Uncertainty Analysis and Application of Construction Diversion Tunnel Discharge Capacity

Transcription

1 International Conerence on Mechatronics, Electronic, Industrial and Control Engineering (MEIC 2014) Uncertainty Analysis and Application o Construction Diversion Tunnel Discharge Capacity Wang Changhong* School o Civil Engineering and Transportation, South China University o Technology; Guangzhou ; China; chhwang1@scut.edu.cn Wang Zhaoli School o Civil Engineering and Transportation, South China University o Technology; Guangzhou ; China; wangzhl@scut.edu.cn Abstract For the construction o the diversion tunnel discharge capacity problem, the dierence o the starting point between the theory o uncertainty and the traditional deterministic theory is analyzed. The discharge capacity is aected by a comprehensive variety o actors, including the cross section area, roughness, wetted perimeter, Chézy coeicient and many other actors, and thus discharge capacity is multiple unctions o these random variables. On the basis o the actual in-depth research and getting a lot o inormation, reasonable statistical analysis and properly determination o the distribution o various types o random variables are made. Establishing a reasonable probability density unction and perormance unction, using Monte-Carlo method to simulate, the more objective and real discharge capacity reliability o the construction o the diversion tunnel can be accessed. Combined with practical engineering examples under the actual monitoring data or comparison, the results are objective and reasonable. The research results can provides a certain degree o programming reerence or the same type or the uture diversion project o the the same river basin. Keywords-discharge capacity; reliability; Uncertainty; diversion tunnel; Monte-Carlo simulation method I. INTRODUCTION Traditional deterministic theory insist that as long as the quality o the project in line with the design and construction requirements, then the actual discharge capacity o the construction diversion tunnel at the design condition should be equal to o the design value. Nevertheless uncertainty theory believes that the actual discharge capacity o the construction diversion tunnel are not the only values and not deterministic; a certain guarantee rate also known as reliability are rather linked. Compared with the deterministic theory, the latter is more in line with the objective conditions o the project. Speciically, the discharge capacity is aected by a comprehensive variety o actors, including the cross section area, roughness, wetted perimeter, Chézy coeicient and many other actors, and thus discharge capacity is multiple unctions o these random variables [1]. Such as roughness n, is an integrated random variables relecting the low resistance o diversion structures, including both over-low surace roughness, but also other aspects o the low regime o actors, and these actors are random; Thus, roughness n is itsel unctions as a multivariate random variable or random unction, other actors aecting the diversion low capacity is also similar. Thereore, the reliability issues with discharge capacity o diversion structures are considering the assurance o the design discharge rate [1] based on the eect o these random variables. II. RELIABILITY DEFINITION & ANALYSIS OF RELIABILITY PROBLEMS A. Reliability deinition o engineering systems System reliability can be deined as: within the system at a given time t, the probability o complete scheduling capabilities under the conditions stipulated [2-5], Commonly risk and reliability index measure reliability in quantitative analysis. I R represents a generalized resistance and S represents the generalized load o the structure, limit state o the structure is expressed as Then, the structure is in reliable state;, the structure is in ailure state According to the deinition o reliability and the principle o probability theory, i the random variable vectors X (X 1, X 2, X 3 ) o the system is composed o by the basic random variable, the corresponding probability density unction (X), whereby the unction represented by this random vector g(x) is called the unction; Perormance unction is a unction that represents the System State, working conditions can be expressed as: g(x) { (1) The probability that the risk o system ailure rate is: (2) In this ormula : * g + is ailure domain The authors - Published by Atlantis Press 375

2 During the study o the diversion tunnel discharge capacity in coerdam conditions, the diversion design low as a determined targeting low, According to equation (1), the basic expressions o discharge risk o diversion tunnel can be established o, the ollowing unctions is: (3) Here in: is diversion design low,m 3 /s; is the actual discharge or tunnel,m 3 /s Obviously, when,the system is in sae or critical state; when,the system is in dangerous or detrimental state. Thereore, the risk rate o diversion tunnel discharge calculation is to determine the probability o an event when the case occurring o. B. Analysis o reliability problems The core issue o reliability analysis is to calculate the random inormation processing and stochastic risk.when normal distribution actors aecting the reliability and perormance unction as a linear unction o time, the risk rate and reliability can be