Mathematical model of electrical system including transformer and electrical drives based on induction motors and synchronous motors

On the basis of interdisciplinary modelling methods the mathematical model of the electromechanical system, including power transformer, induction motors and synchronous motors, is formulated and presented in the paper. The motors are joined with the driven objects through a mechanical power transmission system. In addition, the transformer is loaded by nonlinear circuit RL. Differential state equations of the studied system are given as the Cauchy’s formulas. Numerical simulations of the system operation have been made for the selected cases. The results of computer simulation are presented in a graphic form.


Introduction
An interdisciplinary variational method has been used in order to formulate a general mathematical model of an electrical system. The method is based on a modification of Hamilton's principle [1,4]. The electromechanical system including transformer, induction motors and synchronous motors as well as nonlinear load RL is analysed. Differential equations describing operation state of the system are derived by formulation of modified Lagrangian terms using the abovementioned method [2,3].

Mathematical model of the system
The electric diagram of the analysed system is depicted in Fig. 1. In general case the system is consisted of k electric drives, where k 1 is number of induction machines and k 2 =k-k 1 is number of synchronous machines.
where: k 1 deals with induction motors and k 2 deals with synchronous motors.
Taking into account the following dependencies [1,7]:  1  1  1  1   1  1  ,  ,  ,  ,  ,  , , , where: ψ is matrix of magnetic linkages (for transformer and motors), τ is matrix of reverse magnetization inductances, B is topology matrix, Π is oblique transformation matrix, 1 Π is Park transformation matrix, the mathematical model of the system referenced to the current coordinates has been obtained [1,4,9] From the third dependency in (6) the following relationships may be obtained: which may be used to derive the matrix equation of nonlinear load of transformer: For the load assembly ( Fig. 1) it may be written: As a consequence of time differentiation of equation (27), taking into account initial conditions and dependencies (18), (19), (21) and (26), the following dependency may be derived: which may be used to derive the voltage across the load circuit: The electromagnetic torques of motors are given as follows [1,4,10]:

Conclusion
The application of interdisciplinary variational approaches allows to formulate mathematical models for very complex electromechanical systems and power generation systems including electric load assemblies consisting of transformer, synchronous motors and induction motors with elastic shafts as well as active-inductive load, etc.
On the basis of the presented above results of computer simulation the following conclusions can be drawn: a) complex physical processes occurring in electromechanical systems are dependent substantially on both processes occurring in each element of the system and interactions between all elements b) improperly designed mechanical power transmission systems of electrical drives may lead to the increase of elastic moments as a consequence of resonant processes, which, in turn, may lead to the damage of the entire system.