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Multi-objective management model forwaste-load allocation in a tidal riverusing archive multi-objective simulatedannealing algorithmM. Zewdie a & S. Murty Bhallamudi aa Department of Civil Engineering , IIT Madras , Chennai , IndiaPublished online: 09 Aug 2012.

To cite this article: M. Zewdie & S. Murty Bhallamudi (2012) Multi-objective management model forwaste-load allocation in a tidal river using archive multi-objective simulated annealing algorithm,Civil Engineering and Environmental Systems, 29:4, 222-230, DOI: 10.1080/10286608.2012.710607

To link to this article: http://dx.doi.org/10.1080/10286608.2012.710607

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Civil Engineering and Environmental SystemsVol. 29, No. 4, December 2012, 222–230

Multi-objective management model for waste-load allocation ina tidal river using archive multi-objective simulated

annealing algorithm

M. Zewdie* and S. Murty Bhallamudi

Department of Civil Engineering, IIT Madras, Chennai, India

(Received 30 November 2011)

Management of water quality in a tidal river through appropriate releases from an upstream reservoir, inaddition to removal of pollutants at discharge points, has been addressed in the present work. The multi-objective waste-load allocation (WLA) problem is solved using a simulation–optimisation framework.Archived multi-objective simulated annealing (AMOSA) algorithm is used as an optimiser. The proposedmodel is used to assess the impact of tidal flow conditions and upstream releases on management deci-sions for maintaining water quality. Tidal flow conditions shift the Pareto-optimal (PO) front between theupstream releases and treatment cost towards left, indicating that the dilution from back flow reduces thetreatment cost as well as upstream release requirement. Imposition of a higher DO standard for waterquality shifts the upstream release vs. cost PO front upwards and the same is observed for the inequityand cost PO front. It is found that, for a given inequity measure, the cost of treatment can be reducedsignificantly by adopting an optimal upstream release value. Better equity can be achieved, for the samecost of WLA, when the upstream release itself is varied in time optimally.

Keywords: tidal river; waste-load allocation; multi-objective management; AMOSA

1. Introduction

Although the global water crisis tends to be viewed as a water quantity problem, water qualityis increasingly being acknowledged as a central factor in the water crisis. Even though not allcountries are facing a crisis of water shortage, all have, to a greater or lesser extent, seriousproblems associated with degraded water quality (Ongley 2000). In this context, managementof water quality along a river system (maintenance of acceptable level of water quality for anintended purpose) becomes very important. In reality, the river water quality management problemis multi-objective by nature (Yandamuri et al. 2006) and therefore, it demands a multi-objectiveoptimisation (MOO) framework, where two or more conflicting objectives, subject to certainconstraints, are simultaneously optimised (solved). MOO is characterised by determining a familyof alternatives solutions to the conflicting objective functions (problems) that must be considered

*Corresponding author. Email: tambek72@gmail.com

ISSN 1028-6608 print/ISSN 1029-0249 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/10286608.2012.710607http://www.tandfonline.com

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Civil Engineering and Environmental Systems 223

equivalent in the absence of information concerning the relevance of each objective relative tothe others (Tung and Hathhorn 1989).

2. Formulations of models

Two deterministic, multi-objective waste-load allocation (WLA) models are formulated in thisstudy, namely, (i) cost-upstream release model and (ii) cost–equity model.

2.1. Cost-upstream release model

It may be possible to maintain the water quality in a river either by treating the effluents to anappropriate level before they are discharged into the river at different locations or by increasingthe assimilative capacity of the river through an upstream reservoir release. Therefore, whilesolving the WLA problem for a stream, a decision-maker may wish to determine if there arereasonable treatment cost-release volume trade-off solutions to maintain a specified water qualityin the river. The present work considers a river segment in which flow conditions are affectedby tidal variation at the downstream end and they are inherently transient in nature. The rate ofrelease of water can either be a constant or it can take M different values over a period of 1 day.Moreover, the release rate may be constrained by fresh water availability in the upstream reservoirand/or the limiting hydraulic factors (velocity, channel capacity, etc.). Other constraints includethe specification of a minimum value for DO at all the check points along the river and the equalityconstraint representing the river response. The model formulation is as follows:

Min F1 =NS∑i=1

Ci(xi), (1)

Min F2 =M∑

k=1

Qrel,k ∗ Tk . (2)

Subject to:

xi ∈ xsi ∀i, (3)

(DOj,t)a = f (−→Qrel, �W , �X, �R), (4)

