflow from reservoir i over time

I *^-i +. UUI -LL'LI^l L. and C is the ninimrm mean seven day release allowed for reservoir i

3.3.7.6 Diversion Capacity Constraint Any diversion in the flood control system will have an upper limit conceptionaLly simi.lar to that specified for reservoir storage. In the case of a diversion it is the diversion capacity. This may be expressed mathematicaTly as follows :

t Divr ¿Tì ITT-7 l-1 t where Divr is the mean seven day diversion flow diverted fron channel

! reach i over time period t; D- is the absolute diversion capacitv for 1 the diversion in chamel reach i.

3.3.I.7 llpper Lhit on Diversion Flow Values

As ure11 as the physical diversion capacity described in the preceding section, any diversion within the flood control systen must be constrained so that its flow ís less than or equal to the inflow above it in the river system. That is, it is impossíble for a diversion to divert more flow away from a damage centre than is available to the diversion. With in reference to Figure III=2 this nay be expressed mathematically as

tt t Divr <0 + LOC III-8 11

in aanatll ^rv ¡ f¡r 5v¡rvr sf

++ LL t Div-r

+ where Div-rl is the mear seven ðay diversion flow diverted from reach i I over time period t; I." is the mean Seven day i-nf1ow to reach i over rimc neriod the mean seven day l-:ocal] ínflow to reach i, uarrrv Hvr t:t --fOCl -1 --is expressed as end of reach inflow over tiine period t. All terms are in r.u-rits of mean seven day c.f .s.

3.3.2 Computational Constraints 3.3.2.1 Peak Discharge Constraint This constraint is a computational constraint used simply to identify the peak mean seven day discharge as the largest of all mean seven day discharges during the tiine periods in the arralysis at each damage centre.

It may be represented mathenatically as

f nD >rì-Y ii III-10

1- where Q.t ir the mean seven day dì-scharge at the end of chanrrel reach i, that is, at damage centre i over the tirne interval t; QPi is the peak mean seven ðay discharge at damage centre i. It should be noted that damage centres are assumed to be located at the end of a given reach. 3T The peak mean seven day discharge at the respective danage centres are used ín the obj ective fi-u-rction.

3.3.2.2 Non:negativity Constraint This constraint is flndarnental to the linear progranrning technique and states that all variables in the linear progranming analysis mrst be greater than or equal to zero. No variables may take on negative values.

3.4 ObjéctiVe Fúnction The linear progralnming technique requiTes a linear relationship of the variables involved in the analysis to be minimized. This relation-

ship ís lanown as the objective function. In this thesis the objective

function is the damage cost of flooding at selected danage centres

oçressed in arbitrary damage units. For simplicity these damage units are taken as weighted magnitudes of flooded areas" For the purposes of this thesis, that is the obtaining of preliminary optimal operating schedules for the flood control systern components it is not necessary to ex?ress the damage in do11ars. The weighted magnitudes of the flooded areas serve equally well provided the weights make the dollar values and the areas roughly proportional. Thus the objective of the 1ínear progrannning model constructed in this thesis j-s to obtain optimal operating schedules for the flood control Systen components So aS to minjmize the total flooded aTea, considering all locations being protected, given the nrn-off hydrographs at aII significant points in the systern. The flood damage to be minimized, or objective function, is assr¡ned to be some function of only peak mean \/ weekly discharge. The problern therefore requires, finding the minimr¡n of a non-linear objective firnction subject to a large nunber of linear and/or non- linear constraints.

3.5 Time Dependéncy 0f ObjéctiVè Fúnctión at Damáge Centres In the application of the linear progralmning technique to the determination of optinal operating schedules for the components of the Assiníboine River flood control system it must be recognized that a major portion of the flood prone area in the Assiniboine River Basin is agricultural in nature. Agricultural flood damages are decidedly time dependent in that flooding early in a crop year, before the crop is planted, may cause 1itt1e or no reduction to the crop yield that year and therefore 1itt1e or no damage. However, flooding later during the year will cause steadily increasing damage as the crop yield is reduced due to a shortened growing season until, at some point, it is too late to plant a crop at all during the year in question and harvest the crop during the same year. At this point the entire value of the crop in question is 1ost. It is necessary that this tin,e dependency of agricultlral flood damages be adequately reflected in the linear prograrrning model. It has been noted earlier that the basis of optimization in this thesis is to minimize flooded area within the river basin. By this technique all flooded area would be assuled to have the same value regardless of the time of flooding. What is required to reflect the time variability of agricultural damage is that flooded areas in the river basin may take on different values of flood danage, dependent on the time of flooding. This may be accomplished by including in the objective function n4l\ several flood peak values for each damage centre each for a different tírne period.

By means of introducing this family of tjme dependent peak mean

seven day dishcarges denoted where i is the darnage centre in "s QPLi question with each aq. defined for a different time dependent fanily L1 component 1 over a different tìme period t, t=t1r tz, it is possible to model the phenomenon that agricultural flood damages are time dependent. Each QP. will be the peak discharge at darnage centre i for the tirne Lf dependent family component 1 over tìme period t. to t . Each variable Qf has a d.ifferent coeff i.i".,. :-3 .fr" obj ective L1. function reflecting the relative tine variabiliry of agricultural damages.

All individual tïne periods t must be included once in the analysis and in only one time dependent family component. This concept may be illtistrated as follows with reference to equation ITI-10.

+ L nP .Y:ô IIT-10 ii

For the purpose of this discussion 1et us assume ten time periods.

Therefore t varies from one to ten. Also let us assume that the damase relationship for time periods one to ten is as follows:

Time Period Damage Relationship

One to three Damage = 0.25 QPr-

Four to Six Damage = 0.50 QPZ.

Seven to ten Damage = 0.90 QPS.

There are thTee tíne dependent fanLily conponents in this ovarnrr'l o

Equation III-10 may be modified to represent this situation as follows: +- L np a taìl' .1-= | ( IïI-10a I1 i t QP tQa for t=4r6 III-1Ob 21 I t fnr f='7 1ñ QP rQa v t tLW III-10c 3i a

Each variable QP , L=1,3 is given a different value in the objective Li function as follows:

0.2s QP 1i

0. s0 QP ¿J-

0.e0 QP 3i As shown, equation III-10 has been replaced with three separate equations, each modelling one segment of a tjrrre dependent fanily re-

lationship for flood damage.

3.6 Nonlinearity In the application of the linear prograrrrning algorithm to the flood control problem in this study non-linear relations occur in the objective firnction. These nonlinear relations nrust be replaced by piecewise linear approximations so that the problem may be solved by a standard linear progrann'ning solution procedure employing a separable progranrning analysis.

The application of the separable progranrníng technique will be illustrated in the case of the damage curves for the damage centers in the mode1. These non-linear damage curyes provide the infonnation needed to derive the optimal release schedules for the system. The form of these functions is inportant since the total damage is to be minimized in

35 the optfunal solution. If the curves are conve¡ ìn chena rho nrnÞ'1 êm m'ìz be solved in a straightforward manner. Concave curves or a combination of concave and convex curves, on the other hand, tend to create difficulties as a linear prograÍnìing technique rnay, in this case, arrive at a 1'ocal optimr.un rather than a g1oba1 optimun for the solution. Concave curves mr:st either be approximated in a convex manner, or if this involves an i.rnaccept- able loss of accuracy in the analysis, dealt with through another form of optiniization rather than linear programming. This thesis deals only with convex firnctions.

The nethod of solution, used in this study, involving a rnodification of linear progranu.ning lcrown as separable progranrning will be illustrated with reference to Figure III=4, which is assrmed to represent the peak mean seven day discharge=damage fi-inction for a given damage centre.

A convex cost curve for the damage centre is first approximated by piecewise linear approximation, as shor.rn in Figure ITI-4. This is accomplished by selecting a n¡.¡nber of breaþoints, or points at which the slope of the piecewise linear function changes. Let us define M as the nulber of segments in the pi-ecewise linear function approximati-ng Ui (Qi) and let

U , U ,"", U 7j 2j inj be the ascending values of Q at which the slopes of the piecetr'ise linear segments change value. Next 1et us subdivide the peak discharge at the damage centre Q, into a set of ar:xi1iary variable such that

Q. =5- Q IIT-11 J Á--r mJ m=1

36 wherewlIUIv !',M =- LrL,1 -2- ... lY¡ 4lru^-r çCLLIf^^^L ^VnJ is bounded as follows:

The nonlinear cost function Uj (Qj) can now be defined in terms of its piecewise linear approximations :

Ìt U (Q) ), K A IIr-13 J ) flt=.L mJ ml

r\.J.> difference wíth respect where m-l the partial of the objective fi.¡rction and defined as: to 0'ml is

= ACIAQ III-14 nj urj

trthen the nOnlinear terms ¿¡o ronleeori Ìrv fhe pieCeWiSe I inpqr forrnq the objective function becomes: NM ffi Min()' ) K a -) III-15 LÅ l--á -i ì; J=1 m=l- rvhere M is the nmrber of conponents for a given cost function approximation a¡rd N is the mmber of damage centres in the analysis. The problem has now been transformed into an ordinary linear progranming problen. Since the variables K*, are increasing in character, in a cost mininrization problem the linear prograrnning algorithm will select the cost variables in their correct sequential order, and no add- itional constraints are required. That is, the linear progranming algorithm will naturally e:haust the lower.valued Kr. variables first, which is computationally correct. 77 3.7 Size of the Linear Progranrning Problen The computational burden of solving a linear progranming application

is directly reflected in the cost of doing so on an electronic computer. In order for any model constructed to be operationally applicable it is extremely important that this computational cost be minimized. It should be noted that computational costs involved in the solution of linear programning applications rise as a positively increasing exponential

function of the nunber of variables in the analysis. It rnay be readily appreciated that is is i.nrperatiye to minimize the nunber of variables in the analysis while at the same time constructing arr adequate model of the flood control system under analysis. Two techniques were applied in this thesis to minirnize t]ne nurber of variables in the analysis.

3.7.7 Elimination of Variable Qf

The variabl" al has been defined, previorisly in this thesis as the mean seven day flow at d.amage centre i over tine period t. The variables + Qr" do not appear directly in the linear progranrning model fonnulated in this thesis. Rather, they are calculated as the sunnation of all system inflows, positive and negative, at the damage centre in question. Equations IIi-4 and rrl-10 are repeated at this point for reference.

t 1 LL Y T + LOC - Diyr I]I -4 i i ii t ND >o III-10 -\ í

l{ith reference to equation III-5 and III-10 the following equation be arrived

38 tt + l.ìD > T + T/-\n - Divr III-16 iii i

Variable QP. is the peak mean seven day discharge at damage centre i, which is located at the end of chanrrel reach i, over time period t.

This variable is as defined earlier in this thesis. I: is the mean seven - day inflow to channel reach i over time period t. LOCi is the mean seven day 1oca1 inflow to charrrel reach i over tirne period t, arrd Divrf is the J- mean seven day diversion flow diverted from cha:rnel reach i over time period t. It is to be noted that no new variables have been introduced in the formulation of III-16, rather the variabf" has been eliminated. With QlI reference to equation III-16 it may be seen that the peak nean seven day discharge at any damage centre in question, i,uhich is noted as QP. is defined as being greater than or equal to the sr¡n of all inflow arriving at a damage centre in question for a given time period. The given time period varies over the entire range of tine periods analysed in the model.

It must be noted that both positive and negative flows to the damage centre in question are considered in the formulation of equation III-16.

3.7.2 Elimination of Fixed Valued Variables Certain variables in the analysis are fixed in value by their very nature and thus carurot be allowed to assume different values at the optimal solution. Examples of variables of this type are system inflows or initial reservoir storage 1eve1s. In the interest of computational efficienry a trarsfornation is carried out with respect to these noted fixed valued variables. For any constraints containing fixed valued variables the variables in question are first rmrltiplied by their corresponding

39 coefficient in the constraint equation and then subtracted from the right hand síde of the constraint in question. In this manner the mmber of varj.ables that the li¡rear progranuning algorithm mu.st address is signi- ficantlv reduced.

40 t2 ^2 =16 \U \ U=20 \ \ t) \ \ \ 23. \ \ \ \ \ 5 I \ \ \ \ \ 4 \

A t

F iqure Itr -1

SAM PLE LINEAR PROGRAMMING PROBLEM

4I RESERVOIR INFLOW

I w

ouTF LOW,

DIVERTED FLOW CHANNEL REACH ¡< D ivr I 't

I \ ,.'FLOW AT DAMAGE, CENTER ø UNCONTROLL ED REACH INFLOW, DAMAGE CENTER J LOCiI

Figure III- 2

HYPOTHETICAL R IVER SYSTEM I1- 9t-r I1 or

St- I S¡

iíme period r -1 r +1

Sr=St-1 +It-t-Ot-r

Figure m- 3

DEFINITION OF TIME PERIODS USED IN ANALYSIS DAMAGE COST lt j(a¡)

K.j

PEAK DTSCHARGE QJ

Figure III-4

SEPARABLE PROGRAMM ING EXAMPLE FIGURE

44 4.0

T}IE LINEAR PROffi.ATßIING }4]DEL

APPLIED TO T}ß ASSINIBOINE R]VER

SYSTA,I

45 4.0 TFIE LINEAR PROG.AM{ING T,ODEL APPLIED TO TFiE ASSINIBOINE RIVER SYSTEM

4.1 The Assiniboine River Model The Assiniboine River basin in Manitoba between Shellmouth Reservoir and Winnipeg is the river basin or physical system nodelled in this thesis. It is broken down into six river reaches on the basis of the location of Water Survey of Canada streamflow gauging stations on the Assiniboine River. These river reaches are from Shellmouth Reservoir to Russell, Russell to Miniota, Miníota to Brandon, Brandon to Portage 1a Prairie, Portage 1a Priarie to Headingley and Headingley to l{innipeg. The systen thus described is shown on Figure IV-1, and the locations of the above noted Water Survey of Canada streamflow gauging stations are shown on Fígure IV-Z. There aïe seven points of inflow in this model. These are: the inflow to Shellmouth l{eservoir, inflow fron Shellmouth Reservoir to Russell, ilflow from Rirssell to Miniota, inflow from Miniota to Brandon, inflow from Brandon to Portage la Prairie, inflow from Portage 1a Prairie to Headingley, and the inflow from Headingley to Winnipeg. The inflow from Headingley to Winnipeg includes flow on the Red River in Manitoba not diverted down the Red River Floodway. The model constructed in this thesis was tested using 7974 streamflow data over the tjme period April 15 to August 11. The year 1974 was selected because it was a high flow year and hydrometric flow data was readily available. The ilflows to Shellnouth Reservoir were calculated by the Water Resources Division, Department of l4ines, Resources and Environ- mental Managenent, Government of lt{a¡ritoba. The remainilg irifloiv values were calculated on a inean seyen dav basis as the difference in mean seven

46 day flow values between successive Water Survey of Canada streanflow gauging stations. The exception to the above noted calculation technique involves the inflow at Winnipeg, which includes flow from Assiniboine River and the Red River mimrs whatever flow was diverted through the Red River Floodway.

