A model for the analysis of seasonal aspects of water quality control

JOURNAL

A Model

OF ENVIRONMENTAL

for the Analysis

ECONOMICS

AND

of Seasonal DAN

The Hebrew

University

MANAGEMENT

Aspects

140-1,51

of Water

(1979)

Quality

Control

YARON

of Jerusalem, Rehovot, Chicago, Chicago, Illinois

Received September

6,

Israel, 6’06’37

and The University of

8, 1977; revised July 1, 1978

A multiseasonal mathematical programming model for the analysis of water quality control within a river basin is presented. Phenomena of interseasonal variation in flow intensity and the river’s pollutant assimilative capacity are introduced into the model by defining several seasons and their characterizations by seasonal parameters. An application of the model to a typical case situation is reported. The results suggest that seasonal adjustments in the treatment levels of wastewater treatment plants involve a considerable saving potential in comparison with a situation in which treatment levels are rigidly determined for the entire year.

1. INTRODUCTION In the last decade, researchers have devoted considerable effort to the development of models capable of generating socially optimal solutions to problems of water quality management in river basins. Although the ideal formulation of such problems is to maximize net benefits from pollution abatement, because these benefits are hard to quantify a second-best approach has been to minimize the social cost of achieving predetermined quality standards. To this end, one group of researchers (e.g., Graves et al. [lo], Hass [12], Haimes [ll], Hwang et ~2. [14], Herzog [13]) has applied linear and nonlinear programming techniques. Such an approach has considerable value, in that it promotes understanding of the system and the interaction of its parts and can yield estimates both of alternative costs of various levels of MTater quality and of the trade-offs between variables of the system. A shortcoming of most of the mathematical programming approaches has been their deterministic framework; rather than reflecting the stochastic nature of t’he system, river flows and related parameters have been assumed constant.’ Another group of studies avoids this handicap only at the cost of assuming another. Models with semianalytical approaches like dynamic programming (e.g. Dysart and Hines [7], Hwang et al. [14]) or with heuristic approaches like simulation (Davis [4]) can easily accommodate stochastic elements, but their amorphous st’ructures tend to obscure the economics of the system. Accordingly they lose some of the valuable economic insights gained by mathematical programming, which explicitly incorporates such economic relationships as shadow prices and coefficients of substitution. 1 See Marks [16] for a similar view. 140 0095-0696/79/020140-12$02.00/O 1979 by Academic Press, Inc. Copyright All rights o3 reproduction in any form reserved.

SEASONAL

WATER

QUALITY

TABLE Relative Itiver

Monthly

I Month “a for Several Illinois

Flows with Respect to an “Average

Gauging location

1

2

3

141

CONTROL

4

5

Rivers

6

7

8

9

10

11

12

I.3 1.1 1.3 1.3 1.0

0.6 0.8 0.8 0.5 0.7

0.6 O..j 0.6 0.1 0.4

0.6 0.6 0.5 0.2 O..?

0.5 0.7 0.5 0,s 0.6

0.6 0.9 0.5 0.4 0.7

0.9 0.9 0.7 0.8 0.9

.Du Page, West Branch Fox

Chicago Algonquin Riverside Pontiac Momence

West

Des Plaines Vermillion Kankakee Source. Computed lk Computed and

as (I/T)

0.8 0.8 0.8 1.1 1.1

1.0 1.0 1.1 1.2 1.3

1.7 1.7 1.9 2.2 1.7

from data in U.S. Geological CI

CF,,d(E:,

F,d/121,

1.0 1.9 2.0 1.9 1.7

1.6 1.4 1.4 1.x 1..5

Survey r24]. w h ere

F,,,t

T is the number of years for which data were available

Russell and Spofford

[al]

and Russell [22]

is t’he flow

(generally

in month

m and

ye:tr

t,

I:<).

