Z. Qian et al. (Eds.): Recent Advances in CSIE 2011, LNEE 124, pp. 563–569. springerlink.com © Springer-Verlag Berlin Heidelberg 2012

Optimization Estimation of Muskingum Model Parameter Based on Genetic Algorithm

Sihui Dong, Bo Su, and Yang Zhang

Dalian Jiaotong University, Huanghe Road 794, Shahekou District, Dalian City, Liaoning Province, P.R. China

shdong@djtu.edu.cn, {sunnytrip,shdong}@126.com

Abstract. Trial-and-error method used in ascertaining the model parameters of Muskingum can not get the optimal solution usually, and the calculation is very complex. The calculation precision of other methods, such as non-linear programming and least square method, is also not ideal. To solve this problem, genetic algorithm, which has global optimization ability, is applied to solve the Muskingum model parameters. Study case shows that computation precision of genetic algorithm has obvious advantages compared with traditional methods.

1 Introduction

Muskingum flood routing algorithm was brought forward by G. T. GT Mccarthy at 1938, which is widely applied in channel flood routing calculation (S. Y. Lin, 2001). There are several traditional methods to solve Muskingum model parameter k and x. In which, trial-and-error method and least square method are the most representative methods (Yangzhou Water Conservancy School, 1979).

Trial-and-error needs to calculate and draw graphs largely, and its solving process needs subjective estimation. So this method has some blindness in parameter solving, and the optimal solution can’t be solved usually (J. Q. Guo and Y. Zhang, 1995). Least square method can avoid the anthropogenic indeterminacy of trial-and-error method. But it can only solve the local optimal solution and can’t solve the global op-timal solution usually. So the calculation precision of least square method is not ideal, too (H. B. Zhu and Y. Yu Ying, 2000; S. C. Zhan, 2006; Y. Wang, Q. Y. Tu and H. Chen, 2008; P. T. Chang, K. Yang, etc, 2009).

Genetic algorithm is a stochastic optimization method, which borrows ideas from evolvement disciplinarian of biology. It handles the optimization object problem di-rectly, and needn’t the function consecution or differentiability. Genetic algorithm has preferable global optimization ability and is being applied in combinational optimiza-tion, machine learning and other relevant functional areas (R. Storm and D. Price, 1997; K Price, R. Storn and J. Lampinen, 2005; C. Gong, Z. L. Wang, 2009).

Genetic algorithm is applied to solve the parameters of Muskingum model. The goodness of fix between the inversion outflow process and actual outflow process is taken as optimization objective.

564 S. Dong, B. Su, and Y. Zhang

2 Principle of Muskingum Model

If there is no inflow in middle of channel, the principle of Muskingum channel flood routing model can be described with following equations:

OIdtdW −=/ (1)

)(QfW = (2)

Where, W is the storage of channel, t is time, I and Q are inflow and outflow respectively.

In the flood routing model of Muskingum, the equation (2) was manipulated as follows:

])1([' QxxIkkQW −+== (3)

Where, 'Q is discharge for representative channel storage, k is coefficient of channel

storage, x is flow proportion coefficients. The equation (1) is discretized as dif-ference equation form, which is as follows:

tQQtIIWW iiiiii Δ+−Δ+=− +++ )(5.0)(5.0 111 (4)

Where, Wi+1 and Wi are the channel storage of begin and end of period I, Ii+1 and Ii are channel inflow of begin and end of period i, Qi+1 and Qi are channel outflow of begin and end of period i, tΔ is period interval of hydrological observation.

The following equation (5) can be obtained through simultaneous equations (3) and (4):

iiii QCICICQ 21101ˆ ++= ++ (5)

Where, 1ˆ

+iQ is computational outflow of period i+1, the parameter specification is

as follows:

( ) ( )( ) ( )( ) ( )⎪

⎩

⎪⎨

⎧

Δ+−Δ−−=Δ+−Δ+=Δ+−−Δ=

tkxktkxkC

tkxktkxC

tkxkkxtC

5.0/5.0

5.0/5.0

5.0/5.0

2

1

0

(6)

Muskingum channel flood routing formula is composed of equation (5) and (6). Apparently, the key of model application is the estimation of parameters k and x or C0, C1 and C2.

Trial-and-error method and least square method are commonly used. The estima-tion value for k and x of trial-and-error is the most optimal estimation value while supposing WQ ~' is rectilinear figure. So this method might not ensure the fit error

least between inversion calculation outflow and actual outflow. And the fit error is biggish usually. And the solving results of least square are not global optimal solution usually. So the application of Muskingum model is restricted to a certain extent.