4.2 Studv Operational Rules For the purposes of this thesis the operational guidelines for the two major flood control works on the Assiniboine River, that is Shellnouth

Reservoir and the Assiniboine River Diversion, have been simplified somewhat from those outlined in Chapter II. Shellmouth Reservoir is regulated so that the spillway will not be used. In other words there is no use of live storage above the spillway in flood control operations. This assunption is in accordance with actual operating criteria for the Shellnouth Reservoir in that reservoir live storage is generally not relied on for flood control purposes by the Ïlater Resources Division. As wel1, a mean seven day release from the reservoir of 100 c.f.s. has been assumed for riparian puryoses. That is , during no week during the analysis will the discharge from the Shellnouth Reservoir be allowed to fa11 below 100 c.f.s. The operation of the Assiniboine River Diversion has been constrained to fo11ow the Assiniboine River Diversion operation guideline outlined in Chapter II. The pattern of operation assumed for the purpose of this thesis is as follows:

47 Total Flow at Diversion Point

First 10000 c.f.s. down Assiniboine River 10000 Next 15000 c.f.s. down Assiniboine River Diversion 25000 r-fc

Next 10000 c.f.s. down Assiniboine River 3s000 c.f.s.

Next 10000 c.f.s. down Assiniboine River Diversion 4 s000 c.f.s.

Remainder down the Assiniboine River 45000 + L. r.5 .

The flood stage at Winnipeg is taken into accor.u-rt at all times with regard to the operation of the Assiniboine River Diversion by means of placing a high damage value on flood stages greater tha¡r bank capacity in winninco Tt is to be noted that Winnipeg flows include flow of the Red River not diverted by the lVinnipeg Floodway. The high value of possible dan,age at the Winnipeg damage centre may at any tìme overrule the assumed general pattern of operation of the Assiniboine River Diversion. This in accord with the present actual operating practices with respect to the Assiniboine River Diversion in which flood protection at Winnipeg assumes paramount i-nportance .

4.3 Flood Damage Functions As noted previor-rsly the linear progrannning algorithn requires a linear relationship of the variables in the problem to be mininized. This is lsrom as the objective function. In this thesis the objective firnction is comprised of damage functions at each damage centre in the alalysis. These danage functions are described earlier in Chapter 3.0 in section 3.4, Objective Function

48 There are seven ðanage centres in the analysis carried out in this thesis. These damage centres are l-ocated at Water Survey of Canada streamflow gauging stations. The locations are as follows:

Location Water Survey of Canada Station Nwùer

Assiniboine River near Russell 0SME001

Assiniboine River near Miniota 0sME006

Assiniboine River near Brandon 0sriH001

Assiniboine River near Portage 1a Prairie 05¡4J003

Assiniboine River near Headingley 0SlvLIO01

Assiniboine River Dir¡ersion near Portage Ia Prairie 0sLL019

Red River at Winnipeg 050J001

Each damage centre is assuned to be representative of the reach of river from the damage centre in question upstream to the next damage centre. The location of the damage centre at Water Survey of Canada streamflow gauging stations provided stage-discharge relationships at each damage centre. The only exception to this is at i{innipeg where the lVater Survey of Canada does not calculate a stage-discharge relationship.

Flooded area or damage versus discharge relationships were obtained from the Water Resources Division, the Department of lvtines, Resources a¡d

Environrnental ]r4anagement, Government of Manitoba, for the first four damage centres, Russel1, lt4iniota, Brandon and Portage 1a Prairie. These relationships are based on air photo analysis fron previous flood years.

They are shown on Figures IV-3, IV-4, IV-5 a¡d IV-6. From inspection of the above noted figures it may readily be seen that all of these functíons

49 are convex linear approximations to distinctly non-linear firnctions. This non-linearity is to be e4pected in fi:nctions of this type. All of the fi.rnctions have an initial segment of zero slope which infers zero damage. This represents the cha¡nel capacity for the damage centre in question and is in acûnl fact the t'arìge of flow values that may occur in the river reach between the proceeding danage centre and the centre in question before any flood damage occurs in the reach. It should be noted that this range may be quite different fron the range at the actual site of the Water Survey of Canada gauging station. This phenomenon results from the fact that Water Survey of Canada streamflow gauging stations are generally located at constrictions in the river with high charmel capacities so as to catch all flow in the river in easily rnetered charnels. The flooded area verslr.s discharge relationships are converted to damage relationships by nrultiplying each curve segment slope by an agricultural damage factor as discussed in chapter IIr to yield damage, in agricultural damage inits, verslrs discharge relationships. The agricultural damage factors used were as follows:

Agricultural Damage Date Week'of Analysis Factor

anTft t\ lvIay 26 1 o n <<<

't\,t^., )'7 t'øI L t Jtxte 23 7 10 0.5

Jr-.ne 24 onwards 11 t7 1n

The selection of these weights is rather arbitrary since the purpose of the analysis is to test the model capability rather than to devise practical operating rules. 50 These weights imply that r-ip to May 26 there is a linear agricultural darnage relationship of 0.333 agricultural damage units .per acre of land flooded. For the period of l4ay 26 to J:lr:re 23 there is a linear agricultural damage relationship of 0.5 agricultural units per acre of land flooded. For the period of Jrine 24 onward there j-s a linear agricultural

damage relationship of 1.0 agriculturaL damage i.nits per acre of land flooded. It is to be noted that in the application of the linear progranming model to the Assiniboine River basin all flooded area was assumed to have

the same unit value per acre. Agricultural damage factors were then

applied against these trnit values. This assi..nnption may be easily relaxed

to permit any differention in land values deemed necessary in a given analysis. The final three discharge-damage relationships, namely, at Assiniboine River Diversion, at Headinley and in Winnipeg were not based on flooded area versus discharge relationships. Rather, they are what is termed in this thesis as surrogate damage furctions in which preselected flood control conponent operating schedule characteristics are forced on the linear prograunning analysis by selected fornrulation of the above noted relatíonships. Operating rules for the Assiniboine River Diversion have been established as a matter of policy. To achleve compliaace with these rules surrogate damage functions have been introduced. These relationships are shown on Figure IV-7, IV-B and IV-g. The pre-selected component operating characteristics forced on the linear progranrning analysis by these damage relationships is the previously discussed operation pattern of the Assiniboine River Diversion. The operation pattern of the Diversion desired in the analysis is controlled by

51 the stage discharge relationships for the Assiniboine River Diversion, the. Assiniboine River at Headingley and the conbined Assiniboine River a¡rd non-diverted Red River flow at Winnipeg. The last damage relationship, for the Assiniboine River at Wiruripeg, is not formulated to control the operation charàcteristics of the Assiniboine River Diversion directly but rather to reflect the cha¡nel capacity of the Assiniboine=Red River complex in Winnipeg and the very high potential damage should flooding occur in the City. As noted previously the damage discharge relationship at lVinnipeg may at any tirne overrule the preselected operation pattern of the Assiniboine River Diversion so as to relieve flood damage at Wirmipeg. This is in accord to the present operating practices with respect to the Assiniboine River Diversion and is to be expected when one considers the extremely high potential damage of flooding in Winlipeg. All flood damage relationships, wíth the exception of Winnipeg, reflect the time dependency of agr:cultural damages as noted previously in this thesis. In the case of l{ir.nipeg the damage weights are assuned to be unity for all three time periods. That is, it is assumed that 't{innipeg flood damage in is not time dependent and would be equally seveïe regardless of the time of year in which flooding would occur. As noted previously in Chapter 3.0 the damage functions which comprise the objective functions for the linear program optÍmization technique are non=1inear. Thls situation is addressed by the previously defined technique of separable progranmring. As well the previor:.sly defined technique allowing time variability in the objective function through a time dependent fanily of damage functions is employed in the application of the model to the Assiniboine River basin.

52 4.4 Specific Linear Prograruning l'4odeJ- for the Assiniboine River Syste¡1

4 . 4.1 Physical Constraints 4.4.1.I Storage Constraints

The maximun storage available at Shellmouth Reservoir at the spillway elevation is 388,000 acre feet or approxirnate\y 27572 c.f.s. - weeks. With reference to equation (III-1) the storage variables for Shellmouth Reservoir in the linear progranrning inodel must be Lpper bounded at a value of 27572 that is

f s < 27572 ry-1

+ where S' is the volune of water stored ln Shellmouth Reservoir at the start of time period t and is in r:nits of c.f.s. - weeks.

4.4.I.2 Reservoir- Mass Balance Equation 'l{ith reference to equation (III=2) the reservoir mass balance equation for Shellnouth Reservoir may be stated as follows:

L1 r t-.1 L-.I L-l S -S +J -0 rv-2 where and the Shellmouth Reservoir storage voLumes, in units St St-1 "r" of c.f.s. - weeks, at the beginnilg of time intervals t and t-1. It-l

mean seven day inflow a¡d outfloic from Shellmouth Re- and Ot-1 "t" the servoir over time period t-1. The storage units of c.f.s. - weeks are required il order to maintain dimensional honogenity in the problem in which all flow values are mean seven day values. 4.4.7.3 Reser'¿olr Release Constraint For the purpose of this study releases from Shellmouth Reservoír are only allowed via the reservoir conduit. That is, no flow over the spillway is allowed since the probLem is constrained so that the water level on the reservoir nay not exceed the spillway clest elevation. An elevation=.discharge relationship for the conduit and spillway flow is shown on Figure IV-1-0. The conduit rating curve on this figure j-s a maxim¡n flow curve and based on a Prairie Farm Rehabilitatlon Authority memoraldum of August 25, 1966. An elevation storage relationship for Shellmouth Reservoir is shor¡n on Figure IV-11. Figures IV=10 and IV=11 may be combined to yield on outflow-storage relationship for conduit flow only or conduit and spi-llway combined. An outflow-storage relationship for conduit flow only is shown on Figure IV-IZ. For the purposes of this thesis the conduit flow curve may be approx- inated as shown on Figure IV-1,2. All reseryoir releases must be less than or equal to this lùniting function defined by the reservoir storage for the tirne period in question. With refeïence to equation (III-3) this nay be represented as

+l.LL 0 <0.066S +1930 IV-3

ivhere O' is the mean seven day outflow from Shellmouth Reservoir over time period t; St is the storage volr.me in Shellmouth Reservoir at the beginning of t jme period t.

4.4.7.4 Riparian Constraint With reference to equation (III-6) the riparian constraint for Shellmouth Reseryoir for the purposes of this study may be stated as:

t 0 >100 rv-4

+ wnere u ls tne mean seì,¡en day outflow from Shellmouth Reservoir over time

PUI l-U(l L. r.lrl_> outflow is constrained to be at least 100 c.f.s. for all time periods.

4.4.7.5 Diversion Capacity Constraint

With reference to equation (II-.8) the rrpper ljmit for flow dor'm the

Assiniboine Diversion may be stated as:

t DIVR < 25OOO w-5

t wneïe ulvK ís the mean seven ðay Assiniboine River Diversion flwo over

fimo noriná Lfrrrv yu¡ fvu t and is constrained to be less than 25000 c.f.s. for al1 time

+L^ nerinác in LrlU criICL-L/>-L5.^.^^'l--^-: -

4.4.7.6 Upper Lj¡rit on Diversion Flow Values l{ith reference to equation (II-10) the Assiniboine River Diversj-on

flow m.rst be constrained so that at aJJ- times it is less than or equal to

the flow available to be diverted down the Diversion. This may be rep-

resented as:

tttttt DIVR

The chan¡rel flow constraints are defined for each damage centre in the format shou'n in equation (II--6). With reference to Figure fV-15 they are defined as follows:

Damage Centre A - Russell QPA' Ot * LoCAt III-4 Damage Centre B = Miniota

QPB ' Ot + LoCAt * LoCBt ]II-5 Damage Centre C -.Brandon

QpC ' Ot + LOCAT + LOCBT + LOCCT rIT-6 Damage Centre D - Portage 1a Prairie t QpD r Ot * LocAt + LOCBT + LOCCT + LOCDT . DrVRt rrr-7 Damage Centre E - Headingley

QpE t Ot + LoCAt * LOCBT + LOCCT + LOCDT + LOCET III-8 D]VRt

Danage Centre F - Winnipeg

QPF'Ot + LOCAT * LOCBT + LOCCT + LOCDT + LOCEI III-9 + LOCFT - DrvRt

Damage Centre G - Assiniboi¡e River Diversion QPc ' DIVRT IIr-10 where:

Ot is the mean seven day outflow from Shellrnouth Reseryoir over time period t LOCAT is the meari seven ðay j¡f1ow to reach A, Shellmouth Reservoir to Russell, over time period t 56 LOCBT is the mean seven day inflow to reach B. Russell to Miniota, over time period t

LOCCT is the mean seven day inflow to reach C, Miniota to Brandon, over time period t

LOCDT is the mean seven day inflow to reach D, Brandon to Portage 1a Prairie over time period t + LOCE--^^-L is the mean seven day infLow to reach E, Portage la Prairie to Headingley, over time period t

I I,OCF" is the mean seven day inflow to reach F, Headingley to Winnipeg, over time period t QPA is the peak nean seven day flow at damage centre A, Russell Å^^^-^ QPB is the peak mean seven day flow d-^+ L LadlildBç uurr^^ñ+,^ Lt ç R Mi n'ì nte QPC is the peak mean seven day flow at damage centTe C, Brandon

Å^^^-^ 'l q QPD is the peak mean seven day flow d^+ L Lldtild.BÇ \-çl^^ñ+-^ r Ll E n Þnrtq oo rTa]-r].e

QPE is the peak mean seven day flow at damage centre E, Headingley QPF is the peak nearr seven day flow at danage centre F, I{innipeg QPG is the peak meal seven day flow at damage centTe G, Assiniboine River Diversion

DIVR is the mean seven day Assiniboine River Diversion flow

4.5 Sunrnary of Problem Contraints

A sumnary of all constraj-nts involved in the linear programriing model of the Assiniboine River flood control system is provided herein. This sunnary includes the previously discussed techniques of time dependency and separable progrannning enployed to address the tì-me dependence ard non-lj¡learity situations inherent in the problem r_mder arralysis.