(DOj,t)a ≥ DOstd ∀j and t, (5)

Qmin < Qrel,k < Qmax ∀k, (6)

where Ci(xi) is the cost of wastewater treatment at loading point i, xi the waste removal fractionat loading point i, NS the number of point pollutant source locations, Qrel,k the upstream releaseduring kth period of the day, Tk the duration of kth period, M the number of release segmentsin a day, xsi the set of all waste treatment options for source i, (DOj,t)a the dissolved oxygenconcentration at check point j at any time t for actual treatment, �W the vector of waste inputsto the point sources, �R the vector of parameters describing the pollutant transport process. Otherconstraints considered are those defining the acceptable range of the waste removal fractions.A range from 0.35 to 0.98 for removal fraction of total pollutant input is chosen. Lower limitis based on minimum requirement of 0.35 to ensure prevention of floating solids from beingdischarged in to the river and upper limit of 0.98 is based on the maximum efficiency attainableby practical treatment technology (Tung and Hathhorn 1989, Yandamuri et al. 2006). Equation(4) defines the water quality as a function of waste inputs and stream conditions.

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224 M. Zewdie and S. Murty Bhallamudi

2.2. Costs-equity model

In many instances, attempts to maximise waste discharge would result in an allocation of largequantities of waste to the upstream users, while the downstream dischargers would be requiredto treat their influents at levels of maximum possible efficiency, if equity is not considered in theformulation. Equity between various users can be measured in a number of ways. In this study,the equity consideration of equal percent removals among the various dischargers was utilised(Yandamuri et al. 2006). The cost–equity model formulation, for specified upstream releases, isas follows:

Min F1 =NS∑i=1

Ci(xi). (7)

Min F3 =NS∑i=1

(xi

x− Wi

W

). (8)

Subject to, constraints given by Equations (3–6), where x̄ is the average waste removal level, andW̄ the average waste input over NS number of point sources.

3. Framework for MOO

A simulation–optimisation framework is used for solving the multi-objective management prob-lems formulated in the earlier section. The classical simulated annealing method was reported togive good performance with single-objective problems. However, not many attempts have beenmade in the past to extend it to MOO in general and WLA problems in particular. Scalarisingthe multi-objective problems is a commonly used technique to solve MOO problems. In thiswork, a domination-based energy function is employed instead of scalarising the multiple objec-tives problems. For instance, the number of solutions that dominate the new solution determinesthe acceptance probability in classical multi-objective simulated annealing (MOSA), while inAMOSA it is the amount of domination with respect to the solutions in the archive and thecurrent solution being used as acceptance criteria. Bandyopadhyay et al. (2008) compared theperformance of AMOSA to the most well-known and often used MOO algorithms. They reportedthat AMOSA performed better than MOSA and non-dominated sorting genetic algorithm-II,with respect to performance measures such as purity, convergence, and minimal spacing. Also,AMOSA is less time consuming as compared to NSGA-II for complex problems and it producesmore distinct solutions. In this study, this algorithm (AMOSA, Bandyopadhyay et al. 2008) isused as optimiser for solving the multi-objective non-linear optimisation problems presentedin the earlier section. AMOSA is coded in the MATLAB programming language and validatedusing solution for a benchmark problem (Fonseca two objective minimisation problems). POfront obtained by AMOSA matched closely with analytical solution given by Abraham et al.(2005).

A dynamic water quality simulation model is used to determine the water quality responsesto different external interactions and processes in the water environment. This model consists oftwo sub-modules: (i) flow module and (ii) transport module. Flow module solves the completeunsteady flow equations for water (Saint Venant Equations) using a Preissman scheme. The trans-port module solves governing advection–dispersion–reaction equations for biochemical oxygendemand (BOD) and DO, using an implicit finite difference scheme.

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Civil Engineering and Environmental Systems 225

4. Results and discussion

4.1. Illustrative example problems

In this study, the proposed multi-objective WLA model is first used to illustrate the effect ofdownstream water level variation due to tides on the WLA decisions for a tidal river. The modelsare then used to illustrate the trade-off between the treatment cost, equity and the total volumeof release from an upstream reservoir. These effects are illustrated through example applicationsto a hypothetical tidal river system and with realistic data, shown in Figure 1, and Adyar Riverin Chennai, India (Figure 2), respectively. Operation policy obtained using the water qualitymanagement model is applicable for short-term operation. The hypothetical river system consid-ers up to three Pollutant Loading sites (PLD1, PLD2 and PLD3), at a distance of 5.0, 7.5 and10 km from the upstream end, a time-varying release of water (Qrel) at the upstream end, and atime-varying water level at the downstream end (tidal boundary). Hydraulic particulars are: totallength, L = 15.0 km, bottom width, B = 100 m, side slope, m = 1V:2H, bed slope, S = 0.00005,Manning roughness coefficient, n = 0.02, and the initial flow rate, Qo = 100 m3/s.