Optimization Estimation of Muskingum Model Parameter Based on Genetic Algorithm 565

The parameters estimation of Muskingum model is actually a nonlinear optimiza-tion problem. Genetic algorithm, which is based on natural selection and genomic mechanism, is a new method to handle nonlinear optimization problem. It does not be restricted by whether the optimization problem is linear, continuous and differentia-ble, and does not be restricted by variable amount or constraint condition. Genetic al-gorithm seeks global optimal solution under the guidance of the optimization criterion function directly.

3 Principle and Process of Genetic Algorithm

Supposed the parameter optimization problem of common nonlinear model is as fol-lows:

( )α

∑=

−=n

iiim QPxxxfF

121 ;,,min (7)

mjbxats jjj ,2,1,.. =≤≤ (8)

Where, xj is the model parameter to be optimized, f( ) is the common nonlinear model,

Pi the input observational data, Qi is output observational data, • is to take norm,

α is a constant which is determined with optimization criterion, F is optimization criterion function.

The genetic algorithm process comprises following steps: (1) Model parameter encoding. Bit string is simulated as chromosome. The bits of

string are simulated as gene. A long bit string is to express the tentative solution with-in the feasible zone of optimization problem. The process to express the solution of optimization problem with bit string is called encoding. Each bit string represents an individual. The set of many individual is called population. Genetic algorithm acts on the bit string which length is e, and I={0，1}e. The variable xi can be expressed with the bit string (di1， di2，…，die ) according the following decoding function

xi= Γ i(di1, di2, …, die ) )12/()2)((1

−⋅−+= ∑=

− ee

j

jeijiii daba ,

(i=1,2,…, m). Where, (di1， di2，…，die ) is the bit string of stage i of individual d = (d11，…，d1e，d21，…，dme )∈I e. The operational objective of genetic algorithm is bit string, and not the formal solution of optimization problem, which lays a foundation for being a universal optimization method.

(2) Fitness evaluation. Fitness function, which is directly objective function or is transformed from objective function, is to simulate the individual adaptability to liv-ing environment. The individual decoding function is composed of the decoding func-tion of above m bit stage strings, i.e., Γ = Γ 1 Γ 2 … Γ m. So the fitness of individual d can be set as F(d)=g(f ( Γ (d))), where, g( ) is the objective function or is transformed from objective function. The function of g( ) is to ensure that fitness value is nonnega-tive and the fitness value of better individual is bigger. The familiar forms of g( ) are

566 S. Dong, B. Su, and Y. Zhang

linear proportional function, power proportional function, exponential proportional function.

Fitness function is used to calculate adaptability value of each individual. The in-dividual of bigger fitness value is selected with a larger probability, and is eliminated with a smaller probability, and vice versa. The probability of individual di selected is:

∑=

=S

jjii dFdFdp

1

)(/)()(

New population is composed of S individuals, which are selected with the above probability distribution. New population replaces parent population. Genetic algo-rithm likes a net, which covers the feasible zone of optimization problem. Large num-bers of bit strings compose a continual evolutional population.

(3) Individual cross. Supposed cross probability is pc, so there are S·pc individuals to be crossed in population. Arrange the bit strings of two individuals trimly. A point is selected randomly along the length direction of the string. The sub strings of two individual bit strings left of the point are exchanged. So the two new individuals are produced.

The cross operator of genetic algorithm actually imitates the biology model com-pletely. The new individuals will replace the individuals of smaller fitness value. So, the new population keeps unchanged scale, and the individuals of lower fitness value are eliminated.

(4) Individual mutation. In genetic algorithm, mutation operator is a kind of assis-tant operator, and acts on the bit string of individual. The bit value is changed (from 0 to 1, or from 1 to 0) randomly at a small probability pm. The value of pm takes 0.001 to 0.01 usually. Mutation does not depend on the variable dimension or the string length. From the viewpoint of optimization, simple mutation commonly has not significant progress in the process of optimization. But mutation can ensure that a simple popula-tion will not appear, which can not evaluate. Cross and mutation operators increase the probability of producing new individuals.

(5) Evolution iteration. Change to step (2), and alternate until satisfying solution appears or evolution iteration times reaches a predetermined value. The solution of the individual of biggest fitness value is taken as the result of optimization problem. With the selection, cross and mutation from population to population, the region of high fitness value in solution space is discovered. In the region, the number of indi-viduals increases constantly.

The above is traditional generic algorithm. To raise the optimization speed, many relative studies have been brought forward. In this paper, the strategy of holding best individual in each population and the strategy of encoding with real number are ap-plied.