J/Í- Storase Constraint

St . 27572 t^T Tv = Lrrt| tt

Reservoir lJass Balance Constraint

St - St-,1 + 1t-'1 _. 6t..1 for t = 2177

Reservoir Release Constraint

Ot . 0.06ó St + 1930 for t = 7r\7

Peak Flow Constraint at Damase Centre A Russell

PA1L + PAZL + PA3L tot + LOCAT for t = 116

PA1M + PAZM + PASM rot + LOCAT for t = 7rt)

PA1H + PA2H + PASFI tot + LOCAT for t - 11 17

Peak Flow Constraint at Damase Centre B - Mlniota

PB1L + PB2L + PB3L tot + LOCAT + LOCBT for t = 116

PB1M + PBZM + PBSIT,I tot + LOCAT + LOCBt for t = 7r!0

+ PB1H PB2H + PBSH tot + LOCAT + LOCBT for t = 77 r17

Peak Flow Constraint at Damase Centre C - Brandon

€^-+- 1É. PC1L + PCZL > Ot + LOCAT * LOCBT + LOCCT rul L - r:u 58 PC1M + PCAIT. > Ot + LOCAT * LOCBT + LOCCT for t = 7170

PC1H + PC}H> Ot + LOCAT * LOCBT + LOCCt for t = t1-,l7

Peak Flow Constraínt at Damage Centre D - Portage Ia Prairie

PD1L + PDZL > Ot * LOCAT + LOCBT + LOCCT

+ LOCDT - DrvRt fnrf= 1 6

PD1M + PDZM t Ot * LOCAT + LOCBT + LoCCt

+ LæDt = DIVRT for t = 7,70

PD1H + PDZH > Ot + LffiAt * LOCBT + LOCCT

+ LocDt - DrvRt t^? f = ll l/

Peak Flow Constraj¡rt at Damage Centre E = Headingley

PE1L + PE2L + PESL > Ot * LOCAT + LOCBT + LOCCT

+ LOCDT + LOCET - DrvRt forf='1 6

PE1M + PEzu + PE5T4 > Ot + LoCAt + LoCBt + LOCCT

+ LOCDT + LOCET - DrvRt for t = 7,70

PE1H + PEZH + PESH > Ot + LOCAT + LOCBT + LOCCT

+ LOCDT + LOCET - DIVRT for t = 77,17

Peak Flow Constraint at Damage Centre F - Winnipeg

PF1 + PFZ + PFs > 0t + LOCAt * LOCBT + LOCCT + LOCDT

+ LoCEt + LoCFt --. DrVRt for t = !.rT

59 Peak Flow Constraint at Damage Centre G - Assiniboine River Diversion

PV1L + wzL > DIVRT for f=16u lrv

+ 1 1^ PV1M + PVZM > DIVRT for L - I ,rV

+ + 11 1'7 PV1H PVZH > DIVRT for L - _Ll_rJ-l

Riparian Constraint

Ot t 100 fnrt=" 't"1 77

Divers ion Capacity Constraint

DrvRt < 25000 for to = 7,I7

Upper limit on Diversion Flow Constraint

DIVRT .Ot * LOCAT + LOCBT + LOCCT + LOmt for to = 7rI7

where: f is the storage volume, in c.f.s. - weeks, in Shellmouth Reservoir at the beginning of time period t + I is the mean seven day inflow to Shellmouth Reservoir over time period t

0" is the mean seven day outflow from shellmouth Reservoir over time period t

PA1L is the peak mean seven day flow at damage centre A, the Assiniboine River near Russell. It is section 1 of the linear rnnrnvi¡¿tion to theLr¡v non-linear¡rvrt rf,¡tçel damage\"4l@éç fi.ulctionlutuLlvlt andaru is-L) selected ó0 over time period L which varies from t=1 to t=6

PAZL is the peak mean seven ðay flow at danage centre A, the Assiniboine River near Russe1l. It is section 2 of the linear

approximatíon to the non=linear damage fr-rrction and is selected over time period L which varies from t=1 to t=6. Other peak variables are identified in a símilar ma¡ner. The first character of the variable name, P, indicates a peak mear seven day

flow; the second character, A,B,C,D,E,F or V indicates a danage centre as follows: A - Russell B - Miniota C - Brandon D - Portage 1a Prairie E - Headingley F - Wimipeg V - Assiniboine River Diversion The next character is nuneric and represents the section of the linear approximation to the non.linear damage function the variable represents. The last character represents the fanily of time dependent variables to u'hich the variable in question belongs. This is defined as follows:

L - April 15, to lvlay 26

M - lrlay 27 to Jl;trre 23

H - Jr.me 24 onwards ot is the mearr seven day outflow from Shellmouth Reservolr over time period t LOCA" is the mean seven day loca1 infloiv to reach A, over time period t and is applied to the model at Russell 61 LOCB' is the mean seven ðay local- inflow to reach B, over time period t and is applied to the model at Miniota + LOCC" is the mean seven ðay 1ocal inflow to reach C, over tirne period t and is applied to the model at Brandon + LOCD" is the mean seven ðay 1oca1 inflow to reach D, over time period t and is applied to the model at portage la prairi-e f LOCE" is the mean seven day local inflow to reach E, over time period t and is applied to the model at Headingley + LOCF" is the mean seven day 1oca1 inflow to reach F, over time period t and is applied to the nodel at Winnipeg DIVR" is the mean seven day flow down the Assiníboine River Diversion over time period t

OL 6l WATER SURVEY OF CANADA HYDROMETRIC STATIONS

05MEO0 | ASSIN IBOINE RIVER NEAR RUSSELL 05MEO06 ASSINIBOINE RIVER NEAR MINIOTA 0sMHool ASSINIBOINE RIVER AT BRANDON 05 M,JOO3 ASSINIBOINE RIVER NEAR PORTAGE LA PRAIRIE SHELLMOUTH DAM 05MJ00l AS SINIBOINE RIVER NEAR HEADINGLEY oSLLOtg ASSINIBOINE RIVER DIVERSION NEAR PORTAGE LA PRAIRIE 05ME001 r il OQ /-- ñ LAKE À€ M ANI TOBA t^tÞ L d" 05MEo06 ó ìF ñ\"4 oSLLol9 oSMHOOl MJOOS

SCALE IN KILOMETRES 0l 20 30 40 50 orozo30 SCALE IN MILES

Figure lV - ? WATER SURVEY OF CANADA Þ STREAMFLOW GAUGING LOCATIONS fN THE STUDY AREA l/^e l0 x l0 To d lNcH 7 x l0 tNcHEs ||^\"6 KEUFFEL & EssER co, M^oE rx u.s.^. 46 1322

iti iii iti iti lli,lf Ái lii u'n" K"E ln,T'JPJ"" ié=lïg¿.,1^ä,'f ,'ff 46 1.322 [./*ç l0 x l0 To ]l lNcH 7 x t0 tNcHES l\"16 KEUFFEL & ESSER CO, [^oE ta u.s.^. 46 t322 7 ro tNcHES M*E- l0 x l0 To ]{ tNcH x I ìì lç. KEUFFEL & ESSER CO. ÀrÁD[ td u.s.^. 46 7322

i ¡. i 'l i + -l -l I I

Li I¡

il t/*e l0 x l0 To .lt lNcH 7 x l0 lNcFlEs {-\-.G KEUFFEL & ESSER cO. M^DE I[ U.5,Â, 46 1322

Lr) f.l lrÈ F Ð C) fn ili p. Ë F Þ !/*ç l0 x l0 To 1{ lNcH 7 x to r¡rcHES [l-¿4 KEUFFEL A ESSER CO. ¡r^o[ tH u.s.^. 46 1322

To SHELLMOUTH RESERVOIR llNFr_ow

I Y

SHELLMOUTH RESERVOIR

SYSTEM ¡ NFLOW __-___-___-_@ A RUSSELL AT RUSSELL

SYSTEM INFLOW B MINIOTA AT MINIOTA ------@

SYSTEM INFLOW C BRANDON AT BRANDON

ASSINIBOINE RIVER D IVERSION SYSTEM TNFLOW _-______1. D PORTAGE LA AT PORTAGE LA PRATRIE - PRAI RI E

SYSTEM INFLOW.------L E HEADINGLEY AT HEADINGLEY

SYSTEM lN FLOW *É AT WINNIPEG F WINNIPEG

Figure lV -13 SCHEMATIC REPRESENTATION OF RIVER BASIN ASSINIBOINE 75 5.0

COMPUTATIOML TECI-INTT QUES

76 5. 0 cOrpuTATI0NAL TECFINTQLTES

5.1 The Computer Model The computer model constructed in this thesis consísts of tlrree separate prograrns which are run sequentially but individually il the analysis. That is, each program is run a¡d the output from the program in question is inspected and accepted before the next program is run. The three prograns are: 1. the Matrix Generator 2. the Linear Progrannning Sollution Algorithm 3. the Report Writer These individual programs will not be discussed separateTy.

5.2 Matrix Generator The purpose of the matrix generator prograrn is to transpose the problern under analysis from an algebraic representation of corrstraints and an objective function to the input format required by the linear progranrning solution algoritlun.

Each individual problem r-rnder analysis could be kelpinched sep- arately in the correct i:rput format for the linear programning solution algorithm. Ilowever, one nust realize the volume of input that is required. Tn the case of this study approximately 250 computer cards are required for every single problem anal-yzed. It should be noted that

this mea¡rs that for every single minor chaage r¡ade to a problem under

analysis, a ner^/ deck of approximately 250 cards would be required. A minor change would involve such things as a different reservoír storage starting 1eve1 or changes irr any one of the inflows to the system. These chaages would occur frequently on an operational basis. 77 It rnust also be noted that in the development of the workíng linear progranuning model there were literal.Ly hrmdreds of problem fornulations attenpted before the final l-inear progranrning model was determined. Without a matrix generator to generate this input for the linear progranming solution algorithm, tt may readily be seen that the volume of work required to develop the working linear progranrning nodel would be high enough to have been infeasible. The rnatrix generator written for this thesis is a general matrix generator and is not limited to the problemanalyzed in the thesis. The matrix generator will handle any linear progranmring problem as long as the problem is expressed in a simple algebraic notation. As we11, the matrix generator has the flexibilíty of designating any variables so desired as constants. This tlpe of varia-ble would include 1oca1 inflows or initial reservoir storage conditions. il¡ith the matrix generator developed in this thesis it is a simple matter to investigate different linear progrannning models or the effect of di-fferent inflow hydrographs vrrnn nnf'ìma]vI/ oneration of a gìven flood control system that has already been modelled. As noted previously in this thesis, in the interest of computational efficienry, it is desirable to ljmit the nurber of variables actually in the analysis to as sna11 a nimber as possible. The matrix generator developed in this thesis employs a trarsformation lvith regard to the previorxly noted fj-xed-valued variables. For any constraint containing constarts, these constants are first multiplied by theiT corresponding coefficient in the constraint and then subtracted from the right-hand side of the constraint in question. In this marrner the number of nust address is reduced. varj-ables that the linear progranrning algorithn /ó As noted previously the number of variables in the problem has a direct effect on computational efficiency and will be reflected in the cost of the computer resources required to solve the problem. It should be noted that if the nunber of variables exceeds arl upper bound detenriined by the storage capability of the con'puter installation used for the analysis the particular linear progranrning solution algorithm employed in this thesis is inapplícabl-e. A eomputer source code listing of the natrix generator developed in this thesis is provided in Appendix A. As we1l,

Appendix A contains documentation as to the fornrat of input required by the natrix generator prograln and sample output from the computer pTogram'

5.5 The Linear Programning Solution Algorithm The linear progïarnming soLution algorithn employed in this thesis is

based upon the mutual primal dual simplex method of Michael L. Bolinski

and Ralph E Gomery. The al-gorithm in question was supplied through the courtesy of Charles D.D. Howard and Associates, Consulting Civil Engineers, I{iruripeg, lt4anitoba. The algorithm proved to be quite efficient in the solution of the linear progranrni-ng problens involved in this thesis- Several different algorithms were tried out in the analysis and conrputational difficulties weïe encountered wj.th then. ûn the other hand the ntutual prinal dual

simplex method proved to be computationally practicable and quite

ôn1 ^++aUIIfUfV¡I -a ç.

'Writer 5 .4 The Report An irirportant point in a¡ opti.nization nodel is that the model output nust be presented in a readily r-rrderstatdable concise format. It is not 79 necessary that this output is meaningful from a linear progranrning point of view. Rather, it is important that the output related to the physical

system being modelled especially since the object of the optimization nodel is to assist in the for¡mrlation of subsequent si¡n-ilation analysis. The report writer shows firstly all system inflolv values. Under operational conditions these would be forecast values for all d-amage cen- tTes. Secondly, the natural streamflows are presented at all damage centres. Thirdly, optimal operating schedules for Shellmouth Reservoir

and. the Assiniboine River Diversion resulting from the optimization analysis are presented in a table together with streamflow conditions arising from the optirnal operati-on of the flood control system. All terms are in units of mean c.f.s. - weeks. The output from the report writer is shown in Appendix B. Also

shown is a listing of the computer source code and user j-nfornation to describe input to the report writer prograin.

80 I8

SJTNSTTI 0'9 6.0 RESULTS

6.1 The Optimization lifodel The linear progïatrnning optimization model constructed in this thesis is not intended to provide actual operational decisions with respect to the operation of the Assiniboine River flood control system. Rather, it is intended to provide reasonable starting postulates for a more detailed simulation arralysis of the flood control system in question. It is i:r,portant nevertheless that the optimization model produce starting postulates which are near optìmal with respect to the relative timing of the individual operatíons. The values of the reservoir and diversion releases should also be fairly close to optimal values. In the case of the Assiniboine River flood control system it is important that periods of high and 1oi.v Shellmouth Reservoir release be accurately timed- The tirning of the high and 1ow values of the flood control systen operating components is more important to the postulating of input to a simulation analysis than the actual values of tire individual flood control system conponents obtained fron the optirirtzatton analysis ' The results from the optimization mode1, that is operating schedules for the flood control system, may be input to a more detailed simulation model to fine ilme the operating schedules. This ivill allow the determination of acceptably accurate estjmates of the operating schedules with a minimum of simulation model computer ïuns. This in turn minimizes the cost of determining the operating schedules and, possibly frorn a flood control viewpoint, more important the tjme required for the analysis.

82 The l-ínear prograffnìing optimization model constructed in this thesis was evaluated by applying the nodel spring 1974 conditions on the Assiniboine River in Manitoba. This application ü/as described earlier in thi-s thesis. As noted previously there are six damage centers located along the Assiniboine River; Russell, Miniota, Portage 1a Prairì-e, Headingly and Winnipeg. For each of the above noted damage centers natural flow values ald resultalt optinal flow values from the optimization model constructed in the thesis are shown on Figures VI-1 to VI-6 inclusive. The optimal operating schedules resulting from the linear prograrrning optirnization model for both Shellmouth Reservoir and the Assiniboine Ríver Diversion of the Assiniboine RiveÏ flood control system are shown on Figures VI-7 and VI-8 respectively. The computer printout from the report writer constructed in this thesis is shown on Figures vI-9 to VI-11 inclusive. Inspectíon of the above noted figures will give a complete graphic and numeric sumnary of the results of the linear progranrning optirnization model constructed in this thesis for a given hydrologic event. In this case the hydrologic event is the spring 1974 streamflow conditions on the Assiníboine River as noted earlier.

