477
2008,20(4):477-484
GENETIC PROGRAMMING TO PREDICT SKI-JUMP BUCKET SPILLWAY SCOUR* AZAMATHULLA H. MD., GHANI A. AB., ZAKARIA N. A., LAI S. H., CHANG C. K., LEOW C. S., ABUHASAN Z. River Engineering and Urban Drainage Research Centre, University Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Pulau Pinang, Malaysia, E-mail:
[email protected]
(Received November 2, 2007, Revised April 8, 2008)
Abstract: Researchers in the past had noticed that application of Artificial Neural Networks (ANN) in place of conventional statistics on the basis of data mining techniques predicts more accurate results in hydraulic predictions. Mostly these works pertained to applications of ANN. Recently, another tool of soft computing, namely, Genetic Programming (GP) has caught the attention of researchers in civil engineering computing. This article examines the usefulness of the GP based approach to predict the relative scour depth downstream of a common type of ski-jump bucket spillway. Actual field measurements were used to develop the GP model. The GP based estimations were found to be equally and more accurate than the ANN based ones, especially, when the underlying cause-effect relationship became more uncertain to model. Key words: Genetic Programming (GP), neural networks, spillway scour, ski-jump bucket
1. Introduction The disposal of flood water exceeding the reservoir capacity is normally achieved through provision of spillways in the body of a dam. There are many types of spillways, out of which the ski-jump bucket type is more commonly used (Fig.1). The energy dissipation in such a spillway is in the form of a jet of water leaving away from the bucket lip into the air, and then, falling into the plunge pool formed at the point of impact on the tail water. The jet impact is transmitted through cracks and fissures of the rock in the form of hydrodynamic pressure fluctuations, which might give rise to hydraulic jacking action and which might further get intensified because of air locking. This causes the rock mass to break into small pieces that get swept away at the downstream of the river resulting in large amount of scouring. The scouring or erosion continues up to the point where the impinging * Biography: AZAMATHULLA H . MD. (1972-), Male, Ph. D., Senior Lecturer
jet is no more able to exert breaking pressure on the rock or where the secondary currents produced are less strong to remove the rock blocks[1]. The process of scouring also continues till an equilibrium scour depth is reached, which corresponds to a situation where the increased water depth in the scour hole results in exertion of bed shear stress that is insufficient to cause further bed erosion or to a condition where the rate of bed erosion is balanced by the rate of deposition of material brought back into the scour hole by the eddy flow. There are various hydraulic, morphologic, and geotechnical factors governing the depth of scour, ds, namely, (referring to Fig.1) discharge intensity q, height of fall H1, bucket radius R, bucket lip angle, I, type of rock, degree of rock homogeneity, time, and mode of operation of spillway. Over a period of several decades many investigators in the past have given empirical formulae on the basis of laboratory as well as prototype observations in order to predict the scour depth downstream of the ski-jump bucket spillway. The following formulae are popular to predict spillway scour depth:
478
Fig.1
The ski-jump bucket spillway scour[5]
Veronese formula: (recommended by the USBR[2] )
d s = 1.90q 0.54 H10.225
(1)
Wu’s formula[3]:
ds q 0.51 = 2.11( ) H1 qH13
(2)
Martin’s formula[4]:
d s = 1.5q 0.6 H10.1
(3)
The above formulae are very convenient to use but have a major drawback, that is, they involve idealization, approximation, and averaging of widely varying prototype conditions. As a result, the predicted scour depths can be considerably different than their actual values. Apart from the complexity of the phenomenon involved, this could also be because of the limitation of the analytical tool used by most of the earlier investigators, namely, the statistical regression. The use of a soft computing tool like Artificial Neural Networks (ANN) in place of the regression for the problem under consideration met with large success as shown in Azamathulla et al.[5,6] and Lee et al.[7] . Recently, Singh et al.[8] applied genetic programming for estimation of the Longshore Sediment Transport Rate (LSTR). More details can be found in Singh et al.[8]. This is motivated the present work, in which the scour problem is tackled with the help of another soft computing tool, namely, the Genetic Programming (GP) and hence, it is extremely flexible in data mining. In the current study, the GP is used to predict the relative depth of scour when the ski-jump bucket spillway is involved. The scour depth would be useful in designing the plunge pool.