Values of various transport variables are: initial BOD concentration in the river = 0.0 mg/L,initial DO concentration in the river is DOsat = 9.2 mg/L, BODs generated at PLD1, PLD2 andPLD3 are 2500, 2000 and 1500 mg/L, respectively. The DO in waste discharge is zero, and the rateof wastewater discharge is 1.0 m3/s. BOD removal cost is Ct = 1.587 Rupees (Rs) per kilogram,and value of water based on water productivity Cw = 3.15 Rs/m3. In general, salinity in the wateraffects the re-aeration as well as the BOD decay rate. In this study, the salinity in the water dueto salt water intrusion is considered during the flood tide by using modified decay rate (k1 m) asa multiple of k1 in a range of (0.5–0.95) and DOsat = 8.0 mg/L. In all the cases, in this study,the computational time step, �t = 300 s and the distance step, �x = 100 m. Simulated annealing

Figure 1. Illustrative example river system.

Figure 2. Location map of Adyar River.

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226 M. Zewdie and S. Murty Bhallamudi

parameters are chosen based on the information available in (Rao et al. 2004) and initial numericalexperimentation made to obtain the best results. Standard random number generation method isused to generate the solution sets at each iteration process.

4.1.1. Case-1: effect of tidal conditions on the trade-off

In this section, the effect of downstream tidal conditions on the water quality management isillustrated. For this purpose, the cost-upstream release multi-objective model is used to obtain thePO front between the treatment cost and the volume of water released from the upstream reservoir,under tidal and non-tidal flow cases for single pollutant loading at 5 km and DOstd of 5 mg/L. Itcan be observed from Figure 3 that the trade-off curve for the case with tidal flow lies below andto the left of non-tidal case. For the same cost of treatment, the non-tidal flow scenario requireslarger amount of water to be released from the upstream reservoir as compared to the tidal flowto maintain the same minimum DO requirement at all points. A similar effect is observed in thecase of cost-inequity PO front (Figure 4). This illustrates the significance of properly consideringthe downstream transient effects while arriving at appropriate water quality management strategyfor tidal rivers.

4.1.2. Case-2: effect of water quality standard on the cost-equity trade-off

In this section, the effect of water quality standard on the water quality management in tidalrivers, from the perspective of cost–equity trade-off, is illustrated. For this purpose, the cost–equity multi-objective model is used to obtain the PO front between the treatment cost and equity.A specified constant upstream reservoir release of 100 m3/s and two specified DOstd of 5 and6 mg/L are used. It can be observed from Figure 5, as expected, that it costs more to conform

5

7

9

11

13

15

17

19

1.5 2.5 3.5

Treatment cost (MRs.)

Rel

ease

vol

ume

M m

^3

Tidal Flow Non-Tidal

Figure 3. Effect of tidal flow on cost upstream release volume PO front.

0

1

2

3

4

5

7 8 9 10Treatment cost (MRs.)

Ineq

uity

mea

sure

Dostd 6mg/L Dostd 5mg/L

Figure 4. Effect of water quality standard on cost−inequity trade-off, case-2.

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Civil Engineering and Environmental Systems 227

8

10

12

14

16

18

20

1 2 3 4Treatment cost (MRs.)

Vol

. wat

er (

Mm

^3)

DOstd 5mg/L DO std 6mg/L

Figure 5. Effect of water quality standard on PO front (tidal boundary conditions).

to a better water quality standard. Also, inequity in the treatment burden is reduced if the waterquality standard is lowered, for the same total treatment cost.

4.1.3. Case-3: effect of time-invariant upstream release on the cost–equity trade-off

This example problem is chosen to demonstrate how the inequity in WLA can be brought downsignificantly by releasing an appropriate amount of water from an upstream reservoir, in additionto treating the wastewater at pollution discharge points. PO fronts for two alternative strategies arecompared for this purpose. In the first alternative, wastewater is treated at the pollution loadingpoints while maintaining a specified upstream release. In the second alternative, the upstreamrelease (temporally invariant) is also considered as a decision variable, in addition to removalfractions at the pollution loading points. The pollution loading rates at the three loading points,the cost of treatment, and other parameters are the same as before. It can be observed from Figure 6that for the same inequity, total cost is much less if the strategy of using upstream water releasesin addition to treating the wastewater is adopted.

