parameters estimation of river water quality model...

© by PSP Volume 27 – No. 2/2018 pages 749-756 Fresenius Environmental Bulletin

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PARAMETERS ESTIMATION OF RIVER WATER QUALITY

MODEL USING A DIFFERENTIAL EVOLUTION ALGORITHM

Hongqin Xue1, Xiaodong Liu2,*, Yong Xiong2, Li Gu2, Ting Yang2

1School of Civil Engineering, Nanjing Forestry University, Nanjing 210037, P.R. China

2Key Laboratory of Integrated Regulation and Resource Development on Shallow Lake of Ministry of Education, College of Environment, Hohai University, Nanjing 210098, P.R. China

ABSTRACT

To avoid the shortcomings of traditional opti-mization algorithms, a new multi-parameters estima-tion model based on Differential Evolution(DE) al-gorithm coupled with water quality model was pro-posed in this paper. The computational results of three numerical cases indicate the proposed parame-ter estimation model has high accuracy and good anti-noise properties. It can give precise results un-der both steady flow and unsteady flow. It can be used to identify parameters for both 1D and 2D water quality model, including analytical solution model and numerical model. Comparison results show that the method based on DE has roughly same accuracy as that on GA, but has higher convergence speed and better anti-noise property. This work contributes to the parameter estimation for river systems. KEYWORDS: Differential evolution (DE), Parameter estimation, River, Water quality model INTRODUCTION

River system sustains a complex interaction and exchange of mass and energy within its biotic and abiotic components. It offers resilience to change and also recovers itself from the minor changes imposed on it from the polluting surround-ings. Growing population, urbanization and industri-alization have magnified the amount of total wastewater generated which threatens the health of river ecosystems when discharged into it. Rivers are increasingly getting polluted due to large scale with-drawal of fresh water and increased discharge of wastewater. Many nations have already initiated re-source intensive river quality restoration projects [1]. Knowledge of dispersion of pollutants in streams is necessary for the determination of both the accepta-ble limits of effluent input and the concentration along the river course in these projects. Planning of these projects requires application of river water

quality models. For many decades, river water qual-ity models have been developed worldwide.

When the pollutant enters streams intentionally or unintentionally, it disperses in every direction due to diffusion and spatial velocity distribution. First the pollutant mixes completely in the vertical direc-tion and then in the transverse direction. Most rivers have a high width/depth ratio and pollutants become mixed vertically within a short distance from the source. Vertical mixing is only important in the so-called near field and is often neglected when consid-ering subsequent transverse and longitudinal mixing. When dealing with the practical engineering prob-lems, it is not computationally efficient to use three-dimensional (3D) models. Instead, researchers ap-plied two-dimensional (2D) water quality models to simulate the transverse mixing in channels. After the transverse mixing has taken place, the primary vari-ation of concentration is in one direction; dispersion from that section onwards is referred to as longitudi-nal dispersion and is independent of the geometrical configuration and type of source. In natural channels, 1D mathematical models are commonly adopted to model the longitudinal dispersion process. Which-ever model is chosen, the selection of proper param-eters is the most important and also the most difficult task. The dispersion coefficient represents the rate of pollution and it is the most desired parameter in any water pollution modeling study. In many practical and natural situations, one or two dimensional dis-persion coefficients are often required for modeling. Besides dispersion coefficients, some other parame-ters, i.e. degradation coefficient, also must be ob-tained. It is a relatively simple task to use a measured dispersion coefficient, if it is known. However, for streams where mixing and dispersion characteristics are unknown, the water quality coefficients can only be estimated using theoretical equations, empirical equations or inversion method.

A theoretical method to predict the longitudinal dispersion coefficient was first proposed by Taylor [2] and expanded by Elder [3], who derived an equa-tion to compute the longitudinal coefficient for a uni-form flow in an infinitely wide open channel, assum-ing a logarithmic velocity profile. Fischer et al. [4] found that the transverse profile of the longitudinal


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velocity is 100 or more times as important in produc-ing longitudinal dispersion as the vertical profile in the natural streams. Fischer [5] and Fischer et al. [4] gave the following quantitative estimate of longitu-dinal dispersion coefficient in a real stream by ne-glecting the vertical velocity profile entirely and ap-plying Taylor’s analysis to the transverse velocity profile.

