设计洪水峰量最可能组合法的计算通式
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摘要
对于传统的多变量分析方法,给定某一联合重现期,满足防洪标准的峰量组合形式有无数多种,但不同联合设计值得到的洪水过程线及水库最高调洪水位均存在较大的差异,如何在联合重现期水平下选取合理的设计值尤为关键。本文利用Copula函数建立洪峰与不同时段洪量的多变量联合分布,采用OR重现期作为防洪标准,联合概率密度函数值作为洪水发生相对可能性的度量指标;以满足防洪标准为约束条件,构建峰量最可能组合的通用模型,并通过拉格朗日乘数法进行求解。采用最可能组合方法分别计算了汉江流域丹江口水库的3维和4维最可能联合设计值,并与单变量同频率法、多变量同频率法设计值对比。结果表明:单变量同频率设计值达不到防洪标准,其假设洪峰与洪量完全相关,各个洪水特征量均采用单变量概率分布来描述,未能充分考虑各个特征量的内在相关性,并不符合汛期洪水发生的内在规律;最可能组合法与多变量同频率法相比,洪峰偏小,长历时洪量偏大,丹江口水库具有多年调节能力,洪量起主要作用,所以最可能组合法推求的设计洪水结果偏安全。峰量最可能设计值更符合丹江口水库的应用要求,因此,本文所提方法具有较强的统计基础,更加符合水文现象的内在规律,可用于水库设计洪水计算。
For conventional multivariate frequency analysis methods,there are an infinite number of combinations of flood peaks and volumes satisfying the flood prevention standard under the given joint return period( JRP); however,different joint design values would lead to differences of flood hydrograph and maximal water level which occurs in reservoir routing. Hence,how to select reasonable design values under given JRP is very important. In this paper,the Copula functions were used to construct the multivariate joint distributions of flood peaks and volumes,and the OR JRP was adopted as flood prevention standard. The normalized joint probability density function value was adopted to measure the relative occurrence likelihood of flood events,and the general model of Most Likely Composition( MLC) of flood peaks and volumes were established subject to the flood prevention standard. The Lagrange multiplier method was adopted to solve the model. The trivariate and four-dimension MLC design floods of Danjiangkou reservoir in Hanjiang basin were estimated using the proposed method,and the quantiles estimated by univariate identical frequency( UIF) and multivariate identical frequency( MIF) methods were compared with those calculated by MLC. The results indicated that the UIF estimations could not satisfy the flood prevention standard; the UIF method assumed that the flood peak is fully dependent upon the flood volume and all the variables are characterized based on univariate distributions. The UIF method cannot take the correlations of hydrologic variables into consideration,and it does not satisfy the inherent rule of flood events. It is also indicated that the MLC method had a smaller flood peak and larger long-duration volume than those by the MIF method. The Danjiangkou reservoir has the capacity of multi-year regulation and the flood volumes are mainly responsible for flood prevention safety,and thus the MLC estimation values are safer. The joint design values of MLC method are more rational according to the application requirements in Danjiangkou reservoir,and consequently the proposed method had a strong statistical basis and could reflect the inherent rules of hydrological events,and it could be used to estimate the design floods of reservoir.
For conventional multivariate frequency analysis methods,there are an infinite number of combinations of flood peaks and volumes satisfying the flood prevention standard under the given joint return period( JRP); however,different joint design values would lead to differences of flood hydrograph and maximal water level which occurs in reservoir routing. Hence,how to select reasonable design values under given JRP is very important. In this paper,the Copula functions were used to construct the multivariate joint distributions of flood peaks and volumes,and the OR JRP was adopted as flood prevention standard. The normalized joint probability density function value was adopted to measure the relative occurrence likelihood of flood events,and the general model of Most Likely Composition( MLC) of flood peaks and volumes were established subject to the flood prevention standard. The Lagrange multiplier method was adopted to solve the model. The trivariate and four-dimension MLC design floods of Danjiangkou reservoir in Hanjiang basin were estimated using the proposed method,and the quantiles estimated by univariate identical frequency( UIF) and multivariate identical frequency( MIF) methods were compared with those calculated by MLC. The results indicated that the UIF estimations could not satisfy the flood prevention standard; the UIF method assumed that the flood peak is fully dependent upon the flood volume and all the variables are characterized based on univariate distributions. The UIF method cannot take the correlations of hydrologic variables into consideration,and it does not satisfy the inherent rule of flood events. It is also indicated that the MLC method had a smaller flood peak and larger long-duration volume than those by the MIF method. The Danjiangkou reservoir has the capacity of multi-year regulation and the flood volumes are mainly responsible for flood prevention safety,and thus the MLC estimation values are safer. The joint design values of MLC method are more rational according to the application requirements in Danjiangkou reservoir,and consequently the proposed method had a strong statistical basis and could reflect the inherent rules of hydrological events,and it could be used to estimate the design floods of reservoir.
