Application of multi-regional planning models to the scheduling of large-scale water resource systems development

Journal of Hydrology, 28 (1976) 101--125

101

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

I.

PLANNING THE DEVELOPMENT AND CONTROL OF WATER RESOURCES

APPLICATION OF MULTI-REGIONAL PLANNING MODELS TO THE SCHEDULING OF LARGE-SCALE WATER RESOURCE SYSTEMS DEVELOPMENT D.W. Moody U.S.

Geological Survey

Reston, Va.

22092, USA

INTRODUCTION An important part of the water resource planning process involves the selection and scheduling of structural measures (e.g., water-pricing policies or land-use regulations like flood-plain zoning) to optimize development objectives. The problem of selecting configurations of water resources projects and scheduling their construction has been discussed by a number of authors, deLucia and Rogers (1972), for example, used a linear programming model

(the North Atlantic Regional

Supply Model) to select the minimum annual-cost combination of water sources for meeting future water demands. They assumed that the water-supply and water-use activities in each subbasin are located at single points or nodes in the model. The nodes are interconnected by the natural surface-water drainage and interbasin transfers of water. Facet and Marks (1972) described the use of screening and sequencing models to formulate water resources projects in Argentina. They used a linear programming model to select sites for the development of hydroelectric power and irrigation water and to determine the capacity to be added to each side during each planning period so that the weighted sum of net benefits from each planning objective is maximized. Integer variables were introduced into the model to account for fixed costs associated with initial project construction, to provide additional reservoir capacity for sediment storage, and to include certain policy constraints

(e.g., a given project must

102 be built during one planning priod). Having obtained a configuration of projects, they next used stochastic streamflows to simulate the operation of the proposed system in order to evaluate the system's performance. Finally, they formulated an integer program to sequence the proposed projects subject to budgetary constraints, certain physical interrelationships between projects

(e.g., a dam must be built before power can

be produced at a given site) and the need for population growth to support irrigation. In another approach to the sequencing problcm, Butcher, Haimes, and Hall (1969) used dynamic programming to derive the order in which a given set of water-supply projects should be constructed in order to minimize present cost. Mixed-integer programming was used by O'Neill (1972) to select and schedule for construction the minimum present-cost configuration of water-supply projects which will meet future water demands in the central area of South East England. Demand centers, ground-water sources, reservoirs, and diversions from the stream system from nodes which are interconnected by the natural surface-water drainage or by pipelines. The model selects and schedules specific water sources for construction and determines the capacity of the links between nodes needed to meet demands. This paper describes a water resources planning model that is

also formulated as a mixed-integer program. A given geogra-

phic area is divided into planning regions within which regional water demands are assumed to be located at single points. Water import and export projects interconnect the planning regions. For each planning period, the model determines the water-supply projects to be built and the amount of water to be produced, treated, and imported or exported to meet future water demands at minimum present cost. Model constraints include a response function which relates ground-water withdrawals to aquifer recharge induced from stream segments that form boundaries of the aquifer. The use of this model in conjunction with several water demand models, used to evaluate the impact of nonstructural measures, are also discussed in the context of water resources planning in the Commonwealth of Puerto Rico.

103 Acknowledgements This study is part of a cooperative program between the U~S. Geological Survey and the Puerto Rico Environmental Quality Board. The assistance of the Puerto Rico Department of National Resources,

the Planning Board, the Aqueduct and Sewer Authority,

the Department of Agriculture,

and the Water Resources Authority

in providing access to pertinent data, files, and reports is gratefully acknowledged.

PUERTO RICO WATER RESOURCES PLANNING MODEL The Island of Puerto Rico, although not large in area (3,440 sq mil), has a very diverse physical environment. Owing to interactions of persistent east-northeasterly

trade

winds with an east-west trending mountain range in the interior of the island, mean annual rainfall ranges from over 200 inches in the northeast section of the island to less than 25 &nches in the southwest. Also related to elevation is mean annual potential evapotranspiration which ranges from 35 inches in the mountainous

interior to 65 inches along the coast. The combi-

nation of these conditions leads to abundant ground- and surface water supplies on the north coast and limited supplies on the south coast. Twenty-three major stream systems drain the interior of the island, most of which are less than 20 miles long (Bogart, Arnow, and Crooks, 1964). The domestic water demands municipios

(195 mgd) of Puerto Rico's 78

(political divisions) are now met by 62 urban water

systems and 176 rural water systems operated by the Puerto Rico Aqueduct and Sewer Authority.

