River water quality management model using genetic ... - Springer Link

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

River water quality management model using genetic algorithm Egemen Aras · Vedat To˘gan · 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 used to solve the cost optimization problem for regional wastewater treatment systems. In this study, a river water quality management model was developed through the integration of a genetic algorithm (GA). This model was applied to a river system contaminated by three determined discharge sources to achieve the water quality goals and wastewater treatment cost optimization in the river basin. The genetic algorithm solution, described the treatment plant efficiency, such that the cost of wastewater treatment for the entire river basin is minimized while the water quality constraints in each reach are satisfied. This study showed that genetic algorithm can be applied for river water quality modeling studies as an alternative to the present methods. Keywords Self purification · Genetic algorithm · Dissolved oxygen · Treatment cost optimization · River pollution

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

E. Aras · V. To˘gan · M. Berkun (B) Civil Engineering Department, Karadeniz Technical University, Trabzon 61080, Turkey e-mail: [email protected] E. Aras e-mail: [email protected] V. To˘gan e-mail: [email protected]

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to control non-point source pollution under the actual circumstances in a polluted river, mathematical optimization techniques can be utilized to develop optimal wastewater control strategies. Water quality modeling in a river has developed from the pioneering work of [16], who developed a balance between the dissolved oxygen supply rate from reaeration and the dissolved oxygen consumption rate from stabilization of an organic waste in which the biochemical oxygen demand (BOD) deoxygenation rate was expressed as an empirical first-order reaction (Eq. 1), producing the classic dissolved oxygen sag (DO) model (Eq. 2). y = L 0 [1 − ex p(−k1 t)] Dt =

k1 L 0 [ex p(−k1 t) − ex p(−k2 t)] + D0 ex p(−k2 t) k2 − k1

(1) (2)

where y Dt Do k1 k2 Lo t

: BOD : DO deficit at time t : DO deficit at time zero : BOD reaction rate constant : reaeration constant : ultimate BOD : time

Reliable determinations of the first-order oxygen uptake rate constant (k1 ), ultimate BOD (L 0 ), and reaeration coefficient (k2 ) parameters in this equation are of importance. k1 can be obtained 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 observed effects of a change in the river system. The usual application of a water quality model is for 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 achievement of water quality goals [4]. For this purpose, some optimization methods, such as linear programming [15], non-linear programming [8], dynamic programming [11], were used. On the other hand genetic algorithm were introduced to solve the cost optimization problem for regional wastewater treatment [4,14]. Cho et al. [4] used various water quality parameters such as total nitrogen and total phosphorus in the optimization problem in addition to biochemical oxygen demand (BOD) and dissolved oxygen (DO). Pelletier et al. [14] used genetic algorithm to find the combination of kinetic rate parameters and constants resulting best 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 the application of mathematical techniques, such as linear, non-linear or dynamic programming result in a local optimum which is dependent on the starting point in the search process and showed that genetic algorithm in general provided a lower cost solution. They also discussed the advantages and disadvantages of genetic algorithm with the conventional optimization methods. For example, genetic algorithm deals with a population of solutions which are spread over the solution space. It simultaneously climbs many peaks in parallel during the search so that the probability of trapping into a local minimum is reduced considerably. Genetic

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algorithm uses a rational fitness function to select the members of the next generation while mathematical techniques rely on derivatives of the unconstrained objective function. Some researchers also indicated these points for different modeling problems [4,5,7,20]. Revelle et al. [15] developed a water quality management model (WQMM) using linear programming. The aim of their study was to determine the degree of treatment (% BOD removal) that should be required to minimize the cost of treatment for the system while maintaining defined levels of water quality (dissolved oxygen). They expressed mathematically a simplified problem based on linear programming formulation to show how a linear programming problem is structured and to illustrate graphically the characteristics of its solution using BOD and Sag equations [16]. Then, an optimization problem having more realistic objective function was discussed and developed by them. Oxygen concentration is the prime indicator of water quality. In this study, the degree of treatment (%BOD removal) required of each wastewater discharge source in a given river system to minimize the cost of treatment while maintaining defined levels of water quality was determined. The main objective of this research is to investigate a WQMM using genetic algorithm considering the advantages of genetic algorithm stated above, in comparison with the linear programming 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 located treatment plants on the river can be given as follows [15]. Cost = a1 ε1 + a2 ε2 + a3 ε3 + (c1 + c2 + c3 )

