Küçük bozucu etkiler altında güç sistemlerinin dinamik davranışlarının incelenmesi

Abstract

ÖZET Genel anlamda güç sistemlerinin kararlılığı, bütün enterkonnekte üretici ve tüketici sistemlerin paralel ola rak senkron işletimlerinin sağlanmasıyla ilgilidir. Para lel çalışmadaki aksaklık, -küçük veya büyük bozucu etkiler tarafından oluşabilir. Enterkonnekte güç sistemlerinin lineer dinamik performansı, normal işletim koşullarında güç sisteminde oluşan küçük bozucu etki ler altında sistem makinalarının davranışlarıyla ilgilidir. ilk birkaç salınımda açık ola rak gözlenmemesine karşılık osilasyonlar sistemin işletim karakteristiklerine bağlı belirli bir değere erişince, dinamik kararsızlık ortaya çıkar. Bir enterkonnekte güç sisteminin küçük bozucu et kilere karşı dinamik davranışı, temel altsistem modelle rinin kullanımıyla incelenmektedir. Küçük bozucu etkiler altında genel kararlılık inceleme metodu, sistemi tanımla yan diferansiyel ve cebrik denklem takımının lineerleşti- rilmesi olmuştur. Güç sisteminin dinamiğini gösteren eşitliklerin bir çalışma noktası civarında lineer leştiril- mesi ile toplam sistem, dış girişlerden arınmış bir özerk sistem olarak elde edilebilir. Dinamik kararlılık..analizleri, altsistemlerin oluş turduğu toplam sistem matrisinin özdeğerlerinin bulun masıyla yapılır. Asimptotik kararlılık için toplam sis tem matrisinin özdeğerlerinin reel kısımları negatif işa retli olmalıdır. Dinamik kararlılığın iyileştirilmesinde özdeğer duyarlılığı ve lineer programlama kullanılarak yapılmaktadır. Lineer programlama teknikleri ile sistemin kritik özdeğerleri sol yarı düzleme kaydırılarak sistemin kararlı çalışması sağlanabilir. Lineer programlamayla birleştirilmiş özdeğer duyarlılığından yararlanarak dina mik kararlılığın oluşturulması, farklı dinamik karakte ristiklere sahip güç sistemlerine başarıyla uygulanabile cek bir yöntem olarak görünmektedir. - v -