4.1.4. Case-4: effect of time-variant upstream release on the cost–equity trade-off

In a tidal river environment, the periodic nature of tides affects the availability of water in theriver for dilution purposes. Volume of water increases during high tide, while it is low during ebbtide. It may be possible to minimise the amount of water released from the upstream reservoirfor the purpose of dilution, by varying the upstream release in time. The effect of time-variantupstream release on the cost–equity PO front is studied in this section. For this purpose, thecost–equity PO front for the time-variant upstream release (alternative I) is compared with time-invariant case (alternative II). The BOD loading DOstd and other input variables are same as in

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80

Total cost (*10^6 Rs.)

Ineq

uity

Alternative I Alternative II

Figure 6. Effect of time-invariant upstream release on trade-off between cost and in-equity, tidal boundary conditions,DOstd = 5 mg/L, case-3.

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228 M. Zewdie and S. Murty Bhallamudi

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60 70

Cost (M Rs.)

Ineq

uity

mea

sure

Alternative II Alternative I

Figure 7. Effect of time-variant upstream release on trade-off between cost and in-equity, tidal boundary conditions,DOstd = 5 mg/L, case-4.

case-3. It can be observed from Figure 7 that the trade-off curve for the case using time-variantupstream release lies to the left and below the trade-off curve for the other case. It shows that it ispossible to improve the equity significantly by having a time-variant upstream release, for the sametotal cost.

4.2. Application of management model to Adyar River

The multi-objective model is applied to theAdyar River in Chennai, India, to determine an optimalstrategy for maintaining water quality using releases from an upstream reservoir and treatmentoptions at point pollutant source locations. The results presented in this section should be viewedonly as planning level results for comparing several strategies to maintain water quality in theAdyar River. The river receives a sizeable quantity of sewage after reaching Nandambakkamnear Chennai. The total amount of sewage produced in Adyar River basin, based on the averageamount of water supply to the basin area in litres per capital per day (LCPD) (i.e. 135.0 LPCD),is expected to be 132.7 million litres per day (MLD). Based on this figure and the total amountof treated sewage (i.e. 77.0 MLD), there is an amount of 55.7 MLD untreated sewage that couldmake its way to Adyar River. In this work, we considered that, this amount of sewage with thespecified BOD concentration of 350 mg/L is released into the river system at five prominentsewage effluents outlet points (i.e. Chennai bypass, MIOT Hospital, Sidapet near Parsan Nagar,Kotturpuram bridge and Green Way road) which are located at 10, 14, 19, 20 and 25 km fromthe upstream end. The sewage after treatment is released into the lower river catchment near KasiBridge in the industrial area.

The river is almost stagnant except during the rainy season. It has varying depth with approx-imately 0.75 m in its upper reaches and 0.5 m in its lower reaches. It discharges about 190–940million m3 water annually to the Bay of Bengal (Krishnaveni and Gowri 2008), which is ∼6.02to ∼29.8 m3/s. The peak discharge is about, 200 m3/s, during the northeast monsoon seasonbetween September and December. The lower region ranges between 14 and 26 km from upstreamend has DO saturation ranges between 6% and 40% which is lowest of all reaches (Rajkumaret al. 2008). In this work, we considered 12 km, lower tidal segment of the river, where the BODloading is high and the DO level is worst. During low flow periods (February and August) thedevelopment of a sand bar across the mouth of the estuary due to monsoon-driven long shore drifttypically reduces the tidal range from ∼0.6 to ∼0.2 m (Rajkumar et al. 2008). This explains thatthe dissolved O2 in the lower catchments and estuary must be replenished mainly via air–sea gasexchange and the importance of the tidal flow.

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Civil Engineering and Environmental Systems 229

0

1

2

3

4

5

6

0.11 0.115 0.12 0.125 0.13 0.135 0.14

Total cost (*10^6 Rs)

Vol

ume

of w

ater

(*1

0^6

m^3

)

Figure 8. Cost–water volume trade-off for Adyar River.

0 50 100 150 200 250 300

4.5

5

5.5

6

6.5

Distance from upstream end (*100 m)

DO

con

cent

ratio

n (m

g/L

)

LCSLRVS

Figure 9. DO profile simulated for extreme solution values from PO front.