(3)

where, εi = efficiency of treatment Plant i,ai and ci are slope of the linear portion and intercept of linear portion of the cost curve [15], respectively (i = 1 . . . 3). Since it is assumed that each plant will be required to provide at least primary (35%) treatment, constraints on efficiency, 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. However, this solution, while producing a minimum cost, may not necessarily meet the specific stream quality requirements [15] and the treatment constraints presented above. The relation between plant efficiency and BOD discharge, an inventory equation which is essentially a mass balance and an explicit restriction on water quality written in terms of maximum allowable oxygen deficit are the adopted three groups of constraints in the optimization problem by [15]. All linear programming formulation given by [15] is based on the oxygen-sag equation. The mathematical expression of optimization problem [15] is as follows; Find the minimum value of objective function, (Eq. 3), under the constraints; • Efficiency constraints 1 1 1 ε1 + M1 = 1; ε2 + M2 = 1; ε3 + M3 = 1 P1 P2 P3

(5)

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Environ Fluid Mech (2007) 7:439–450 Reach 1

Reach 2

Q2 Q2

Reach 3

Community 3 Q3

Community 1

Q3

Community 2

Q1 Q1

Stream flow = Q

Sea or Lake

Fig. 1 River basin

• Inventory constraints 1. At beginning of first reach: On deficit Q D1 − (Q − Q 1 ) E 0 = T1 Q 1

(6)

Q L 1 − (Q − Q 1 )F0 − Q 1 M1 = 0

(7)

2. At beginning of second reach: On deficit E 1 − α I I L 1 − (e−r1 xıı )D1 = 0

(8)

On BOD

Q D2 − (Q − Q 2 ) E 1 = T2 Q 2 On BOD

F1 − (e−k1 x I I )L 1 = 0

(10)

Q L 2 − (Q − Q 2 )F1 − Q 2 M2 = 0

(11)

3. At beginning of third reach: On deficit E 2 − β I I L 2 − e−r2 yıı D2 = 0 Q D3 − (Q − Q 3 ) E 2 = T3 Q 3 On BOD

(9)

(12) (13)

F2 − (e−k2 y I I )L 2 = 0

(14)

Q L 3 − (Q − Q 3 ) F2 − Q 3 M3 = 0

(15)

• Quality constraints 1. in first reach D1 ≤ D A ; α I L 1 + e−r1 x I D1 ≤ D A ; α I I L 1 + e−r1 x I I D1 ≤ D A (16) 2. in second reach D2 ≤ D A ; β I L 2 + e−r2 y I D2 ≤ D A ; β I I L 2 + e−r2 y I I D2 ≤ D A (17) 3. in third reach D3 ≤ D A ; γ I L 3 + e−r3 z I D3 ≤ D A ; γ I I L 3 + e−r3 z I I D3 ≤ D A

123

(18)


443

where E 0 = known oxygen deficit, in the stream just above the top of Reach 1, mg/l; E i = deficit at end of Reach i (i = 1, 2), mg/l; D j = deficit in the stream at the beginning of Reach j ( j = 1 . . . 3), mg/l; M j = BOD concentration released from the treatment plant at the beginning of Reach j, mg/l; T j = known deficit of wastewater flow, mg/l; L j = BOD concentration 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; P j = 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 . α I,I I , β I,I I , and γ I,I I are coefficients defined as follows; αI =

k1 k1 (e−k1 x I − e−r1 x I ) α I I = (e−k1 x I I − e−r1 x I I ) r1 − k1 r1 − k1

(19)

βI =

k2 k2 (e−k2 y I − e−r2 y I ) β I I = (e−k2 y I I − e−r2 y I I ) r2 − k2 r2 − k2

(20)

γI =

k3 k3 (e−k3 z I − e−r3 z I ) γ I I = (e−k3 z I I − e−r3 z I I ) r3 − k3 r3 − k3

(21)

3 Genetic algorithms Since 1960’s the researchers are interested in imitating living beings to develop powerful algorithms for difficult optimization problems. A term now is in common use to refer to such techniques is evolutionary computation. One of the types of evolutionary computation methods is genetic algorithms (GA) which is a stochastic method inspired by the theory defined as survival of the fittest briefly. In order to apply the genetic algorithm, a population of solutions within a search space is initialized on the contrary of the traditional optimization methods that starts from a single point solution. The population can be viewed as points in the search space of all solutions to the optimization problem. Each individual in population has a fitness value defined by a fitness function. Then the artificial evolution processes called the genetic 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 and revolution is repeated until a satisfying solution to the problem is obtained ensuring certain design 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 by a string of genetic factors called chromosomes. A set of chromosomes make up a generation. The generation evolves through the genetic operations called selection, crossover and mutation.