0 0 0

1 1E ( ) ( )( )

y yW

xt

U h y U h y dydydyA h ye

¢ ¢= - ò ò ò (1)

where A is flow area; y is distance in transverse direction; W is width of channel; U ¢ is deviation of local depth-averaged velocity from the cross-sec-tional averaged velocity U; and te is local transverse mixing coefficient. Equation (1) is rather difficult to use because detailed transverse profiles of both ve-locity and cross-sectional geometry are required. Some researchers developed different equations us-ing different velocity profiles for the Equation (1) [6-10]. Thus, the theoretical method is not encouraging in terms of data requirement. A number of investiga-tors have developed empirical equations for predict-ing the dispersion coefficient based on experimental and field data [11-14]. The longitudinal dispersion coefficient can be estimated by using flow and chan-nel parameters. But since most studies have been conducted based on specific assumptions and chan-nel conditions, the behavior of existing equations varies widely even for the same flow conditions and streams. Comparatively, the parameter inversion method, which is based on concentration profile, is thought to be more accurate and reliable. The method would be called tracer test method if the con-centration data is obtained by using tracer test. Many tracer test methods have been published. For exam-ple, the longitudinal dispersion coefficient can be ob-tained by using curves of the temporal variation of concentration (C–t curves) at two or more stations downstream of the tracer injection point. The mo-ment method [4, 15], routing procedure [5], fitting method [16], and the optimization method [17] are available for the determination of longitudinal dis-persion coefficient using C–t curves and other prop-erties of the stream. Besides tracer test method, sometimes, the concentration data is from field water quality measurement without expensive tracer test. For example, trial-and-error is the most common method used in parameters identification. With this approach, an initial group of estimated values for the parameters is assumed and inserted into the numeri-cal model, and then the simulated results are com-pared with field data. If the deviation is significant, the parameters are repeatedly adjusted until the de-viation is acceptable. It is not feasible if the studied river has a complicated topography. To address this, researchers have proposed a number of auto-calibra-tion models [18] by using optimization method. Its

optimizing process largely avoids the calculation er-rors caused by field measurements. Traditional opti-mization algorithms sometimes cannot get satisfac-tory coefficients with observation data with noise for their limitations, such as high demand for objective function, local convergence, and the convergence speed decreasing by geometric progression with the increase of the number of parameters. This limits their application in multi-parameter identification for complicated models. To address this, artificial in-telligence techniques, such as artificial neural net-work (ANN), adaptive neuro-fuzzy inference sys-tem, genetic algorithm (GA) and Bayesian net-works(BA) have shown promising performance in predicting longitudinal dispersion [19-22].

Recently, as another evolutionary technique, differential evolution (DE), has been proposed for unconstrained continuous optimization problems in 1995 [23]. Although the original objective in the de-velopment of DE was for solving the Chebychev pol-ynomial problem, it has been found to be an efficient and effective solution technique for complex func-tional optimization problems [24]. In 1996, DE algo-rithm has been proved to be the optimization algo-rithm with the highest convergence speed during the First International Contest on Evolutionary Compu-tation. Due to its simple concept, easy implementa-tion and quick convergence, the algorithm has at-tracted much attention and wide applications [25]. However, until now, there have been few published work on DE for parameters estimation of water qual-ity models.

In this present study, a DE approach is applied to estimate the parameters of river water quality models. Numerical cases and comparisons demon-strate the effectiveness and robustness of DE.

MATERIALS AND METHODS Parameter Estimation Method Based on DE.

The problem of parameter estimation could be con-verted to an extremum problem by defining a object function J . The agreement between the measured results and simulated values is one of the most im-portant indicators as to how well a model is cali-brated [26]. The object function J , which is Ex-pressed with the norm, is defined as follows:

2*

0

1( , )2

TJ C X C C dt= -ò (2)

where C is calculated value vector, C* is ob-served value vector, X is unknown parameter vector. In parameter identification, model parameters are adjusted to achieve minimize J for all variables sim-ultaneously. The extremum problem can be de-scribed mathematically as follows:

Minimize 1 2( ) ( , , ..., )nJ f X f x x x= = (3) Subject to 1, 2, ...,ilow i iupx x x i n£ £ = (4)


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where, 1 2( , , ..., )nX x x x= is the parameter vector,

n is the number of parameters to be identified, ilowxand iupx represent the lower bound and upper bound

of the parameter ix . The J is a non-linear function, and its minimization problem can be solved by mul-tidimensional non-linear optimization methods. DE algorithm coupled with water quality model was used here for its high efficiency and robustness.

