Environ Fluid Mech (2007) 7:439–450DOI 10.1007/s10652-007-9037-4

ORIGINAL ARTICLE

River water quality management model using geneticalgorithm

Egemen Aras · Vedat Togan · Mehmet Berkun

Received: 29 March 2007 / Accepted: 27 August 2007 / Published online: 19 September 2007© Springer Science+Business Media B.V. 2007

Abstract Conventional mathematical programming methods, such as linear programming,non linear programming, dynamic programming and integer programming have been usedto solve the cost optimization problem for regional wastewater treatment systems. In thisstudy, a river water quality management model was developed through the integration of agenetic algorithm (GA). This model was applied to a river system contaminated by threedetermined discharge sources to achieve the water quality goals and wastewater treatmentcost optimization in the river basin. The genetic algorithm solution, described the treatmentplant efficiency, such that the cost of wastewater treatment for the entire river basin is mini-mized while the water quality constraints in each reach are satisfied. This study showed thatgenetic algorithm can be applied for river water quality modeling studies as an alternative tothe present methods.

Keywords Self purification · Genetic algorithm · Dissolved oxygen · Treatment costoptimization · River pollution

1 Introduction

Over the last decades river basin management has become increasingly complex. Increas-ing demands of society regarding ecological and chemical quality of river reaches, use andprotection of water bodies and pollution with many different substances lead to new viewsand strategies towards policy making for river basin management. Although it is difficult

E. Aras · V. Togan · M. Berkun (B)Civil Engineering Department, Karadeniz Technical University, Trabzon 61080, Turkeye-mail: berkun@ktu.edu.tr

E. Arase-mail: egemen@ktu.edu.tr

V. Togane-mail: togan@ktu.edu.tr

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440 Environ Fluid Mech (2007) 7:439–450

to control non-point source pollution under the actual circumstances in a polluted river,mathematical optimization techniques can be utilized to develop optimal wastewater controlstrategies.

Water quality modeling in a river has developed from the pioneering work of [16], whodeveloped a balance between the dissolved oxygen supply rate from reaeration and the dis-solved oxygen consumption rate from stabilization of an organic waste in which the biochem-ical oxygen demand (BOD) deoxygenation rate was expressed as an empirical first-orderreaction (Eq. 1), producing the classic dissolved oxygen sag (DO) model (Eq. 2).

y = L0 [1 − exp(−k1t)] (1)

Dt = k1L0

k2 − k1[exp(−k1t) − exp(−k2t)] + D0exp(−k2t) (2)

where

y : BODDt : DO deficit at time tDo : DO deficit at time zerok1 : BOD reaction rate constantk2 : reaeration constantLo : ultimate BOD

t : time

Reliable determinations of the first-order oxygen uptake rate constant (k1), ultimate BOD(L0), and reaeration coefficient (k2) parameters in this equation are of importance. k1 can beobtained from BOD data using some mathematical techniques discussed by [2,3,6,12,17].k2 can be determined under field or laboratory conditions.

Water quality modeling is the development of abstractions of phenomena of river systems.The main objective of river water quality modeling is to describe and to predict the observedeffects of a change in the river system. The usual application of a water quality model isfor forecasting changes in water quality parameters resulting from changes in the quality,discharges or location of the point or non-point input sources [1,13].

In water quality management, the treatment cost may be as important as the achievementof water quality goals [4]. For this purpose, some optimization methods, such as linear pro-gramming [15], non-linear programming [8], dynamic programming [11], were used. Onthe other hand genetic algorithm were introduced to solve the cost optimization problemfor regional wastewater treatment [4,14]. Cho et al. [4] used various water quality parame-ters such as total nitrogen and total phosphorus in the optimization problem in addition tobiochemical oxygen demand (BOD) and dissolved oxygen (DO). Pelletier et al. [14] usedgenetic algorithm to find the combination of kinetic rate parameters and constants resultingbest fit for model application. Genetic algorithm was also used in many types of models by[10] and [20].

Gupta et al. [10] reported that the algorithms used for minimizing the cost through theapplication of mathematical techniques, such as linear, non-linear or dynamic programmingresult in a local optimum which is dependent on the starting point in the search process andshowed that genetic algorithm in general provided a lower cost solution. They also discussedthe advantages and disadvantages of genetic algorithm with the conventional optimizationmethods. For example, genetic algorithm deals with a population of solutions which are spreadover the solution space. It simultaneously climbs many peaks in parallel during the searchso that the probability of trapping into a local minimum is reduced considerably. Genetic

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Environ Fluid Mech (2007) 7:439–450 441

algorithm uses a rational fitness function to select the members of the next generation whilemathematical techniques rely on derivatives of the unconstrained objective function. Someresearchers also indicated these points for different modeling problems [4,5,7,20].