2. Genetic programming The concept of GP is borrowed from the process of evolution occurring in nature, where the species survive according to the principle of “survival of the fittest”. GP, a branch of the Genetic Algorithm (GA) [9] , is a method for learning the most “fit” computer programs by means of artificial evolution [10]. In other words, its behavior forms a metaphor of the processes of evolution in nature. GP, similar to GA, initializes a population that compounds the random members known as chromosomes (individual). Afterward, fitness of each chromosome is evaluated with respect to a target value. The principle of Darwinian natural selection is used to select and reproduce “fitter” programs. The main difference between GP and GA is the representation of the chromosomes and final solution. A GA creates equal length strings of numbers (chromosomes) in the form of binary or real, which represent the solution. However, GP creates equal or unequal length computer programs (Fig.2) (a symbolic expression that consists of variables (terminal) and several mathematical operators (Fig.3) (function)) in the LISP language or other computer languages as the solution.
Fig.2
A parse tree of the expression ( + a ( / bc ) )
Therefore, unlike GA, in GP there is no need to define the form of the objective function a priori. In fact, it is the GP that determines not only the coefficients and parameters of the objective function
479
but also, and more importantly, the form of the objective function itself. This is one of the advantages of GP as compared to GA. Although research on GP techniques dates back to the 1960s and 1970s, GP emerged as a distinct discipline [11].
Fig.3
An illustration of a mutation operation for genetic programming
The applications of GP to solve problems in hydraulic engineering are very few. There are, however, some studies dealing with the solution of general civil engineering problems with GP. Unlike the ANN these works have been made recently, typically, about eight years ago and further, they are restricted to relatively fewer areas and include optimization and designing of truss and other structures [12], rainfall-runoff modeling[13], modeling of waste water treatment plants[14], ultimate shear strength of reinforced concrete deep beams[15], estimation of concrete strength [16], prediction of stability of slopes[17], and evaluation of resistance to flow by vegetation[18]. 3. GP modeling of ski-jump bucket scour A GP software, GPLAB[19] in conjunction with subroutines coded in Matlab were used in this study. From previous experience[5], grouped variables produced good results. The parameters, namely, Froude number, q/(gH13)1/2, and the relative scour depth, ds/H1,.were selected as input terminal and the output terminal , respectively. To find the optimum formulation, five functions, namely, plus, minus, product, division, and power were used. A large number of generations were needed to find a formula with minimum error. First, the maximum depth of the tree and the length of the branch were assigned. With these constants, a large number of generations were required to minimize the error. These constants were changed and the program was executed to search for a formula with minimum error and as short as possible in length. The optimum GP structure had the following characteristics: (1) Selection Method: Selection is done by the Lexictour method. In this method, similar to the tournament approach, a random number of individuals are taken from the population and the best fit is chosen. The main difference is that if two individuals are
equally fit, the shortest one tree with fewer nodes is chosen as the best. (2) Operations: The operations that were used in this study were crossover and mutation. They were selected by adopting a rule with a minimum probability of 0.1. (3) Fitness Function: The sum of absolute differences between the obtained and expected gravimetric water content for all sets of data in the database was used as a measure for fitness. (4) Population size: 300 members. (5) Maximum depth of tree representation allowed during generations: 5. (6) Total generations: 4000. The majority of previous research work on scour predictions were based on hydraulic model studies[20]. While hydraulic model studies have advantages like repeatability, they have helped more in exploring the scour mechanism than in obtaining more accuracy in the depth estimation. Scale effects, inability to correctly model certain field conditions like bed morphology and loss of flow energy in aeration, and failure to consider a variety of causative factors simultaneously are some of the deficiencies associated with the model measurements. It was, thus, decided to calibrate the neural networks[6] and GP with the help of realistic field conditions only although it is recognized that prototype measurements might also suffer from instrumental uncertainties and inaccuracies, and lack of availability of data on all causative parameters.