did endeavor

to blend the t\vo

approaches by using mat,hematical programming to analyze the socioeconomic part of the system (e.g. firms, wastewater treatment plants), along with a simulation model, to analyze the wat’er quality ecological parts of the system. :11though conceptually quite appealing, t’heir approach scrms to rcquirc considerable

research resources for finding an optimal solution, in the itwativt: proccsn cmploycd, and in practice, its application is subjwt to the availabilit,y of sucl-1 rcsourcw. The strat,cgy of our study is to separate the analysis into t.wo stagcx. At the first stage a mat)hcmat’ical programming mod& involving some simplifying assumptions, is applied to the whole system (including both its economic and ecological parts) in order to gain with its aid the understanding of the intclrwlationships bct~wcen its elements and arrive at approximate solutions and policy altwnativw. At’ the second stage simulation is applichd to twt alternativrs for decisions lvithin a more accurate framework. Xote that, simulation alone, xvhilc c4FirGnt in twting alternative policies, has limitrd cwativity and capability for tho formulation of sound altcrnat,ivcs. This paper is restricted to the discussion of th(l mathctmatical programming approach. Wo prrsent an attempt, to analyze water qualit’y control problems b\-ith a multiscasonal mathematical programming model (following Dantzig’a [:%I work on linear programming under conditions of uncertainty) into which elcmcnt~s of chance-constrained programming (Charnes and Cooper [2]) are incorporated. The idea is to capture phenomena of interseasonal variation by introducing into the model wvcral “seasons” which are charactcrizcd by natural factors affecting watt>r quality, and are roprcwntcd in the model by ‘5wasonal” parameters. ‘I‘hc: intrawasonal variation is incorporated into the analysis 1,~ formulation of “chanw constraints.” The sprcification of th(h variables as deterministic or stochastic is discussed later in the paper. Note that “seasons” can be defined in tclrms of months rather than longer periods, thcrchy rtducing the: intrawasonal variation. The idea of the multisrasonal approach arosc following scrutiny of empirical data wlatc>d to water yualit’y in DuPagc County, northcastc>rn Illinois. St>asonal variation in tjcmporature is a well-known phenomenon, and there is no nwd for corroboration of this statcmcnt. The observed, apparently systematic variat8ions

142

DAN

Relative River DuPage, West Branch DuPage West Branch DuPage East Branch DuPage East Branch DuPage, East Branch DuPage

Monthly

YARON

TABLE II DO Concentrations with Respect to an “Average for the DuPage River, Illinois

Month”a

1

2

3

4

5

6

7

8

9

10

11

12

West Chicago

1.2

1.3

1.2

1.1

0.9

0.8

0.7

0.9

1.0

0.8

1.1

1.3

Naperville

1.3

1.5

1.2

1.0

1.0

0.6

0.8

0.9

0.8

0.9

1.0

1.3

Glen Ellyn

1.1

1.5

1.7

1.6

0.9

0.7

0.9

0.6

0.7

0.5

1.1

1.4

GlendaleHeights

1.2

1.1

1.5

1.5

0.8

0.5

0.6

0.7

0.7

0.6

1.3

1.5

Lisle Channahon

1.4 1.1

1.5 1.1

1.4 1.1

1.3 1.1

1.1 1.1

0.5 0.9

0.7 1.0

0.6 0.8

0.7 1.0

0.7 1.1

1.1 1.1

1.3 1.0

Gauging location

Source. Computed from IEPA (unpublished) data. w h ere D,t is the DO concentration (mg/liter) a Computed as (l/T) Ct CDmtl(Cm &J/121, in month m and year t, and T is the number of years for which data were available. The readings were taken sporadically over the years 1964-1975, with the number of observations for computing the monthly averages over T years ranging from 2 to 15 (in summer months).