83 6.2 Discussion of Results l\s note

Agricultural Date I{eek of Analysis Damage Factor

April 15 - May 26 1 6 0.333

May 27 - J:;irle 23 7 10 0.5

J:urrle 24 onwards 11 L7 1.0

With reference to the above table it may be seen that least damage j-s caused in time periods 1 to 6. More severe damage is caused in tine periods 7 to 10, md the most severe damage is caused in time periods tI r.o 77 . With refelence to Figure VI-11, tt may be seen that for each danage centre in the a:ralysis the peak mean seven day discharge is greatest during time period 1 to 6, less in tine periods 7 to 10 and least in time periods 11 to 77. As noted previously, flood danage in tenrs of agricultural damage units has been assuned to be a si:rple fu¡rction of peak meân Seven day dischaTge. Tl-lus, for each damage centre in the analysis, flood damage is greatest in ti:ne periods 1 to 6, less in time periods 7 to 10 a¡d least in tjme periods 17 to 17 - This result is as e>çected because of the previously noted time dependency of agricultural damages in the objective fi.¡nction. I,Vith refelence to Figure VI-11 it may be seen that the first week l-ll.!¡ +trL.tIç a ct.lld-Lj'/>-LJ----r.,-;- involve-sfl¡vvlvvJ a4 JPrrrlrósnì11ing ofvI Shellmouthu,rur¡rr Reseryoir. J|is is 84 realistic when one considers that the second week of the analysis is the peak mean seven day inflow to Shellmouth Reservoir. With respect to the formulation of input for a subsequent si¡n;lation analysj-s it should be noted that the model indicates that a postulated generating schedule for shellmouth Reservoir that spi11s early in the spring period is in order. The balance of time periods 1 to 6 indicate that the first tirne frane in the analysis, that is tjme peri-ods 1 to 6, generally be one of Shellmouth Reservoir spi11ing. This is in accord with the time dependency of agricultural damage functions r-rsed in the model developed in this thesis ' With reference to Figure VI-11, it may be noted that the mean seven

day Assiniboine River Diversion flow rate for time period 2 is 16,100 c.f.s' This is greater than the previously noted primary ljmit for flow down the Assiniboine River Diversion of 15,000 c.f.s. It is anticipated that ín- vestigations through the r.se of subsequent si¡rulation analysis would as- certain whether or not the Assiniboine River Diversion discharge could be held to 15,000 c.f.s. without undue flood damage downstream on the Assiniboine Ríver. In overview Figure VI-11, the output from the report writer constructed in this thesis, provi-des the necessary tining relationships between the operation of individual flood control system components of the Assiniboine River system and the relative magnitude of the same operating schedules to formulate reasonable starting postulates for a subsequent sjmulation analysis. With reference to Figure \{-11 it may be seen that the operation of Shellmouth Reservoir has been detennined by the optimization model to be one of generally early spilling of the reservoir followed by a period of moderate releases and finally by a period of ninjmal release as deternined by riparian flow constraints. 85 The operation of the Assiniboine River Diversion may be seen to be one of initially high diverted flow values followed by a period of steadily decreasing diversion flow values and finally by a period of non use of the diversion. The above noted tining of the operation of the individual flood control components together with approximate nagnitudes of the operating schedules for the same flood control system components is what is required to formulate initial postulates for a subsequent si¡irlation analysis. This is the objective of the optimization model constructed in this thesis.

6.3 lr{odeI Run Cost

The three comPonents of the model constrLlcted in this thesis, that and iL-' c flro m:trir{---^-- Þgllglo^-^?4+^? 4LLrr t the linear progranming solution algorithm

the report writer are run on a CDC Cyber , 77 0 computer located in lVirmipeg at Cybershare Ltd.

Run costs r^/ere as follows:

I4atríx generator $1. 04 Linear prograinning solution algorithm 1. 15

Dannrfr\uyvr r^rri tar .36

Total run cost 2. s5

Fron the above it nay be seen that the total cost for the analysís of one set of forecast inflows for the systern as a whole is $2.55. It is argued that at this pri-ce rate it is quite feasible to evallrate the effect on system component operating schedules of different inflow values 86 to the system. In this manner it is thus possible to deal with the stochastic aspect of systen inflow values in a manner which.is acceptable from an operational if not theoretically elequant point of view.

6. 4 Sr¡nnary Conclusions In sunnary it is concluded that the linear progranu'ning optimization model constnrcted in thís thesis successfully provides feasible operating schedule postuXates for a subsequent sjmulation analysis. The model constT-ucted in this thesis has been successfully applied to a rather simple flood control network on the Assiniboine River in l.{anitoba. It is recogntzed that the application of the model in this specific instance may well be a form of overkill in that the system is in fact sinple enough that acceptable starting postulates for a subsequent simulation analysis may be determined without the optimization mode1. However, it ís argued that th-is would not be the case in a more complicated system. lrbreover, it is the simpliciry of the Assiniboine River flood control system that a11ows one to properly assess the applicability of the mode1.

The model can be used at a reasonable cost. This factor augers well for the feasibility of using this model on an operational basis for

more colupLex ftood" contrÇ1 systems.

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(o oo Figure VI-11 OuÈpuE froq Reporc wrlcèr APPENDIX A

MATRIX GENERATOR

99 APPENDIX A - T4ATRIX GENERATOR

A-1 Introduction The matrix generator program transfers the problem under analysis from an algebraic representation of constraints and an obj ective fr.mction to the input format required by the linear progranmring solution algorithm. The natrix generator allows the flexibility of designating any variables as constants" In this nanner different variables in a given analysis may easily be held to constant values in the analysís.

A-2 Program Logic The matrix generator trarsforms tine related families of constraints to the input format required by the linear progrannning solution algorithm.

A tjme related fanily constraint is a constraint which has the same form with respect to the variables involved and the same coefficients over a finite m-unber of time periods. Only the variables change in that a new variable is defined for each time period. This is illustrated as follows:

Constraint Family Name LESS fimc ncr-iods defined over 1to5

Constraint Name Constraint

LESS 1 Q1 < 5000

IESS 2 Q2 < 5000

LESS 3 Q3 < s000

LESS 4 Q4 < 5000

LESS 5 Q5 < 5000

100 In this example tt nay be seen that the same form of the constraint is maintained over the five time periods involved but a ner,r variable is introduced for each tjme period.

The matrix generator reduces constraints of the family related type to the input matrix format required by the linear progrannning solution algorithn. An internal function of the matrix generator prograrn is to transfer each constraint name and each variable in the analysis to a unique integer value in the interests of progranrning efficiency. A cross reference table is built for both constraint names and variables in the analysis. In the interest of computational efficiency the matrix generator employs a transformation with regard to the previously noted constants. For any constraint containing constants, these variables are first multiplied by theiï corresponding coefficient and then subtracted from the right hand side of the constraint in question. In this manner the mmber of variables that the linear prograrTrning solution algorithn nmst address is reduced. The ntmber of variables in the problem under analysis has a direct effect on computational efficiency. A listing of the program is provided on Figures A-1 to A-7 inclusive.

A-3 Capacity and Limitations

The following limitations are i:r.posed: 7. A maxirni¡n of two hundred unique variables in the problen under analysis.

2. A maximl¡n of sixty unique variables i-n any one constraint. Problems involving either a greater nunber of equations or a greater ni¡nber of variables in any given constraint than noted above will requir" r0l minor progran modifications. A-4 Input

Sample Input is shor^m on Figures A-8 to A-14 inclusíve. An input

'r deck consists of the f^'rruf Ivwrtr5.^,.,; --.

1. System Initial Condition Cards

These cards define initial system conditiors in the analysis, constant per card.

Colur¡rs DescriÞtion Example

2-5 Variable Fanili'Name S

7-1 ñ Variable Time Period I

77-20 Variable Constant Value 72743

2. Constant Value Cards

These cards define constants in the analysís, one constant per card.

Colimns Description Example

2-5 Variable Family Name I

/-IU Variable Time Period 1

77-20 Variable Corstant Value 2900

3. Run Title Card

This card contains the alpha-ni.¡neric description of the problem under analysis " NZ Colun¡s DéScription Example

1- B0 Run Title Assiniboine Rlver L/P Flood Control Qptimization Study.

:Constráiúts 4 " Obj ectivei.Fúnction'and The objective function.and the constraints are input to the matrÍx generator in the same format. The objective function is input first, prior to the problem constraints. All constraints are identified by a constraint type as follows:

TyPe Description

N Objective Function

E Equality Contraint: =

T Less than constraint;

LE Less than or equal to constraint; < G Greater than or equal to constl'aint;

G Greater than constraint; >

4"1 Constraint Náme Card

Coh¡nns Description Example

2-5 Constraint fan,ily name OR]T

9- 10 Constraint type N

T7-20 Right Hand Side Value Blank of corstraint

103 4.2 Row Definition Card

Colunns DescriÞtion Exanple

1- 10 Starting time value -L

for row names

]-'7-20 Nr¡nber of time periods

constraint is to be generated for

4.3 Variable Counter Card

Coltunns Description Elanrpl-e

1- 10 Ni.nrber of variables in 54 constraint

4.4 Variable Fanilv Name Cards

Eight variable family nanes may be input per card"

Colunns Tlaqerinf i nn Example

2-5 Variable family name PAlL

7 -I0 Variable family name DA 1I

37- 40 Variable family name PAZFI

4"5 Coefficient Card Eight coefficients may be input per card to correspond to the variable fanily names on the preceding Variable Family Name Card.

L04 Colunns DéScriÉtion E_xanple

1- 10 Coefficient value 0.0

LI-20 Coefficient value 0. 06

71- 80 Coefficient value 0" 18

+.A/ o Variable Lower,Time Qualifier Card

Th-is card defines the lower time qualifier for each variable input

the preceding Variable Fanily Name Card.

Coh.un¡s Description Exanple

I 1- 10 Variable lower time I qualifier 2L-30 Variable lower time

ñlr1 lf tf aT

71- 80 Variable lower time

or¡el i fi or^

4.7 Variable ljpper Ti¡le Qualifier Card

This card defines the i-rpper time qualifier for each variable input on the preceding Variable Family Name Card.

Colwms Tlec.r-ri nf i nn Example.

1- 10 Variable i-rpper ti:r, e 1

n¡ra-l-ifior

Variable upper time nrre'lificr

71- 80 qpper Variable tíme 105 qu¿rrrrur^.-^1;.c:^- 4.8 Fixed Válue Cârd

This card defines whether or not the variables input on the preceding Variable Family Name Card are constants. Two t¡rpes of constants are allowed as follows:

F - Fixed value variable or corstant for all time periods. F1 - Fixed value variable or constant for only the first time period.

Coltmms DeScripti.on ExaFple

1- 10 Constant indicator 77-20 Constant indicator

71- 80 Constant indicator

4.9 Variâb1-e JJpper Bóund Câ-rd This card is used to input an upper bound on all variables introduced on the preceding Variable Fatnily Name Card.

T\^^^--:*+-: Coh¡nns rruJ\-r ry L-LUtt^* Example

1- 10 IÞper bound value 1 500

17-20 lÞper bound value 1300

71- 80 tÞper bound value 1300

card t¡'pes 4.1 to 4.9 are repeated as necessary to introduce and define all the variables for a given farnily of cor¡straints.

106 0utput

The resultant output from the ltÞtrix generator program ís shown on Figures A-16 to A-24b inclusive. The printed output is in two parts. The first part is a listing of each constraint family in the analysis. Each constraint fanily is identified by a one to four character alphameric name. To this narne an i-nteger is added to denote lvhich tìme period the constraint applies to. For example with reference to Figure A-17:

T,IXAL 1-6 This corresponds to the following constraints:

TDGLl

NDGL2 IfiAl5

lvfrAL4

¡,frAT,5

NI(AL6

Each variable in the constraint is identified by a one to four character alphameric name and an integer denoted time period. With reference to Figure A-17 it nay be seen that a range is given for the time period. For exanple:

PAlL 1-1

Corresponds to: PA1L1 L07 while:

L- 6

Corresponds to: 01

For each constraint defined by the time period indicator the variables in the constraint are defined according to their family name and their time period qualifier. With reference to Figr-u'e A-17 for corstraint IfiAt 1-ó the following constraints are i:lplied:

MXALI: -1.0 PA1L -1.0 PA21 -1.0 PASL + 01 + LmÆ < 0"0

MXAL2: -1"0 PA1L -1.0 PA2L -1.0 PA3L + OZ + L0CA2 < 0.0

I,XAIS: -1.0 PA1I -1.0 PA2L -1.0 PASL + 03 + LOCAS < 0.0

ti{KAL4: -1.0 PA1L -1.0 PAZL -1.0 PA3L + 04 + L0CA4 < 0.0

Ir4üI5: -1.0 PA1L -1.0 PA2L -1.0 PASL + 05 + LOCAS < 0.0

IfxAt6: -1.0 PA1L -1.0 PAZL -1.0 PASL +,06 + L0CA6 < 0.0

As shown above each variable in the constraint is nrultiplied by its corresponding coefficient as shoran in the row labe11ed "coefficient". As noted earlier any variable in the analysis may be held constant. Any variables held constant are shown in the next row label1ed "Fii(ed Va1ue". A I'Fr indicates that the variable in question is fixed over the entire range 108 of the constT'aint in question while a "F1" indicates that the variable in question j-s. only held constant for the first time period in the analysis.

All varíables are ræper bounded in this analysis. The rpper bound specified for each variable is shown in the next row of the output labelled "llpper Bounds".

As noted earlier all variable names and all ïow names are converted

to integer values to facilitate the computer progranming required. A

cross-reference of variable names and integer representation and row names and integer representation is shown on Figures A-25 to A-31 inclusive.

The output fron the matrix generator in the format required for the linear progranmring solution algorithm is shown on Fisures A-3?. to A-S5 inclusive.

109 o:OGtÁY clrc!.t 73/,L 0or:1 FTft L.6+!52 7y/jq/?q. 20.3?.'1 2 ÞLct i

ãp^CrA v YÀrGi N (i NpllT, l¡P a5 =I\Ft" 7, CUtp'Jf i I Â Þ:r =OUTpUT, rr : Â p -: . Ê-F o.- c_ F1' r t :Þ: F 1 r Z I f ã 2=P F 2, f a 22, T Â c : 2 1 | 1 A o F 2 2 I Þp 1 Z J, +r | !Þ:¿1, 1t c; gA I f r B ¡ri lX r f ÂF-: B:Hl^e rC IX. F JXV ÁL r r Âpg9=trI XVÂLt C ^ôtZ5 c tÂ19IY Gi'l¿o¡ ¡¡: oP0GqÁH c C rHIS Þeôto¡v fc4¡lSpOSaS 1N ALG!rqÁJC SYSTiI{ Otr CONSTaAINTS Â{O C ù N OIJECIIV! FUIIiTICN INIO IHE INOUf FCPPôT OEOUIPqS 8Y THÉ c Lrlctc o?eGolprI'lG soLut IcN aLG0iIr!il{. IT Is Â GENipaL usE HÄiFrx 10 C CgÌ'9?¡rr0Þ ô¡tC IS ÁppLICÂqLE TC {Ny paOBLEr,,EypçESSE0 IN rHa ç CCC?:C-T IIIPUT FOqHAT' c I NT:¡;iD ')fl Jf I Àlfi6ÉF 1, CC!r¡ T? t2 I NT5':E ]4HC'¡T,?OH,VA RI9L,FILEl.FIL!¿,FILE3,VAC, VÂRCNT I ¡ir:19ç'IILY: TCNFAF' PE.!00 I N-l:1Ép :. INT:CEÞ TFSl,E:\I1 o .ì : N T : t E f Y P : , T Y e o E L , E O u A L , L E 5 , ¡{ G G P I c ' ' r , ?t) tIts1\5Io.r FtxE0(5t I, upe.t0s (5 ¡ ), IqcH { 6û ),IVÁp (4 ), CoEFF (e }, å-lcu'lts (ã q (q (60t r , ì ,INVûq , HIL.T 160 ) i LOHLM'r (óO ) , CO:F (60 | ,TITLE( e ) 0IÈq.rslc.¡ vôJ ( zt 0 ) ,p:q I0r_r ( 2c 3 ) , vaLUg ( 2c 0 | , TEST ( 20 C,3) lÂTq llat-Á/F / -)arÁ I¡,,.!cuf/r,5/ ?5 0ô'tô i,/Lq c_/tl/4H L/,LESS/qH LEl,EOU¡L/LH ç-O/tN/LH ù/l Li / t,rI Gl I Gof P /,.H G:-/ lAlÂ E,/ûu a/ taîñO/\tl:-.-^tO/,f.CUrL/I/ tlOUf?/Z/ )511 CeJ'/t ax9Jf / \Ãf^ F/I)H Fl 'lórÄ Fl,/t!H Fl/ c c INJT:ÁLr7g t:c

