Mathematical model of a part of an opened extra-high voltage electrical grid

. In the paper, based on interdisciplinary approaches to modeling, a mathematical model of a part of an opened extra-high voltage electrical grid, which key elements are two long power transmission lines with distributed constants is presented. Within this framework the analysis of transient processes in power transmission lines in a single-line arrangement is carried out. The results of transient processes are displayed by means of figures; they are under ongoing research.  ; 1, 2 k  (8) where L S 1 , L S 2 , L S 3 – inductance of the systems 1, 2 and 3, respectively; R S 1 , R S 2 , R S 3 – active resistance of the systems 1, 2 and 3, respectively; e S 1 , e S 2 , e S 3 – electromotive force of the systems 1, 2 and 3; i S 1 , i S 2 , i S 3 – current of the systems 1, 2 and 3, respectively; i


Introduction
The main objective of electrical grids is transmission and distribution of electricity, provided that its quality indicators are preserved. It is quite clear that the latter cannot be maintained without a reliable information on progress of transient processes in electrical grids. One of the most effective approaches to obtaining the above-mentioned information is use of mathematical modeling apparatus, which is based on the fundamental laws of applied physics; in our case it concerns electrodynamics. The key elements of extra-high voltage electrical grids are long power transmission lines with distributed constants, which leads to formation of boundary and mixed problems in mathematical terms. This, in turn, causes the problem of searching for boundary conditions for the telegrapher's equation. Therefore in the present paper we offer a technique of effective searching for boundary conditions for the wave equation, which, as a result, makes it possible to study transient wave processes in electrical grids.

Analysis of recent research
There are many scientific papers devoted to analysis of transient processes in power systems. Let us consider some of them which are related to the current paper.
In the article [1], the questions of mathematical models and macro-models of power transmission lines development are discussed. A discrete macro-model of a single-phase power transmission line with the use of "black box" is developed. The model description, the order of its development and research findings are presented.
In [2], mathematical models of two -and three-wire alternating-current transmission lines are developed, with the help of which different nature of transient processes is analysed using the software complex АТР-ЕМТР.
It is obvious that not all these approaches to development of mathematical models of transient processes in long power transmission lines enable to take into account influence of other elements of a power system on processes in a line. Yet models including this feature are implemented in АТР-ЕМТР and MATLAB/Simulink software complexes, which does not always provide for more adequate result, especially when electromagnetic models of Ψ -A-type elements are used as models of power systems elements, and especially when the circular and field mathematical models of the latter are used [3,4].
The aim of the paper is analysis of transient processes in the part of opened extra-high voltage electrical grid whose key elements are two long power transmission lines with distributed constants power systems.

System description
The present paper describes a model of an opened extra-high voltage electrical grid (Fig. 1). Power transmission lines are presented in the form of equivalent circuits with distributed constants. They combine power systems for parallel operation, which are presented by their own electromotive force, internal resistance and inductance.
We study only symmetrical modes and this allows us to consider a three-phase symmetrical electrical grid in a single-line arrangement. For the purposes of analysing transient processes in the part of the opened electrical grid, shown in Fig. 1, we suggest using a modified Hamilton-Ostrogradsky principle [5] The extended functionality of the mathematical operation for the studied system according to Hamilton-Ostrogradskii will be similar to that presented in [5][6][7][8]: where S -action according to Hamilton-Ostrogradskii, L* -extended Lagrange function, L l -linear density of the modified Lagrange function, I -energy functional. The extended Lagrange function is presented in [5].
Let us write down elements of the extended Lagrange function [5] where L S1 , L S2 , L S3 -inductance of the systems 1, 2 and 3, respectively; R S1 , R S2 , R S3 -active resistance of the systems 1, 2 and 3, respectively; e S1 , e S2 , e S3 -electromotive force of the systems 1, 2 and 3; i S1 , i S2 , i S3 -current of the systems 1, 2 and 3, respectively; i(x,t)current in the line; R 0 , g 0 , C 0 , L 0 -distributed constants of the line; Φ l3 -linear density of external energy dissipation; Φ lB -linear density of internal energy dissipation; Q(x,t) -the line charge; k -the line number.
One can acquaint oneself with the method of the Similar equations are presented in [3,10,11], for example. Therefore, to articlee synthetic, we suggest finished equations of the electromagnetic state of the object, Fig. 1. where u 1 , u 2 , u 3 -substation busbars voltage of the power systems 1, 2 and 3 respectively.
Kirchhoff's Second Law for electrical circuits with distributed constants is used as boundary conditions for (9): We will write down (9) and (13) in discrete space for the j-node of the line (using the concept of the central derivative): Having analyzed the equation (14), it can be seen that in order to find the voltage on the first and the last nodes of discretization, it is necessary to find the unknown voltages in the fictitious nodes The next step is to differentiate (17) and (18) with respect to time: Now, based on (15) and Fig. 2, we will write: Substituting (10) Substituting (14) to (24), which is written for the first discretization node for the first line and deriving the voltage from there, we will obtain: Equations (22) and (23) against each other will produce: Voltage in the fictitious node Due to space constraints, we will not provide findings for voltage in fictitious nodes of the second line, but will provide the finite expressions: The current in the line can be found by discretizing (13) by the finite difference method, using the concept of the right derivative [5]: The following system of differential equations is subject to joint integration: (10)

Computer simulation findings
A computer simulation was conducted to study transient processes during system switching in a steady-state mode [10,12,13].
The parameters of the equivalent circuit elements in Fig. 1 correspond to the parameters of the real part of 750 kV electrical grid. The line No. 1 is 282 km long, and the line No. 2 is 476 km long. The parameters of the grid elements are as follows.
The parameters of two long lines with distributed constants are as follows: e S1 = 622 sin(ωt + 10.3º) kV, e S2 = 581 sin(ωt) kV, e S3 = 600 sin(ωt + 4.9º) kV, R S1 = 2.           2. The mathematical model of the long power transmission line with distributed constants allows to analyze complex transient processes in such lines, which are almost impossible to take into account on the basis of a circular equivalent circuit of a long line scheme, because the partial differential equations help to describe physical processes taking into account environment continuity, in particular, to calculate propagation velocity of electromagnetic waves in long lines, to consider different wave processes, et caetera.