accurately calculated; But in practice many o the basic random variables are not normally distributed, and unctions are mostly nonlinear unctions, thereore a simple method can not calculate the reliability directly and accurately. Commonly used approximation methods including: Center point method, checking point method ( method) and mapping transormation method [6],these analytical methods are established based on the theory o probability and numerical analysis. Speciically to calculate the risk o diversion tunnel discharge rate, using the analytical method irstly to analyze the actual discharged, getting probability distribution (unction) situation, then probability distribution unctions on the risk o the diversion tunnel overlow by means o mathematical analysis can be got.this approach requires knowledge on quantitative distribution characteristics o, due to the complexity o the hydraulic distribution uncertainties, the probability distribution o is not easy or diicult to get, Wanting to get analytical solution is very diicult, so as to aect the risk o diversion tunnel discharge rate calculation, including the impact on the accuracy o the results. Analytic method by using numerical simulation methods can overcome the "bottleneck" problems, this paper will ocus on the method o numerical simulation. Representative methods o numerical simulation, such as Monte-Carlo(MC) its basic idea is to construct the probabilistic model o random variables and probability sampling simulation model, when the simulating count is large enough, simulation results can be used as approximate solution o the original problem, and meet certain accuracy requirements. Although there are deiciencies o analog computational complexity o this method, but in case o better hardware and justiied algorithms, it is an ideal way to gain reliability. Through multiple simulations o the diversion tunnel outlet low, getting computation and statistical unction result, diversion tunnel ailure events probability or discharge risk rate can be obtained. III. MATHEMATICAL MODEL OF RISK OF DIVERSION TUNNEL DISCHARGE RATE A. Establishment o the unction o discharge risk rate Because o discharge low o diversion tunnel showing a alternation o pressurized semi-pressurized and non-pressure in the lood, in order to research its reliability problem o discharge capacity o the diversion, examining the adverse low patterns, that is pressure low pattern. The tunnel pressurized discharge capacity stream low (here ignoring the rictional head loss) can be expressed as: g (4) 1 Here in:, is water depth above 1 the tunnel inlet loor,m; is the sum o three local loss coeicient including the imports section the exit section and center section; R is hydraulic radius,m; C is the Chézy coeicient 1 1 ; A is tunnel section area,m 2 ; is gravitational acceleration,9.81 m/s 2 ; h is imports and exports rising o the tunnel, ; i is tunnel slope rate; L is tunnel length, ; is roughness rate B. Uncertainty analysis o the eect o the diversion tunnel discharge capacity Then considering uncertainty eects o the parameters: in equation (4), X (A,R,n,h,i,L) is a random vector relecting the discharge low ability o diversion tunnel, unction is expressed as a unction o the random vector, namely =(A,R,n,h,i,L). The uncertainty o A,R,n,h,i,L aects the mean and variance o, ollowing a simple analysis o uncertainties o hydraulic actors: 1) As or hydraulic structures, the hydraulic elements: such as tunnel section area A, the hydraulic radius R, tunnel slope rate i, tunnel length L, on the one hand the uncertainties o such hydraulic elements are caused by measurement, construction materials and loting error; on the other hand due to low cavitation and riverbed scour change. Here the ormer reason is only taken into account. This pure geometry (measurement, loting) error, mathematically is proven to be approximately normally distributed [7]. 2) Randomness o roughness n: arising primarily rom the construction and operation o the uncertain and irregular changes in the structure o the tunnel wall and, in addition, cavitation phenomena and actors such as depth changes also have dierent eects on n, it is diicult to determine rigorously the distribution o n approximation can only be made: Applying stochastic simulation and assumed distribution method, scholars study o the distribution [8] o roughness coeicient n. Based on the inormation o n values. given the normal value n b, minimum value n a and maximum value n c, and then assuming n approximately triangular distribution is easible. 3) Randomness o h (the upstream and downstream water level dierence): on one hand comes rom the uncertainty o water level (the actors inluencing the level are complex), on the other 376