The cost and volume of water released are for 1-day operation. In the Adyar River basin, it maybe a problem to get fresh water from upstream, especially in the dry season. In such a case, it isadvisable to go for higher treatment level in order to maintain the in-stream water quality. However,the flow rate could rises up to 200 m3/s in the monsoon period. This provides an opportunity to thedischarger, to reduce the cost of treatment in such particular periods. The trade-off curve presentedin Figure 8 helps in making a decision regarding when and how much of the wastewater needsto be treated. It can also be observed from the trade-off curve that one can reduce the amount ofextra water to be released from the reservoir (over and above the 10 m3/s average base flow) tonearly zero level, by treating the wastewater discharged into the river system. In other words, theexisting in-stream flow is sufficient to maintain the water quality if the wastewater is treated to97% before it is discharged into the river. It should be noted that this is possible only after thepresently polluted water in the river is flushed out. DO profiles simulated for extreme solutionvalues (least cost solution point, LCS; and the least release volume solution, LRVS) from PO frontare presented in Figure 9. It can be observed from the DO profiles on the figure that the minimumwater quality requirement is met almost at all points. Values of the decision variables (upstreamrelease (Qrel) and the removal fraction levels at each pollutant source location) for some selectedpoints on the PO curve (i.e. corresponding to LCS, LRVS and selected compromise solution,SCS) are summarised in Table 1.

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230 M. Zewdie and S. Murty Bhallamudi

Table 1. Extreme value of solution sets on the PO front.

Major outlet points along the river (km)

TC (∗106Rs) TRV (106 m3) Qrel (m3/s) 10 14 19 20 25

LRVS 0.134 0.006 10.07 0.458 0.595 0.723 0.651 0.477LCS 0.116 5.48 73.33 0.376 0.724 0.398 0.608 0.402SCS 0.121 1.331 25.40 0.410 0.369 0.366 0.816 0.714

Note: TC, total cost; TRV, total release volume.

5. Conclusion

In this study, a multi-objective management model is proposed for water quality in a tidal river.The management model is based on a simulation–optimisation framework, in which a dynamicsimulation model for BOD and DO is embedded. The optimisation problem is solved using theAMOSA algorithm. It is found from the illustration runs of the model that the effect of time-varying boundary conditions should be considered appropriately while arriving at managementdecisions for water quality in a tidal river. The flood tide has a considerable impact on the waterquality improvement through increasing the volume of water that increases the dilution capacity ofthe river. Optimal release of water from an upstream reservoir, in addition to removal of pollutantsat discharge points, reduces the total cost (including the cost of water), and increases the equityin treatment burden among polluters. Temporally varying optimal upstream reservoir releasesreduce the inequity further. Application of the proposed model to the Adyar River in Chennai,India, is demonstrated. Under the present scenario of Adyar River treating each wastewater beforebeing discharged into the river improves the water quality of the river.

References

Abraham, A., Jain, L., and Goldberg, D.E., 2005. Evolutionary multi-objective optimization: theoretical advance andapplications. Germany: Springer.

Bandyopadhyay, S., Saha, S., and Maulik, U., 2008. A simulated annealing-based multiobjective optimization algorithm:AMOSA. IEEE Transactions on Evolutionary Computation, 12 (3), 269–283.

Krishnaveni, K. and Gowri, V., 2008. Application of GIS in the study of mass transport of pollutants by Adyar and CooumRivers in Chennai (Madras), Tamilnadu. Environmental Monitoring and Assessment, 138 (1), 41–47.

Ongley, E.D., 2000. Water quality management: design, financing and sustainability considerations-II. World Bank’s WaterWeek Conference: towards a Strategy for Managing Water Quality Management, 3–4 April, Washington, DC, USA.

Rajkumar, A.N., Ramesh, R., Purvaja, R., Barnes, J., and Upstill-Goddard, R.C., 2008. Methane and nitrous oxide fluxesin the polluted Adyar River and estuary, SE India. Marine Pollution Bulletin, 56 (12), 2043–2051.

Rao, S.V., Bhallamudi, S.M., Thandaveswara, B.S., and Srinivasulu, V., 2004. Planning groundwater development incoastal deltas with paleo channels. Water Resources Management, 19 (5), 625–639.

Tung, Y.K. and Hathhorn, W.E., 1989. Multiple-objective waste load allocation. Water Resources Management, 3 (3),129–140.

Yandamuri, S.R.M., Srinivasan, K., and Bhallamudi, S.M., 2006. Multiobjective optimal waste load allocation models forrivers using non-dominated sorting genetic algorithm-II. ASCE, 132 (3), 133–143.

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