Abstract

• Introduction
• 1. Goal work
• 2. Method
• 3. Criteria for evaluating the sign of the roots of the characteristic equation
• 4. Software for evaluation of static stability
• 5. Evaluation sign roots using algebraic criteria
• 6. Example controlled study, SS ACD proportional type
• Conclusion
• References

Introduction

Static stability (SS), or steady state stability — is the ability of the electrical system back to its original position after his small perturbation (deviation regime parameters) [1].

In any electrical system established mode does not mean constancy of all its parameters. The electrical system has a large number of loads that change, and these changes appear and disappear. In this connection, on generators systems appear some additional minor points, also stochastically, which increase or decrease the moments that act on the shaft generator rotors and shifting them into small corners [6].

Thus, the electrical system constantly occurring small perturbations , and the reason that the city of not fixed. These are some free perturbations causing free movement, which may be growing or decaying, oscillatory or aperiodic. His character determines the static stability, which is a prerequisite for system health. Static stability is checked when working perspective and design, development of special devices of automatic control (calculations and experiments), the commissioning of new elements of the system, the operating conditions [7].

Study of static stability are given in terms of addressing the problems of analysis and synthesis.

In solving the problems of the analysis verified the stability of the regime that was established is determined by the maximum steady state power system, given all the parameters evaluated several indicators of the quality of the transition process.

In solving the problems of synthesis determined by the type of excitation system and its regulation, the regulation function, the parameters of the excitation system and regulators. This is based of the specified requirements to the maximum steady state or quality of electricity at a steady state

The task of determining the conditions of static stability and transient nature difficult enough for the real electric systems that have automatic excitation regulators and speed. To facilitate the solution of this problem using various mathematical techniques and methods. One of the most common methods of investigation of the static stability is the method of small vibrations using specific criteria that evaluate the signs of real roots, or the real parts of complex conjugate roots of the characteristic equations, and thus determine the nature of the transition process (TP).

1. Goal work

The purpose of the master's work is the development of software for research SS electrical systems of varying complexity by small vibrations through practical algebraic and frequency criteria in a computer.

To achieve this goal will address the following objectives:

• analysis of the theoretical research positions static stability of electrical systems using algebraic criteria (Hurwitz, Routh) and frequency criteria (Mikhailova, D- partitioning);
• Development of software for implementing these methods on a PC ;
• Develop guidelines for the study of static stability using software.

2. Method

Perturbations in the system, which cause small deviations discussed in the study of static stability is not defined not by their place of origin, not in magnitude. That is, the origin of the perturbation such that establish absolute values of the regime in their variances (initial) values impossible. Thus, the task of investigating the static stability is reduced to the problem of determining the nature of the change only mode settings.

In establishing criteria for practical answer was only a yes-no, stable-unstable mode from the initial state under small perturbations. Set coca in any case will be the nature of the transition process (aperiodic oscillatory, decaying, growing), or perhaps deciding analyzed the system of nonlinear differential equations.

Russian mathematician AM Lyapunov was proved that the judge may on static stability of the linearized system of equations, since small deviations — are those in which the system behaves as a linear. They were two theorems the essence of which lies in the method of the first approximation (the first Lyapunov method): the system is stable in a small (static) if sustained its linear approximation.

The method of studying the static stability for the linearized equations called the method of small oscillations or small deviations [4]. Linearization is as follows. Any independent variable and portrayed as a constant amount in the original mode and infinitesimal approximation:

Expanding the nonlinear function in a Taylor series and keeping only the linear terms of this series, we obtain a system of linear differential equations, but with respect to infinitesimal increments. Solution of this system has the form [9], [4]:

where p – the roots of the characteristic equation, which is in the canonical form looks like this:

Number of components equal to the order system of equations. Since the coefficients of the characteristic equation, which determines the actual parameters of the system — the real numbers, its roots can be real or complex conjugate. Thus, the characteristic equation and determining its roots may determine the nature of changes in small increments over time mode settings. Real roots corresponds to a member of the species . A pair of complex conjugate roots . When all the roots and the real parts of the complex roots are negative, all components of the transition process modulo exponentially damped. Overlooked mode in this case is statically stable. When the real roots of there at least one positive, the corresponding component of the transition process will exponentially grow indefinitely. Statically unstable output mode (aperiodic stability fault or slipping). When the complex roots of the pair will be having a positive part, then the corresponding component will be in the form of growing oscillations in time.

Thus, a necessary and sufficient condition for static stability of the original mode of the electrical system is the requirement unlike all real roots and real parts of the complex roots of the characteristic equation. There are methods (criteria ), which allow to determine the signs of the roots and thus to judge the stability of the system without identifying themselves roots of the characteristic equation. Order of operations for the study of static stability of electrical systems by small oscillations is as follows [9]:

1. Sostavit mathematical description of transient processes in the system in the form of nonlinear differential equations.
2. Proizvoditsya linearization of the original equations to obtain the linearized equations.
3. Sostoit characteristic equation and the characteristic determinant.
4. Otsenivaetsya stability using criteria that have signs of roots.