Fig.4
Observed scour depths against varying values of q and H1
4. GP model development and validation A majority of earlier works on scour predictions used the hydraulic model studies, and the publications reporting such observations indicated that only three types of information, namely, scour depth below tail water level ds, discharge intensity q, and head drop H1 are uniformly reported in all references and that the
480
Table 1 Observations of the prototype[5] Sl. No. 1
Discharge intensity q (m3/s/m) 34.20
Head H1 (m) 31.72
Scour depth below tail water level ds (m) 12.19
Damle et al.[23]
2
25.10
20.29
8.08
Damle et al.[23]
3
72.49
30.42
18.29
Damle et al.[23]
4*
42.76
46.18
19.51
Damle et al.[23]
5
21.37
12.55
19.51
Damle et al.[23]
6
3.62
24.85
10.37
Damle et al.[23]
7
75.80
85.00
28.00
Wu[3]
8
113.6
180.00
43.00
Wu[3]
9
68.80
49.00
20.00
Wu[3]
10*
40.00
34.00
20.00
Wu[3]
11
25.00
31.00
19.00
Wu[3]
12
95.20
97.00
30.00
Wu[3]
13
2.60
1.80
2.50
Martins[4]
14
1.80
1.90
2.40
Martins[4]
15
17.00
6.30
14.30
Martins[4]
16
60.00
7.30
16.20
Martins[4]
17
32.00
26.00
11.00
Martins[4]
18
50.00
14.00
18.00
Martins[4]
19
14.00
9.00
6.40
Martins[4]
20
34.00
32.00
12.20
Martins[4]
21
25.00
27.00
8.10
Martins[4]
22
72.00
36.00
18.30
Martins[4]
23
43.00
50.00
19.50
Martins[4]
24
21.00
19.00
19.50
Martins[4]
25
3.60
25.00
10.40
Martins[4]
26
170.00
53.00
55.00
Martins[4]
27
60.00
17.00
17.00
Martins[4]
28*
48.00
19.00
24.00
Martins[4]
29
70.00
19.00
32.00
Martins[4]
30
10.00
30.00
9.00
Martins[4]
31
32.00
6.00
11.50
Zvorykin and Akhmedov ( 1975)
32
31.40
4.00
11.00
Zvorykin and Akhmedov ( 1975)
33
25.00
8.00
16.50
Zvorykin and Akhmedov ( 1975)
34
14.00
1.00
6.35
Zvorykin and Akhmedov ( 1975)
Data source
35*
83.30
115.44
47.00
Lowe et al. (1979)
36
112.71
212.90
37.20
Sen[24]
37
39.30
115.74
10.60
Sen[24]
38
51.30
86.53
11.40
Sen[24]
39
69.50
92.35
17.00
Sen[24]
40
39.00
49.00
27.40
Sen[24]
41*
47.6
26.6
24.70
Sen[24]
42
143.43
19.45
16.00
Sen[24]
43
48.00
90.00
70.00
Sen[24]
44
78.00
88.50
88.00
Sen[24]
45
26.50
96.00
23.00
Akhmedov[27]
481
Table 1
Continue Discharge intensity,
Head
q (m3/s/m)
H1 (m)
Scour depth below tail water level ds (m)
Data source
48
47.80
220.00
62.00
Akhmedov[27]
49*
96.50
32.00
35.40
Khatsuria[28]
50
42.56
83.50
32.00
Khatsuria[28]
54
25.86
83.50
32.00
Khatsuria[28]
55*
41.00
49.00
18.00
Khatsuria[28]
56
41.20
83.50
32.00
Khatsuria[28]
57
55.99
84.00
32.00
Khatsuria[28]
58
48.98
83.50
41.00
Khatsuria[28]
59
56.20
84.00
41.00
Khatsuria[28]
60
61.33
83.50
41.00
Khatsuria[28]
61
46.50
23.00
18.00
Khatsuria[28]
62*
97.54
47.85
15.00
Khatsuria[28]
63
97.54
47.84
23.00
Khatsuria[28]
64
42.6
56.70
19.50
Khatsuria[28]
65*
21.5
21.8
28.20
Khatsuria[28]
67
46.5
25.00
10.00
Khatsuria[28]
68
275
101
68.00
Yildiz and Üzücek[29,30]
69*
57.58
163
27.50
Yildiz and Üzücek[29,30]
73
20.51
102
13.50
Yildiz and Üzücek[29,30]
74
31.40
27.00
15.00
Yildiz and Üzücek[29,30]
75
14.00
12.00
6.35
Yildiz and Üzücek[29,30]
76*
96.30
148.00
37.50
Yildiz and Üzücek[29,30]
77
32.62
143.00
23.00
Yildiz and Üzücek[29,30]
78
12.10
97.00
12.00
Yildiz and Üzücek[29,30]
79*
275
91.00
68.00
Yildiz and Üzücek[29,30]
80
26.50
96.00
23.00
Yildiz and Üzücek[29,30]
81
53.10
97.80
37.00
Yildiz and Üzücek[29,30]
82*
79.60
98.50
49.00
Yildiz and Üzücek[29,30]
83
116.66
65.00
36.00
Yildiz and Üzücek[29,30]
84
79.26
47.00
29.00
Yildiz and Üzücek[29,30]
85
116
64.92
35.96
www.ferc.gov/industries/hydropower/ safety/eng-guide/chap11.pdf
86
79.06
154.2
8.82
www.ferc.gov/industries/hydropower/ safety/eng-guide/chap11.pdf
87
52.95
97.53
36.88
www.ferc.gov/industries/hydropower/ safety/eng-guide/chap11.pdf
88*
79.33
98.45
48.76
www.ferc.gov/industries/hydropower/ safety/eng-guide/chap11.pdf
89
57.50
122.8
27.43