引文
[1]Guo Shenglian,Liu Zhangjun,Xiong Lihua.Advances and assessment on design flood estimation methods[J].Journal of Hydraulic Engineering,2016,47(3):302-314.[郭生练,刘章君,熊立华.设计洪水计算方法研究进展与评价[J].水利学报,2016,47(3):302-314.]
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[3]Sraj M,Bezak N,Brilly M.Bivariate flood frequency analysis using the copula function:A case study of the Litija station on the Sava River[J].Hydrology Process,2015,29(2):225-238.
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[7]Li Tianyuan,Guo Shenglian,Yan Baowei,et al.Derivative design flood hydrograph based trivariate joint distribution[J].Journal of Hydroelectric Engineering,2013,32(3):10-14.[李天元,郭生练,闫宝伟,等.基于多变量联合分布推求设计洪水过程线的新方法[J].水力发电学报,2013,32(3):10-14.]
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[9]Liu Zhangjun,Guo Shenglian,Li Tianyuan,et al.General formula derivation of most likely regional composition method for design flood estimation of cascade reservoirs system[J].Advances in Water Science,2014,25(4):575-584.[刘章君,郭生练,李天元,等.梯级水库设计洪水最可能地区组成法计算通式[J].水科学进展,2014,25(4):575-584.]
[10]Xue Jinping,Hu Zhigen,Liu Quan.Risk analysis of the river diversion under construction of cascade hydropower stations[J].Journal of Sichuan University(Engineering Science Edition),2014,46(1):75-80.[薛进平,胡志根,刘全.梯级水电站建设施工导流风险分析[J].四川大学学报(工程科学版),2014,46(1):75-80.]
[11]Zhang Q,Xiao M Z,Singh V P.Uncertainty evaluation of copula analysis of hydrological droughts in the East River basin,China[J].Global Planet Change,2015,129:1-9.
[12]zban A Y.Some new variants of Newton’s method[J].Applied Mathematical Letter,2004,17:677-682.
[13]Zhang Zhenghao,Zhang Qiang,Xiao Mingzhong,et al.Reservoir-based ecological water operation in the Liaohe river basin characterized by synchronous occurrence of wet/dry events[J].Acta Ecologica Sinica,2016,36(7):1-10.[张正浩,张强,肖名忠,等.辽河流域丰枯遭遇下水库调度[J].生态学报,2016,36(7):1-10.]
[14]Song Songbai,Nie Rong.Asymmetric archimedean Copulas for multivariate hydrological drought frequency analysis[J].Journal of Hydroelectric Engineering,2011,30(4):20-29.[宋松柏,聂荣.基于非对称阿基米德Copula的多变量水文干旱联合概率研究[J].水力发电学报,2011,30(4):20-29.]
[15]Wu Tao,Chen Xi,Yan Yusong.Study of the binary correlation quantum-behaved PSO algorithm[J].Journal of Sichuan University(Engineering Science Edition),2014,46(4):103-110.[吴涛,陈曦,严余松.2元相关性量子行为粒子群优化算法研究[J].四川大学学报(工程科学版),2014,46(4):103-110.]
[16]Zhou Ying,Song Dedun.A model for the optimal design of flood control reservoir[J].Advances in Water Science,1994,5(3):167-173.[邹鹰,宋德敦.水库防洪优化设计模型[J].水科学进展,1994,5(3):167-173.