These systems have little, if any,

excess capacity, and as a result, some 30 municipios had water shortage during a dry period in the Spring of 1973. Irrigation water demands

(194 mgd), primarily used for the production of

sugar cane, are supplied by a complex of reservoirs,

tunnels,

hydroelectric power plants, and canals, some of which transfer water from river basins on the north side of the island to those on the south. These complexes serve three irrigation districts administered by the Puerto Rico Water Resources Authority.

104 Rapid economic development of the Commonwealth of Puerto Rico during the last 15 years has utilized much of the avilable flat land and water resources in some areas. Future expansion of the economy, particularly the establishment of industrial centers along the south coast in association with natural harbors, will require new water supplies to be developed or existing supplies to be reallocated from agricultural uses. The commonwealth is currently attempting to implement techniques with which to assess the adequacy of regional water supplies to support projected patterns of economic development and to screen inventories of water resources projects proposed by consultants. In this manner, plans developed at the regional or local level can be coordinated on a multi-regional or island . wide basis. The Puerto Rico Water Resources Model is a mixed-integer program that aids the planner in screening large number of proposed water resources projects. Given a set of future water demands, the mathematical program selects and schedules for construction, the configuration of water-supply projects which will meet demands at minimum present cost. In formulating the model, it was assumed that: i.

The Commonwealth of Puerto Rico is divided into a number of planning regions, each containing one or more river basins. Similarly, the period of time under study (the planning horizon) is divided into a number of planning periods of equal length.

2.

Because most of the stream systems are short, once water has been withdrawn for use, it is lost to the system. No consideration is given to the possibility of water reuse or artificial recharge of ground-water supplies.

3.

Estimates of optimum sizes of individual projects and their associated yields and operating rules are determined during preliminary engineering studies.

4.

Project operating cost functions are approximately linear.

5.

Water demands are of two types: raw water, primarily used for irrigation, and treated water for domestic use. They

105 are also assumed to be located at a single point in each planning region. 6.

The future regional demands by region are stated as known requirements.

7.

The minimization

of water resources

the planning horizon and discounted satisfactory

investments

summed over

to present value is a

criterion for evaluating

alternative project

configurations.

In any given planning region, water demands may be met from a number of sources

(fig. i). Raw water may be obtained from run-

off of the stream diversions,

from reservoir

imported from other planning regions. directly from well fields,

storage,

or it may be

Treated water may be obtained

(assuming that ground water is chlori-

nated at the well), from desalination

plants,

or it may be

imported.

Any raw water, produced or imported, may also be

processed

in a water-treatment

plant. Thus, a region's raw-

water demands may be met from either raw water or treated-water supplies. demands,

Finally,

in addition to meeting its own internal

a planning region may assist other regions in meeting

their demands by exporting either raw or treated water.

Mathematical

Formulation

The planning model's objective function minimizes

the present

cost of meeting future water demands over the planning horizon. The objective

Minimize

function is expressed as

5 N I Lik ~ % ~ Z (~ni~~ (k) Qni~(k) + FCni~(k) Ini%(k)) k=l n=l i = 1 % = i

7

N

I

I

Lik.~(k)

~(k)

.~(k) l(k) ) + r~nij% nij~ (eq. i)

k=6 n=l i=l j=l ~=i

i#j where k = project

type: 1-diversion,

4-desalinization 6-raw-water

2-reservoir,

plant, 5-water-treatment

import, and 7-treated-water

3-well field, plant, import;

106

RAW WATER .j th REGION

I

REGION

i

IMPORT (UW)

RESERVOIR

)RT

WATER

DIVERSION DAM INS)

I

WASTE WATER

Q

I

(WP)I.

~

WATER TREATMENT l PLANT

~ UPLANDS --

.

.

.

.

WELL FIELD ~ (GS)

WATER

O_

I~ EXPORT . . ~ ~

j IMPORT (TW)

TREATED WATER th REGION I

!