in the intensity of river flows are shown in Table I. Several examples of the pattern of seasonal variation in water quality are presented in Table II. Moreover, the DuPage River basin has potential for flow augmentation by storing storm runoff and flood water. The years 1961-1967 record at least one annual flooding of the river’s West Branch (U.S. Geological Survey [24]) and show a range of one to eight. The available preliminary estimates (U.S. Corps of Engineers [23]) of flood damage caused by the more severe floods in the years 1948-1972 in the West Branch of the DuPage River suggest that the total damage caused by those floods was between 5 and 8 million dollars. The current water quality policy notably concentrates on relationships derived for the critical conditions, setting its regulations accordingly. Indeed a summary of the Illinois rules and regulations for effluent standards (NIPC, [17, p. 211) states that these standards “are based primarily on dilution available during low flow conditions which are assumed to be most critical. Critical low fiows are defined as those associated with an occurrence which may be expected to extend over a I-i-day period at a frequency of one in 10 years. The dilution ratio results from comparing the expected volume of effluent during the design year to the cited low flow.” An examination of effluent permits issued by the EPA to industrial plants and wastewater treatment plants in the DuPage River basin in Illinois shows that effluent standards throughout the year are in fact constant, with no attempt to benefit from the high-flow conditions. However, institutions and individuals seem willing to explore the possibilities of seasonal adjustments in the system. 2. OUTLINE

OF THE

MODEL

The model concerns a river divided into IM reaches. The typical elements linked to the nzth reach are industrial plants and a municipal wastewater treatment plant. The latter collects and treats wastewater discharged by the house-

SEASONAL

WATER

QUALITY

CONTROL

143

holds and industrial plants linked to it, and the former have the additional option of discharging their pollutants directly into the river. The unit of t,ime is 1 year, divided into S seasons. A season is characterized by variables such as temperature, intensity of river flow, and the river’s pollutant assimilative capacity.2 The probability that season s will occur, P(B,), is assumed to be known and equals the expected number of days in the season divided by 365. The model is formulated as a two-stage mathematical programming problrm (Dantzig [S]) of minimizing t!he expected social cost of maintaining watc>r quality at or above given standards with a predetermined probabilit’y 1~~1.~ Population and indust,rial output levels are also given. The decision variables endogenously incorporated into the model arc industrial production technology (including on-site pollution abatement) and the levels of pollution emitted directly into the river and indirect,ly through the municipal plants. Incorporated into the model as exogenous parameters arc decision variables related to the location, scale, and service area of t,he municipal treatment plants ; land use regulations ; and other measures concerning location. Some of the endogenous decision variables are seasonally adjustable; th(>ir seasonal flexibility is supposed to reflect the real-life situation. In t,he first version of the model,

these variables

arc limited

to the level

of treatment

treatment plants, but in the second version industrial can also vary to t’ake advantage of seasonal changes. 3. MATHEMATICAL

at the municipal

production

technology

FORMULATION

In the interest of brevity, but without loss of generality, only one pollutant4 and a typical reach of t’hc river will be considered. A transfer row and a t,ransfcr variable will represent the open-end link to the next> reach. The problem is thus to minimize x,

(1) subject to C Xii= + C XirN 2 bi, - < C UtjX,;I i

+ d”” = 0,

- $ C ai,XirN + t” + dNs = Hs, r -12 7 C aZrXiTN + t” 5 kH”, r paddy e.dD”

-

+

psdN”

dN”

+

e”

2

@,

=

0,

(2,

and Xii=, XirN, 09, dNs, t”, es 2 0,

(3,

2 An operational characterization of a season is presented below in the mathematical formulation of the model. 3 The first stage relates to decisions with respect to variables which remain constant throughout the whole year; the second stage relates to decision variables which may be seasonally varied. For details see the mathematical formulation in the following section. 4 Introducing additional pollutants raises empirical questions concerning specification of their interrelationships and will be discussed elsewhere.