DE is a population-based evolutionary compu-tation technique, which uses simple differential op-erator to create new candidate solutions and one-to-one competition scheme to greedily select new can-didate. The general steps of the estimation method are as follows:

(1) Parameters set for the DE: the parameters include the number of parameters (n), population size (NP), scaling factor (F), the probability of cross-over operator (CR), the maximal number of genera-tions (Tm). NP is at least 4 so that the mutation can be applied.

(2) Initialization: the initial population is gen-erated from specified ranges of parameters by ran-dom methods, which is usually preferred to keep a suitable balanced distribution in the initial popula-

tion. 0 ( )i ilow iup ilowx x rand x x= + ´ - , where

rand is a random number between 0 and 1. (3) Selection of water quality model: Accord-

ing to the river flow condition and pollutant emission mode, suited water quality model is necessary to be established correctly.

(4) Calculation of fitness: since the DE can only make the individual evolution in the direction where the fitness increases, so the fitness function can be designed as follows:

2* 1

0

1( ) 1/( ( )) ( )2

TFit X J X C C dt -= = -ò (5)

Where, C is calculated by using water quality model, *C is obtained by measurement. Calculate

the fitness of the current X. Check mT T> , where T is the number of generation. If so, output the current value of X as the final parameter estimation results. Otherwise continue.

(5) Mutation. The purpose of mutation is to broaden the search scope. After a current individual

jX was selected, the new individual jY is gener-ated as follows.

1 2 3( ) ( ( ) ( )) 1,2,...,j r r rY X t F X t X t j NP= + - = (6)

Where 1r , 2r , 3r are randomly chosen and 1 2 3r r r i¹ ¹ ¹ . [0,2]FÎ is a constant called

scaling factor which controls amplification of the differential variation 2 3( ) ( )r rX t X t- .

(6) Crossover. The crossover operator is ap-plied to increase the diversity of the population.

Thus, for each target individual

1 2( ) ( , ,..., )j j j jnX t x x x= , a crossover vector

1 2( , ,..., )j j j jnZ z z z= is generated by the follow-ing equation:

( )),

ji iji

ji

y rand CR j k iz

x else£ Ú =ì

= íî

，if(

(7)

Where irand is the ith independent random number uniformly distributed in the range of [0,1]. ( )k j is a randomly chosen index from the set

{ }1,2,...,n . [0,1]CRÎ is a constant called proba-

bility of crossover operator that controls the diversity of the population.

(7) Selection. An individual of the given popu-lation is selected according to its fitness to form the next generation by selection operator. After the crossover operation, the member of the population of the next generation is generated by the following one-to-one based greedy selection criterion:

, ( ) ( ( ))( 1)

( ),j j j

jj

Z Fit Z Fit X tX t

X t otherwise³ì

+ = íî

i f （ (8)

where ( 1)jX t + is the individual of the new popu-lation. After this go back to step 4.

The flow chart of the parameter identification model can be seen in Fig.1.

RESULT AND DISSCUSION

Case 1. Test data are taken from the literature [20, 27]. The case was a numerical test not an actual test. The river cross-section area A was 20 m2, and the average velocity u was 0.5 m·s-1. The tracer (10 kg) was released in the upper reaches of the river. Assuming that the longitudinal dispersion coeffi-cient Ex was 50 m2·s-1 and degradation coefficient K was 0.3 d-1 in the case Tracer concentration (to three decimal places) at different times in the downstream 500 m was calculated by 1D water quality model, and are shown in Table 1. Treating the data in Table 1 as observational data and A, u, Ex and K as un-known parameters to be identified, this case can be used to verify the reliability of parameter estimation method.

Observation Data with Noise. No matter how

high accuracy measuring instruments are used, ob-servation noise can not be avoided in measurement. Since the noise is usually random, some random noise was added to the values in Table 1. The obser-vational data with noise can be expressed as:

*(1 * )j jC Cd b d= + . Where *jC is true concentra-

tion, b is random number between -1 and 1, d is noise (disturbance) level.