3 hand comes rom construction and measurement error; These actors can be considered independent by each other, the inluence is diicult to distinguish between primary and secondary, so they are treated as approximately normally distributed [9]. These general parameters o hydraulic distributions o random variables can be determined in this way: assuming that design values is reliable, taking design values or random variables mean, then average error o construction or measurement can be used to acquire mean-square deviation [9]. Such as is construction error o tunnel length, That is, tunnel length is Postulated in normal distribution, in the probability theory, we know that is also normally distributed, the expectations o is, the mean-square deviation o is, the average error is [1] : (5) Getting the quadrature o ormula (5), and noting,getting the ollowing equation: 2 (6) With the estimate o average error, then by equation (6) to seek mean-square deviation o random variable and probability distribution unctions o tunnel length Lcan be obtained. Similarly, other random variables and, assuming that they are normally distributed, taking its design values as mean value, ollowing the estimates o their errors, using similar to equation (6) ormula, there will be mean-square deviation, then their probability distribution unction expressions can be obtained. For roughness n, where assuming the triangular distribution, triangular probability density unction and its distribution unction is: (7) { 2 2 { Taking the design value or the mean value o roughness, denoted, and then estimating or its maximum value and minimum value o, you can get three triangular distribution parameters o roughness : and, resulting in expression o the probability distribution unction. C. Simulation process 1) Pseudo random number generating The program perormed by MC, large amounts o random numbers need to be generated irstly; Uniormly distributed pseudo-random numbers is the basis or analoging random number o another distributions, there are algorithm methods, such as cache method, iterative method, method o residues and shiting method etc, this [1, 10] paper is using random number generator program, (8) recommended by related literature, it is veriied with better uniormity and independence, and has suiciently long periods o time. 2) Each random variable sample o eecting discharge capacity o diversion tunnel a) Normal distribution random number simulation method n random samples o the uniorm distribution 1, 2, are generated in [0,1]. On the basis o them, the random number is treated as the probability o an event occurring, the sampled value o the random variable X at a given distribution is calculated. For a normal random variable can be generated using the ollowing method: (10) 1, 2 are independent o each other, and obedience,, -, thus 1, 2 can be treated as two independent standard normally distributed random variable samples; random variables 1, 2 o normal distribution,, - can be obtained by linear transormations: x u u (11) 1 1 x u u 2 2 b) simulation method o other distributed random number For other types o distributions, when generating random variables that are required, the commonly used method is the inverse unction o probability distribution unction must be exported, this is very diicult or some complex distribution; Thus it applies only to simple distribution, speciic practices are as ollows: Let X be random variables and be subject to continuously distributed, whose distribution unction is ; Given, corresponding to r the value x is: 1 X F ( r) x (12) 1 In the ormula: F ( ) x is inverse unctions o F ( ) x ; When r is uniormly distributed random number in [0,1] made on machine, the value x calculated by equation (12) is the random variable samples o X. Application o inverse transormation method, the random variable o uniorm distribution, exponential distribution and triangular distribution can easily be got. Due to the roughness coeicient n ollowing triangular distribution, generation o random variables can be used with the inverse transorm method, Assuming r 1 as random number in U[0,1], then (9) (13) D. Calculation o the risk The sample value or each random variable is substituted to equation (4), then calculating Q i and comparing Q i and Q d. i Q i < Q d, Meaning the ailure o 377

4 statistical modellin, assuming total number o simulation to be N, number o ailures to be N, Thereore, the estimated value or P, i.e. ailure probability o diversion tunnel discharge capacity is : (14) From Bernoulli's law o large numbers: N lim P P 1 N N (15) From the above ormula: Converges to. The simulation results can be obtained rom ormula(15), it is clear that analog precision increases with the number o N. Reliability analysis o discharge capacity o diversion structures, not only wants to get the single value o reliability, but also wants to know the eect o hydraulic actors on the reliability. In particular, it is necessary to analyse the various hydraulic actors mean-variance change impacting on reliability. Its practical signiicance is sel-evident, because o reducing hydraulic actors mean-variance deviation means reducing construction errors, which can be carried out to improve the construction quality o standard quantitative guidance, namely the decrease o mean-variance deviation causes reliability increase, among them once the most obvious hydraulic actors are ound, thus the most critical actors o quality control in the construction are ound. Keeping the others constant, respectively let mean-variance deviation o the hydraulic radius R, tunnel slope rate i, tunnel length L, roughness n occur some increase, the resulting change o discharge capacity reliability relects the discharge capacity reliability sensitivity to changes with the inluence o these actors. march), the corresponding peak low at the dam site is m 3 B. Selection o the original random variables According to the ormula (4), the random variable is choosed: Flow coeicient and elevation o upstream water level H 0 are random variables, approximately normally distributed; the mean o is 0.748, mean-variance deviation o is 0.05; the mean o H 0 is 119.0, mean-variance deviation o H 0 is 0.8. C. Algorithms & low chart This example using VB language program, low chart as shown in Fig. 2, the basic steps are as ollows: 1) n random number o low coeicient are generated, its mean is 0.748, mean-variance deviation is 0.05, and they are the normal distribution. 