4 The Application of Genetic Algorithm for Muskingum Model Parameter Estimation

According formula (7) and formula (8), the problem of Muskingum model parameter estimation can be expressed as follows:

Optimization Estimation of Muskingum Model Parameter Based on Genetic Algorithm 567

( ) ( )∑=

+ +Δ+−−Δ=n

iiItkxkkxtF

215.0/5.0min

( ) ( ) ( )tkxkItkxktkx i Δ−−+Δ+−Δ+ 5.05.0/5.0

( ) α15.0/ +−Δ+− ii QQtkxk (9)

0.. ≥kts (10)

In this paper, take α =2, i.e., least square criterion is adopted. Genetic algorithm is used to estimate the parameter Muskingum model.

Example 1. The flood process data of the channel from Chengou bay of Nanyun river to Linqing river in 1961 are listed in column 1 and column 2 of table 1 (referred from example 2 in the literature (G. J. Zhai, 1977)). tΔ takes 12h. To compare the rever-sion calculation results of genetic algorithm in this paper, trial-and-error method, non-linear programming and least square method. The optimization results of genetic algorithm are k=1.125(×12h) and x=-0.605. The inversion outflow of genetic algorithm is listed in column 3 of table 1. The inversion outflow of other methods is referred in the literature (G. J. Zhai, 1977), and listed in column 4 to column 6 of table 1. From table 1, the optimization results of genetic algorithm are better than other methods obviously.

Table 1. Inversion outflow of different methods

Inflow / I(i)

Actual outflow / Q(i)

Inversion outflow / QC(i) Genetic

algorithm Nonlinear programming

Lear square

Trial-and- error

261 228 228.0 228.0 228.0 228.0

389 300 300.0 300.8 305.2 288.1

462 382 376.0 380.7 384.8 381.0

505 444 438.7 444.0 446.9 451.1

525 490 480.7 486.2 487.9 495.3

…… …… …… …… …… ……

182 234 229.4 226.1 223.8 222.5

167 193 203.8 195.3 194.0 189.3

152 178 174.0 173.2 172.3 169.0

Average absolute errors / m3/s

4.08 5.16 5.61 7.38

Average relative errors / %

1.04 1.16 1.29 1.78

Example 2. Junan county reservoir is a big reservoir in Shandong province, which lies in lower reaches of Xun river. The data in column 1 and column 2 of table 2 are the actual observational inflow and outflow of one flood of the channel at July, 2001 (H. G. Zhang, H. W. Zhang and S. P. Ji, 2005). tΔ =12h. The optimization results of

568 S. Dong, B. Su, and Y. Zhang

genetic algorithm are k=1.0105×12h and x=0.3635. The data in column 3 and column 4 of table 2 are the inversion calculation results in the literature (H. G. Zhang, H. W. Zhang and S. P. Ji, 2005). The data in column 5 are the inversion calculation results of genetic algorithm. From table 2, we can know that the fit precision of the method in this paper is higher than trial-and-error method and search method.

Table 2. Inversion calculation results of different methods

Actual Inflow / I(i)

Actual outflow / Q(i)

Inversion outflow / QC(i)

Trial-and-error Search method

Genetic algorithm

75 75 0.0 0.0 75.0

407 80 194.5 183.0 113.5

1693 440 878.6 834.7 515.3

2320 1680 1904.0 1738.0 1608.8

2363 2150 2224.5 2050.0 2244.8

1867 2280 2135.5 1980.0 2278.8

1220 1680 1519.0 1535.1 1843.6

830 1270 1116.0 1131.4 1232.3

610 880 785.5 835.0 859.6

480 680 610.0 655.0 628.7

390 550 494.0 499.6 494.6

330 450 408.0 412.0 403.1

300 400 390.0 368.5 341.5

260 340 360.0 314.8 307.9

230 290 290.0 271.0 266.5

200 250 232.5 234.3 234.0

Average Absolute Error / m3/s 108.07 99.27 52.09

Average Relative Error / % 22.90 20.92 9.99

5 Conclusion

Muskingum method is widely applied in channel flood routing. The parameter opti-mization precision of traditional methods isn’t satisfying. A new method based on ge-netic algorithm is brought forward to estimate the parameters of Muskingum model. The method takes the goodness of fix between actual outflow and inversion calcula-tion outflow as optimization objective, and optimizes the model parameters. Case study shows that the method in this paper has high goodness of fix relative to tradi-tional method.

Optimization Estimation of Muskingum Model Parameter Based on Genetic Algorithm 569

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