www.ferc.gov/industries/hydropower/ safety/eng-guide/chap11.pdf
90
32.6
102.1
13.41
www.ferc.gov/industries/hydropower/ safety/eng-guide/chap11.pdf
91
7.67
24
5.53
Lopardo et al. (2002)
Sl. No.
* Validation set.
482
information on other factors affecting the scour was not commonly available across them. Although there are many factors that affect the scour depth, only some of them are of primary importance[21]. In addition considering that many traditional prediction formulae, including those because of Veronese[22], Damle et al.[23], Wu, and Martins, are based only on q and H1. A neural network with 2 input nodes and one output node only was developed in Azamathulla [6] . In total, there were 91 input - output pairs formed from the published data reported in Damle et al. [23], Wu [3], Martins[4], Sen[24], Spurr[25], Wang[26], Akhmedov[27], Khatsuria [28], and Yildiz and Üzücek [29,30]. Table 1 presents the compiled measurements. They are graphically shown in Fig.4 which shows the ordinates of the observed scour depths against the varying values of q and H1. Presence of a wide scatter and absence of fixed or regular and simple relationships between these input-output variables can be noted, which justifies application of the GP for the prediction problem under consideration. From the 91 data sets used in this study, 70 data sets (approximately seventy five percent of these patterns) chosen randomly till the best training performance was seen were used for training, whereas remaining ones were used for testing or validating GP model. 5. Analysis and results The scour prediction in the present study has been made on the basis of field data reported earlier in Azmathulla[5]. The past publications uniformly report input values of (referring to Fig.1) discharge intensity q, height of fall H1, and scour depth below tail water level, ds although, apart from these, the type of rock, degree of rock homogeneity, time, and mode of operation of spillway also influence the scour process. The GP model was, therefore, developed with the former set of values as input (Froude number) in order to predict the relative scour depth. Unlike the ANN and ANFIS models reported earlier[5] , in case of these, GP based forecasts were found to be necessary to predict the scour depth. This appears to be consistent with the fact that, in the end, the GP gives a program (or a set of mathematical algorithm) for only one output parameter unlike a matrix of connection weights and bias like an ANN that maps the entire input vector with the output vector. Figure 5 shows the outcome in the form of scatter plot for the testing set of data, not involved in training the GP. For use in practice, the predictions are given in terms of scour depth, and an excellent prediction made by the GP can be seen. This is quantitatively reflected in the error statistics of the Correlation Coefficient (CC), the root-mean-square error, RMSE,
and the average absolute error (deviation), į. These values, respectively, were 0.977, 0.861, and 0.177 for the scour depth (Table 2). Azamathulla et al.[6] had presented a variety of alternative ANN models of feed forward and cascade correlation types in their work. A look into all of their scatter plot and error statistics revealed that the prediction accuracies of the present GP models are generally comparable to the various ANN models. Additionally, all the GP models have the absolute errors (į) much lower than the ANN models although the CC and RMSE are relatively somewhat lower and higher, respectively. For engineering applications, the average percentage errors reflected in the statistic, į, could be more attractive, indicating better acceptability of the GP. However, the results