Revelle et al. [15] developed a water quality management model (WQMM) using linearprogramming. The aim of their study was to determine the degree of treatment (% BODremoval) that should be required to minimize the cost of treatment for the system whilemaintaining defined levels of water quality (dissolved oxygen). They expressed mathemat-ically a simplified problem based on linear programming formulation to show how a linearprogramming problem is structured and to illustrate graphically the characteristics of itssolution using BOD and Sag equations [16]. Then, an optimization problem having morerealistic objective function was discussed and developed by them.

Oxygen concentration is the prime indicator of water quality. In this study, the degree oftreatment (%BOD removal) required of each wastewater discharge source in a given riversystem to minimize the cost of treatment while maintaining defined levels of water qualitywas determined.

The main objective of this research is to investigate a WQMM using genetic algorithmconsidering the advantages of genetic algorithm stated above, in comparison with the linearprogramming proposed by [15] using the Sag equation.

2 Water quality model

The objective total cost function to be minimized for a river basin (Fig. 1) having three locatedtreatment plants on the river can be given as follows [15].

Cost = a1ε1 + a2ε2 + a3ε3 + (c1 + c2 + c3) (3)

where, εi = efficiency of treatment Plant i,ai and ci are slope of the linear portion and inter-cept of linear portion of the cost curve [15], respectively (i = 1 . . . 3). Since it is assumedthat each plant will be required to provide at least primary (35%) treatment, constraints onefficiency, εi , are

0.35 ≤ εi ≤ 0.90 i = 1 . . . 3 (4)

It is seen that the minimum cost occurs when all plants provide only 35% treatment. How-ever, this solution, while producing a minimum cost, may not necessarily meet the specificstream quality requirements [15] and the treatment constraints presented above.

The relation between plant efficiency and BOD discharge, an inventory equation whichis essentially a mass balance and an explicit restriction on water quality written in terms ofmaximum allowable oxygen deficit are the adopted three groups of constraints in the opti-mization problem by [15]. All linear programming formulation given by [15] is based on theoxygen-sag equation.

The mathematical expression of optimization problem [15] is as follows; Find the mini-mum value of objective function, (Eq. 3), under the constraints;

• Efficiency constraints

ε1 +(

1

P1

)M1 = 1; ε2 +

(1

P2

)M2 = 1; ε3 +

(1

P3

)M3 = 1 (5)

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442 Environ Fluid Mech (2007) 7:439–450

Q1

Q1

Q2

Q2

Q3

Q3

Sea or Lake

Stream flow = Q

Community 1

Community 2

Community 3

Reach 1 Reach 2 Reach 3

Fig. 1 River basin

• Inventory constraints1. At beginning of first reach:

On deficit Q D1 − (Q − Q1) E0 = T1 Q1 (6)

On BOD QL1 − (Q − Q1)F0 − Q1 M1 = 0 (7)

2. At beginning of second reach:On deficit E1 − αI I L1 − (e−r1xıı )D1 = 0 (8)

Q D2 − (Q − Q2) E1 = T2 Q2 (9)

On BOD F1 − (e−k1xI I )L1 = 0 (10)

QL2 − (Q − Q2)F1 − Q2 M2 = 0 (11)

3. At beginning of third reach:On deficit E2 − βI I L2 − (

e−r2 yıı)

D2 = 0 (12)

Q D3 − (Q − Q3) E2 = T3 Q3 (13)

On BOD F2 − (e−k2 yI I )L2 = 0 (14)

QL3 − (Q − Q3) F2 − Q3 M3 = 0 (15)

• Quality constraints1. in first reach

D1 ≤ DA; αI L1 + (e−r1xI

)D1 ≤ DA; αI I L1 + (

e−r1xI I)

D1 ≤ DA (16)

2. in second reach

D2 ≤ DA; βI L2 + (e−r2 yI

)D2 ≤ DA; βI I L2 + (

e−r2 yI I)

D2 ≤ DA (17)

3. in third reach

D3 ≤ DA; γI L3 + (e−r3zI

)D3 ≤ DA; γI I L3 + (

e−r3zI I)

D3 ≤ DA (18)