_a I e INÊU- FIYiC VALIJE VAOTAqLES ç r-|'O(9. Éc 3A ?2 ) 1-1 Z ) VÁR I IN ), p:cI O0 ( IN ),VÂLUE ( INI 'i.F tat)F (cr | 3020r 30u 1 30¿1 JN=:\+1 G0 fo 3a2? 30?c îcNrI{Uq c) c INPIJf OU\¡ ':'JTLE CÂÞD c ?EôD (INDâTA,TC-A} -TITL:

77 c INFUT CO'ISIRÂINT gOUÂTIO'IS.NOIÉ fHAT TH5 CBJFCTTVE FUNCfIOh I< T?Ê¡f:D ÂS IH= F]?ST CONSTFAINT ÍOUÂTICN FOF INPUT FUPPOSES c

r¡10 lgôo(Ih0ÀTÄ,1e¿.t cNFÂH,TyFE,RÈ¡5 JF(5cF fJ¡t05T¡ I ) 4999.410 I r+101 :CNr:NUF ¡lrIv:5:N-fwEs+1 IF(f'TIyqs.:1.¿t HqIrS tI0uT,?76t ): a0f lNcÂ ta, I 0 6l I sTÂ RT, T 80 'EADtf\0r.rar 106) NV 1, c JNFUr 1Âra F:ÂSIqILITY IEsT r) Itr(llv.6T.6!l c0 T0 9,J1 rcf=t ::Nn:q Dq 3.1 J=rr200 9tt :É(J.N:.11 JS'r=Isf +8 TF IJ. N:.I I I:ND=IEI\O}8 lF (l:N0.úT.NV ) lE¡ln=t¡V r:ÂB( INolTô, t0 5) ( It'VôR lIt, I=IST, I:¡r0) a ( ( É ? : 3 I N t_r À f Á , I 0 7 I C C : ( I ) , I : I S T , I : N 0 ) ?: 6l_ìf TN0¡ lÂ, ! 0 E ) (LCt{L¡.T (I ), I= ISr r IENO r ?:ÄC(IflC0 rÁi 106 t (HILHI (I!, I:IST, I:t O) tEô0IIN¡lrÄ,1C9t (FIxECrIt,I=ISr,IE{Dt ?:Áa ( I ¡.rlirÄ, tcTt ( up3N0s( J t,I=lst, IE{O) IÊ tI5ND.q.j.rtv) 60 rc JP1 t00 ]É1 CONTÌ NT,,E 38t ì0NT1r¡95 () c cu'rour sril{vÁóy 0F c0NsTÞtINl FCr¡.tS c 105 IF(Cr¿FÁv. u?ITg(I0uT,27q) (lC¡l=\lc\+lîo.oPJT) H IF(\iN.Gr. St tlFIrE (IOUTrZTel H :FfrllN.Gr.Jl ¡lCU=1 qîIl: (Ioltf ,260' DNFÁ H, IsfÂFfrl 110 ,¡tol7.- l7ltJf ,2691 IF ( rYPí.:0.¡Il TYe0:ç0U¿L :F f ryoE.ã0.L) TypO:LÍSS Flgure A-2 ÍF I'vo:.80.f l TYP0=.1UóL ¡tatrix Generator Progrü LlEtlnB :F tryñi.S0.cl TyÞO:cclF Þ?oG? 4Y vô T6.\l 73/aI¿ 0P-=1 tr-N r*.6+15? 7t/)u/?a. ?0.\?,9¿ F'G'.

119 lSr=L -I].l 15I'D= I '10 1á0 J=r,2C0 IF(.1.R:.1t IST=lSTrq | ?t) '!F (J. NZ.t I I:Àr0=IEr0 +8 IF (I: H0.GT.r¡Vl IENC:flV ì{OIT: IIOI'T,262I (INVÂP (I' r I:IST,IENO) r{RIri ( I0U.,295) (L0r.rL v I (I ), HtLHr ( Il, I= IST, I:Nn } f (C0EÉ ( "{c JI: ICUT,?6r' } It, I=15T, I.N0t l?5 l,lÞIr: ( I0',r.¿70 ) (FIxED ( Il, I=IST,IEN0t HeIT: ( I0itLUZt | ( Up9NrrS (I t, I:JSl. IcN0) :FfryÞ:.t0.N.ôN0.I:X0.E0.ñvt,,{pIf:(IcuTrz6J) Typo IF f lYo:.!lE.N.ÂN0. IEN0. EO.ñVl !loIl: ( IcU7i265) Typo rFHs IF(I1r,rD.:0.¡lv) G0 T0 3e0 130 3ÉC DourlngS 380 n0Nrl¡:uE ICH-=1 c 135 I COOTifÁP C';5 TPìI-J IHIS ôO-LOOP N\IC: ÉOP EVgRY FOI{ IN fHE FÁfCIX. c 10 120 I=ISTÂRT,I "0H:D0Y+1cHcs:cus 11.1 c c c.r )09 ryp! C IF (IYPq.;1.Lt Iq0H( I"HC^t:) =-1 ç : (TYoE.Í0.61 Iqt}l ( IÞHCNr ) =1 lu5 IF lryr5..r¡. gt JloH(Iqrc:t¡)=0 IFlf YoE..rc. Nl fRHCNT: IFI{CNl+1 IF(l"HC{f .LE.20} G0 r0 3010 IcPCNT=1 valf =. | 2?, 296t (IqOH r II ),f t=1,2C I 150 5010 lGr'fIrJUS ICt'Í=ICNr+! c c PACIJÁP lOES THOU IÞIIS OO-LOOP O¡]CE FOP fVEEY VÁ9IÂBL€ IN c iÀCH F0P 0É 'lr! ¡'ÂrqIx. i.59 c r0 r'1 J:li\V IF lI^\r.:0.1 ) COl,rlfÞ=LCHLHI (.J ) tF lIîNf .|lÉ.1 ) Courtr.= LCt¿Lpf (Jt +ICNT-t IF (ç1Uft19.. GT. HILHr tJl t CCUxTR:HILFt ( J I 160 e C cu'Þ'rf ç?cs5 FEF:qENCE OF F0u ñ:H:S c lFl t..-O r1l \¡0H=NoCH+[ (¡tccH.Gr.5 g P IF I HqIf: f IcuTl. 2qe) 165 H IF tNo 0r¿.cT.5 ! Ì HR I15 { I0UI1. ¿r.0 ) t\) IF tÀ¡'0H.CT.ç0 I NqCH: L IF(.r.E0.ll vqlri(Icrrr1,zs1r c\FÁH, LpoH C FO? FIYED Flgure A-3 Á VALUE VAFIÀEL1 Ma!rlx Generator Pro8ru Llstlng t7q c 'I{FC( lF (FIxEq( Jl.E0.F ) CÂLL F IxVÂL ( IÀVÂq ( J ),C0tlrlTÞ.FHSCHG, VAc,pEpIOC, otcGcau v¿f G:.1 7jl1tr CPr=1 FTil 4.6+L52 7E/J\/?0. ?0.J?.0¿ FÂ6= q -ì +vÁLl,i, IcHc(,cc!F (J) ) IF(ÊIXq3f J'.IE.F) G0 fe J1C0 (Jì IF(IîqC<.:O.t l Helll(6,?5 Jl INV¡e 'C0Utlfq t75 IFfI,ìrrc(.E0.1r sT0P ?PSS=ÞcSc-cHSCHG -t', G C 321 31qg ceNt INU: lF (FTXEn( Jt .:0.F1.AN0.COUt'rTP. ÊC.LCHLqT ( Jì ) CrLL cIxVÂL(INVnP(J)' 180 sC OUN fo,o¡¡ gÇHÇ, VÂ e roE? I CD, VÁLUE' iCHCK'rl CEF ( J I I JF (FIx Erl ( J ) .rlE.Ét.0R.CC|JNTF.NE. LoHLHT ( J t I G0 T0 31C1 TF(IlHC'<. :C.1) r{cI'rE (6,2î31 INVÁa(J},CoUNTç IF(Iît]C(.E0.1) Sr0P oHçs=cps!-çHscHG 1e5 G 0 1'',) 321 l1e1 cctif !\uf c I.IÂS c 'ESI ÌO -CÉ5 IF VÂCI,1 3Lí ÂLF:ATJY -?EEN IEFINqO c 191 ,racctlf=c 30 | 1 VAÞrl\tf :vÄ ccrtr+1 Itr(Vrec\tr.Gt.?03t (;0 re 902 Ir(rqSf (VÄç:ilÍr1).Eo.EE¡lÐ) ç0 TO l¡00!1 IF(TiST(VÁFCrif 'l).80.1¡\VÁÞ(J)l GC f 0 3rl0C L95 10 f-l lccl 30cc IÉ (r:sr(vÁPcÀlr,2l.:rl.ccuNril G0 'I0 3î07 ,;n f1 ,r0û1 40C0 fîSrlVÀclNr'1¡=!r¡l,tÁÞ(Jl .-Sr I Vd.Cilt, ¿):q0UNTq zAA VÂçÏgL:Vd?I9L+1 r=!f ( vÁoîNTr I )=vaR IBL C c OUTÞ1.II C"C5S PEFEqg!CE L TSTING OF VÂEIÀ€L:S C ICNrZ:JC\t2+t ( f¡D 3 C Utl D S Iî N f Z I = Uo B S I J ) lF(ICl,tf?.Lr.¡l Gî r0 3C01 R ( 3 ( e o U ñ f | I r I I f ¿ 1 . 2 t ) 0 S I I ' I I = 1 . I lCNr¿=G 270 3008 ilv¡e=Nvacfl IF olv0c .1,f .5ql HeIlÉ f IcUr?,2rrl' 1F (!Vôç.Gf .5q ì lI0uf z,zr¡Ll IF(\VAc.CT.g0ì ¡tVÀc:"l?IlÉ ! ''rÞ tp fJ I VÁp I4L If1 r lotjT ?, ¿41 I IÀV r ClU¡{fR ' elc HoIri ( ?1. 29?t Jr'tvÂe ( J),CcuNfÊ,vÂoIBL 1C 07 t CNr IÀlljÍ ICCU¡{l--ÌC llJ\f .1 IF(IîOUilr.6r.rt G0 T0 lct6 IVôP f lCO,rNrl=f EST (VA2CÀri 5 t ??q C0EFF r f 11l,Nf Ì =C0:F (J ) T1 J?1 H 5C c (.^l c 0u'rprtT Y01ÞIx a ??5 J0 0É HqIf 1. t?1,. 2L¿l IIVAÞ r II LC OEFF (II I, II=1 | ß I Flgure A-4 I CCl, {f = t- Ha crlx Cenerator Progræ Lletlng IVÂo (I10rt.tT t:f E5t (VÂeCt,T i 3 I aCç-FË | Jr-1U¡JTr =CCiF lJl o¡0G:lY $ÄtG:! 71t7q Ctf-l FTN 6 .6+ L5? 7r/)4/?0. ¿0.3¿.OZ PA65 t Ial G0 TO lzt 2 3r C 32L CCI\Y'INUE I 4UfPlrT i{1frIY 2!ç I F (Ii:ouftr. Eo. e I HpI f q lzi.2t? ) ( I vÂq ( I I ) , c0EFF ( I I ) ' lI=1r8) IFIIî0U\r.Eo. 5) r{pII: (?L' ?901 IF(ICOUNT.N:.I| '{plf i 124 t2L? t (IVÂq (II},CCEFF( II). I T=1 IC cUNTI IC0Utlf=0 ' c e ..0 c rF(TyÞE.ie.r,tt c0 f0 320 IFFSîl=I"HSCr+1 IF(I?r-tSCr.Gr.8l 60 f0 30 11 q (I?qscTt:tFsç z.t't te '0 3¿ J0 11 r{qIT---l2.tt2-9at (B(Il)' !I=1' 8l IcPSCI=! ¡ (Irr{çcTt:rr.tss JZC ^¡prl \Ut c G0 l1 ¡+!l c c tufp'Jf pteqL:p F0 L/P c zl. r. g_og H Þ IT . (?4. ?? il r o Ir: ( ¿1, ¿971 qalf : (¿0, 7-.)2t Jlf Lç. r{eIf : (¿1 ,?1nl l{ÞIri ( zf,.29cl VAFIIL=Vô ol9L+1 ¡,rcIr: (?c, zoòl P0l.t,vÂ?IsL

c otJtprrr vôoIÁaLE -oou¡¡0sr9.r.¡.s.vtLuESro0H IYP:s fì r¡ìlc¡lrz.Eo.!) Go ro Jccq F ( 1 N t 2 . L r . , ll o I T Ë ( ¿ I Z 1 3 I ( I C U N D S ( I I I r I I I I C N T ? t I I .Àtua t ' = ' l00o ^CNf . IcH: IoHC'lT-l IF (I?HC.t-.L9. 2et qPlfa l??.,?9É) (IR0!l ( I l) | I I:1. Icxr ¿71 IF (I'HSCr.:0. 8) r{PIr: l?1.?971 (e(lI}' II=1.8) IF (I5HSCr.Ní. n ) r.tpTf : t23,?97 I (p (IIt r II:1r IÇt{SCr I 3 0 T'l gql