144

DAN

i = 1, 2, . . . , I; j = 1, 2, . . . , J; T =

YARON 1, 2,

. . . , R,

a&

s =

1, 2,

. . . , S,

\vhcrcx

XijD

Level of production output i, using technology j, involving direct discharge of the pollutant into waterways (units/day) Level of production output i, using technology r, involving discharge of XiIN the pollutant into the municipal sewage system (unit/day) fij” @iiD>, &TN (XirN) CO& of production (financial outlay) of input i expressed as functions of XijD and XirN, respectively ($/day) Predetermined level of daily output i (units/day) bi Quantity of the pollutant discharge per unit of product i, using techaii, ai, nology j or T, respectively (lb/unit) t Level of removal of the pollutant by the municipal treatment plant (WW Cost of pollutant’s removal as a function of t ($/day) s(t) dD, dN Quantity of pollutants discharged, respectively, directly into the waterways and indirectly through the municipal sewage system (lb/day) H Predetermined discharge level of the pollutant by households int,o the municipal sewage system (lb/day) k Maximum fraction of the pollutant removed at the municipal treatment plant using the present technology (scalar; 5 1) Maximum reduction in water quality not violat)ing ,8 percent of the time & the required water quality standard in the reach (mg/liter) (see t.he following section for the derivation of 0) “Transfer coefficient,” the level of reduction in water quality due to the P discharge of one unit of the pollutant into t’he reach [(mg/liter)/lb] e Quantity of the pollutant flowing into the following reach (lb/day) Note that due to the natural digestion of the pollutant in the reach only a portion of e, say Lye, 0 < Q! < 1 is assumed to enter the head of the following reach. The discrepancy between e and eyeis due to the artificial segmentation of the river into discrete reaches. The objective function represents the cost of production of the predetermined industrial outputs, plus the expected cost of t’reatment of wastewater by the municipal treatment plant over the S seasons. The second and third restrictions in (2) are definitions of clD* and clN* which represent the toOa quantities of the pollutant discharge, respectively, directly into the river and indirect’ly via the sewage syst’em. The fourth restrict’ion represents the upper limit on the level of removal of the pollutant. The water quality restriction (row 5) is derived from the relationship (4) Pi&n&s - ps@ - p@‘Js 2 Q-s. 8) 2 p, water quality when cZD8 where P denotes probability, QnatFais the “natural” = dN8 = 0, &reqS8is the water quality required in season s, and ,8 is a prescribed probability level. We introduce some simplifying assumptions, namely, we refer with expectation to p* as deterministic and to &nat,* as normally distributed E(&natss) and standard deviation us. Under these assumptions (4) becomes equivalent t#o (5) :5 or 5

Note

that

p8 and & nst.s in (5), and (~8 in (6) fully

characterize

the natural

conditions

in season

S.

SEASONAL

WATER

QUALITY

14.3

CONTROL

where xp is t,hc value of the standard normal varia~~h~ rorrcsponding to the 13 probability level. The assumption regarding p’s (= the transfer coefficients) as deterministic seems t,o be proper under condit,ions in which a large share of the variance of (Qnil,,8 _ pj”s _ psdN8) is due to the “natural” water qualit,y. This is apparently the case in the West Branch DuPage River, for which the model was designed, and where approximately two-thirds of the BOD loadings into the river originate in non-point, sources [lS]. If the above simplifying assumption is not valid, thr variance of t,hc whole linear expression (Qnat,Y - pydn8 - pSrlNS)should be considered and restriction (5) becomes nonlinear, implying that the solution of the problem becomes a formidable task.” The last restriction in (2) and the variable e8 constitute the link to the following reach. Denoting by 1 and 2 t,hr: consecut,ive reaches, the connc&on bet wean the t#wo reaches is expressed by -l@s

-

l,jNs

+

l,.q

=

0, (6)

- las.les - C C 2a1,2u2X1,cD+ ?dD,?= 0. u 0 In the following, t,he only aspect of relationship (6) that will be specifically mentioned is t,he linking shadow price yS of the pollutant (relating to the first TOWof (6)). The cost functions f$, firN, and q8(t8) are assumed to be convex, a reasonable assumption for industrial and wastewater treatment plants mit)h predetermined capacit#y and a given minimum removal level7 Reformulating gs(ts) into CJ~(C, ta) in order to accommodate capacity (c) as a second argument may lead to loss of t,he convexity prop&y. To avoid this difficulty the model considers the capacit,y of t,he treatment plant,s as exogenously predetermined parameters. Accordingly, an empirical applicat>ion of the model should test several runs with alternative configurations of treatment plants (location and capacit,y), but there is some solace in that’ fern configurations will be a priori practical in general. Let the dual variables corresponding to restriction set, (2) be denoted by ~~ (i = 1, 2, . . .) I), Xi’“, XNs, ?, 6”) and yS. At the optimal solution of (I), (a), and (3) the following Kuhn-Tucker conditions are satisfied (since fizz), fijN and g(t) arc convex, and the rest.rictions are linear, the Kuhn-Tucker conditions art’ ncwssary and sufficient) :

XoNs

+

p”6””

-

y””

<

0

all s;

6 A different formulation of “chance constraints” for a similar problem can be found in Deiningel [5]. His formulation leads to nonlinear relationships; no attempt to solve an empirical problem is presented. 7 8ee Frankel [!I].