4 Optimization of a WQMM using GA In the optimization study given by [15], the selection of εi as a design variable is appropriate and reasonable. Therefore, a chromosome represents the efficiency of treatment, εi , at each plant and it consisted of the combined string of real values of the treatment level in the given ranges (Eq. 4). The fitness value of the chromosome is evaluated from the results of the water quality and treatment cost. The fitness value is the sum of the total treatment cost, Eq. 3, and the penalty for a chromosome, Eq. 22.

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f(ε) = Eq(3)(1 + penalty)

(22)

In genetic algorithm, the penalty is used when the constraints given in the optimization problem (Eq. 5–18), are violated. Thus the problem, constrained and defined by [15], is transformed into an unconstrained problem, which enables by genetic algorithm suitably. The penalty term given in Eq. 22 is computed in the following manner: If a particular model which runs with the set of design variables εi violates Eq. (5–18), then the penalty term=10 × total violation 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 the optimization problem given by [15] and summarized above. JGAP is a genetic algorithms component written in the form of a Java framework. It provides basic genetic mechanisms that can be easily used to apply evolutionary principles to the solutions of the optimization problems. JGAP has various kinds of genetic operators and code scheme for the chromosome. As mentioned before, double or real code scheme is preferred for representing the treatment level 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 total of the number of the design variables, DoubleGene(0.35,0.90) specify the lower and upper bound of design variables (genes). The initiating of the design variables are formed by JGAP randomly. Since the double code scheme is used, there is no need to decode of the chromosome. 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 supports both fixed and dynamic crossover rates. The mutation operator runs through the genes in each of the chromosomes in the population and mutates them in statistical accordance to the given mutation rate. For this study the adopted crossover and mutation operator among the support 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 number of pairings of parents in a particular generation. The mutation rate is automatically determined 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 obtained from the solution of WQMM using GA were given in Table 2. Table 2 also showed the results obtained by [15] using linear programming. In genetic algorithm process, the population size and maximum iteration number are adopted as 150 and 500 respectively. At least 20 runs are performed and Table 2 presents the best result of 20 runs. In addition Table 2 and Fig. 2 present other five possible optimum values of the efficiency parameters and the related objective function values obtained from across all 20 of the evolutionary runs. Since each evolution could be considered to give an acceptable solution, all results given in Fig. 2 are true and valid. However the best one is drown attention among the optimum results obtained from over

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Table 1 Data for the optimization problem Parameter

Value Reach 1

Reach 2

Reach 3

Bio-oxidation constant (days−1 ) Reaeration constant (days−1 ) Half reach length (days) Reach length (days)

k1 = 0.30 r1 = 0.40 x I = 0.40 x I I = 0.80

k2 = 0.27 r2 = 0.45 y I = 1.00 y I I = 2.00

k3 = 0.25 r3 = 0.65 z I = 0.60 z I I = 1.20

Cost function Slope of cost curve ($) degree of efficiency Intercept of cost curve ($) Discharge flow (mgd) BOD concentration entering plant (mg/l) Deficit of discharge (mg/l)

Plant 1 y1 = a1 ε1 + c1 a1 = 425000 c1 = 347000 Q 1 = 31.3 P1 = 284 T1 = 7.00

Plant 2 y2 = a2 ε2 + c2 a2 = 352000 c2 = 425000 Q 2 = 36.8 P2 = 408 T2 = 7.00

Plant 3 y3 = a3 ε3 + c3 a3 = 451000 c3 = 28000 Q 3 = 12.9 P3 = 121 T3 = 7.00

Stream flow Deficit above first reach BOD above first reach Allowable deficit Saturation concentration of oxygen

Q = 400 mgd E 0 = 0.50 mg/l F0 = 1.00 mg/l D A = 4.0 mg/l Cs = 8.5 mg/l

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

5 Results The results obtained from the linear programming and the genetic algorithm were given in Table 2. There is a small violation of the standard in the third reach (D3 is greater than DA ) as mentioned by [15]. However, there is no violation on the results obtained by genetic algorithm. The total cost obtained from linear programming is cheaper than the total cost obtained in this study with genetic algorithm. However, when linear programming showed the violation on the graphical representation of the solution, it is not encountered violation on the graph drown the concentration of dissolved oxygen versus time of flow in days according to the results obtained in this study (Figs. 3 and 4). The result obtained in this study using genetic algorithm implies that genetic algorithm performs more trials than the linear programming within the design space of WOMM and genetic algorithm simultaneously can reach many peaks in parallel during the search so that the probability of trapping into a local minimum is reduced considerably. Figure 5 shows the histories of the genetic processes of the optimization of WQMM for the design variables and the value of the objective function respectively. JGAP finds the minimum value of objective function for the optimization problem among the candidate solutions which are created and tested by genetic algorithm, until the algorithm reaches the maximum iteration number adopted as 500 in the design. Due to this, although JGAP finds the optimum design 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 algorithm and the convergence plot in JGAP.

