Simulation of Two-Dimensional Distribution Laws of Random Correlated Quantities of Natural-Climatic Factors in Context of Probabilistic Assessment of Reliability of Hydraulic Structures of Cascades of Hydroschemes

. When performing calculations to assess reliability of hydraulic structures of cascades of hydroschemes on the basis of probabilistic methods, the necessity to simulate random natural-climatic phenomena producing loads and effects on hydraulic structures arises. In particular, statistical series of random quantities of such important natural-climatic phenomena are considered: annual lowest average monthly temperatures, annual maximal amplitudes of average monthly temperatures. Each of the enumerated natural-climatic phenomena is characterized by presence of close correlation connections between random quantities when passing from one hydroscheme of the cascade to another. The necessity to consider correlation connections requires construction (simulation) of joint distribution law of random quantities system. The purpose of the work is simulation of joint distribution law of system of random variables that do not satisfy the normal distributions, taking into account correlation connections between random variables when passing from one hydroscheme of the cascade to another. Methods of the theory of correlation and methods of mathematical statistics with the use of software package MathCad were used in the course of the investigation. Simulation of joint law of distribution of system of random variables that do not satisfy normal distributions, taking into account correlation connections between random variables when passing from one hydroscheme of the cascade to another, and also assessment of accuracy of results, that were performed, have shown advantages of this approach from the viewpoint of accuracy of results obtained by different procedures. The results can be used in probabilistic calculations of reliability of hydraulic structures and cascades of hydroschemes.


Introduction
Assessment of safety and reliability of hydraulic structures on the basis of probabilistic methods is regulated by normative documents [1][2][3][4][5][6][7][8][9]. Taking into account the extremely high potential danger of hydraulic structures, improvement of methods of assessment of their reliability is an important and relevant problem. During performing calculations on assessment of reliability of hydraulic structures of hydroscheme cascades, necessity to simulate distribution laws of random natural-climatic phenomena that create loads and effects on hydraulic structures arises. In this investigation the approaches that allow simulating a joint law of distribution of system of random quantities that do not satisfy the normal distributions in the closed form, and also obtaining the conditional distribution laws of random quantities of natural-climatic phenomena taking into account correlation connections, are realized.

Analysis of recent researches
Statistic series of random quantities of such important natural-climatic phenomena, obtained by direct measurements in dam sites of hydroschemes of the Dnieper cascade of hydroelectric stations: annual maximal flood discharges Q max,i , annual maximal ice thickness h max,i , annual lowest average monthly temperatures t min,i , annual maximal amplitudes of average monthly temperatures t min,i were investigated by methods of probability theory and mathematical statistics with substantiation of the proposed distribution laws in investigations [10][11][12]. Each of the enumerated natural-climatic phenomena is characterized by presence of close correlation connections between the random quantities when passing from one hydroscheme of the cascade to another. Investigations [11,13] deal with revealing correlation connections between random quantities of natural-climatic phenomena in hydroschemes of the Dnieper cascade of hydroelectric stations. The necessity to take into account correlation connections between natural-climatic phenomena requires construction (simulation) of joint distribution law of random quantities system, which is realized in investigation [11]. In the mentioned sources, distribution laws of random variables of natural-climatic phenomena, that enter into the system, do not satisfy the normal distributions, therefore approaches to transform them into the normal laws by way of the use of the corresponding transformations were used. Principles of construction of the joint distribution law of system of random variables that satisfy the normal distributions are widely presented in present-day investigations [14][15][16]. Investigations of two-dimensional and multidimensional joint distribution laws of systems of discrete and continuous random variables that do not satisfy the normal distributions are proposed in investigations [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].
The performed critical analysis of the present-day investigations and publications made it possible to formulate the purpose and determine the objective of the investigation. The objective of the investigation is development of the algorithm of construction of joint distribution law of random variables system taking into account correlation dependences between the natural factors: between annual maximal flood discharges of the watercourse (r. Dnieper); between annual lowest average monthly temperature at hydroschemes of the Dnieper cascade; between annual maximal amplitude of variations of temperature of outdoor air at the hydroschemes of the Dnieper cascade; between annual maximal ice thickness at the hydroschemes of the Dnieper cascade.
The purpose of the work is simulation of joint distribution law of system of random variables that do not satisfy the normal distributions, taking into account correlation connections between random variables when passing from one hydroscheme of the cascade to another.
Methods of the theory of correlation and methods of mathematical statistics with the use of software package MathCad were used in the course of the investigation.
Distribution (1) presents lognormal distribution on the plane. In this case each of random variables Х 1 or Х 2 has density of lognormal distribution: Conditional law of distribution of random variable Х 2 at a fixed value of variable Х 1 has form [31]: But in practical calculations it is more convenient to use expressions (2-3) in closed form [14]: Conditional mathematical expectation of random variable Х 2 at a fixed value of variable Х 1 has form [14]: and conditional dispersion and standard deviation of random variable Х 1 are determined by expressions: r of density of distribution of continuous system of random variables (Х 1 , Х 2 ), that satisfy lognormal distributions.
The value of random variable Х 2 is determined by Let us illustrate the presented approach by an example. We simulate the joint law of distribution of two-dimensional system of random variables (Х 1 , Х 2 ), that satisfy lognormal distributions. Analysis of statistical data on annual maximal amplitudes of average monthly temperatures at hydroschemes of the Dnieper cascade, and also determination of parameters of their distribution functions is performed in investigation [12]. It is presented in Tabl By results of correlation analysis of statistical samples of maximal amplitude of average monthly temperatures of outdoor air, °C, correlation coefficient of two samples at t. Kaniv and t. Kremenchuk is r = 0.871. It is presented in Fig. 1.
The linear regression equation is taken as where   Sample correlation coefficients, sample covariance, standard errors are calculated. It is presented in Tabl. 3, 4, 5.  Fig. 2.  . 2. Function of density of distribution f( 1 (t 1 ),  2 (t 2 )) of system of two correlated variables t 1 , t 2 , that satisfy lognormal distributions.
The value of random variable t 2 is determined by conditional distribution law (9) with parameters m( 2 (t 2 ) 1 (t 1 )), ( 2 (t 2 ) 1 (t 1 )). It is presented in Fig. 3.   Conditional distribution laws of lowest monthly average temperatures and of maximal amplitudes of monthly average temperatures according to [40] correspond to normal law if values of expression (14) are within the confidence interval: where  It is presented in Fig. 4.

Conclusions
Taking into account the great diversity of distribution laws of random variables of natural-climatic factors connected by correlation dependencies, and mathematical complexity of construction of joint distribution laws, method which is based on transformation of distribution laws into normal form has advantage in the further use. Assessment of accuracy of the results obtained by different procedures is performed. The results can be used in probabilistic calculations of reliability of hydraulic structures and cascades of hydroschemes.