146

DAN

YARON

with and The superscript o denotes the variables’ values at the optimal solution. Rearranging the terms of (7) and referring to the absolute values of the nonpositive Lagrangian multipliers we can proceed with the economic interpretation of the social cost of pollution. For to8 > 0:

where ] V] 5 IT”~//P(~?~) is the marginal cost of the technology constraint (at k) in season s, while I?* 1 is weighted by the probability P(8J. The weighted marginal social cost of pollution discharged in season s via the treatment plants is accordingly equal to the sum of the marginal cost of treatment plus the marginal cost of the technology constraint, with the sum weighted by P(0,). If we define PNa = AoNs/P(r98), (8) becomes

so that the unweighted marginal social cost of pollution via the treatment plant in season s is equal to the marginal financial cost of treatment plus the marginal cost of the technology constraint. (Note that 7 can be zero if the restriction is not binding.) On the other hand, eons = p [ 60s[ + To8 w-u or, dividing

by P(OJ, j;oNs

= paIs”“I + yes+

(11)

The marginal social cost of pollution in a given reach in season s is the sum of the marginal cost of maintaining water quality in that reach plus the marginal cost of polluting the following reach.* Note that either yo8 or Pa alone can be zero. If both are zero, XoNs = 0, in contradiction to the assumption that to8 > 0. The marginal social cost of pollution due to direct waste dumping is ),oDa = p16”*1 + ,A-, (12) or PDs = ,8/,-o, 1 + ,yo,. (13) Considering the first row of (7) and rearranging the terms we find that any process included in the optimal solution satisfies the following: (14) 8 Referring

to (6) we have, for leoa > 0, *-pa = la* 90~“.

SEASONAL

Upon subst,itution

WATER

QUALITY

147

CONTROL

of koDs = P (Oa)j;ODs,we find : (15)

The marginal social cost of production of product i is equal to the marginal financial outlay plus the expected value of the social cost of pollution due to direct discharges. Similarly, for any industrial process producing product i and discharging wastewat’er into the municipal system: (10,

The interpretation of the last term in (16) becomes clear by substituting into (16) and obt,aining

(9)

(171 Recall that k is the maximum fraction of the pollut’ant removed at the waste treatment plant and that kai, is the increased treatable quantit#y of the pollutant in pounds due to one unit of Xi?* in season s adjusted by a correction term h-a,, 1PJ I. In concluding this section, I should note t)hat alternative formulations of t,he model are possible to reflect different situations with respect, to the flexibility of seasonal adjustments of other elements of the system. If, for example, all indust’ries have the opt’ion of varying t,heir technologies of production and on-site pollution abatement over the seasons, the objective function (1) would be rrformulated to (18)

with the seasonal index s added to the XijD and XirN variables. In such a case the marginal social cost of production of product i may vary from one season to another. A further modificat’ion of the model may involve seasonal variation of the industrial output of some of the heavily polluting products. 3. IMPLICATIONS

AND

POSSIBLE

APPLICATIOSS

OF THE

MODEI,

The importance of seasonal variability and the importance of policies intcndrd to benefit from seasonal adjust,ments are empirical issues to be examined within an empirical domain. A case study analysis will be presented in a later section. I shall discuss in this saction, however, some general ideas regarding thr implications and the capability of the model. Implications

with Respect to User Charges and Pollutioll

Taxatiorr.