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a Revelle et al. [15]

Efficieny BOD concentr. Dischar. (mg/l) Total cost ($)

BOD at begin. of reach (mg/l) BOD at end of reach (mg/l) Deficit at begin. of reach (mg/l) Deficit at middle of reach (mg/l) Deficit at end of reach (mg/l)

D2 = 3.17

D(y1 ) = 4.25

E 2 = 4.42

Plant 2

ε2 = 0.90 M2 = 40.7

D1 = 1.01

D(x1 ) = 2.04

E 1 = 2.78

Plant 1

ε1 = 0.53 M1 = 133.0

1501000

F2 = 9.03

F1 = 8.90

ε3 = 0.35 M3 = 78.7

Plant 3

–

D(z 1 ) = 4.34

D3 = 4.50

–

1539850

ε1 = 0.624 M1 = 106.784

Plant 1

E 1 = 2.40

D(x1 ) = 1.827

D1 = 1.01

F1 = 7.298

L 1 = 9.28

L 2 = 11.83

L 1 = 11.31

L 3 = 11.27

Reach 1

Reach 2

Reach 1

Reach 3

Genetic Algorithm

Linear programminga

Table 2 Results of the optimization problem

ε2 = 0.90 M2 = 40.799

Plant 2

E 2 = 3.89

D(y1 ) = 3.767

D2 = 2.837

F2 = 6.049

L 2 = 10.38

Reach 2

ε3 = 0.350 M3 = 78.65

Plant 3

–

D(z 1 ) = 3.669

D3 = 3.9968

–

L 3 = 8.39

Reach 3

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

Environ Fluid Mech (2007) 7:439–450 1 Efficieny values

Fig. 2 Possible optimum solutions for WQMM. (a) Efficiency values for possible solutions; (b) BOD concentration discharged values for possible solutions; (c) The objective function values for possible solutions

447 2

3

a1 0.5 0 1

2

3

4

1

0.683

0.653

0.69

0.627

5

6

2

0.868

0.889

0.862

0.899

0.9

0.9

3

0.359

0.356

0.36

0.355

0.35

0.35

0.625

0.624

B O D co n c e n t r . d i s c h a r g . ( m g / l )

Possible optimum solutions

b

M1

120 100 80 60 40 20 0

M2

M3

1

2

3

4

5

6

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


c

1565000 Total Cost ($)

1560000 Co st ($ )

1555000 1550000 1545000 1540000 1535000 1530000

1

Total Cost ($) 1557720

2

3

4

5

6

1551009

1559034

1543028

1540275

1539850


6 Conclusions A water quality model based on the cost optimization of a described river water quality control system using genetic algorithm gave comparable results to the linear programming based on the Sag equation. Genetic algorithm solution described the treatment plant efficiency to be provided by each of the three communities such that the cost of wastewater treatment for the entire river basin is minimized while the water quality constraints in each reach are satisfied. This study also showed that genetic algorithm provides a convenient technique in performing more trials

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

DO

Saturation concentration of oxygen Increasing deficit

Maximum allowable deficit VIOLATION

SAG Curve

DO Standart

TIME

Fig. 4 Meaning of violation and allowable deficit

in comparison with linear programming used by [15] in order to obtain effective design of WQMM. Moreover genetic algorithm is capable of climbing many peaks in parallel during the evolutionary search so that it gives many acceptable solutions. In contrast to the mathematical techniques which rely on derivatives of the unconstrained objective function, genetic algorithm 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 about to the optimization problem, which sometimes needs considerable computation effort. Genetic algorithm can be used as an alternative method and JGAP is an effective tool for the river water quality modeling studies.

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449 1

a

2

3

1.00

Efficiency

0.80 0.60 0.40 0.20 1

31

61

91 121 151 181 211 241 271 301 331 361 391 421 451 481 Generations Total cost ($)

b

1800000 1700000

Co s t

1600000 1500000 1400000 1300000 1200000 1

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

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

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