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Environ Fluid Mech (2007) 7:439–450 443

where E0=known oxygen deficit, in the stream just above the top of Reach 1, mg/l; Ei=def-icit at end of Reach i (i = 1, 2), mg/l; D j=deficit in the stream at the beginning of Reachj ( j = 1 . . . 3), mg/l; M j=BOD concentration released from the treatment plant at thebeginning of Reach j , mg/l; Tj=known deficit of wastewater flow, mg/l; L j=BOD concen-tration in the stream at the beginning of Reach j after mixing with the wastewater effluent,mg/l; F0=known BOD concentration in the stream just before the beginning of Reach 1,mg/l; Fi=BOD at end of Reach i , mg/l; Pj=concentration of BOD entering Plant j , mg/l;r j= reaeration coefficient in Reach j , days−1; k j=bio-oxidation rate constant in Reach j,days−1. αI,I I , βI,I I , and γI,I I are coefficients defined as follows;

αI = k1

r1 − k1(e−k1xI − e−r1xI ) αI I = k1

r1 − k1(e−k1xI I − e−r1xI I ) (19)

βI = k2

r2 − k2(e−k2 yI − e−r2 yI ) βI I = k2

r2 − k2(e−k2 yI I − e−r2 yI I ) (20)

γI = k3

r3 − k3(e−k3zI − e−r3zI ) γI I = k3

r3 − k3(e−k3zI I − e−r3zI I ) (21)

3 Genetic algorithms

Since 1960’s the researchers are interested in imitating living beings to develop powerfulalgorithms for difficult optimization problems. A term now is in common use to refer tosuch techniques is evolutionary computation. One of the types of evolutionary computationmethods is genetic algorithms (GA) which is a stochastic method inspired by the theorydefined as survival of the fittest briefly. In order to apply the genetic algorithm, a populationof solutions within a search space is initialized on the contrary of the traditional optimizationmethods that starts from a single point solution. The population can be viewed as points in thesearch space of all solutions to the optimization problem. Each individual in population hasa fitness value defined by a fitness function. Then the artificial evolution processes called thegenetic loop which mimic natural evolution are applied to produce new candidate solutions.At the end of the process, the newly created generation replaces previous generation andrevolution is repeated until a satisfying solution to the problem is obtained ensuring certaindesign criteria are satisfied or a maximum number of generations are reached [9,18,19].

In genetic algorithm, each possible solution of an optimization problem is represented bya string of genetic factors called chromosomes. A set of chromosomes make up a genera-tion. The generation evolves through the genetic operations called selection, crossover andmutation.

4 Optimization of a WQMM using GA

In the optimization study given by [15], the selection of εi as a design variable is appropriateand reasonable. Therefore, a chromosome represents the efficiency of treatment, εi , at eachplant and it consisted of the combined string of real values of the treatment level in the givenranges (Eq. 4). The fitness value of the chromosome is evaluated from the results of the waterquality and treatment cost. The fitness value is the sum of the total treatment cost, Eq. 3, andthe penalty for a chromosome, Eq. 22.

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444 Environ Fluid Mech (2007) 7:439–450

f(ε) = Eq(3)(1 + penalty) (22)

In genetic algorithm, the penalty is used when the constraints given in the optimizationproblem (Eq. 5–18), are violated. Thus the problem, constrained and defined by [15], istransformed into an unconstrained problem, which enables by genetic algorithm suitably. Thepenalty term given in Eq. 22 is computed in the following manner: If a particular model whichruns with the set of design variables εi violates Eq. (5–18), then the penalty term=10× totalviolation value of constraints; or if it does not violate the constraints, then the penalty term=0.Genetic algorithm is used to minimize Eq. 22.

Java Genetics Algorithms Package (JGAP), which is a free software, was used for theoptimization problem given by [15] and summarized above. JGAP is a genetic algorithmscomponent written in the form of a Java framework. It provides basic genetic mechanismsthat can be easily used to apply evolutionary principles to the solutions of the optimizationproblems.

JGAP has various kinds of genetic operators and code scheme for the chromosome. Asmentioned before, double or real code scheme is preferred for representing the treatmentlevel at each plant called design variables as follows.

Gene[] sampleGenes = new Gene[3];for (int i=0; i < sampleGenes.length; i++) {

sampleGenes[i] = new DoubleGene(0.35,0.90);}

where sampleGenes represent a chromosome composing of three genes which is the totalof the number of the design variables, DoubleGene(0.35,0.90) specify the lower and upperbound of design variables (genes). The initiating of the design variables are formed by JGAPrandomly. Since the double code scheme is used, there is no need to decode of the chro-mosome. Linear scaling is adopted for the proper selection of the individuals. In the JGAP,the crossover operator randomly selects two Chromosomes from the population and "mates"them by randomly picking a gene and then swapping that gene. Crossover operator supportsboth fixed and dynamic crossover rates. The mutation operator runs through the genes ineach of the chromosomes in the population and mutates them in statistical accordance to thegiven mutation rate. For this study the adopted crossover and mutation operator among thesupport of JGAP libraries are as follows.

conf.addGeneticOperator(new CrossoverOperator());conf.addGeneticOperator(new MutationOperator());

It is set the default crossover rate to be populationsize/2 and this rate describes the numberof pairings of parents in a particular generation. The mutation rate is automatically deter-mined by the mutation operator based upon the number of genes present in the chromosomes.Single point crossover is adopted in genetic algorithm process.