c 'Eg!IIIôT:ON CON9IIION¡ c H 901 lrcr?ÍlI?r'T,¿511 llv H 40 ro 9Ca Þ 902 ,{cIl: lIcUT,2i¿} G0 T1 gqf ?e1 c 999 S rgo c FLcure A-5 n É0ov1r sfÄÌ:!ENfs r{atrlx Generator ?rogre Lfstlng c ?a5 100 Écç:|11f2Y,Â41 ÞoDGoÂY y¡7c9'l 73/7t qor=1 ÉTN ö.6+L52 72/1\,tZO. ?E .32 .0? r-1 II l0t Fgc'1 1T (ÂL' Ilrr FtZ' 0) !02'0.r'l 1r (tÍ'Àt{' tX' Ir¡rF10.i I 10L E3Þ'¡rT f 1X. ôt, lXrÂ¿¡,F10.C I 1C5 F00r{âl( 3 flxra4} t 291 106 60F!.ttT( qMt 107 ÉCÞ¡{tf (qÊ11.0} 109 F0ÞYs1(qltc) ?LZ rgo"lÍ f 8( t3,F7.J) ì z1J trcqHdf (¡É1c.ll 295 Z110 çCPY:'(+-¡'l0vrraOH NÂP5¡'2X.+çCl{ NO.+./) Zr¡1 Ê0o{ I I (.- r, l: x, TVAoIABL: +, OX r 4NtJyBEq',/) ?\2 çCq{IT(fI¡I'CPOSS OEFEâqUCE OF COH NÀHES ANC L/P MAIqIX FOI{ r\uP3!?srl I¡IPUT 2q3 FCÞYTfIIl¡'{CFOSS P¿ÉEC:NCE OF VAqTÄEL5 NÄHES ÂND L/P INPUT YÂTRIX 3el s ccLUeN \trJPo.cs' l ?50 ÊQçY'lT(r1¡.+VÂLUE OF T=".I10i2x.qGP:AfEq rHÀN TqFHINA rT!0rt 4C.EXECUTION 251 -0çYil (¡:- d.¡VÁLtJE CF NV: ''I10'2X'sGP :ÁTEc. lHâÀl 2C. gxECUlICN T:RtlI '.1ÀTg!'.rl ?5? cDçuJ. ('t r,¡.tup9ag cÉ IN0EFEN0;'tT vÄ?IAeLEs IN ¡NÀLySIS EXC!:0S ¿0 ¡n. f Y:ctjrI0N T€?tINrT9C. rt z53 trCc{tT(r!.,.fFÇ0q...FIx10 vÂFIÂ1LE F0uNf't F ( r I r p | y ¡ '',14rI4,1y,!N0T ¿ 5 5 0 C a I T t r V A I I I L I , I , A 4 , T 6 , 1 X r N 0 T F C U ¡1 D. ) 25€ FCÊYrrr¡1"tVÀPIÂ8L="1Yiô4'1x'Ir{'lY,rN0r FCIJt'C.EXECUTI0N TqqHINAf J1: {lC.ATfEY'lING TO ESfôBLISH UPP:? qOUNO*) r, pô 260 FCcYI I lr- rî0¡tsf I\T :! ¡t¡c... r.a4, I Q r lx r 4-ã, TL ) 241 ÉCf;trlf (r ¡r'çE¡¡ç?ôL C0ÀSTRAINT ÉCoìl¡l 2É? rcç"lTl¡'tr,+vÂçIlBL: ",1cx,8(r¡Xrar¡,Lxlt ¿43 FCó9âr (" r,1tCy,..r,¡¡¡.2y,r..) r15 ¿4rr E0cYl: (' ICIENIr.lCxrFl).i,7 l?X.F1C,J) l 265 FocHlìT f r "'C0EFÉ¡ r1t0X' !.r'Ât¡'2X r r. +'tYrF10. f l 269 F0eylT(¡ ¡,r------_-___-__r) ?70 FCcHrTfr .rrsIxEc vaLUE',1cx,al0.7t2xrô1ù)ì ?.7! ;CoY.\a (r rrrrtFp.t eNrSr.11X,F13.,l,7( ZX,F!0.Ct ) 321 ?78 ÊOev1 r{at ¡,rSUFqAqY 0F CD¡rSÌ?AI¡tT F0CBS.t ?79 FOpATr (rtr,'CFJ:CTIV¿ FUNCTIC¡r(iFEArÉ0 ÂS FIçSf C0NSTR¡INT gy yATp frx G:ttqolT0o"t 2ß1 F0oY¡t(¡ r,lCYrÀt¡rIr¡,I101 ?82 FeoH ÂT (lY. Ar¡.I5.I51 290 F0cH l1 lr 4 .l 291 Fco'l 1T(' 1'I zoz ÉcoH.1r (At !c I 29b ÊCclJll (?:¡¡) ., !'Il¡c 3r! 2o5 çî.titf t. agRI0c+, 16X rt r,l?ta -a ¡ I2r1xr 7 f 6y r IZ, +-", I¿,1Xt ) ¿96 É.c.l lrt2^1qt 297 ÉCce tf(Âc10.tl 294 É0cH1r(¡ t") î N0 H H Lrr

sYfrqrìLIe oEFEoE¡tDE,r1P (q:Jl Plgure A-6 HÁtrfx CeneraÈor Program Lletlng yvrL su?trîufl N: FI 73/7L Oof:1 FTft 1.6+r¿5¿ 7" t)t'/?9. 20.3?,02 F3Gq

r 1 SUP?,lU rIrt5 FIxvâ L t I¡tvÀ:., rcN Íp,: HscHG, vaê , pEâ vÄLu -lt_ tccEcl Io0, É,IcHcK. c c lHIS SU9?CUTTNE trJNOS T']5 INPUT VÄLU: IHÂT A VÁFIÁBL5 TS IO CE SEI -O Tqq VlCIFIES THî PCH q.H.S. V¡LU9 OY fHE ¡IEGAIIVE OF THE a FIY:I VÀìIÂ?LE VALU: TIC5S TI-IE VARIÂELE CCSFFICJÊNT.

I NIEGEO V ÁA,CEOI OO, DIvl'tsIoil vÂe (2tc l,q!?I00'5NC (?00 ) rfaLUE (¿00 ) 10 D Af Â :Ert')/qrr:ENti/ c tCr¡ÇK:0 I ÇNI--O t001 ICNr=JC¡lrÈ1 lF fJ¡lvao.:0.vÁ?lIcNfì t Go l0 3000 IF(Vô?tfcÀtT).E0.EEN0l c1 fc _o99 ao r1 :3t!1L .10c0 IF(I:Nfc.:0.ÞEÞl0D(ICNrl ) co fc 100¿ î0 rl r!:l- 20 30r2 o cSCq13:VôLUc( JC¡¡T) rCOtrF e:fu)N 9co IrPl(=1 r.leIT: (8Ar 2C0l VAq 20C FOÇur; (1'l (1X, Ái{} | ,¡eI-: lAo. ?01t J.tvÁo,Icr.To 241 tretYl- lt .,¡Lrlrll .: f t,, \l ENO

qlp sYq90Ltc "EFEp:\CE (c=l) ENTPY POJIITS DEF LI\q O9É:P:NCES , 3 FIYVôL L ZT ?7

V¡ P IÄ PLE S S¡I TYPE Þ:LOCÁÍION ___!L -:r J ?0 OgFINED 1 3l¡ ::¡rn I\TEG5? PfFS I 76 DEFIN9O trl 0 I cFC( INfgGÍq ç,P. OEFINEI t lz 22_ t3 fc¡Jr INTCGEq 1L L5 L.) L8 ?J OEFIN:TI ll IC¡llÞ I¡ITEG:? É.P. 18 29 0EF:N9r) I IÑVAÞ Ir]f f G:? F.e. ':F: 25 OEFJN3D p10I00 î,Q, I I¡ITEG:? ÁFCIY È:r: 3 9 13 0qFIÀ¡E0 t o pFAL FccH 6 É,p. DEFIN:''Ì I ?0 A V LIJf P:ÀL ÂEatY ç,p. PSFS I 20 0EFIHf0 1 |-ìv ¡" INT:Gq? ÂÊ9IY ç.P. F:F: I 915 t6 z3 ô 0€FIri:0 I ÊILE I'IHE! YODS tÂoqcg Ft{r '.tpI lES ?3 Plgure A-7 lfÁÈrix Generetor Progræ Lletlng S i?141.

I I 29n0. I b30n. rlr I 3 60r)0. f L2û0. I Þ ?900. I 2500. I 7 2900. I 2fltl). I 1A00, I o e00. I tl ô 33. I l2 ?ô I . I t3 I I I5 I lo 50. I I7 LOCA 0l t2')0. L(ìCÁ n¿ l?00. LOCA 03 lll'). L0cÁ 0ô 300. L0CÁ 0c 300. LOC ô 06 300. Lr)CÂ o7 lùù. Lnc Á 0q I00. LOCA 0c 100. L()Có lo I0u. LocÄ 1l Lt/C a t? l). LOCÂ l3 L0cÀ lô n. L0c Â lf, LOCÂ ¡6 LOC-À l7 Lf)c q NI 1500. LOCB o? 6t00. L0l^t¡ 03 ¿Ht)0. L0 cn 0ô ¿400. LOC F o5 32fJlì. LOCq 0b 3400. LOCÉ¡ o7 6700. LOCc 0¡ì 370t1 . L()Cc 09 30r)0. L0cc ì0 7+0t). LqCF tt L0cc l? l3hì. L0crì I ?oo. LlJCfr 14 In7l. L0r¡: l5 I {ì56. LoC,ì lÉ lI')0. L()CF LI 1056. L0cr 0l JAOO. Lr)c c Hil? 1t00. L0cr H03 ArìllO. {ou L(.¡ac ?100. Plgure A-8 L()CC 05 1500. InpuÈ Èo YÂCrLx ÇeneraÈor L{)CC 0/' 13n0. ri LOCC o'l 2()o . L' LOCC 08 lh0f). L0cc n9 l7oo. =j t- i

C á Õ ô ô î ô O 1 O I a O ô ô - I ô O ô ô - O ô O Õ ô Õ .î - Ô 3 ¡l ô O ô !a ô a I ô i O O a I Õ O Ô Ô ,- O O ô O O ô O ô - - ¡ 1 ¡ i! ì q iI ¡ n ¡q n M J ? Q A 2 ? a ) z a o 3 2 2 D o Õ o o Õ o i Õ ¡ Z I ¡ ¡ ¡ n r I I ! I I I ¡ I ¡ ¡ T I I ¡ = = 3 BTT f SOCH -/eJ ÐJO355Ð5 OOOOôã5CO/-r Z Z {}ú ¡UN-o -!31-tf C I {ùú Þ UaP{Oi Þu!-c Cl {}ú F !NÞ{f i }UNF5Ú I {OÚ Þ!tuF{OU èUN-o

E : -Fr.!!L!ùu ¡N r !ruu.! F¡ !Éûu r c ¡î E''¡ ÈNNÈ -/NruÞ ÈU¡{CF -É ¡c.È J-rX O(J C!= 3{aNN Þ !L'þ@O r }t ù{{N{lúO ¡f Þ}J ! I ¡F È -}3{N}U ÞúF{ f Ê H ãcU-3 F¡ ÞÚo Þ ¡ù € i !dÞ= Ðcóc >o5J Flors tÞ: = -oc o- ¡ €O O ¡oc oÐ ?roa r { oc ¡ o f } 5 Þ oc ¡ o: o J o ó c 3 c -.!Þc - c ? i: r )a ¡ ou {,¡}uúo 3 o c o J f i -- 5E È bu 6 {Fr \...E ¡ f, o o

= c r C T I

c I

o o> l,b

x o a o

l-r I r'l ll pÂlL pâ2L ea3L PÀlM PA?M pô-r'¡ PAIE P¿aF 0.0 0.0b 0.73 0.t/ 0.r)9 1.09 0.0 0.1ð ilr tttr trt I till tll I lSofì. I300. lo0n0o, 1500. t300. 100000. 1500. I300. ÞÂ3H PùIL Þ52L PC3L ÞRIM Pq2Fì PtJ3U PbIts ¿.15 o.o o.r')5 0.6? ().ù 0.oh 0.93 0.0 lill III I llll III I l0oo0{). 3000. 1000. llj000ll. 3rJ0U. IrJoo. I0U000. 3000. eRZF Þq3'l PCIL PCZL PCIþ PC?v PClr-t PCets 0.15 1.85 0.0 l.:i3 0.0 ¿.0 0.0 4.0 lttI Itt I Itll III I l0¡0. lr)000ô. 600t). l.000utr. 4000. IrJ00{)0. 40u0. 100000. p0lL pf)tr pu?r.r Þ0lH FD2rr PEIL PE2L 'r0¿Ln.0 0'¿ð 0.0 lj.a 0.0 0.79 0.0 0.tJ7 lrlt lrl I Itll lrl I l00ufr. Ir)0c00. llìr)00. l0u0(10. ì0r)0u. lu00oo. 10u00. I0000. eE3L ÞFlH PF2È PE3H pFlq PF2ts pE3k PFIÂ 0.1 0.r) o.l o.l: rJ.o r). ¿ 0.3 0.0 llll Itr llll ItI i 100000. Irl00rr. l00l)0. loor¡Ur-r. lur,u0. lù001. l00uu0. 70700. pF?L eFlL þFlþ pF2-Å pF3M PFIP PF?ts pFJF 0.a 0.ô 0.0 0.À 0.* 0.0 0.4 ttlt III I llll tlI t c00nn. 1110000. 707n0. 9r)0f)(r. 10u000. 70700. gu00r). 10000t). eVlL Þv2L pvlv ÞV?È¡ pVlti PV2H 0.03 (r .ltl l).05 0.15 0.I 0.3 lìll lt llll ll 15000. lfì000. lbo0o. l00oo. It0L)ll. lú0on. sxôL L 0.0 lh

PÀIL PÀ?L P^3L O L('CÀ ,ì1.0 -1.0 -l.o I.0 l.r) Hllrr I (Olllô õ F l5nt'. 1100. l0l)0n0. I0l/r)(,0. l0D00u. Þ¡óv L o.0 7 I0 5 ea lÞ Pô2M PÂ ìv l.) LOf:Â Plgure A-10 -1.0 -1.0 -1.0 I.(' 1.0 Inpuc to HaÈflr Generåtor ¡ll7 1 I I I I'r l0 F 1500. lf0{t. 10000n. It)u0{rû, l0r)00ù. MXÀt¡ L 0.0 It t7 5 -iI PAll¡ PA?H PÂ3h O LOCÂ -t.0 -t.(' -t.0 1.0 1.0 I I I lt tl I I I 17 LT 1500. 1300. lnr)0n0, I0000U. 10rJ000. .IBL L O.O lo b eÊlL Pq?L PF.¡3L Cr L0Cô LoCÊ -1.0 -1.0 -1.0 1.0 l.f) I.0 IIII It ll16 6^ 3000. loofr. 100000. 100000. 1000u0. 100000. vxFp L 0.0 t to () rqle pp2y PÊ-lþ 0 LOCÁ LOC¡ -1.0 -l.r) -1.0 l.l' 1.0 1.0 ltlT t I I l0 l0 lr) FF 3000. 100o. l0(Jo0r). l0rJ00'). l0U{r00. I00000. .xFP L 0.0 ll t7 A ÞqlF pqAH PÈlr O LOCA LOCÈ -1.0 -1.0 -1.0 1.0 l.¡) 1.0 I I t lr lt ll I I i 17 17 17 300f1. l0r)0. l0ooot). [l)0(J00. 100000. I00(J00. lxcL L 0.0 lh ô eclL pc2L 0 L0ca L0cÊ Locc -1.0 -t.o t.0 ¡.c¡ 1.0 I.0 tttl It llítl th F FF ô000. 1100t)ù. ì00t,ôr). t00000. I0r)u0rJ. l0ùn00. ulc! L 0.0 7 L0 ã .c I r. pc2M r) L'-ìcÀ L0rc Locc -1.0 -1.0 I.0 l.{, l.f.) 1.0 Hl I 7 7 r'Jt I lo lo I0 l0 OF FF 400n. 11r0000. l0f)illrlr. l0ur)1,0. l0rìr)00. 10lJ000. ufcF L 0.0 ll 17 6 FIgure A-11 "clH Pc?H o L0cÂ LocF Locc InpuÈ to r{åtrfx Generator -1.0 -1.0 1.0 l.lr 1.0 t.0 .l I ll ll tl tt ll1717 t7 17 t À000. Ir)00r)0. l0r)rl(r0. t0rtìcL'u. l000rJ'J. )'00000. er'FL L o. t) I6

