Engineering networks simulation and assessment of the mathematical model accuracy

. The article shows the way to implement a quasilinear mathematical model of flow distribution in pipeline engineering networks that is effective in a wide range of changes in the multidimensional random vector of loads at network nodes and provides reliable determination of the parameters of the probability distribution functions of flows in active and passive network elements. The proposed model consists of determining the matrix of generalized network parameters-the load distribution coefficients along the branches of the circuit, calculated at the point corresponding to the mathematical expectation of the node loads. Based on the obtained model, the convergence of the results obtained with the results of simulation of engineering networks is proved using a numerical experiment on an electronic computer. The effectiveness of the developed model, the corresponding algorithms and a set of programs for an electronic computer is shown - the value of the criterion of reduced costs for parametric optimization of engineering networks can be reduced by 5-7% compared to the methods used in practice. The possibility of obtaining at the design stage the equivalent hydraulic characteristics of engineering networks in the form corresponding to the data of experimental measurements of pressures at the nodes of real complex engineering networks is proved.


Introduction
The main task of mathematical modelling of engineering networks in this work is to assess the reliability of the mathematical model of stochastic flow distribution proposed in [1,2]. For simulation, three calculation schemes of the utility network are used, shown in Figures  1, 2 and 3. It is easy to see that these calculation schemes differ in their dimensionality, the number of nodes and branches, which allows us to objectively reveal the advantages of the mathematical model on networks of varying complexity.

Materials and Methods
*Corresponding author: umarxodja@bk.ru   Block А1 built according to section data 2.1 [2]. Since the parameters of this model, in the general case, change every day of operation of the utility network, for the operation of the block, the initial information contains not only the values of the mathematical expectations of the amplitude and phase shift for each of the two harmonies but also the values of the coefficients of their variation. Also, the mathematical expectation and the coefficient of variation are also characterized by the value in (2, 3) [2]. Such a volume of initial information makes it possible to fairly reliably simulate the process of consumption of the target product at any node of the calculation scheme. In this work, the process of consuming a target product in water supply systems is modelled, based on the results, the study of which is used for the initial information necessary for modelling. In the block А1, a sensor of pseudo-random numbers distributed according to the normal law with zero mathematical expectation and unit variance is provided [6].
Considering the block А2, it should be noted that almost any of the known algorithms and programs for calculating the steady-state flow distribution can be used here [7,8]. The only requirement for them from the standpoint of the features of simulation is the need for a fairly convenient software replacement of the values of nodal loads based on the results of the block А1. The block algorithm used in this work А2 detailed in [2] section 3.3 Block А3 the simulation algorithm is quite simple, and its essence boils down to the fact that for all elements of the network design scheme, including active elements, mathematical expectations are calculated, variance, standard deviations and coefficients of variation for each of the distributions of interest of random variables (P) by well-known [6] formulas: Initial information used to operate the unit А1 is presented below. The algorithm for simulation modelling of the utility network Figure 3, described in [9], is given in Table 3.1. When modelling a network, Figure 1 used data for nodes 1, 2, 3 from table 3.1, but for the network Figure 2 received data corresponding to nodes 1 9 from table 1. In table 3.1, amplitude values harmonics and standard deviation are given in relative unitsin shares for each of the nodes of the design scheme.

Results and Discussion
The results of the simulation modelling of three engineering networks are presented in tables 3.2  3.4. Figure 4 shows the relationship between the coefficients of variation of flows in the lines of networks ( q  ) and head losses ( h  ), obtained from modelling. Also shown here is the line corresponding to the above [1]) the relationship between these coefficients. Good agreement between the experimental and theoretical data confirms the correctness of the latter and the possibility of calculating the parameters of the distribution functions of the pressure delivery in passive elements from the data on the parameters of the flow distribution functions. When calculating the parameters of the distribution function of the total loads in the network ( 1 ) and head losses in the network (1) by formulas [1] and [2], [1] a single value of the correlation coefficient between the process of consumption of the target product in the nodes of the utility network was adopted ij r =0,25. The quantity ij r obtained from the graph of figure 3.5, where the change in the variance of the total network load is shown depending on the value ij r in [1]. For all three considered networks, the value ij r at which the calculated value of the variance of the total load (in simulation modelling) coincides with the value obtained from [1]) approximately equals 0.25. The same meaning ij r is also used in calculating the dispersion of head losses in the network, which is quite acceptable since the discrepancy between the simulation data and the calculation by the mathematical model For the correct selection of pumping equipment and tormented points, it is necessary to find the limits of the possible change in head losses in the network at different values of the total load  j Q . This can be done by considering a system of two random variables  H and  j Q assuming for each of them the normal probability distribution law. Let's consider the correlation coefficient between the values of these random variables as known, for example, take it equally as before 0.25. We can find the so-called conditional distribution  H , that is, the laws of its distribution for various fixed values  j Q , known [6,9,10], that the density of the conditional distribution of two correlation normally distributed random variables is determined by the expression:  Fig. 6. Change in variance of total load network depending on the value of the coefficient correlations between the target consumption process product at network nodes. a is the network in Figure 1; b is the network in

Conclusions
In this article, we propose ways to implement a quasilinear mathematical model of flow distribution in pipeline engineering networks, which determines the matrices of generalized network parametersload distribution coefficients along the branches of the scheme, calculated at a point corresponding to the mathematical expectation of node loads. Based on the model obtained in the work, the convergence of the obtained results with the results of simulation modelling of engineering networks is proved by numerical experiment on an electronic computer. The effectiveness of the developed model, the corresponding algorithms and the software package is proved. The values of the criterion of reduced costs for parametric optimization of engineering networks are given by comparing the results obtained, it is shown that they can be reduced in comparison with the methods currently used in practice. The article indicates the possibility of obtaining equivalent hydraulic characteristics of engineering networks at the design stage in the following cases.