Matrix methods and algorithms for steady–statestability analysis of electric power systems

The relevance and demanding of the subject of the thesis. Today, in the world practice, in the sphere of ensuring the stable operation of electric power systems (EPS), pride of place goes to the creation of high-performance control systems of power generation and consumption processes, involving intelligent technologies. One of the most crucial tasks of rapidly developing modem EPS is ensuring its steady-state stability based on real-time operational data processing and analysis. In this area, the leading countries of the world focus on the improvement of control systems to ensure the stability of electric power systems, taking into account fluctuations of mode parameters. «Expenses for the creation of intelligent electric systems, including Smart Grid, arc as follows: USA - 7.1 trillion, China -7.3 trillion, Japan - 0.8 trillion US dollars. The use of the Smart Grid system in 2020 will allow the USA to save about 1.8 trillion US dollars»1.
Large-scale activities on the effective organization of power generation and improvement of the EPS stability arc held in the Republic of Uzbekistan. A number of research works arc carried out in this area, including high-performance reequipment of the steam-gas and gas-turbine technology, providing electricity generation, the development of effective control system of technological objects and improvement of methods and algorithms of research of control systems.
The world's attention is paid to the development of more advanced methods for determining the stability of electric systems, in particular, matrix methods and algorithms that allow to more deeply explore the modal properties of electric power systems, taking into account modem control devices. The implementation of targeted research is a priority in this field, while researches arc more relevant in the following areas: the development of matrix methods and algorithms for determining the stability of the electric power system based on automatic generator field regulators; the development of simplified criteria for determining the limit mode for the stability of complex EPS; the development of a model of synthesis of automatic field regulators on the basis of the system inclusion technology. Ongoing researches in the above-mentioned areas show the relevance of the subject of this thesis.
This thesis research is designed, to a certain extent, to perform the tasks stipulated in the Decree of the President of the Republic of Uzbekistan ПП-2343 dated 5 May 2015ycar “On the Program of measures to reduce energy consumption, to implement energy saving technologies in the fields of economy and social sphere for 2015-2019 yy.”, in the Decree of Cabinet of Ministers of the Republic of Uzbekistan No.238 dated 13 August 2015 year “On approval of the regulation relating to the republican commission on energy efficiency and development of the renewable energy resources”, as well as other legal documents adopted in this area.
The objective of the research is to develop matrix methods and algorithms of analysis of steady-state stability of complex EPS, to develop simplified methods of estimation of steady-state stability limit and synthesis of controller models based on the systems’ embedding approach.
Scientific innovativeness of the dissertation study consists in the following:
on the basis of Lyapunov’s function in the quadratic form, the simplified criterion of steady-state stability of the electric power system was obtained, which consisted in positivity of the first major minor of the quadratic-form matrix (<7n > 0), ensuring necessary and sufficient conditions of the EPS stability;
the combined use of the method of Lyapunov’s function in the quadratic form and node voltage equations was developed, which allowed for reducing studies of steady-state stability of a complex electric power system to the circuitry of a simple electric power system, or the “generator-bus” system;
a promatrix of the complex unregulated and regulated electric power system was developed on the basis of the system’s embedding method, which allowed for the study of dynamic properties of EPS;
the algorithm of pole transfer from the spectrum of the matrix of own dynamics of the electric power system was developed, which ensured stability and damping of fluctuations of the operation condition parameters at small oscillations in electric power systems;
mathematical model of the class of regulators of the complex electric system was constructed on the basis of the system’s embedding method, which ensured stability and damping of fluctuations of the operation condition parameters at small oscillations in EPS.
CONCLUSION
The theoretical and computational-experimental studies of steady-state stability of complex electric systems, carried out on the basis of the developed matrix methods and algorithms, using Lyapunov’s functions in the quadratic form and system’s embedding method, allow for the following conclusions.
1. Lyapunov’s functions in the quadratic form arc recommended as the effective method for studying the linear dynamic systems, including the electric power system described with the help of linear zed differential equations.
2. Adequacy of conditions of the steady-state stability loss of the electric power system, obtained on the basis of positivity of major minors of the quadratic-form matrix of the Lyapunov’s function in the quadratic from, to the same conditions provided by Hurwitz criteria, was proven.
3. Obtained theoretical and computational results allow one to investigate the stability of EPS “in the small” by means of analysis of the positivity condition of the first major minor of the Lyapunov’s function in the quadratic formq11; > Oand consider it to be the practical (simplified) criterion of steady-state stability of EPS, providing for both its necessary and sufficient conditions.
4. The combined use of Lyapunov’s functions in the quadratic form and nodal equations allows one to reveal the generator (station), operating in the complex EPS, which is coming to the limit in terms of steady-state stability. The mathematical condition of this proposition is dqnj /dFl -» max, i.e. the maximum of the derivative of the first major minor of the quadratic-form matrix by the regulated parameter for the j-th generator. It is evident that in this case, the study of the steady-state stability limit of the complex EPS turns into the study of the “generator-bus” circuit.
5. On the basis of the systems embedding method, promatrices of the unregulated and regulated complex EPS arc developed, which provide a full description of all kinds of characteristics of transitional processes, including the possibility of studying dynamic properties of electric power systems at small oscillations of their operation condition parameters.
6. The model is proposed, in which the regulator of the complex electric power system has been synthesized on the basis of the systems embedding theory to describe analytically the class of regulators ensuring stability and damping of oscillations of the studied EPS.
7. The conducted computational-experimental studies aimed at analysis of steady-state stability of complex EPS on the basis of the systems embedding method have shown the qualitative match of the obtained results with the results, checked in the course of practical running of electric power systems on the basis of Classical Methods that confirms the adequacy of the models developed to the already existing ones.
8. The mathematical model of the electric power system, resolved relative to deviations of absolute angles of synchronous generators, is developed, which can be used autonomously for the study of small oscillations of complex EPS. This model of small oscillations must be used together with node voltage equations, determining voltage modules of nodes and their arguments, representing absolute angles relative to the balancing node.