summary power system dynamics under small perturbations In recent years, electric power systems, worldwide, have grown markedly in size and complexity. In order to maximize efficiency of generation and distribution of electric power, the interconnections between individual utilities have increased and the generators have been re quired to operate at maximum limits for extensive periods of time. In addition, the most economic sites for genera tion plants are often remote from load centers and power must be transmitted over long distances. The majority of power system interconnections are made through AC trans mission lines arid the interconnected system, there may be thousands of synchronous generators in service to supply the load. With the advent of in ter connection of large elec tric power systems, many new dynamic power system prob lems have emerged. A problem of much interest in the study of large electric power systems is the elimination of low- frequency oscillations, which often arise between coherent areas within a power network. These oscillati ons are related to the dynamics of inter area power trans fer and often exhibit little damping. With the utilities increasing power exchanges over greater distances and in greater amounts, the use of conventional power system stabilizers may not provide sufficient damping for these interarea modes. The reliability of the interconnected system is enhanced by virtue of the capability of transferring po wer readily from one area to others within the system. But in the meantime, the multiple interconnections of multi^areas make the system much more vulnerable to in stability, not only because of the complexity of multi- area interconnections but also because of the drastic reduction of spinning reserves of individual areas. The stability characteristics of large electric power systems is too varied to permit a simple classifi cation of all behavior, and it is not possible to separa te all analyses into simple categories^ Because of their essential nonlinearity, the stability of power systems depends on the severity of the applied disturbances. Criteria for power system design specify the types of fault the system must be able to withstand without major loss of synchronism and consequent breakup. It is also - vi -critical that the power system remains stable while ope rating with no faults. Power system analysis refer to these separate, but related, stability problems as tran sient stability and dynamic (small-signal) stability. Transient stability studies are limited to rela tively short time intervals, typically is or less. They are most often used to determine the stability of a sing le unit or plant during the initial period of high stress immediately following a nearby fault. On the other side, dynamic stability studies cover longer real-time inter vals. In general the system operating conditions are res teri c ted most by the need to maintain transient stabi lity. In recent years, however, as power systems have been operated with higher power transfer levels to meet economic constraints, dynamic stability problems have become apparent. In order to achieve the required high transfers of power, the controls associated with the ge^ nerators have become critical. The dynamic stability of electric power systems has been a subject of major theoretical and practical interest since the advent of interconnection of large electric power systems, and it continues to grow in import tance as the control requirements of the power plants become more sophisticated and demanding. This thesis is concerned with the representation of steam and hydraulic turbines, their speed-governing systems, synchronous generator models, control system of the various voltage regulators and excitation systems now available. Basic models for turbines in power system sta bility analyses are presented. These models provide ade quate representations for mechanical and electrical sub systems of an interconnected power system in most stabi lity analyses. In this study, the mathematical model of a multi- machine system has been obtained by linearizing the sys tem equations about an operating point. Dynamic performance for small perturbations of an interconnected power system is studied using basic models which are given in this report... The general method of studying stability under small perturbations has been to linearize the set of differ antial and algebraic equations describing the interconnected system. The mathematical description of such model before linearization is made up of the following two types of equations. 1. Subsystem equations (Dynamics).~.' V1Xx. = f.(x., u., t) (1) Yi - <3±{x±, u±, t) (2) where x. : state variables of subsystem i x. : time derivate of x. a. i u. : input variables of subsystem i y. : output variables of subsystem i f.,g. : time-varying nonlinear functions. 2. Interconnection equations (Topology) U - FY fGV (3) W = JY f KV (4) where U : vector of all u. Y : vector of all y. V ; overall system input variables W : overall system output variables.F-/C3./J/K : tiftie*- varying interconnection {incidence) matri ces. after linearization of the equations representing the power system dynamic around an operating point, which will represent the behavior of the system under small perturbations, and accounting for the interconnection equations, the total system of equations can be represent ted by an autonomous system of equations without external inputs in the state space form: X = AX Where X : vector of state variables öf the autonomous system X : vector of time derivate of X -- viii -A : coefficient matrix of the total system. The solution of Equation (5) have the general form: A-, t /2t Xnt X = C,M,e + C,M,e +... + C Me n (5) xl. z z n n ' where a. : eingenvalues of the matrix A U. : corresponding eigenvectors of the matrix A G. : arbitrary constants determined from initial conditions Analysis of dynamic stability is done by finding the eigenvalues of the matrix A which can be obtained by the techniques studies in this report. Improvemens of dynamic stability can be done using eigenvalue sensitivi ties and linear programming. This method using eigenvalue sensitivities and linear programming is developed to select the relevant parameters of a linearized dynamic power system in order to maximize shifting the critical eigen-values to the left at a given operating condition to improve system stability. In this method, parameters of interconnected sys^- terns can be selected without neglecting interaction bet ween dynamic subsystems. The method is general and can be used to select the best parameters for any linearized interacting dynamic systems, optimizing cost and dynamic performance. The sensitivity matrix relating the change of system eigenvalues with respect to its parameters is utilized to develop an iterative algorithm employing li near programming techniques to shift the system's criti cal eigen-values to the left. The flow chart of this algorithm is given in the following figure. The method developed in this study to optimize shifting the system's critical eigen-values using linear programming techniques has been successfully applied to power systems with different dynamic characteristics. ix -K=K + I Q START INPUT dc'0' ) FORM WE SYSTEM A-MATRIX ÂK,*A^W>, COMPUTE,K, # OF 4K) COMPUTE i SOLVE THE LINEAR PROGRAMMING PRO& MiN.£ %îaoŞ' Jal SUCH THAT: $!<**^ UPDATE ¦o^%<^^ OUTPUT K, ^ Figure. Flow Chart of tne linear Prog jl airlifting Algorithm used to Shift the System's Critical Eigen-Values.