3. Criteria for evaluating the sign of the roots of the characteristic equation

Sustainability criteria are classified as direct, which require finding the roots, and indirect, that do not require calculations the roots. These include algebraic (Hurwitz method, Roth) and frequency (method Mikhailova, Nyquist, D-partition) [9].

4. Software for evaluation of static stability

The program is created in the environment of mathematical integrated package Mathcad. This is explained by the fact that an integrated package is the possibility of combining fragments of various systems together while using built-in features Windows. These include:

• Use all the tools Windows;
• many modern window interface;
• color registration documents;
• creation of dynamic animation files;
• creation of sound.

These systems are easy to use interface — the means of communication with the user in the form of windows that move, keys and other items. They have effective means of scientific graphics. Given that most of the time required for preparation of the text, it is advisable to work in an environment text editor Word, and paste objects with MathCAD [3]. This insert objects into a word processor with a mathematical system MathCAD gives full access to all the features and tools of the latter. It is also possible using hyperlinks cause a file created in an environment MathCAD. Using special keys when using hyperlinks easily carried turn to the previous or next document. Text, formulas, and simple drawings (output circuits, the equivalent circuit) is formed directly in the editor Word. Illustrations that are the result of calculations graphic (vector diagrams, functional dependencies, and others) are formed in MathCAD environment with the subsequent introduction of the resulting image in a document Word. Animated drawings (rotating vector diagrams and coordinate axes, functions and graphs, as well as consistency of results for different solutions of initial data, etc.) are also among the MathCAD. To create a sequence of animation frames we use the standard technology for creating animation files with the extension Avi. Double push the left mouse button when the cursor is on the drawing made by means of MathCAD allows to activate and use the latest opportunities for analysis, the influence of various factors on the character of the transition process. If you close the window MathCAD, with turning to document word, which was studied. Animation drawings can be called directly from a Word document using hyperlinks, or temporarily created in the medium MathCAD. Perform various kinds of calculations is possible to do, if you call from a text file on the basis of existing links MathCAD documents and vice versa when performing the calculations may refer to the text material to obtain the necessary assistance.

Software (SW) to assess the static stability is developed for a simple electrical system, which is shown in Figure 1

5. Evaluation sign roots using algebraic criteria

Hurwitz criterion.

For a regulated system with ARV proportional type of strong action values of the coefficients of the characteristic equation as follows [4], [9]:

where

Hurwitz determinants calculated and all additional determinants. Displayed on the screen for the output value of said Hurwitz determinants. Calculate the roots of the characteristic equation of the system is studied by means of symbolic mathematics Mathcad [3]:

Performed comparison of the values of all Hurwitz determinants and roots of the characteristic equation obtained. On the basis of this comparison, conclusions about the conditions of violation of various kinds of SU (aperiodic, periodic and self-excitation). Analysis allows us to determine the conditions of positive stability at different values of the gain ARVs. It is also possible to identify the impact of the value of the time constant ku regulation. Calculated as determinants Hurwitz, all additional determinants and roots of the characteristic equation. Analyzes all of these parameters and conclusions about the conditions of violation of static stability.

Routh criterion

For the simplest of the controlled system with a proportional counting ARV term coefficients table. Elements of each of the following terms found by the formula [5], [10]:

where k — number of the column; i — the row number in which the coefficient. The screen displays a table compiled by Routh. Based on the amount of change of the sign in the first column of the conclusions about the conditions of different types of violations SS.

6. Example controlled study, SS ACD proportional type

For the system to work statically stable at high angles of 90, you must enter the excitation control [8], It is known that statically stable system will work if the following condition.

where — the minimum required gain; — maximum gain [2]. Changing said coefficients depending on the starting angle of the load shown in Figure 2

For the given parameters of the system, which explores the maximum permissible ratio is independent of the initial angle and is 1.855. The minimum required ratio depends on the output mode see that the system can operate statically stable, the introduction of certain gain.

When the system will be aperiodic stability fault type.

Consider an example where the system is stable

From an analysis of the 3 that for given input parameters mode oscillation process is damped, which corresponds to a stable system. Mikhailov hodograph while also corresponds to a stable system.

Conclusion

Thus the development of software for the evaluation of the static stability of controlled electrical systems, the definition of signs of real roots, or the real parts of the complex roots of the characteristic equations, graphical interpretation using Mathcad, a number of advantages: clarity, simplicity, possibility of definition of sustainability, without additional calculations.

This master's work is not completed yet. Final completion: December 2014. The full text of the work and materials on the topic can be obtained from the author or his head after this date..

References

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