Data used to start the optimization process were given in Table 1 [15]. The results obtainedfrom the solution of WQMM using GA were given in Table 2. Table 2 also showed the resultsobtained by [15] using linear programming. In genetic algorithm process, the population sizeand maximum iteration number are adopted as 150 and 500 respectively. At least 20 runs areperformed and Table 2 presents the best result of 20 runs. In addition Table 2 and Fig. 2 pres-ent other five possible optimum values of the efficiency parameters and the related objectivefunction values obtained from across all 20 of the evolutionary runs. Since each evolutioncould be considered to give an acceptable solution, all results given in Fig. 2 are true andvalid. However the best one is drown attention among the optimum results obtained from over

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Environ Fluid Mech (2007) 7:439–450 445

Table 1 Data for the optimization problem

Parameter Value

Reach 1 Reach 2 Reach 3

Bio-oxidation constant (days−1) k1 = 0.30 k2 = 0.27 k3 = 0.25Reaeration constant (days−1) r1 = 0.40 r2 = 0.45 r3 = 0.65Half reach length (days) xI = 0.40 yI = 1.00 zI = 0.60Reach length (days) xI I = 0.80 yI I = 2.00 zI I = 1.20

Plant 1 Plant 2 Plant 3Cost function y1 = a1ε1 + c1 y2 = a2ε2 + c2 y3 = a3ε3 + c3Slope of cost curve ($) degree of efficiency a1 = 425000 a2 = 352000 a3 = 451000Intercept of cost curve ($) c1 = 347000 c2 = 425000 c3 = 28000Discharge flow (mgd) Q1 = 31.3 Q2 = 36.8 Q3 = 12.9BOD concentration entering plant (mg/l) P1 = 284 P2 = 408 P3 = 121Deficit of discharge (mg/l) T1 = 7.00 T2 = 7.00 T3 = 7.00

Stream flow Q = 400 mgdDeficit above first reach E0 = 0.50 mg/lBOD above first reach F0 = 1.00 mg/lAllowable deficit DA = 4.0 mg/lSaturation concentration of oxygen Cs = 8.5 mg/l

all runs. So Table 2 illustrated the best one in accordance with the literature. The possibleoptimum solution 6 in Fig. 2 represents the best solution.

5 Results

The results obtained from the linear programming and the genetic algorithm were given inTable 2. There is a small violation of the standard in the third reach (D3 is greater thanDA) as mentioned by [15]. However, there is no violation on the results obtained by geneticalgorithm. The total cost obtained from linear programming is cheaper than the total costobtained in this study with genetic algorithm. However, when linear programming showedthe violation on the graphical representation of the solution, it is not encountered violation onthe graph drown the concentration of dissolved oxygen versus time of flow in days accordingto the results obtained in this study (Figs. 3 and 4).

The result obtained in this study using genetic algorithm implies that genetic algorithmperforms more trials than the linear programming within the design space of WOMM andgenetic algorithm simultaneously can reach many peaks in parallel during the search so thatthe probability of trapping into a local minimum is reduced considerably.

Figure 5 shows the histories of the genetic processes of the optimization of WQMM forthe design variables and the value of the objective function respectively. JGAP finds the mini-mum value of objective function for the optimization problem among the candidate solutionswhich are created and tested by genetic algorithm, until the algorithm reaches the maximumiteration number adopted as 500 in the design. Due to this, although JGAP finds the optimumdesign in early generation, it continues the optimization process to the maximum iteration(Fig. 5b). So, there is a difference between the classical convergence plot in genetic algorithmand the convergence plot in JGAP.

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446 Environ Fluid Mech (2007) 7:439–450

Tabl

e2

Res

ults

ofth

eop

timiz

atio

npr

oble

m

Lin

ear

prog

ram

min

gaG

enet

icA

lgor

ithm

Rea

ch1

Rea

ch2

Rea

ch3

Rea

ch1

Rea

ch2

Rea

ch3

BO

Dat

begi

n.of

reac

h(m

g/l)

L1

=11

.31

L2

=11

.83

L3

=11

.27

L1

=9.

