Mathematical models investigation of electrotechnical problems solution on PC

   Scientific adviser: Sivokobylenko Vitaliy F.

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Abstract of masters final work

BASIC MAINTENANCE of WORK

Algorithm of calculation

In basis of numeral methods, applied in this work, a multistep non-obvious method in which carry out replacement of derivatives an algebraic polynomial on the formula of differentiation back FDN will be fixed. A derivative by (p+1) known value of function is here as

Here - permanent coefficients the values of which are found from equation (2), and S - number of preceding values of function with an interval between them dx=h

In particular, for the second order method

From FDN can be got also formulas of numeral differentiation of functions on the known values of function in a few points.

For three points of function

Сonsider a discrete mathematical model for the calculation of PP in an electric network, which the method of FDN is underlaid. In the beginning we will consider a chart, consisting of linear impedors with the generalized parameters and not containing the sources of current.

Figure 1 - Chart of substituting for the branch of electric chart with the generalized parameters

Designate tension on the branch aswhere U₂ and U₁ are tension of knots of begin and end of branch accordingly. Matrix equations, describing the state of electric chain in the moment of time of t+h (h - is a step of integration), in obedience to the law of Ohm will look like the following:

where R - resistance matrix of chart branches, L - matrix of own and mutual inductances of branches;

vectors of tensions of branches, E.M.F. of branches, currents of branches and tensions on the capacities of branches accordingly (in the moment of time t+h);

pi⁽ⁿ⁺¹⁾- vector of derivative of currents of branches in time (in moment t+h);

Figure 2 - Discrete chart of substituting for the generalized branch of electric chart

The derivative of current at times can be also expected a close numeral method on two and more points of current i(t). For these points we take the values of currents on current, following and a few previous steps of integration of DU. For providing of possibility of change of step and order of calculation will take advantage of general formula of method of FDN.

With the indicated approach macromodels can be created AD, SD and other elements.

Differential equations of elements of network and their discrete models

Equations for elements, possessing phase symmetry (LEP, transformator, cable), will be written down in the three-phase system of co-ordinates a, b, c or in the diphasic system of x, y, in immobile in relation to stators of generators. For symmetric electric machines (asynchronous engines [AD]) the axes of x, y are used, and for asymmetrical (synchronous generators and engines) the system of co-ordinates, tightly coupled with the axes of d, q of rotor. That's why on every step of calculation for synchronous machines the count of regime parameters is carried out from the axes of d, q to the axes of all other elements a, b, c or x, y.

In the models of the revolved electric machines (engines and generators) the charts of substitution are used with a multicontour rotor, that allows more exactly to take into account the effect of expulsing of current in a rotor.

For synchronous machines in connection with a presence on the rotor of puttee of excitation on the longitudinal axis of d DU it is expedient to decide in the axes of d and q, tightly coupled with his rotor (fig. 3). Only in these co-ordinates in equations it is succeeded to avoid periodic coefficients, depending on the angular turn of rotor. We will consider that the damper array of rotor on every axis is presented the twocontour chart of substitution.

Figure 3 - Currents of starting of synchronous engine in the axes of d and q

At the known parameters of chart of substitution, mechanical permanent time, will realize the program of calculation in the package of MathCAD for the non-obvious method of Euler. For an asynchronous engine a similar calculation is realized the non-obvious method of Euler (fig.4a) and FDN of 3rd order for a linear discrete model (fig.4b).

Figure 4 - Chart of speed of rotation at starting AD a) by the method of Euler b) by the method of FDN of the 3rd order
(animation consists of 5 shots with a delay in 50 мс between shots; the quantity of cycles of reproduction is limited of 5)

Also from known catalogue information, we will make a chart of substitution and find its parameters, it is possible to find the currents of stationary and transient processes for other elements of chart, for example for a transformator. For it a calculation will be conducted on the simplified chart, without the account of branches of magnetizing.

Here for finding of value of streamtripping a numerical method of Runge-Kutta is used. For the increase of the productivity of calculations, and also maintainance of stability of decisions during commutation in this program it is planned to substitute an existent method by the method of Euler which allows to get incremental decisions.

Now there is the program ASDG-1 for the design of transient processes in the system of power supply with asynchronous and synchronous machines, in which a decision is realized drafting of differential equations for the separate elements of network, programms of calculation of tensions of key and branches and calculation of DU is made by the method of Runge-Kutta.

CONCLUSIONS

1. Discrete charts are developed substituting for the basic elements of power electric charts, based on the non-obvious methods of numeral integration of DU and intended for the analysis of transient and emergency regimes.

2. With the use of the developed methods the calculation of transient process is executed in the system of own needs of the electric station, caused a short circuit and subsequent actions of devices of relay defence and automation. Design of the indicated mode at the use of the known methods executing is difficult.

3. The nearest task is an improvement of the program for possibility of calculation of the system with the use of discrete methods.

REFERENCE

1. Sivokobylenko V.F., Mejenkova M.A. Povedenie sinhronnyh generatorov s sistemami samovozbujdeniya pri blizkih korotkih zamykaniyah // Sb. nauch. trudov Doneckogo gos. tehn. univ-ta. Seriya: "Elektrotehnika i energetika". - Doneck: DonGTU. - 1998. - Vyp. 2. - S. 20-23.
2. Sivokobylenko V.F., Mejenkova M.A. Raschet na PEVM tokov korotkih zamykanii dlya vybora ustavok releinoi zaschity elektrostancii // Sb. nauch. trudov Doneckogo gos. tehn. univ-ta. Seriya: "Elektrotehnika i energetika". - Doneck: DonGTU. - 1999. - Vyp. 4. - C. 186-190.
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11. Perhach V.S. Matematichni zadachi elektroenergetiki. "Vischa shkola", L. - 1989, 464 s.

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