'

"l

kDEMAND~. ~ WASTE DESALINIZATION WATER PLANT (DS) I

OCEAN

I

Figure i. -- Schematic representation of the i-th region in the Puerto Rico Water. Resources Planning Model: NS = diversion; GS = well field; DS = desalination plant; RS = reservoir; WP = water treatment plant; UW = raw water import; TW = treated water import; Q = upland runoff into stream segment P; R = flow requirement downstream of stream segment P; small arrows represent the recharge of well field GS by stream segment P.

n = planning-period number

of p e r i o d s

the p l a n n i n g % = project

index:

(each p e r i o d

horizon

number:

of k - t y p e

n=l...,N;

extends

%=1 ..... Lik;

projects

and N is the

is T y e a r s

so t h a t

NT years); and L i k

in p l a n n i n g

region

is i.

the n u m b e r

107 i,j = planning-region

indexes: i=l,...,I; j=l,...,I;

and I is the number of planning regions in the model. Note that in the following variables the subscript j indicates a water import project transferring water from planning region j to i.

c(k)

~(k) operating, maintenance, and replacement (OMR) costs ni~'~nij% = per unit of water produced (dollars per million gallons) by project number % of type k in planning region i reduced to present value from planning period n.

~(k) Qni%('k~nij )~

=

continuous column variables representing the total quantity of water produced, processed,

or trans-

ported during the planning period n by project of type k in planning region i (millions of gallons per planning period).

per

(k) F ~ (k) the present value of the construction costs of FUni %' Unij~ project number % of type k in planning region i. The payment is assumed to be made at the beginning of the planning period when water is first delivered from the project.

1ni% ( k'Inij )(k)

integer column variables;

if I=l, then the construc-

tion of project number % of type k in planning region i is completed at the beginning of planning period n. Otherwise,

I=O.

The first term of the objective function (eq. i) represents the operating costs and fixed costs of diversions, reservoirs, well fields, desalination plants, and water-treatment

plants.

The second term represents the operating costs of raw- and treatedwater imports. The objective function is subject to the following constraints i.

Project initiation constraints. from project

-- No water may be provided

of type k in planning region i until the

project has been constructed.

Furthermore,

once constructed,

the project cannot provide more water than its yield or capacity.

108 For water production projects, nw

, (k) r

~(k)

>

~,(k)

ui% nLl= Inil

-

qni%

k=l,

,5;

n',i, and

iELik (eq. 2)

n',i,j, and

IgLik (eq. 3)

"'"

and for water import projects, n'

Tuijl (k) n~l (k) > Q(k) k=6,7; i#j; = Inij~ - nil where

n' = planning period being evaluated. b(k) . (k) = yield or capacity of project number ~ of type k i% 'uij£ in planning region £ (millions of gallons per planning period).

2.

Project limitation constraint.

-- Project % of type k in

planning region i may be constructed once and only once during the design horizon. For water production projects, N n=l

~(k)=~ _< 1 ±n±~

k=l,

5; " ""'

i and iELik

(eq. 4)

and for water import projects, N n=l

(k) < Inij£ - i

If project

k=6,7; i#j;

i,j, and

Z~Lik

(eq. 5)

of type k in planning region i already exists

at the beginning of the first planning period, this constraint is unnecessary.

3.

Treated-water

production

(or processing)

constraints.

-- The

volume of treated water produced by well fields, desalination plants, and water-treatment

plants and the volume of treated-

water imports minus exports must be equal to or greater than the treated-water demands in planning region i during planning period n. eik [ ~ ~=i [k=3

Q(k) + ni%

I-~(7)

^(7) )]

= £-qnj i£ J > j~l(qnij

i#j

TW . V n and i nl

(eq. 6)

109 where TW . = demand for treated water in planning region i during nl planning period n (millions of gallons per planning period). In the context of this constraint,

a treated-water

import

(or export) represents a pipeline connecting a treated-water demand center in region i with another treated water demand center in region j.

4.

Total water demand constraint.-- The total volume of water produced

(the outputs of diversions, reservoirs, well fields,

and desalination plants) and the imports minus the exports of raw and treated water must be equal to or greater than the total water demands in planning region i during planning period n.

Z=i

k=l

~(k) + qni£

~

~

k=6 j=l

(Q(k) ~(k) nij~ - qnji% j

(eq. 7)

i#j > -

TW . + RW . n l nl

V n and i

where RW . = the raw-water demand in planning region i during nl planning period n.

5.

Flow-requirement

constraint.-- The sum of the natural

(unregulated) upland streamflows in planning region i that discharge to stream segment p, minus the withdrawals from the stream system by diversions and reservoirs upstream of segment p and the quantity of water lost from the stream to the underlying aquifer due to ground water pumping, must be greater than or equal to the required flows downstream of segment p. 2 (Qip

k 1 pk

n ni

ip 3

t i fn-t+l'i'%

~(3)qti > R - nip

V n,i, and

pgP. 1

(eq. 8)

110 where

p = index to stream reaches: p=l,...,Pi;

each planning

region may have one or more stream segments and associated flow requirements. R

.

nxp

= the required volume of water at the downstream end of stream segment p during planning period n in region i (millions of gallons per period).