seem to be not as certain in high-value predictions as reflected in somewhat lower RMSE and CC, which are the measures sensitive to errors at larger observations. It is also possible that the flexibility in the data mining approaches incorporated (ANN, GP) might have reached a saturation level for the given sample size and hence, very large variations in the accuracy levels by either method might not be possible to achieve. However, it is felt that in general that, in the analyses involving larger sample sizes, the GP can be expected to be fairly relatively better than the ANN since the GP can have very large degrees of freedom and hence, more flexibility in modeling and further, it is not governed by many and fixed mathematical functions (like sigmoidal function and various learning algorithms) unlike the ANN. This aspect needs to be further explored by applying the GP method to solve many other types of problems, like the spatial and temporal mapping apart from the present cause-effect type. The current work might inspire such applications in future.
Fig.5 Observed versus predicted relative scour depths by GP
It is to be noted that the previous study[6] had already reported superior performance of the ANN models than the traditional statistical regression schemes. Considering the higher levels of accuracy attained by adoption of soft tools like ANN and GP,
483
the same can be advocated for regular use in future although the regression is very easy to apply. With advances in computer hardware and software, the application of soft tools should not pose problems in even routine applications.
[6]
[7]
Table 2 Network – yielded and true relative scour depths [8] Figure No.
5
Method
CC
RMSE
Av. abs. deviation,
G GP
0.977
0.861
[9]
0.177 [10]
6. Conclusions Accurate predictions of the scour depth at the base of ski-jump bucket spillway are essential for the stability with regard to the dam . This article has proposed an alternative approach of GP in the estimation of relative scour depth using field data. The comparisons between the present GP model with previous works of Azamathulla et al.[6], also found that the GP model has good ability of forecasting the scour depth. In scour estimation, there are several influencing parameters, such as the head, discharge intensity, and median diameter of bed material, classified as rock quality, rock mass rating by using GP model is underway. Acknowledgement The authors wish to express their sincere gratitude to University Sains Malaysia for funding a short term grant (304.PREDAC.6035262) to conduct this on-going research. The authors thanks to Prof. Deo M. C., IIT Bombay for his help in this manuscript preparation by providing useful literature.
[11] [12]
[13]
[14] [15]
[16]
[17]
[18]
References [1] [2] [3] [4] [5]
MASON P. J., ARUMUGAM K. Free jet scour below dams and ski-jump buckets[J]. Journal of Hydraulic Engineering., ASCE, 1985, 111(2): 220-235. UNITED STATES BUREAU OF RECLAMATION. Design of small dams[M]. 1987. WU C. M. Scour at downstream end of dams in Taiwan[C]. International Symposium on River Mechanics. Bangkok, Thailand, 1973, I(A13): 1-6. MARTINS R. B. F. Scouring of rocky river beds by free jet spillways[J]. International Water Power and Dam Construction, 1975, 27(4): 152-153. AZAMATHULLA H. Md. Neural networks to estimate scour downstream of ski-jump bucket spillway[D]. Ph. D. Thesis, Bombay: Indian Institute of Technology,
[19] [20]
[21] [22]