2) n random number o elevation o upstream water level H 0 are generated, its mean is 119.0, mean-variance deviation is 0.8, and they are the normal distribution. 3) Above parameters are put into ormula (4) to calculate(simulation), i.e. one time calculation is a test, i the calculated value is less than the amount o water m 3, meaning ailure, counter +1; 4) Repetition goes on until the end o trial; the ratio o the total number o ailures and the total number o simulation is the ailure probability. IV. ENGINEERING EXAMPLES Being taken diversion tunnel o Lechang Gorge hydroelectric project or example, risk o its discharge capacity is analysed and calculated. A. Basic parameters o the diversion tunnel Elevation o upstream coerdam crest is 120.8m, elevation o downstream coerdam crest o 108.0m; elevation o imported loor o the diversion tunnel or 94.0m, elevation o export loor o the diversion tunnel or 92.0m, pressure segment length o tunnel or 572.0m [11]. Tunnel body section is shown in Fig. 1: Figure 1. Diversion tunnel body section Calculation parameters : A=121.6m 2, L=42.3m, R=2.87m, C=85.15 Coerdam retaining water standard using 10 year lood standard within 4months period (October-ollowing Figure 2. Simulation process o MC method D. Calculated results when a simulating count 10,000 tests, ailure probability P =23.24%;When the number increased to 20,000 times, P =23.17%; Again increased to 100,000 times, P =23.18%. It can be seen that with the increase o number o simulation, in particular more than 20,000 times, probability o ailure P tends to be stabilized. The standard this diversion tunnel is 10-years lood, using period is 3 years. The Yen B.C. classical probability ormula [12] is or reerence, getting the risk n=0.271, compared with the calculated results o MC method in this paper, the ormer is slightly larger The Yen B.C. ormula ocuses on the inluence o two actors on the risk: diversion standard (especially water requency) as well as the service lie o diversion structures, while this chapter ocuses on the various actors o hydraulic uncertainty impacting on tunnel discharge 378

5 capacity, the are two completely dierent angles. The results o dierent study point, can support each other, to some extent, the approach in this paper is reasonable. Diversion tunnels were put into operation on August 2, 2009, to November 2011, experienced a total o 6 times lows exceed peak test o 1000m 3, the diversion tunnel and import & export structure remained stability. Practice shows that the reliability o diversion tunnel discharge capacity is in line with the actual requirements. V. CONCLUSION Research on discharge capacity reliability o construction diversion tunnel has practical signiicance, involving a new subject o study o diversion standards. Some objective and human actor would have an eect on reliability o tunnel discharge. With the proper analysis o random variable distribution types, proper statistical analysis, establishing a reasonable probability density unctions and perormance unction, and rational choice o numerical simulation method, thus the research results tend to be more objective and actual status. Study on discharge capacity reliability o diversion tunnel can provide a certain reerence value or the uture, especially with the diversion project o the same type and same river basin. REFERENCE [1] MI Ziming, ZHONG Denghua, and LIU Donghai, Reliability analysis on discharge o diversion structure, Journal o Tianjin University,Vol.34,No.2, pp , Mar [2] Hu Zhigen, Liu Quan,Chen Zhiding, Risk analysis o construction diversion, Science Press, Beijing, [3] Li Qijun, Chen Zhao, Theory and application o dam overtopping risk, China water & power press, [4] Zhang Junzhi, Existing engineering structure reliability theory and its application, China water & power press, [5] Zhao Guopan, Engineering structure reliability theory and its application, Dalian University o technology press,1996. [6] Ma Rongyong, Earth-rock dam risk analysis method and its application, Science Press, Beijing, [7] Chu Xiangyuan, Study on hydraulic model o uncertainty, Journal o Hydraulic Engineering, vol.5, pp , [8] Li Rong, Discussion on the hydraulic actors inluencing the Man-Ning roughness value, Journal o Hydraulic Engineering, vol.12, pp ,1989. [9] Wang Zhuou, Risk analysis o construction diversion, Journal o Hydraulic Engineering, vol.5, pp , [10] Xiong Guangleng, Xiao Tianyuan, Zhang Yanyun, Continuous system simulation and discrete event system simulation, Tsinghua University Press, Beijing, [11] Liu Lijie, Yao Limin, Construction diversion design o Lechang Gorge hydroelectric project, Guangdong Water Resouces and Hydropower, vol.1, pp , [12] Yuan Guangyu, Hu Zhigen, Hydraulic engineering construction, China water & power press, pp.28,