Q*p = sum of unregulated stream flows discharging from the uplands to stream segment p in planning region i during any planning period

(millions of gallons

per period). ~pk

= a vector

(list) of project numbers of type k which

regulate the flows into stream segment p. Three such lists are used in this constraint; k=l,2, and 3 (see definition for eq. i). fn-t+l,i,%

= the fraction of ground water pumped from well field % in planning period t that is derived from stream segment p in planning period n.

Q(3) tig

= the quantity of ground water pumped by well field in planning region i during planning period t.

This constraint utilizes a response function to relate wellfield pumping to the interactions between the aquifer and the stream. The intent is to provide a means of linking reductions of inflows to the valley caused by upland surface-water development to the development of the ground-water supply and downstream flow requirements. The computation of the f coefficients makes use of a digital ground water model to estimate the interactions between well pumping and drawdown. Assuming that vertical flow of ground water is insignificant and that the head boundary of the aquifer is constant, Maddock (1973) has derived an expression relating the amount of water withdrawn from ground-water storage by a set of wells to the storage coefficients of the aquifer, the amount of drawdown at the wells, and the surface area of the aquifer. The difference between the amount of water pumped and the amount

III removed from storage is the amount of water removed from the surf surface water stream. Maddock and Moody for aggregating the short-term

(1973) discuss procedures

(monthly) interaction coefficients

of individual wells into long-term

(5-year) interaction coeffi-

cients representative of an entire well field.

6. Bounds on integer variables.-- All integer variables are bounded between zero and one.

0 J Ini ~ ~ 1

V n,i, and

0 J Inij% ! 1

%ELik

i#j; V n,i,j, and

(eq. ii) %gLik

(eq. 12)

7. Non-negative constraint on continuous variables. %gLik

(eq. 13)

Qnij~(k) _> 0 k=6,7; i#j', V n,j, and ~gLik

(eq. 14)

Q(k) > 0 ni~ -

k=l,

"'"

,5; V n,i

'

and

Additional constraints may be necessary for a specific planning region because of the physical interrelationships between projects. For example, two projects of different capacities may be proposed for the same site. Obviously only one of them can be built unless staged capacity expansion is permitted. A constraint such as N n=l

~(k) + I(k) < 1 ±ni~ ni~' -

%#%'

(eq. 15)

will prevent project % and project %' of type k from both being built during the time period of interest to the planner. Similarly, if project %of type k must be built before or at the same time as project %' of type k' then a constraint of the form nv

_(k) i . - T(k') n=l

nl

> 0 ~ni%' -

k#k'

•

V n'

(eq. 16)

will be necessary.

MODEL PARAMETERS The Puerto Rico Water Resources Planning Model defined by

112 equations i to 14 contains eight explicit types of parameters: I.

Project fixed costs

(FC)

2.

Project operating costs

3.

Project yields or capacities

4.

Surface water-ground water interaction coefficients

5.

Upland runoff estimates

(C)

Downstream flow requirements

7.

Raw water demands

8.

Treated water demands

coefficients,

(f)

(Q*)

6.

The cost coefficients,

(U)

(R)

(RW) (TW)

surface water-ground water interaction

downstream flow requirements,

and water demands

are functions of time. Yields from reservoirs and well fields are estimated from available hydrologic data as are the upland runoff estimates and the surface water-ground water interaction coefficients. Each of these coefficients is dependent upon the project capacity (physical size of the reservoir or the well), the seasonal pattern of demands

(withdrawals),

project operating rules, and the

way in which project reliability is defined. Finally, because of the uncertainty associated with estimates of the model parameters,

they may best be interpreted as expected

values.

WATER-DEMAND MODELS Uncertainty,

a predominant characteristic of the planning

process, arises from the planner's imperfect knowledge of future levels of population, mixes of economic activity, water availability, behavior of the physical system being studied, and political decisions relating to water-pricing policies, subsidies, and social and economic development goals

(Sewell and Bower,1968).

These uncertainties affect the estimation of the planning model parameters,

and consequently,

they introduce uncertainty into the

planning model results. A major source of uncertainty in the Puerto Rico Water Resources Planning Model is introduced by the water-demand forecasts. The planning model assumes that future water demands are

113 known with certainty for each planning period.