Baumol and Oates [l] have shown that’ by imposing proper user charges and pollution emission taxes it is possible to induce competitive industries to behave in a socially optimal manner. Estimates of such taxes have been derivrd in

148

DAN

YARON

several empirical studies of water quality control (Hass [la], Haimes [ll], Herzog [13], and others). Our model suggests that these taxes should be seasonally adjusted in order to reflect seasonal variations in the marginal social cost of pollubion. Assume that the marginal social cost of pollution in season s is XoDsand XoNS for “direct” and “indirect” emissions, respectively, and that the shadow price of the technological restriction on the level of the pollutant removal is 17°8I. Assume, further, that the producer of product i is charged for his emmission XoDs and (XoNs - kl?l) per unit of direct and indirect discharge. The optimizing (cost minimizing) problem of the ith industry can be expressed as follows : Minimize

+

(PNs

-

k) To8 I) c Ui,XiTN1 (19) r

subject to c xi:, i

+ c XirN 2 bi, r XijD, XiTN 2 0.

For a solution of (19) and (20) to be optimal Kuhn-Tucker conditions be satisfied :

it is sufficient

(20) that the following

(21)

where 4; is a nonnegative number. But (21) is satisfied by letting 44 = pi0 as in (7). Accordingly, the optimal private solution of the industry producing the ith output is equivalent to the social optimum. While the above argument is straightforward from the technical-formal point of view conceptually it raises several questions (as does the Baumol-Oates approach). It assumes that the tax-imposing authority and the private producers agree upon the relevant functions and parameters, a condition violated in the real world where numerous elements of the model are uncertain and variable. Under conditions of uncertainty Fishelson [S] and Roberts and Spence [20] have shown that a mixed water quality control system based on a combination of taxes and emission standards is more efficient than the pollution tax only. Although a discussion of the effectiveness of user charges and pollution taxes vs emission standards is beyond the scope of this paper, knowing the approximate values of the seasonal marginal costs of pollution and understanding the factors which determine them may be useful for policy decisions regarding water quality control, regardless of whet,her it were based on user charges and emission taxes, emission standards, or a combination thereof. Other Possible Applications With proper modifications, in which seasonal variability

of the Model this model may also be used to analyze other issues is an important factor.

SEASONAL

WATER

QUALITY

CONTROL

149

One application, for example, involves nonpoint sources of pollut’ion and its effect on &“““.s. Another application concerns an evaluat,ion of possible savings in water pollution abatement costs using river flow control by means of reservoirs (low-flow augmentation). Alternative policies of flow modificat,ion and/or instream aeration and their effects on the cost of pollution abatement can also b(l chasily tested with this model. A still more ambitious and comprehensive study can involve the joint analysis of flood control, flow regulation, and water qualit! cont,rol, overcoming the problems of “dimensionality” by a decomposition approach. 5. CASE STUDY

ANALYSIS

The case analysis described here is confined to two reaches only, wit,h one town, one treatment plant, and two industrial plants in each reach. The economic data arc based on information from NIPC (Nort,heastern Illinois Planning Commission), IEPA (Illinois Environmental Protection Agency), USEPA, Hydrocamp, Inc., and information obtained from wastewatcr treatment plants direct’ly. Our evaluation is that they are representative of the conditions prevailing in the area. At the same time the two-reach system was synthetically constructed in order to contain enough elements of interest. Water quality paramet,crs w(‘r(’ takr>n from Hydrocomp, Inc. [15] and from Dorfman and Jacoby [6] and were somewhat arbitrarily adjusted to the size and flow intensity of the DuPage River, Western Branch. On the whole it, seems that the following case cxamph generally reflects the t,ypical elements of the problem and the results of the analysis provide evaluat’ion of the order of magnitudes involved. In the application of the model, BOD is a pollutant of interest and t’he DC) concentration is t.he measure of wat,er quality. The treatment, cost, functions {r’(F) were approximat’ed by linear step functions corresponding to three levels of ROD removal: Ll (<90.5%), L2 (90.595.2%), and L3 (95.2-97.670).y Two seasons were dist,inguished, namely, “High flow” and “Low flow,” with rcbspectivc probabilitirs of 0.75 and 0.25. The levels of @ (maximum reduction in water quality not violating the required water quality in the reach, 4 mg/littar DO, 997’ of the time) were assumed to be 3.96 and 1.82 mg/litcr DO, rcspcctivc>ly for the High-flow and Low-flow seasons. They wclrc dcrivcd using rrlationship (5), with h’(&“lltVs) and U” respectively 11.50 and 1.52 for the High-flow and 7.82 and 0.86 for the Low-flow seasons, and zo.y9 = 2.326. The objcctivc function was to minimize, the total variable cost of pollution abat.clment, with the construction and tht equipment of the treatment faciliticks predetermined. Thr variabl(h cost includes operating and that part’ of maintenance cost which is avoidable on a seasonal adjustment basis. Note that, all thn>c trc>atmr>nt lev& w(tre includrd in thca model! and t’hc question was what, levels to operate during the two s(lasonF. The (lssentials of the results are summarized in thta following : (a) During tho High-flow season the first level (Ll) of wastewatcr trcatmc>nt only should bc applied in the upper reach and both Ll and L2 in the lowcxr reach. y These levels correspond to 20, 10, and 5 mg/liter BOD Ll roughly corresponds to what is generally called secondary t,o subsegments of tertiary or advanced treatment (Parker process Ll was not specified because it was redundant.