[2]Qin Guanghua,Song Kechao,Zhou Zejiang,et al.Research on annual runoff prediction based on WA-GRNN model[J].Journal of Sichuan University(Engineering Science Edition),2013,45(6):39-46.[覃光华,宋克超,周泽江,等.基于WA-GRNN模型的年径流预测[J].四川大学学报(工程科学版),2013,45(6):39-46.]
[3]Sraj M,Bezak N,Brilly M.Bivariate flood frequency analysis using the copula function:A case study of the Litija station on the Sava River[J].Hydrology Process,2015,29(2):225-238.
[4]Ganguli P,Reddy M J.Probabilistic assessment of flood risks using trivariate copulas[J].Theoretical and Applied Climatology,2013,111(1/2):341-360.
[5]Volpi E,Fiori A.Design event selection in bivariate hydrological frequency analysis[J].Hydrological Sciences Journal,2012,57(8):1506-1515.
[6]Feng Ping,Li Xin.Bivariate frequency analysis of non-stationary flood time series based on Copula methods[J].Journal of Hydraulic Engineering,2013,44(10):1137-1147.[冯平,李新.基于Copula函数的非一致性洪水峰量联合分析[J].水利学报,2013,44(10):1137-1147.]
[7]Li Tianyuan,Guo Shenglian,Yan Baowei,et al.Derivative design flood hydrograph based trivariate joint distribution[J].Journal of Hydroelectric Engineering,2013,32(3):10-14.[李天元,郭生练,闫宝伟,等.基于多变量联合分布推求设计洪水过程线的新方法[J].水力发电学报,2013,32(3):10-14.]
[8]Xu Changjiang,Yin Jiabo,Guo Shenglian,et al.Deriving design flood hydrograph based on conditional distribution:A case study of danjiangkou reservoir in Hanjiang Basin[J/OL].Mathematical Problems in Engineering,2016.http://dx.doi.org/10.1155/2016/4319646.
[9]Liu Zhangjun,Guo Shenglian,Li Tianyuan,et al.General formula derivation of most likely regional composition method for design flood estimation of cascade reservoirs system[J].Advances in Water Science,2014,25(4):575-584.[刘章君,郭生练,李天元,等.梯级水库设计洪水最可能地区组成法计算通式[J].水科学进展,2014,25(4):575-584.]
[10]Xue Jinping,Hu Zhigen,Liu Quan.Risk analysis of the river diversion under construction of cascade hydropower stations[J].Journal of Sichuan University(Engineering Science Edition),2014,46(1):75-80.[薛进平,胡志根,刘全.梯级水电站建设施工导流风险分析[J].四川大学学报(工程科学版),2014,46(1):75-80.]
[11]Zhang Q,Xiao M Z,Singh V P.Uncertainty evaluation of copula analysis of hydrological droughts in the East River basin,China[J].Global Planet Change,2015,129:1-9.
[12]zban A Y.Some new variants of Newton’s method[J].Applied Mathematical Letter,2004,17:677-682.
[13]Zhang Zhenghao,Zhang Qiang,Xiao Mingzhong,et al.Reservoir-based ecological water operation in the Liaohe river basin characterized by synchronous occurrence of wet/dry events[J].Acta Ecologica Sinica,2016,36(7):1-10.[张正浩,张强,肖名忠,等.辽河流域丰枯遭遇下水库调度[J].生态学报,2016,36(7):1-10.]
[14]Song Songbai,Nie Rong.Asymmetric archimedean Copulas for multivariate hydrological drought frequency analysis[J].Journal of Hydroelectric Engineering,2011,30(4):20-29.[宋松柏,聂荣.基于非对称阿基米德Copula的多变量水文干旱联合概率研究[J].水力发电学报,2011,30(4):20-29.]
[15]Wu Tao,Chen Xi,Yan Yusong.Study of the binary correlation quantum-behaved PSO algorithm[J].Journal of Sichuan University(Engineering Science Edition),2014,46(4):103-110.[吴涛,陈曦,严余松.2元相关性量子行为粒子群优化算法研究[J].四川大学学报(工程科学版),2014,46(4):103-110.]
[16]Zhou Ying,Song Dedun.A model for the optimal design of flood control reservoir[J].Advances in Water Science,1994,5(3):167-173.[邹鹰,宋德敦.水库防洪优化设计模型[J].水科学进展,1994,5(3):167-173.