Obviously,

this

is not the case, for future water demands will depend upon a variety of social, variables

economic,

political,

and technological

such as population growth, migration patterns,

of economic development, use technology,

land-use patterns,

and hydrologic variability.

estimates -ased on population projections

levels

water prices, waterSimple water-demand

and assumptions

about

per capita water-use provide the planner with little insight into the complex interrelationships affect water demands

between the variables

(Sewell and Bower, 1968).

In an attempt to improve the water-demand for residential,

that

industrial,

,=o~ercial,

forecasts,

models

and agricultural

(irri-

gation) water demands have been proposed for use in Puerto-Rico (Moody, Attanasi,

Close, Maddock,

and Lopez, 1973).

Residential water demand.-- Regional per capita water-demand for metered customers is expressed as a function of (i) water price,

(2) personal

income, and (3) housing quality as a proxy

for wealth. Water price is omitted from a second model for unmetered customers.

With these models the planner can investigate

the impact of water metering and water-pricing regional population projections,

policies given

estimates of future personal

income, and estimates of future housing quality. Industrial and commercial water demand.-- Regional

industrial

or commercial water demand for metered customers is expressed as a function of (I) water price, trial code or by commercial portation,

construction,

(2) employment data by indus-

sector, e.g., trade, services,

and finance,

denoting water intensive

industries

and (3) a zero-one variable (not present in the commercial

demand model). Water price is omitted from the industrial commercial demand models for unmetered forecasting

industrial

of forecasting Commonwealth

customers.

and

Implicit

in

and commercial water demand is the problem

industrial

location and growth for a given region.

policies relating

resource development,

trans-

to public investment,

natural

land use, and taxation will have a signif-

icant impact on regional industrial

structure and, thus, on

water demands.

and commercial water-demand

Using the industrial

models the planner can investigate

the impact of water metering

114 and water-pricing policies given employment projections by industry and the future location of water intensive industries. To explicitly include the effects of industrial development policies on employment will probably necessitate the construction of regional econometric models. Agricultural (irrigation) water demand.-- Nearly all irrigation water in Puerto Rico is used for sugar cane. Since sugar cane production is subsidized by the Commonwealth government, the acreage devoted to sugar cane is largely a policy decision. Thus, if the average water requirements of sugar cane in a~ given region are known, the water demand can be determined from projections of the acreage to be planted. This requirements approach, however, ignores economic consideration unless the demand for irrigated land is related to the crop's production function and future market conditions. The final form of the demand models will be determined by analysis of i0 years of water demand data and related economic information. The use of water-demand models to estimate future water demands for the planning model serves two purposes. First, the models explicitly separate parameters subject to government control such as water pricing and metering policies from those that are not such as population growth and employment. Secondly, they reduce uncertainty in de demand forecast for any given set of government policies. Of course, which set of government policies will be in effect i0, 20, or 30 years in the future is quite uncertain. The process could be taken several steps further by developing demographic and econometric models to forecast the independent variables in the water demand models. Again, the principal advantage to the planner of the more sophisticated models would be to separate out additional policy variables under governmental control so that the economic consequences of government policies with respect to industrial development, tax rates and subsidies, public investments, etc., can be evaluated. Unfortunately, the types of disaggregated data needed to construct regional demographic and econometric models are rarely available. In summary, the water resources planning model aids the

115 planner in evaluating alternative configurations of water-supply projects while the water-demand models aid him in evaluating nonstructural measures aimed at managing water demand.

EXAMPLES OF MODEL USE The Puerto Rico Water Resources Planning Model has at least three principal applications in the planning process. i.

Initial allocation of regional water resources--using regional cost curves for water development,

the marginal

cost of water in each region can be established.

If the

marginal cost plus transport costs in one region are less than in an adjacent region, then interregional transfers of water should be considered. 2.

Screening and sequencing proposed projects--Because proposals for water-resources projects originate from a number of sources in Puerto Rico, a means is needed to coordinate and evaluate projects on a regional or multiregional basis. The planning model provides such a tool.

3.

Revision of data collection program--although

the planner's

principal interest is in the selection of projects during the next planning period, he is also concerned about improving the estimates of design-parameters

of the projects to be

built in future planning periods. Having estimates of the sensitivity of planning decisions to errors in the model's parameters,

the planner can periodically revise his data-

collection program in attempts to reduce the uncertainty in parameter estimates. For example, knowledge of the probable location of future reservoirs and when they are likely to be built would be of great value to the designer of hydrologic-data-collection

program.