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Parameters set

Initialization

Water quality model

If T>Tm

Mutation

crossover

selection

G=G+1

Calculation of fitness

Output

Observed value

Y

N

FIGURE 1

Flow chart diagram for the parameter estimation model

TABLE 1 Tracer test data in instantaneous source

tj/min 2 6 10 12 14 cj/(mg·L-1) 0.001 0.254 0.583 0.649 0.663

tj/min 16 20 24 36 40 cj/(mg·L-1) 0.642 0.552 0.444 0.197 0.147

TABLE 2

Results under different noise level

Parameter method GA DE

noise level mean value

standard deviation

relative standard deviation

mean value

standard deviation

relative standard deviation

A

1%d = 19.9282 0.0699 0.35% 19.9533 0.0728 0.36%

5%d = 19.9299 0.1708 0.85% 19.9808 0.2114 1.06%

10%d = 19.8866 0.3818 1.91% 19.9750 0.4431 2.22%

30%d = 20.0991 1.3008 6.50% 20.0672 1.2877 6.44%

u

1%d = 0.4993 0.0012 0.24% 0.4995 0.0011 0.22%

5%d = 0.4997 0.0049 0.98% 0.4997 0.0046 0.92%

10%d = 0.4992 0.0092 1.84% 0.4984 0.0094 1.88%

30%d = 0.4983 0.0284 5.68% 0.5021 0.0282 5.64%

Ex

1%d = 50.0087 0.2519 0.50% 49.9913 0.2573 0.51%

5%d = 50.0905 1.2288 2.46% 49.9721 1.2468 2.49%

10%d = 50.1091 2.4866 4.97% 50.0237 2.3365 4.67%

30%d = 49.6296 8.0001 16.00% 49.6516 7.7054 15.41%

K1

1%d = 0.6167 0.3056 101.87% 0.5184 0.2394 79.80%

5%d = 0.5860 0.3869 128.97% 0.4838 0.2346 78.20%

10%d = 0.5791 0.4196 139.87% 0.5013 0.2318 77.27%

30%d = 0.5691 0.4323 144.10% 0.5067 0.2377 79.23%


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FIGURE 2

Parameters evolution process by DA

Parameter Estimation. Under the condition of instantaneous source and steady flow, analytic solu-tions of 1D water quality model are used here. Using the parameter estimation method on DE, the result compared with the results on GA [20] can be ob-tained in Table 2, where F is0.8, CR is 0.8, Tm is 500. The result shows that both methods have high accu-racy and good anti-noise property at a similar level. The parameters evolution process can be shown in Fig 2. From Fig 2, it can be concluded that DA has better convergence speed than GA.

Case 2. The numerical test is given to test the

parameter identification approach when the flow is

unsteady [20]. The length of studied river is 3 km. The variations of the stream discharge with time can be expressed as Q(t)=10+0.001*t m3·s-1. The river can be separated into six stream sections. Upstream concentration C0=1.0 mg·L-1, and the initial concen-tration in the studied river is assumed to be 0 mg·L-

1. Denoting the longitudinal dispersion coefficients of each section as 50 m2·s-1, 70 m2·s-1, 90 m2·s-1, 110 m2·s-1, 130 m2·s-1 and 140 m2·s-1 respectively, the concentrations of the final section of six segments (to three decimal places) can be obtained by a FDM wa-ter quality model, shown in Table 3. The longitudi-nal dispersion coefficients of the six sections are sup-posed to be unknown parameters, which should be

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90

100

evolution generation

iden

tifie

d pa

ram

eter

ExA

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

evolution generation

iden

tifie

d pa

ram

eter

Ku


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identified by parameter identification model if the proposed model is correct. The test has been used to verify the method based on GA by Liu et al. [20], so Comparison results between GA and DE for param-eter estimation are discussed here.

Parameters Identification on Observation

Data without Noise. The data in Table 3 are calcu-lated concentrations without noise added. The pa-rameter estimation results by parameter identifica-tion model on DE and GA are shown in Table 4. The values by using the two methods are very close to the denoting values, and the errors are 0.01%~0.97% and 0.03%~1.38% respectively. There are no signif-icant different precision between the two methods.

Parameter Identification on 10%d = Ob-servation Data with Noise. Some random noise is added in Table 3, where the data are assumed to be measured concentrations with noise. Using the Matlab program for 500 iterations, the statistical re-sults by DE and GA are shown in Table 5. From Ta-ble 5 it can be concluded that satisfactory results can be obtained by DE or GA when 10%d £ , which in-dicate that both of them have good global conver-gence and robustness. However, the relative stand-ard deviation of the parameters by DE are less than those by GA, which indicate that DE has better anti-noise abilities for parameter estimation.