2005. AZAMATHULLA H. Md., DEO M. C. and DEOLALIKAR P. B. Estimation of scour below spillways using neural networks[J]. Journal of Hydraulic Research, 2006, 44(1): 61-69. LEE T. L., JENG D. S. and ZHANG G. H. et al. Neural network modeling for estimation of scour depth around bridge piers[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(3): 378-386. SINGH A. K., DEO M. C. and SANIL KUMAR V. Neural network – genetic programming for sediment transport[J]. Journal of Maritime Engineering, 2007, 160(MA3): 113-119. HOLLAND J. H. Adaptation in natural and artificial system[M]. Ann Arbor Mich.: University of Michigan Press, 1975. JOHARI A., HABIBAGAHI G. and GHAHRAMANI A. Prediction of soil-water characteristic curve using genetic programming[J]. J. Geotechnical and Geo-environmental Engineering , ASCE, 2006, 132(5): 661-665. KOZA J. R. Genetic programming: On the programming of computers by means of natural selection[M]. A Bradford Book, MIT Press, 1992. SOH C. K., YANG Y. Genetic programming based approach for structural optimization[J]. Journal of Computing in Civil Engineering, ASCE, 2000, 14(1): 31-37. WHIGHAM P. A., CRAPPER P. F. Modeling rainfallrunoff using genetic programming[J]. Mathematical and Computer Modeling Canberra, 2001, 33: 707-721. HONG Y. S., RAO B. Evolutionary self-organising modeling of a municipal wastewater treatment plant[J]. Water Research, 2003, 37: 1199-1212. ASHOUR A. F., ALVAREZ L. F. and TOROPOV V. V. Empirical modeling of RC deep beams by genetic programming[J]. Computers and Structures, 2003, 81: 331-338. CHEN Li. Study of applying macro-evolutionary genetic programming to concrete strength estimation[J]. Journal of Computing in Civil Engineering, ASCE, 2003, 17(4): 290-294. YANG C. X., THAM L. G. and FENG X. T. et al. Two-stepped evolutionary algorithm and its application to stability analysis of slopes[J]. Journal of Computing in Civil Engineering, ASCE, 2004,18(2): 145-153. BABOVIC V. Computer-aided knowledge discovery in hydraulic engineering[C]. Proceedings 15th Congress of the Asia-Pacific division of International Association of Hydraulic Research. Madras: Indian Institute of Technology, 2006,. IV: 65-72. SILVA S. GPLAB, a genetic programming toolbox for Matlab, ITQB/UNL[M]. {http://gplab. sourceforge. net}. 2007. AZAMATHULLA H. MD., DEO M. C. and DEOLALIKAR P. B. Neural networks for estimation of scour downstream of ski-jump bucket[J]. Journal of Hydraulic Engineering, ASCE, 2005, 131(10): 898-908. HOFFMANS G. J. C. M., VERHEIJ H. J. Scour manual[M]. A. A. Balkema, Rotterdam/Brookfield. 1997. VERONESE A. Erosioni de Fondo a Valle di uno Scarico[J]. Annali dei LavoriPublicci, 1937, 75(9): 717-726.
484
[23] DAMLE P. M., VENKATRAMAN C. P. and DESAI S. C. Evaluation of scour below ski-jump buckets of spillways[J]. Proc. CWPRS Golden Jubilee Symp. Poona, India, 1966, I: 154-163. [24] SEN P. Spillway scour and design of plunge pool[J]. J. Irrigation Power, 1984, 41(1): 51-66. [25] SPURR K. J. W. Energy approach to estimating scour downstream of a large dam[J]. Int. Water Power Dam Construction, 1985, 37(7): 81-89. [26] WANG S. Scouring of river beds below sluices and dams[C]. Design of Hydraulic structures - Proc. International Symp. on Design of Hydraulic Structures, Fort Collins, Colo.: Colorado State University, 1987, 295-304.
[27] AKHMEDOV T. Kh. Calculation of the depth of scour in the rock downstream of a spillway[J]. Int. Water Power Dam Construction, 1988, 40(12): 25-27. [28] KHATSURIA R. M. State of art on computation, prediction and analysis of scour in rocky beds downstream of ski-jump spillways[C]. CWPRS, Platinum Jubilee Symp. Poona, India. 1992. [29] YILDIZ D., ÜZÜCEK E. Experience gained in turkey on scours occurred downstream of the spillways of high dams and protective measurements[C]. Proc. 18th ICOLD. Durban, 1994 , Q.No. 71, R 9, 113-135. [30] YILDIZ D., ÜZÜCEK E. Prediction of scour depth from free falling ski-jump bucket jets[J]. Int. Water Power Dam Construction, 1994, 46(11): 50-56.