To demonstrate the use of the planning model, data were obtained for i0 municipios at the eastern end of the island of Puerto Rico. Estimates of future water demands for each municipio were acquired for 1975-94. These demands are essentially linear extrapolations of past trends. An inventory of existing

116 and proposed water-resources

projects for each municipio was

also created. Estimates of project costs, adjusted to 1970 prices, and of project yields or capacities were obtained from consultants'

reports or from the files of the Puerto Rico Sewer

and Aqueduct Authority. The i0 municipios were aggregated into six planning regions, each of which contains one or more major rivers with the exception of one region which represents an offshore island. The six planning-region program

(Region-6) model was set up using a computer

(Moody, Karlinger, and Lloyd, 1973) to produce control

cards and data cards for the Mathematical Programming System Extended

(MPSX)

(IBM, 1971 a and b). MPSX was then used to mini-

mize the model's objective function. A two planning-region (Region-2) model was also built to explore the behavior of the model

(fig. 2). The characteristics of two of the models are

shown in table i.

RESULTS OF SEARCHES Figure 3a and b show the results of optimizing the Region-2 model. Twenty-eight water resources projects were screened and sequenced over a 20-year planning horizon

(1975-1994). The best

integer solution found during a 5-minute search had a cost of $13.1 million.

If the search were resumed, an integer solution

better than $10.8 million could not be obtained. All of the available ground water projects

(in addition to those already

built) and the single diversion project

(already built at the

beginning of the planning period) were scheduled for the first two planning periods. water sources

In planning region B (fig. 3b), treated-

(ground-water projects GSBOI and GSB02) were used

to meet the region's treated- and raw-water demands until a reservoir

(RSB01) was scheduled in the third planning period.

A treated water import project was scheduled for the second planning period to carry ground water from region B to A (fig. 3a). Both the import project and the ground-water fields operated at capacity during the third and fourth periods. Large amounts of raw water were imported to region A from B during the third and fourth periods,

(projects UWAB02 and

water

Wa ste water

0

2.

TWAB01,2,3

Waste water

Jl=:e?eec

J

. .\

I

PA01,2,3,4

~

UWAB01,2 3 Raw \ ............ water ~ ( pool

|

~

~RSB03

Waste wafer

R

0 GSB02

Wasj

LL~WPB01,2,3 ~/ /' Water I ~| demand

Raw ~

,~

~' ~.sBo2 B

-- S c h e m a t i c r e p r e s e n t a t i o n of two p l a n n i n g - r e g i o n m o d e l : NS = d i v e r s i o n ; R$ = r e s e r v o i r ; GS = w e l l field; WP = w a t e r - t r e a t m e n t plant; DS = d e s a l i n a t i o n p l a n t ; T W = t r e a t e d - w a t e r i m p o r t ; U ~ = r a w - w a t e r import; Q = u p l a n d r u n o f f into s t r e a m s e g m e n t ; R = f l o w r e q u i r e m e n t do~-nstrea~ of s t r e a m s e g m e n t .

GSA02

Figure

demand

RSA03

A

118 Table i: Characteristics of two water-resources-planning models and summary of computational experience on an IBM 360/91 computer.

Characteristics

Region-2

Region-6

2 1975-1994

i0 1975-1994

Plannin$ model Number of municipios Planning horizon Number of planning regions Number of planning periods Length of planning period (years) Total number of water resources projects Diversions Reservoirs Well Fields Desalination plants Water treatment plants Raw water imports Treated water imports

2 4 5

6 4 5

28 1 6 5 3 7 3 3

60 i0 7 15 0 18 3 7

165 224 112

303 424 184

Mixed-inteser program Number of rows Number of columns Number of integer variables Time to continuous optimum solution (min) Time to first integer solution (min) Time to best integer solution (min) Optimality "proved"- total time (min)

0.06-0.07 0.12-0.31 0.22-1.56 2.09-10+ 1

0.26 0.80-0.83 1.34-1.44 35+ 2

Two runs "proved" best solution to be optimal within 3 minutes. No better integer solution found after 35 minutes search time.

UWAB03) some of which was treated in a water-treatment plant scheduled for construction in the third planning period. Finally, a desalination plant was scheduled in region A during the fourth planning period. Some excess capacity exists in the system throughout the planning horizon, especially in well fields GSB02 and GSA02 in the first planning period, in well fields GSB01 and GSA02 in the second time period, and reservoir RSBOI in the third and fourth time periods. The diversion had been operating at

119 capacity, but with the introduction of raw water imports the water nroduction of the diversion was cut back. Ideally, the planner would like to know the sensitivity of the system's cost and the sensitivity of the planning decisions, e.g., which projects to select and when to build them~ to variations in the model's parameters. Given the probability dist~iDution of model parameter estimates, a probabilistic interpretation could be given to the system's cost and the planning decisions.