28L

2=

10.3

8L

3=

8.39

BO

Dat

end

ofre

ach

(mg/

l)F

1=

8.90

F2

=9.

03–

F1

=7.

298

F2

=6.

049

–

Defi

cita

tbeg

in.o

fre

ach

(mg/

l)D

1=

1.01

D2

=3.

17D

3=

4.50

D1

=1.

01D

2=

2.83

7D

3=

3.99

68

Defi

cita

tmid

dle

ofre

ach

(mg/

l)D

(x1)=

2.04

D(y

1)=

4.25

D(z

1)=

4.34

D(x

1)=

1.82

7D

(y1)=

3.76

7D

(z1)=

3.66

9

Defi

cita

tend

ofre

ach

(mg/

l)E

1=

2.78

E2

=4.

42–

E1

=2.

40E

2=

3.89

–

Plan

t1Pl

ant2

Plan

t3Pl

ant1

Plan

t2Pl

ant3

Effi

cien

yε

1=

0.53

ε2

=0.

90ε

3=

0.35

ε1

=0.

624

ε2

=0.

90ε

3=

0.35

0B

OD

conc

entr.

Dis

char

.(m

g/l)

M1

=13

3.0

M2

=40

.7M

3=

78.7

M1

=10

6.78

4M

2=

40.7

99M

3=

78.6

5

Tota

lcos

t($)

1501

000

1539

850

a Rev

elle

etal

.[15

]

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Environ Fluid Mech (2007) 7:439–450 447

Fig. 2 Possible optimumsolutions for WQMM. (a)Efficiency values for possiblesolutions; (b) BOD concentrationdischarged values for possiblesolutions; (c) The objectivefunction values for possiblesolutions

0

0.5

1

Possible optimum solutions

seulav yneiciffE

1 2 3

1 0.683

2 0.868

3 0.359

1 2 3 4 5 6

a

020406080

100120

Possible optimum solutions

)l/gm( .grahcsid .rtnecnoc

DO

B

M1 M2 M3

M1 90.028 98.548 88.04 105.93 106.5 106.784

M2 53.856 45.288 56.304 41.21 40.799 40.799

M3 77.561 77.924 77.44 78.04 78.65 78.65

1 2 3 4 5 6

b

1530000

1535000

1540000

1545000

1550000

1555000

1560000

1565000

Possible optimum solutions

)$( tsoC

Total Cost ($)

Total Cost ($) 1557720 1551009 1559034 1543028 1540275 1539850

1 2 3 4 5 6

c

0.624

0.9

0.35

0.9

0.350.3550.360.356

0.8990.8620.889

0.6250.6270.690.653

6 Conclusions

A water quality model based on the cost optimization of a described river water qualitycontrol system using genetic algorithm gave comparable results to the linear programmingbased on the Sag equation.

Genetic algorithm solution described the treatment plant efficiency to be provided by eachof the three communities such that the cost of wastewater treatment for the entire river basinis minimized while the water quality constraints in each reach are satisfied. This study alsoshowed that genetic algorithm provides a convenient technique in performing more trials

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448 Environ Fluid Mech (2007) 7:439–450

Fig. 3 Dissolved oxygen profile of river basin in the optimization problem. (a) Obtained by Revelle et al.[15]; (b) Obtained by GA

Saturation concentration of oxygen

Maximum allowable deficit

Increasing deficit

DO Standart

VIOLATION SAG Curve

TIME

DO

Fig. 4 Meaning of violation and allowable deficit

in comparison with linear programming used by [15] in order to obtain effective design ofWQMM. Moreover genetic algorithm is capable of climbing many peaks in parallel duringthe evolutionary search so that it gives many acceptable solutions. In contrast to the mathe-matical techniques which rely on derivatives of the unconstrained objective function, geneticalgorithm uses a more rational fitness function to select the members of the next generation.The evolutionary process of genetic algorithm doesn’t need any derivation information aboutto the optimization problem, which sometimes needs considerable computation effort. Ge-netic algorithm can be used as an alternative method and JGAP is an effective tool for theriver water quality modeling studies.

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Environ Fluid Mech (2007) 7:439–450 449

0.20

0.40

0.60

0.80

1.00

1 31 61 91 121 151 181 211 241 271 301 331 361 391 421 451 481

Generations

ycneiciffE

1 2 3a

1200000

1300000

1400000

1500000

1600000

1700000

1800000

1 31 61 91 121 151 181 211 241 271 301 331 361 391 421 451 481

Generations

tsoC

Total cost ($)

b

Fig. 5 Histories of genetic process. (a) Variation of design variables values; (b) Variation of objective functionvalue

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