TABLE 3 Concentration values used in parameter identification (mg·L-1)

t/min The number of stream section

1 2 3 4 5 6 5 0 0 0 0 0 0

10 0.213 0.028 0.003 0 0 15 0.399 0.100 0.020 0.003 0 20 0.555 0.203 0.058 0.014 0.003 0.001 25 0.684 0.323 0.121 0.039 0.011 0.005 30 0.787 0.448 0.205 0.081 0.028 0.014 35 0.867 0.570 0.306 0.142 0.059 0.035 40 0.925 0.681 0.417 0.222 0.105 0.070 45 0.965 0.777 0.530 0.317 0.168 0.126 50 0.991 0.856 0.638 0.422 0.245 0.204 55 1.000 0.917 0.737 0.530 0.334 0.305

TABLE 4

Inversion result of parameters by DE and GA under no noise

The number of stream segment 1 2 3 4 5 6 true value 50 70 90 110 130 140

GA Identified value 50.0042 70.0193 89.9807 109.8955 130.4208 141.3565 relative deviation 0.01% 0.03% 0.02% 0.10% 0.32% 0.97%

DE Identified value 50.0163 70.0350 90.0162 109.9824 130.5442 141.9283 relative deviation 0.03% 0.05% 0.02% 0.02% 0.42% 1.38%

TABLE 5

Inversion result of parameters by genetic algorithm under 10% noise level

Stream segment 1 2 3 4 5 6 true value 50 70 90 110 130 140

GA

mean value 48.7907 69.4851 97.8049 111.9494 125.2322 110.2591 standard deviation 3.7558 9.2667 20.9430 26.1876 25.4049 53.3514 relative standard

deviation 7.51% 13.24% 23.27% 23.81% 19.54% 38.11%

DE

mean value 50.2546 64.5020 84.6603 117.8943 127.2391 131.7230 standard deviation 3.7156 5.4940 11.6601 23.4915 18.4143 29.4398 relative standard

deviation 7.43% 7.85% 12.96% 21.36% 14.16% 21.03%


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TABLE 6 Result of parameters under different sampling time scale

Iteration

steps Ex (m2·s-1) Ey (m2·s-1) K1 (d-1)

Value Relative deviation Value Relative deviation Value Relative deviation 47 0.995745 0.43% 0.100104 0.10% 0.295544 1.49%

100 1.000115 0.01% 0.099997 0.00% 0.300266 0.09%

Case 3. This numerical test [26] is given to test the parameter identification approach for 2D water quality model. The length of the studied river was 2 km. The river width B was 100 m. The upstream dis-charge was 60 m3·s-1. The variations of downstream water level with time can be expressed as follows:

02sin tz z ATp×æ ö= + × ç ÷

è ø (9)

where wave amplitude A is 0.1，z0 is 1.0 m, cycle T is 12 h. A point source released pollutants continu-ously with the speed of 15 g·s-1. Upstream concen-tration and the initial concentration in the studied river is assumed to be 0 mg·L-1. 200 meshes were distributed in the studied area which can be seen in the literature [26]. Denoting Ex as 1.0 m2·s-1, Ey as 0.1 m2·s-1, and K as 0.3 d-1, the pollutant concentra-tion of five points (to three decimal places) every 1 hour in the downstream 1000 m can be calculated by water quality ADI model [26]. Treating the data as observational data and u, Ex, Ey and K as unknown parameters to be identified, this case can be used to verify the reliability of parameter estimation method.

Parameters Estimation for 2D Water Qual-

ity Model. Using target function 0.000001J £ as termination condition, the parameter estimation re-sults by parameter identification model are shown in Table 6（T=47）. Using maximum iteration step is 100 as termination condition, the parameter estima-tion results are shown in Table 6. The calculated val-ues are very close to the denoting values. By this method, successful calculation of parameters of 2D water quality model can be achieved.

CONCLUSIONS To avoid the shortcomings of traditional opti-

mization algorithms, a new multi-parameters estima-tion model based on DE algorithm coupled with wa-ter quality model was proposed in this paper. The computational results of three numerical cases indi-cate the proposed parameter estimation model has high accuracy and good anti-noise properties. It can give precise results under both steady flow and un-steady flow. It can be used to identify parameters for both 1D and 2D water quality model, including sim-ple analytical solution model and complicated nu-merical model. Comparison results show that the

method based on DE has roughly same accuracy as that on GA, but has higher convergence speed and better anti-noise property.

ACKNOWLEDGEMENTS This research was supported by the National

Nature Science Foundation of China (Grant No. 51479064, 51379058, 51379060), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions（PAPD）and Qing Lan Project. REFERENCES [1] Sharma, D. and Kansal, A. (2013) Assessment

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Received: 21.12.2016 Accepted: 27.12.2017 CORRESPONDING AUTHOR Xiaodong Liu College of Environment Hohai University (HHU) Xikang Road 1 Nanjing 210098 – P.R. China e-mail: [email protected]

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