MPSX

provides a number of procedures for post-optimal analysis. They are only applicable, however, to the continuous-variable section of the problem since all integer variables are fixed at their integer-solution values prior to using the procedures.

In other

words, the impact of variations of a given reservoir's operating cost on the value of the model's objective function can be assessed but not variations of the reservoir's fixed cost. Another problem in using the available procedures is that only one variable can be varied at a time. If more than one variable is to be varied jointly, separate problems must be run for each combination of new variables. To observe what would happen to the selection and scheduling process if demands and project yields were increased or reduced, eight additional runs were made (table 2). Both rawand treated-water demands of the original problem for all planning periods were increased and reduced by i0 percent. The center entry in table 2 represents the solution to the original problem. Of the total of nine solutions found, one was proven optimal and four are believed to be optimal or near-optimal. The optimality of the remaining four solutions is questionable. A reduction of demands allowed a large and expensive reservoir in region B (RSB01) to be replaced by a smaller reservoir; deferred the construction of the treated-water import project (TWAB03) to planning period three, and eliminated the need for one of the raw-water-import projects

(UWAB03). These cost reductions, how-

ever, were offset by the need for a larger desalination plant. An increase in demands replaced reservoir RSBOI with two smaller reservoirs but required a larger raw-water import project to replace UWAB02 and UWAB03 and scheduled an 8-million dollar

120

Planning region A

Raw water

Treated water
03 ~t

30-

,<

<

v o

Q.

25

25

"E o. "o

20

20

o=

15

c a E

10

;

5

o v

£0

v

ID

15

/ 1

10

5

/ 2

3

4

Planning period

Treated water transfered to meet raw water demand

o <(

~

Raw water processed by treatment plant to meet treated water demand

1

2

3

4

Plonning period

~

Import of raw or treated water to Region A from Region B

Figure 3a. -- Construction sequence of projects and disposition of water produced in planning region A given by near-optimal integer solution of Region-2 model. Cost of system proposed equals $13.1 million. NS = diversion; RS = reservoir; GS = well field; WP = water treatment plant; DS = desalination plant; UW = raw-water import; TW = treated water import. For project locations see figure 2.

121

Planning

region B

Treated water

Raw water 23 .-L

RSB01

-0 o~

E

12 v a Q. a v

o

10

10

u')

0 8

0 a.

o 6

c a

4 E

2 a

1

2

3

4

Planning period

Treated water transfered to meet raw water demand

~

1

2

3

4

Planning period

Raw water processed by treatment pla~t to meet treated water demand

Import of raw or treated water to Region A from Region B

Figure 3b. -- Construction sequence of projects and disposition water produced in planning region B given by near-optimal integer solution of Region-2 model. RS = reservoir; GS = well field.

122

desalination

project in the first time period. This change in

project configuration

explains the increase in the value of

the integer solution from 13.1 million to 20.9 million dollars. The same desalination solutions

project appears in the three high demand

in table 2. An increase in project yields reduced

number of reservoirs

the

and raw water import projects and deferred

Table 2: Integer solutions (in millions of dollars) to p l a n n i n g model Region-2 showing effects of increasing and decreasing demands and yields by i0 percent. All solutions not footnoted are believed to be optimal or near-optimal. Optimal continuous solutions are shown in parentheses.

Water

j%.o,.~,-4

Low (-10%)

cD

•,-4,-4

~

23.83 (12.1)

Expected ii.0 (8.2) (0)

13.1 (10.5)

20.92

18.32 (9.2)

13.9 (11.5)

22.52 (14.2)

Low

2 3

9.21 (7.6)

High (+10%)

(12.9)

o

o m

1

Expected (0) 11.8 (9.6)

High

(+I0%)

~.~

Demands

(-10z)

Solution proved optimal. Optimality of solution in doubt. Definitly not optimal as the project configuration of the expected-yield problem could have been a feasible solution.

the construction

of several projects while a decrease in yields

had the opposite effect. had considerable

In summary,

changes in demand levels

greater effects on the project configurations

and the present value cost of the system than changes in project yields for the inventory of projects screened.