in the effluent of the treatment plants. treatment, and L2 and L3 correspond [19]). The lower bound for t,reatment

DAN

YARON

(b) During the Low-flow season all three treatment levels should be applied in both reaches. (c) Accordingly, the marginal social cost of BOD removal varies significantly from one season to the next as shown below (units are dollars per pound) :I0 High Upper Lower

reach reach

flow

0.59

Low

flow 1.27 1.59

0.66

(d) The total variable cost per day of the two treatment plants is $595 in the High-flow season and $1097 in the Low-flow season, a difference of $502 per day. (e) Abandoning the constant effluent standard policy and allowing for higher efhuent standards in the High-flow season results in a potential cost reduction of $137,046 per year (= $502 X 273 days). One way to evaluate the relative importance of the above saving potential is to relate it to a situation in which effluent standards and treatment levels are rigidly determined for the entire year, with reference to the critical low-flow conditions, and no seasonal adjustments in t,he treatment levels. In such a case the total variable cost of treatment is $400,405 per year (=$1097/day X 365 days), and the potential saving ($137,046) due to the seasonal adjustment comprises 3470 of the total variable cost. The above results suggest that attention should be paid to the saving potential derivable from seasonal adjustments in the treatment levels of wastewater treatment plants, in response to the flow conditions and the river’s assimilative capacity. When considerable differences between Low-flow and High-flow conditions prevail the design of the treatment plants should be adapted to seasonal flexibility. Such a design will emphasize the variable (operating) cost component and reduce the capital cost component (construction and equipment). The model presented in this paper can be useful in studying the trade-offs between these two components. To attain this goal alternat,ive location and capacity configurations of treatment plants should be incorporated into parametric runs of the model. Alternatively, the model could be modified into a mixed-integer programming model to accommodate integer variables, representing location and capacity of treatment plants. Finally, it should be noted that the model approximates some nonlinear relationships by linear functions. Accordingly, the analysis with the aid of the model can lead to a good understanding of the system and provide approximate solutions. These solutions should be tested later on by simulation within a more accurate framework. As previously mentioned, simulation alone has limited creativity and capability for t,he formulation of sound alternatives. ACKNOWLEDGMENTS This paper is part of a project on the Economics of by an NSF grant to the University of Chicago, with G. assistance of Bradley G. Lewis, graduate student in acknowledged. Helpful comments by D. Carlton, the workshop on Economics of Natural Resources, and an to say, the responsibility for errors and omissions rests lo Note

that

the

model

and

the

results

refer

to

Water Quality in an Urban Setting funded S. Tolley as the project leader. The efficient economics at the University of Chicago, is participants of the University of Chicago unknown referee are appreciated. Needless with the author.

variable cost only.

SEASONAL

WATER

QUALITY

CONTROL

1.51

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