ODIL ÞD2L (l LOCÂ LOCr¡ L0CC L0CD ¡l I ! k -1.0 -1.0 1.0 I.c t.0 t.0 I.0 -I.0 t tll lil I I tô! 66b 6 t FFF 10000. 10000ô. l00lr0r,. I000tr(). 1r)00urJ. l00ur)0. lût,000. ¿50u0. þXDH L 0.0 7 IO

TJ e0lu P02H o Loca L0cÊ Locc Loco DIvp -1.0 1.0 l.r) 1.0 t.0 1.0 -1.0 I 171 777 7 I Ì l0 ¡0 l0 tù I0 TO F IFF 10000. lu0ooo. lnr)000. l0o0Dù. 100000. 100000. I0rJtJ{J0. ¿50{J0. 0.0 tl Â ;t¡ I l-r P02l-' r) LOCÁ L()CfJ L0CC L0CD CI!rr -1.0 -1.0 1.0 I.L. 1.0 1.0 I.0 -1.0 I I rt lr it lt It II F IFF 10000. lr]oooo. looorìr). lo0(,0u. I0u0u0. 1000r1il. I0ri000. ¿50ù0. VXFL L 0.u I lo :ËlL PF?L pÉ,lL O LoCÂ L0Co L(,CC Lt,Cl, -1.0 -1.0 -1.0 I.t¡ t.0 t.0 1.0 1.0 Ittl III I lllr, 6c.t1 6 IFF I 10000. 10000. 1000fr0. t001100. I0rl()00. I00000. 100rr00. |'00000. LOCE D IV9 1.0 -1.0 66 100000. 25000. !¡Eþ L 0.0 t to IO pE lp pE2M PF_ìÈ O LOCÂ LôCÊ L0CC LOCt -1.0 -1.0 -l.o 1.0 I.0 t.0 I.0 1.0 tttT tf7 7 I I I LI, to lo l0 l0 ftts F 1000(lq I0000. lrrù000. l0t)00t-'. I0r)0{JrJ. I00000. l0u0{J0. IU0000. LI)CF O IVF lJ 1.0 ¡'J -1.0 7H r l0 t0 F lo0n00. ?5000. UIEH L O.O Flgure A-12 I I t7 Input to Mstrlx CenereÈoE l0 eElH pF2H PFJts O LOCÀ LOCc L0C,C L(,(.t (: -1.0 -1.0 -l¡rl l.ù 1.0 t.0 l.o l.tl C_ I I I I] It tt tt Il I t I 17 t7 t7 t7 t 7 ft r!-l 10000. I0lt00. 100000. l0rit)00. l000oo. I0ouofi. I000rJ0. Iu0000. L0cE D I Ve 1.0 -1.0 ìl tl 17 17 t00000. 25000. ctFL L 0.0 I6 ll 9FIL PF2L PF3L O LOCÂ LOC9 LOCC LUCO -1.0 -1.0 -1.0 I.0 1.0 1.0 1.0 I.0 llll II rì lll6 b6 66 fl tf 70700. 900u0. I00000. I00000. l000urJ. I00000. !0ù000. I00000. LOCF LOCF OI V{ 1.0 1.0 -1.0 til 666 100000. 1000{J0. 25000. uxFM L 0.0 7 t0

ÞFIM PF?F PF3\ O LOCÁ LOCH LOCC LOCD -l.o -1.0 -1.0 1.0 1.0 1.0 1.0 1.0 ItlT I I I l0 ¡0 l0 l0 l0 FF FF 70700. e0000. 100000. I00000. 100000. 100000. 00000. 0 LOCE LOCF DIVK 1.0 1,0 -1.0 77f l0 l0 l0 FF 100000. 100000. 25000. u¡FH L 0.0 ll 17 ll EFIH PFUH PFJH O LOCô LOCI¡ LOCC LUCI) -1.0 -t.0 -1.0 l.t¡ I.0 I.0 1.0 I.0 I I r ll lI lt ll lI I I I t7 t7¡¡ L7 t7 17 TOfOO. q0000. 100000. I00000. 1000U(r. I00000. 1000ûr). 100000, LOCF LOCF DIVC 1,0 1.0 -1.0 ll H tl tl 17 t"J 17 ': f'J ': l? 100000. 1000f)0. ?50(,0. kovL L 0.0 lö Plgure A-13 J PvlL PV2L ()lvr-r InpuÈ to HatrLx GeneraÈor -1.0 -1.0 1.0 rtl lI4 1500rì. I0000. 25000. qDvM L 0.0 7 lo 3 Èvtv Pv?a DIvfr -¡.u -I.u 1.0 tt 7 II l0 15000. 10000. ?5000. sovH L 0.0 It 17 J evlt"r pv?ts DlvH -1.0 -l.fì l.lì lt tl It l7 15000. 100n0. ?cfì00. .ÂrF L 1930. I t7 os 2 t.0 -0.446 ll t7 t 7 FI l0nn00. ¿_1572. ;[Þó (i 100. t t7 ¡ o 1.0 I l7 100000. Sl'nP E 0.0 ?t7 ssf0 -1.0 t.0 1.0 -1.0 ¿l ¡ I t7 16 Ib Ih

¿r1t.. ctlt¿. 100000. l0lJ(1u0, ilvp G 0.0 I t7

J I vP (i LúCô LOCH LOCC LoC0 1.0 1.0 I.t¡ I.0 1.0 -1.0rN) w III l7 17 17 t7 F 100000. lrr0rÌ(J0. lrJ0ô0î. tr,ùu{J0. 1t,0000. I04t'00. l0uuu0. lur¡0r)0. Flgure A-14 Input Èo Hatrlx GeneraÈor |--

c- 0qJ:c'IvE Err¡tcfI0rttf"EÂf:î Âs FIÞ.,-r cctrc:QÁINf ?y t,ÉTÞIt ç:Àl:?Â'0â ill coftcrpÂ I'tr fJnw f.. . oB Jl 1-: vdçIÄeL5 ÞÀ1L e ôÂ3L eal{ P A?H PASY PlLH p¿?.H 'fIY: Ptrcr0D 1- 1 ^2L1- 1 1- I l- 1 1- t 1- I 1- 1 L- 1 CCEFFICIEN' c.t'lc t.000 .090 1.090 1.00c .160 FIXfO V¿LUq upotrÞ 8¡t05 150 3 . 1300. 10 0 00 0 . 1300. 1C0000. 1500. 13û[.

VÁ9IABLE OûJH OB!L PB?L o33L PBlf PB2r',1 p33H P91q fIvE PEeI0C l- ! 1- I !-1 1- 1 1- 1 1- t l- t 1- 1 CCEFFJC I:¡II 2.1C'1 !.000 0.00ú .080 .9J0 !.0:0 FIYED VÀLUE UPP:O. 9NOS 10 0 00't. l00rl . 1000. 10c000. 3000. 1000. toJ000. 300:. vûe:ÁeLE D q2-A oq3q PClL 9/¡.21 PClH PC2H PC 1 H PC?H T!9E Pf9TDC 1- t r-1 1- 1 t- r 1- 1 1- 1 1- I 1- 1 CC'FFICIZNf .150 0.00c 1.330 0.000 J.00r À.0ü0 FIYqO V¿LUq upÞio qÀrs 10J1. rtc04c. l{C00. [0!00!. ¿.009. 100c00. r000. 100cc:.

VlOIAALF o 01L PO?L Þ0lv e t)?Y ODlH PD2H 9.1 | Þa 2l Tr|¡Ê prcloc l- L 1- I t- I 1- 1 1- I 1- 1 1- 1 clEqFIeI!À1. c.0'10 .?40 G.000 ,Llc 0.00 0 1.000 .û7c FIXEC VILUç u9pFo Þ¡rns 1100 1. 1:0 c0 0 . 1CC0C. r-000cc. 10c00. 1300c. Ltocc. vÁelÂqL. o:lL oglY Þ5?v Ç93¡ PElH o92H p:-JH Ps1Â t- ! !- ! TI}¡: P9PIOC l- 1 J- I 1- I t- 1 COETFICI:¡IT 0.010 .130 .190 0.000 .200 .30û !.0ú6 FTY:O VôLU1 UPPEg B NO S 1000c1. r0000. r.0000. t0100c. 1C000. 1000?. 10c000. 7.170,.

VÂOIABLS OF¿L PF 3L oFly oî2A PF3*,| PF!H Pç ¿A PF 3H T'THE OfEIOC 1- t 1- ! l- I 1- 1 1- I 1- 1 t- 1 COEFFIl IENI .bc0 .l+00 0.000 .r+C0 .¿+00 c.000 .1.00 .l+00 Ffx50 vaLtJS UPPFP qNOS ? 0 ofj l. rr0000, 70700. 9000c. 100000. 7070ù. 90c00. 10c000.

VÂPIÂRLE ovtL pv?L Pv1¡' ev?v PVIH Þv2H flFE plcI00 1- t t- 1 1- 1 l- 1 1- 1 t- I CI]'FFICIINf .1 rq .t00 .050 .150 .10 0 .JC0 FJXFN VúLUÉ uÞpEÞ Bllîs t5c11. t e 001. r5c00. 10000. 15000. 10c00. at)

H N) Þ Ffgure A-16 ) ConaÈratnÈ FanllY 0utput su¡,vÀÊY 0F c0r.rsf.: I\l qC?uç

--t I corrslÞaJllT \ÂqE... vxÂL 1 -

VôCIÂqLE t^tt oÁ2L Þ¡3L 0 LocÄ rTUE PEçIO¡ 1- t 1- L r- I L-Ê !-6 COEFFICI:NT -1.r10 -1.û!0 -1.00c 1.0c0 1.000 F:XEO VÄLI'i UÞOçO RNDS 150 r. 13¡0. 100000. 100000. 100c00. LC c3ils lc Â Jl.rf NÂ'{ 9. . . .{xÂ H 10 vrolÀBLF c Ä lrt oÀ21.1 0 Loc¡ rIts5 ÞÊÞI0n t- L 1- ! 1- I 7-10 7- r0 c0ÊFFIC J:Nf -1.C tC -1.000 -1.000 1.000 1.010 F:YEO V¡LU9 F uÞÞEc ÞN0ç 15C"1 . 13!i. t00c00. !00000. 10cc00. LE . c.-1

CoNSÍe¡' Irlr t¡ÂP2.. . {YAH LL- L7

VTCIABLF . a lq o a2,t oÀ3H 1lv5 pEoI0f r- I 1- 1 tt-!7 11-17 Íf NI -1.0 tl -1.110 -1.0cc 1.00c 1.0c0 '?CFFIf:trf YgO VÂLUq uppFc Bt'rl_ç 150 l. 13¡ 0 . t0cccc. t900c0. 10000c, LÊ

H È.J (tì Flgure A-17 ConsÈrslnÈ Ful1y OuEpuc SlttsMðFY OÉ tCllçT. ¿ f\l ç0rqs c0NSfFA r¡¡Ì .rav:.. . YaL I-

vrçIÂeLF o 91L 292L P B3L 0 L0cÂ L OCB IIPE PEETOC 1- 1 1- 1 1- 5 1- 6 1- È CCFFFICIE\T -l.0lc -1.01! -1.C00 1.000 1.00c 1.000 ÉIXED VÂLU9 UPP:P gNOS 300:t. 1000. tJ0000. t01000. 10c000. 100000. LE . 0.0

C?NSTF A IìIT NÂH E.. . YX9H 10

V4FIÁBLE P B 1',I OB2Y o B3ll 0 LOCA L 0c9 II14E PEFIOq t- I 1- 1 7-lr) 7-L0 7-10 COFFFIC IqNf -1.010 -1.000 -1.000 1.000 1.030 FTXED V¿LUg uoÞra q¡r0s 304 I. 10!t. r-!00c0. tlJ00ù. 1C000õ. 100t00. LÈ . !.1 c0uslFA I¡tr \la'i8.. .vY9H It - L7

VÁPJAELg o?tq o ezq oe3H 0 L 0cÁ L CCq lTu. p..I0ñ t- I l- I 1- 1 11-17 COqFFIC Jí\ T -1.0 r0 -1.1'J0 1.000 1.0¡0 1.000 FTYqO V¿LU5 F uooFo a N0s 3011. 100000. 111000. 100000. 10!c00. Li . U.,

H N) o\ Plgure A-18 ConsÈralnÈ PællY OutPut SUHHâPY 0F 30r.lSrt ! (Nf F0?tS r_ !! 'f c0NsTRA Jrlf rlôqE...vY1L

VÂAIÂ9L5 r c1L FC2L 0 L0cÂ L0c9 LOC0 OIVq l- Â TI HE PgÞIOO t- 1 1- É L.- 6 1- 6 l- 6 COqFFIC IgNI -1.q10 -1.010 1.00c 1.0,.1C 1.0!0 1.000 1.000 -1.0c0 F]XED V¡LUg ts F upÞco qÞ0s 10M. 110!04. 10000J. t10c00. l00G0c. 10c000. 100000. 25a\t'. . LE

col.lsTFÁ Illr r.lÄ'¡1.. .HX0M t0

VÂPIÀBLE o 01Y Pgul L0cÂ L0c0 fÏME ÞEFIOE 1- I 1- 1 7-10 7-10 7-10 f-L7 7-10 7-10 COEFFICI:N' -1.010 -1,000 1.Ct0 1.0ec 1.0!0 1.000 1.000 -1.ûcc FIX:O V¿LUE Ê ts F F 1000 1. 1,1 e03c. t00BG0. 110000. 10c009. l-L¡0C90. t01000. ¿500û. UPPEP. NDS . LE . 0.0

coNsrcÀI\l \ôtsc...vYDH 11 -

VAPI ÂBL: o 01q oC2'l 0 L 0c9 L 0cc LCC 0 0lvc rIyE PtaJ0l !1-17 11-17 1!-17 LL-t7 t!-17 11-t; crJEFçIClfNr - t.1'tc -r.110 l.Ctfl l.DCC 1.000 1.00t 1.000 -1.CC3 FTX tr0 V ALUq F UÞPEN BNDS tc00'1 . rc0c0!. ri0cc¡. 10t000. 100c09. 100ct0. r0c000. 2560C. . LE . c.]

H N) { Flgure A-19 ConBtrãlnt P@lly OutDut SUFHAÊY OF CC}IST?¡IN1 FOã'.1 5 'tl c?NsrF A IrlT ¡ ôv tr.. . YYCL

VlPIÄ8LE Þatl P CZL n L0c3 L 0cc TI!'E FEPIOC 1- I 1- ! 1- 6 1-6 L- î, COSFFTCI:NI -1.C10 -1.0c! 1.000 1.0G0 1.000 1.000 FTXED VALUS F UPPEq g N0 s q001. r50010. 1C0C00. t0c06ir. 10000c. 10c000.