Even so, all of

the well fields and the diversion project were scheduled construction

early in the planning horizon regardless

for

of the

123 changes in yields or demands which suggests that detailed study of these projects should begin at once. Changing all of the project yields and all of the demands by a uniform percentage is, of course, on over simplification. The uncertainty of project yields will vary from one project to another.

Similarly,

the uncertainty of treated-water and raw-

water demands will also vary and tend to increase in future time planning periods.

Even if the probability distributions of

each model estimate were known, however,

the number of combi-

nations of parameter estimates that would have to be run to define the model's response surface and, thus, quantify the sensitivity of a given model to parameter variations,

appears to be prohibi-

tively large.

CONCLUSIONS The mixed-integer programming model described in this paper appears to be a viable planning tool for selecting and scheduling for construction the minimum-cost configuration of water-response projects e.g., diversions, reservoirs, well fields, desalination plants,

treatment plants, and import/export projects, which will

meet future water demands. The evaluation of nonstructural measure~ such as water-pricing and water-metering policies can be handled by using water-demand models to forecast alternative sets of water demands. The increases or reductions of costs of waterresource systems needed to meet the demands can then be traded off against the political,

social, and economic costs of the

non-structural measures. The incorporation of uncertainty into the planning model by developing probability distributions of the estimates of project yields, cost coefficients,

and water

demands and utilizing them to give a probabilistic interpretation to the decisions for selecting individual projects for construction does not appear to be a feasible approach to the sensitivity analysis of this type of model. Other techniques will have to be developed to evaluate the sensitivity of planning decisions to variations in the estimates of the planning variables except in the simplest cases.

124

REFERENCES CITED i.

Bogart, D.B., Arnow, TED, and Crooks, J.W., 1964, Water resources of Puerto Rico -- a progress report; Commonwealth of Puerto Rico Water Resources Bull, No. 4, p. 102.

2.

Butcher, W.S., Haimes, Y.V., and Hall, W.A., 1969, Dynamic programming for the optimal sequencing of water supply: Water Resources Research, V. 5, No. 6, p. 1196-1204.

3.

deLucia, R.J., and Rogers, Peter, 1972, North Atlantic Regional supply model: Water Resources Research, V. 8, No. 3, p. 760-765.

4.

Facets, T.B., and Marks, D.H., 1972, Scheduling and sequencing in water resources investment models: Internat. Symposium on Water Resources Planning, Mexico City, 1972, proc., V.I.

5.

Fiering, M.B., 1963, Use of correlation to improve estimates of the mean and variance; U.S. Geol. Survey Prof. Paper 434-C, p.9.

6.

IBM, 1971a, Mathematical programming system extended (MPSX) linear and separable programming program description (program No. 5734-XM4): IBM Corp., Manual No. SH 20-0960-0, p. 153.

7.

IBM, 1971b, Mathematical programming system extended (MPSX) mixed integer programming (MIP) program description (Program No. 5734-MX4): White Plains, N.Y., IBM Corp., Manual No. SH 20-0960-0, p. 153.

8.

Maddock, Thomas, III, 1973, The operation of a streamquifer system under stochastic demands: Water Resources (in press).

9.

Maddock, Thomas, III, and Moody, D.W., 1973, Surface water and ground water interaction phenomena in planning models: Symposium on Use of Computer Techniques and Automation for Water Resource Systems, Washington, D.C., 1974, Proc. (in press).

125 10.

Moody, D.W., Maddock, Thomas, III, Karlinger, M.R., and Lloyd, J.J., 1973, Puerto Rico Water Resources Planning Model program description: U.S. Geological Survey, Openfile Report, p. 49.

11.

Moody, D.W., Attanasi, E.D., Close, E.R., Maddock,Thomas, III, and Lopez, M.A., 1973, Puerto Rico Water Resources Planning Model study: U.S. Geological Survey, Open-file Report, p. 114.

12.

O'Neil, P.G., 1972, A mathematical-programming model for planning a regional water resource system: Jour. Inst. Water Engineers, V. 26, No. i, p. 47-61.

13.

Puerto Rico Department of Natural Resources, 1973, Puerto Rico's Water Resources, an assessment report: Commonwealth of Puerto Rico, Department of Natural Resources, Office of Resources Planning (unpublished draft).

14.

Sewell, W.R.D., and Bower, B.T., 1968, Forecasting the demands for water: Symposium on Methodologies for Forecasting Water Demands, Vancouver, Canada, 1967, Ottawa, Queen's Printer, p. 261.