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.

Field approaches are employed to develop a mathematical model of a power network section. The facility consists of two electric power subsystems described with ordinary differential equations and presented as concentrated parameter systems connected with a three-phase power supply line, presented as a distributed parameter system. The model of the electric power line is described with partial differential equations. Mathematically, the supply line model is described utilizing a mixed problem with explicitly indefinite boundary conditions. All electromagnetic state equations of the integrated system are introduced in their matrix-vector forms. The equation of the three-phase long supply line is expressed untraditionally as a system of two first-order differential equations as a function of long line voltage. Since the power supply line is part of the integrated system that includes two subsystems, the boundary conditions at the line's start and end are implicitly defined, avoiding the traditional application of the Dirichlet first-type boundary condition. An expanded system of ordinary differential equations that describe physical processes in both the supply and loading subsystems is used to calculate the boundary conditions. To this end, third-type boundary conditions, or Poincaré's conditions, serve to describe the wave equation of the electric power line. Such an integrated model of an electric power system helps analyse transient processes across the supply line when the electric power system is switched on and is single-phase short-circuited at the final point of the electric power line. A comparison of computer simulation results with well-known software packages shows a convergence of approx. 96%.

Within the scope of the presented work, a mathematical model of a prototype of a complex motion transmission on a ship was developed. The abovementioned motion transmission includes long elastic elements with distributed mechanical parameters. The system, containing the motion transmission under consideration, is driven by an engine via epicyclic gearing. The torque is transmitted via a long drive shaft to a propeller working with a variable blade geometry. The rotor of a synchronous generator is mounted on the ship’s long drive shaft. This shaft generator produces electricity that is fed to the ship’s electrical network. With the use of the developed mathematical model, electromechanical transients occurring during the transmission of mechanical power are analyzed. This paper analyzes the motion transmission with the use of computer simulation and presents the results of research.

This paper presents a mathematical model of an electric power system which consists of a three-phase power line with distributed parameters and an equivalent, unbalanced RLC load cooperating with the line. The above model was developed on the basis of the modified Hamilton–Ostrogradsky principle, which extends the classical Lagrangian by adding two more components: the energy of dissipative forces in the system and the work of external non-conservative forces. In the developed model, there are four types of energy and four types of linear energy density. On the basis of Hamilton’s principle, the extended action functional was formulated and then minimized. As a result, the extremal of the action functional was derived, which can be treated as a solution of the Euler–Lagrange equation for the subsystem with lumped parameters and the Euler–Poisson equation for the subsystem with distributed parameters. The derived system of differential equations describes the entire physical system and consists of ordinary differential equations and partial differential equations. Such a system can be regarded as a full mathematical model of a dynamic object based on interdisciplinary approaches. The partial derivatives in the derived differential state–space equations of the analyzed object are approximated by means of finite differences, and then these equations are integrated in the time coordinate using the Runge–Kutta method of the fourth order. The results of computer simulation of transient processes in the dynamic system are presented as graphs and then discussed.

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