CDNS TR A I'I' NAY E. . . YYCH 10

VAOIÂELE rc1{ Þc2Y 0 L0cÄ L0c9 L occ TJlt9 ÞÉÊI0n 1- I 7-10 f-10 7 -L0 7-LÛ COSFFJC J:N1 -1.t'10 -1.000 1.000 1.C0C 1.900 1.00c FIXEO VÂLUf o UÞPEÞ ì10 S 1.001, tJ c 01J . 140C00. !0!00r:. 1000ûc. 10000,1 . LE . û.¿

COIISTRÂ III' T'AH:.. . CYCH t1 - L7

V¡PIÂÞLF otlH ac?tl 0 L0cÁ L0c0 L occ f ÌH: FFc I9C r- | L1-!7 1L- 17 IL-L7 COEFFIC I:Nf -r.0.10 -1.000 1.00É 1.000 1.000 1.000 F:YgO VôLU: F uopEp 8¡r0s \0u t!. 14e00¡. 100000. t!1000. t00000. 100c00. Le . 0.c

H F.J Flgure À-20 Conscralnt Pæi.lY 0ucPut suvYÂFY 0F co¡tsrì¡!lll rg:15

CîflSfeA I{r NAHe... YxEL 1- -r

odll VICTô8LE e Ê_?L PE3L LOCÀ L OCB L0cc L CC0 PEFIOO TT!E 1- 1 1- 1 1-l L-6 1-6 t- o 1- 5 CCEFFICJ:NT -1.010 -1.0!0 -1.0û0 1.0û0 1.000 r.000 1.0cc FTXEO VÁLU: F q UÞPqP NOS 100!t. 1000c. 1l0 00 0 . 1!C000. 100000, 1e0600. 10J000. 10JCCC. vôcJÄ8LE L CC: OIVC TIHE'PEFIOC 1- 6 1- 6 CCsFFIC J5N1 -1.000 FIYEO VÁLU5 UÞPER BNDS 100eci. 29090. LE . 0.1 ccì¡sTa¡ I\f NÂrjE.. . {YEH 7- t0

j glY o Þ5 VsCIÁELE € ?..t 3H 0 L 0cÂ L 0c3 C pSFICC LOC LOCD rIPS 1- 1 1- t 7-1C 7-10 7.LIJ 7- 1o 7-LC cogFFIcIS¡¡I -1.010 -r.000 -1.e00 1.004 1.000 1.010 1.000 1.0cû FTYEO V!LUq F U9ÊEo 8N0ç t00!1. rc04l. 10cc00. 10rc0c. 1000fJ 0. 10c000. 10'J000. L0l,j0ú. VrçJÂPLE L CC: 1Iv? TIFi PgCIOE COqFFIC IENf 1.C',t0 -r.900 FIY5O V¡LUg uPpEp e ilos 1: 0 0c l. LE

CONSTFÂ INf {AHÉ...¡{X:H It - L7

p52q V,1RTÀFLE P5tfl ÞE3H 0 L0cÂ LCCB Locc L CCo TIHE FFçIOD 1- t l- f t- I 1l-17 11-17 LL-17 1 t-17 COEFFICIENf -1.040 -1.0c0 -1.000 1.0c0 1.000 1.0J0 1.000 1.JCû FJXEO VÀLUE F uooFo qllnq 1C000. tc0c00. 101000. 10000c. 1C0000. t0c000. 10,100c.

VTEIAELF L CC: .]IV9 1:!{¿ PFç I00 t 1-1- t1-t7 c0çÉFIcI:rrT 1.C1C -1.iÌì10 FIXEN VÀLIJÊ UPPEÞ 8N0S t00cl1. ¿5000.

H N) Flgure A-21 Conscraint Fefly outpuÈ sljPcÀeY oF ctftsreârilÌ Fc?Ys c1¡'STFÂrrlt fiÂY9...YytrL 1 - Iil

ÞF3L VAÞI¡ELE AF2L o L 0cÄ LOCB L0cc L CCn PEçTO9 TTHE 1- 1 t- | 1- 1 L- ç. t-6 1- 6 L-5 L-a clEFFICfS¡tf -1,010 -1.000 1.000 1.000 1.000 1.000 1.0r1 0 FIYfO VÂLIJ9 uPo!Þ 9liDs 7077'l . 90031. 110C00. t00000. 1000c1. 100c¡0. 103000. t0!00c. VÄOIÂBLE L ecr 0l ve TIHS P:PJOO L-i 1- 6 r- c FôÉEÉf'fCTI? 1.010 -1.00c FIXED V¡LUE uooEP Bp4s rq0000. rc0001. 25000. LE . fl.J

CCNSTPA INT ^IÂ-IE.. . YYtrY L0

oÉl}'¿ pF OF3Y vÂcIÂ815 ¿:t 0 LCCÂ L0c9 L0cc L CC0 1f'.,tE ÞEcI00 1- ! 1- I 7-10 7-LC 7-10 7 -4 t C0EFçICIÉÀlT -1.1q0 -1.0t3 -1.000 1.000 1.000 1.000 1.00c 1.uC0 FfYEO VÁLUq F IJPÞEÞ 8¡]1S 7 0,0 1. 9000c. 100000. t0010c. 100000. l0cc0rl . 0. c.

VÁOIAqLí L CCE LCCF OIVC II¡¡q PEÞIOO 7 -t'l 7-L0 c0!FFICt:{r 1.0itc 1.0!0 -1.000 FTYqD V¿LUÉ uppEP q r¡!s 1000r1. t4000c. ?5000. LC . q.J

CON STF À INT TIÂY E. . . HYFH !t- !7

VÂçIÂOLE PFTL{ ÞF3H L0cÂ L 0c9 L0cc L0c0 TIYE PEÞIOC 1- I 1- 1 11- 1; t L-L7 LT- L7 1l-17 CCFFFICI:III -1.0'10 -1.0c0 -1.000 1.00c 1.000 t.000 1.0c0 1.000 FIYFO V!LUf F F UopFe pNDS 7 070 l, c000J. 100c00. t10c00. 100000. 10000J. 100000.

VÂPJACLE I nae LCCF lIVP TIvE P.FI0f_l t l-t - l1-17 11-17 COFFFICIENf 1.ttc 1.000 -1.G00 FIYEO V!LU5 IJPPFO R ¡'NS 190001. L9CC9l. ?5000. LE . Þ.J

H (¡¡ Flgure A-22 Cons tEaint F@lly OurDut - suyPÂcY DF CerlSTÞ ! I{r FC?YS

-t rI GoNSlcÁ I\r ¡lÀ.{E...Y1VL 1 -

VÂeIAqLF Þv1L ov2L 0Ivc lIxç pfF I00 1- 1 1- t t- 6 CCEFFICIg\f -1.010 -1.000 1.000 FTxE0 V ÂLU9 UpÞcÞ E tr0 S 1504'! . t0000. 25C00. lc ñ 1

C0{SlFô INf \tA¡lE'..qnVH t0

VAcfÀÞLÊ pvlY ovzT DIVP TI!. ÞF=J0C 1- 1 7 -tx COgFFIC IEN'T -1.',]r10 -1.C10 1.0c0 rIxE0 vôLUE uPeSc 9 N0s 15011. 1C000. 25C00. LE . 0.0 c0NsfqÁ I¡lf \Àx E.. . {1vH l7

VTPJÂPLE o,r1H ov¿H OIVq lIvE ÞrEI0r 1- 1 1- 1 I l-!7 CCEFÉJCI:!f -1.0:f) -1.000 t.0cc FTY5Ð VÂLUq uoÞcã ollDs 1500 J. 10c50. 25000. LÈ . v.C

H (¡l

Pigure À-23 Constralnt FælIY OutPut sltH{Ácv 0F cc'tsf2ûI¡tf E0'Ys c3r¡51ôaI\r ¡lÂY5...!afE L - !7 !I vlcJÀ8LE 0 T!H9 PEPIOO 1-L 7 I-1_7 COEFFICIãI'JI 1.0'lc F:YEO VÂLUi Ft upeço Frt0s 1C010!. ?7t72. LÊ l-qJc.0 cDNSroÂINf {Auf.,.?IPÂ 1- L7

VÂDJÁ8L: 0 TIFq P5pI0! t-t. CCFFFIC IE¡IT 1.]]10 ÉIX9O V¡LU: uoÞco p N05 10000J. GE tcÊ.0 ccNsTPÂ r¡11 'lÂs 5. . . sloP z- t7

VêOIÄ'JLE I 0 rlvs P:FI0t: 2-r" f -i Ã c0í FFIC IErrT -1.1rr 1.000 1.0e0 -1.00rj F!X11 VÀLUf F1 upÞ5q oN9s ?7142-. 10c000. r0Jc00. EO . 0.J

H (^¡ t\) Plgure A-24 I Con6traLnt Fmily Oulput l-.. Sllxu¡lÞY OF FO,IS?" c. INT FO?"{S --!l co¡rsTcaIr.J- lJÀY8...11vÞ ! - L7

VÁOIÂELE O IVq 0 L0cÂ L0c8 LOCC L 0C0 TIPE PloI0n 1-1 ' 1-17 L.L7 L. L7 L- L7 L-t7 c0EFFIC I1¡tf -1.C',lC 1.000 1.000 1.000 1.0e0 1.000 FIXEO V¡LIJ: fF UOP ER 8 i¿D S 1C6001. lCCt00. 150000. 100000. 1000c0. 100000.

bl ut Plqure A-24b Coástraint Farnlly outpuc cÞoçs ÞEEÉpiÀt:0F q0H Nr'{:s lNc L/o Itle'rT qÂTRJY RCH NUqqgoS

ECH NÁe1 q0H ¡10. -it_ |

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(tl Flgure A-26 Row Nane Croas Reference cc0ss ÞEF5Þ?NtE.oÉ c0H ¡r0Yqs lrt0 L/p I¡cUT HÂTRIx ?Cti Ì.UY't:qS -l o0H \avq P0!t xo. -'l

PY Fq 1q t01 ¡¡ Y Frl 16 102 PXFH 103 r,¡ ! vL I 10r. Y T]VL ? Lq' ¡¡¡ VL 3 10É I''VL 4 L17 POVL ç 103 FOVL 6 109 P IIVH i 110 8 111 P CVq 9 11? VDVH 13 t 13 f¡ 0vH 11 114 yrlvH t? 115 P') VH 13 LT1 P OVH 15 I77 P 0 vr.l 15 118 P CVH !6 t19 HnVH 17 LzA oÀfg 1 t 2l Z L?? côri 3 r 23 $ I ¿{ 5 L?5

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=l- oEFcc:NDg oF a0H NÀtgS FCH Nuqq:P ct0sç 1NC L/o INpl,T HAIqIY 'q

--l i CNH NÂH: poy 10.

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Ul -_l Flgure A-28 R@ Nue Croas Reference c?ess çEFE;:{cE 1F vaç.IÁ.tLE ¡t rqEs Â\e L/Þ INPUf tsIÂTFIT CCLUY\ ÀIUtsO5çS

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ls t 5L FV t Êv lc I 53 ?¡¿ 1 5L 1l 59 o¿ 0-t 0l ,F o9 U' at) of Êr 08 c¿ o9 É3 r t¡ 6\ 0 lt 4= oLz c: 0 t3 0 1i. a1 n tr o L5 70 o 17 7l t: vot 7? CI VÞZ 7'3 c: 7l/ ci ?5

CI vo Ê 77 OI v" 7 74 cl 79 II v"9 80 1I VÞ Tf] 81 cl vc ! t 5? cï VÞ T2 e3 CI vc 13 411 Il Ve lt¡ Ò, DT vp 75 öÞ EI vq 16 87 CI vo L7 e_q lL l ¡g sz 91 53 91

YJ 94 Y? 5ô _44

< lF g¡ s tL ç9 H S 1? t0! UI Plgure A-30 vsrlable NaEe Cross q.eference C?0SS oFÉ;s--\C: OF VOPIÂ:]L: NIH:S lNO L,/O I¡IoUl .1ÂTFIx ccLUYN ¡¡Uvp::s

VAPJÄELi NUYgSO s t3 L9t S 1l+ 102 s 15 103 s 16 10t s 17 105

èH Flgure A-ll Varlable N@e Cross Reference

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REPORT ]4R.ITER

145 APPENDIX B - REPORT I\IRITER

B-1 Introduction

Thì.s prograrn translates the output from the linear programning solution algorithm into a readily readable and irnderstandable format.

B-2 Prograrn Logic

The report writer program assembles and reduces the output from the lj-near progranmring solution algorithm into a concise meaningful format" Optimal operating schedules for the components of the Assiniboine River flood control system are presented. As we11, the program calculates flows that would occur without regulation or diversion by the Assiniboine River Flood Control System. These flow values are referred to as natural flows in the report writer pïogratn. All flow values are in terms of mean c.f.s. - weeks. A listing of the program is provided on Figures B-1 to B-4 inclusive.

B-3 Capacitv and Limitations 1. A naxi¡iuun of twenty time periods in the analysis. 2. A rnaxjmr¡n of one hundred twenty five unique variables in the analysis.

Problems involving either a greater ni.¡nber of time periods or a greater ntnnber of unique variables than noted above will require minor program modifications.

146 B-4 Input San'ple input to the program is shown on Figure DF

An input deck consists of the following:

1" Date Card This card identifies the first calendar date of the analysis. It is printed on each page of output as a sub-heading.

Coh¡nns Description Exanple

1- 80 First date of analysis April 17, 1974

2. Nr.unber of Ti¡rer Periods Card This card inputs the n¡nber of tjme periods in the analysis.

Colunrs T)cqcrinf i nn Example

1- 10 l.tr¡rber of time periods L7

3. Required Variables Card(s) This card ínputs the variables required for output. One fanily of variables is described per card. 0n1y non-constant variables maY requested for output.

Colimms Description Exarple

2-5 Variable family naine S

o- r_u Initial tíme period ¿

11- 15 Final time peri-od t7

L47 B-5 Output

The resultant output from the report writer program is shown on Figures B-6 to B-8 inclusive.

The first section of output, shown on Figure B-6, shows all inflows to the Assiniboine River System for the problem under analysis.

The second section of the ou@ut, shomr on Figure B-7 shows the natural flow values throughout the river systen for the problem under

analysis "

The third section of the output, shown on Figure B-8, shows the optirnal operating schedules determined for the individual flood control system components, Shellmouth Reseryoir and. the Assíniboine River

Diversion, and the resultant flow throughout the river system.

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H Flgure B-8 Report f'¡rlter OutPut

lJl t :- BIBLEOGRAPHY

L57 B]BTEOGRAPHY

Maass, Hufschmidt, Dorfrnan, Thomas, It4arglin, Fair. Design of Water Resource Systems, Harvard University Press, Cambridge, It{assachusetts, 1962.

Windsor, James S. "A Prograrmning Model for the Design of Multireservoir Flood Control Systems", Water Resources Research, Vo1. II, No. 7, February 7975, pp 30-36"

Windsor, Janes S. "Optirnlzatíon Model for the Operation of a Flood Control Systern:, hrater Resources Research, Vo1. 9, No. 5, October 7973, pP 7219-7226.

Burton, J.R., HaI1, W.4., Howe11, D.T. "Optirna1 Design of a Flood Control Reservoir", Congr. Int. Ass. Hydrol. Res., 1963.

Linsley, R.K., Franzini, J.B. '[\rater Resources Engineering, McGraw-Hill,

New York, L972.

Wilde, D.J., Brightler, C.S. Foundations of Optimization, Prentice-Hall

Inc., Englewood Cliffs, N.J., 7967 "

Siddal1, J.N. Analyti-cal Decision-lt{ak_ing in Engineering Design, Prentice-Hall Inc., Englewood, N.J.

OtLasghaire, D.T., Him.ne1b1au, D.M. Optimal Expansion of a i{ater Resources 158 System, Academic Press , 7974.