Olasılığa bağlı kısadevre analizi
Probabilistic short circuit analysis
- Tez No: 46501
- Danışmanlar: PROF.DR. NESRİN TARKAN
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Kısa devre, Short circuit
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Elektrik enerji sistemlerinin tasarımında arıza akımı önemli bir büyüklüktür. Arıza akımını etkileyen başlıca parametreler, sistem durumu, arıza yeri ve tipidir. Deternıinistik arıza analizinde bu parametreler sabit tutulup, genelde en kötü durum için arıza alamı hesaplanır. Artan maliyetler, bu parametrelerin değişkenliğini gözönüne alan olasılığa bağlı yöntemlerin geliştirilmesine yol açmıştır. Olasılığa bağlı kısadevre analizinin amacı bir bölge ya da ilgilenilen bir bara için arıza akımının olasılık dağılımını bulmaktır. Bunun için olasılık teorisinin analitik ifadelerini kullanan bir yöntem ve Monte Carlo Yöntemi kullanılmıştır. Analitik yöntemde tek kaynak ve battan oluşan en basit sistemden başlayarak daha karmaşık sistem yapılan için arıza akımının olasılık dağılımı çıkanhnıştir. Sayısal bir benzetim yöntemi olan Monte Carlo Yöntemi sistem parametreleri değiştirilerek, arıza yeri ve tipinin rastlantısal olarak atanmasına dayanır. Her benzetimde arıza akımı klasik kısadevre analizi bağıntıları yardımıyla hesaplanarak yeterli sayıda benzetim sonucu arıza akımının olasılık dağılımı elde edilir. Bu çalışmada her benzetimde bara empedans matrisini değiştiren bir algoritmaya uygun olarak örnek bir sistemde arıza akımının olasılık dağılımı histogram şeklînde elde edilmiştir.
Özet (Çeviri)
Electric power systems are designed and operated to supply uninterrupted energy to customers. When an electric fault occurs in the system, a fault current of magnitude mucit higher than the normal load current flows in the system components. The magnitude of this current depends on the structural system characteristics, fault location system conditions just prior to the fault and type of fault. Computations of the short-circuit currents fkrvving in the system during power system faults is the domain of the short-circuit analysis. The short-circuit current in power systems is a very important parameter affecting the design of bus systems, grounding systems, circuit breakers, substation apparatus, rotating machines and in fact almost ali aspects of power system design, Fault levels continue to increase and utility companies try to uprate facilities in order to get the most out of existing systems and equipment. As a resuh, the use of probabilistic methods is becoming increasingly attractive. Fault location, system conditions and fault type are the three majör components which affect the magnitude of the short-circuit current. These components are random in nature. in deterministte short-circuit analysis, the system conditions are postulated and the fault currents are computed for various combinations of generator dispatches, netvvork configurations and fault types. Usually, the worst-case conditions are assumed in order to limit the number of studies to be performed. For deterministic short circuit analysis, a variety of computerized techniques have been described. Öne of them is to form bus admirtance matrix and then taking the inverse of this matrix to from bus impedance matrix, By using bus impedance matrbc, short-circuit currents are calculated by solving the network equation, V=Zl(1) where I is the column vector of nodal currents, V is the column vector of nodal voltages and Z is the bus impedance matrbc, The follo\ving simplifying assumptions are used in short-circuit calculations. i) Ali load currents are negligîble ii) Ali generated voltages are equal in phase and magnitude to the positive sequence prefault voltage E which is equal to 1.0 pu. iii) The networks are balanced except at fault points iv) AH shunt admittance is negligîble.Wtih these assumptions and using symmetrical components for the unbalanced faults at the bus m, the fault current is given as, ir=(^+zrrE^ (2) where Z^4,2 is the symmetrical component fault impedance matrix. Similarly, the voltages at the busses other than the faulted bus line currents are calculated with the below equation.. V0W = E0,U _ ZjU,2 (Z0.U + Z0X2 yl EW (3) i i im v t mm / m >> / The fault current in the three phase elements i-j is given i?A2 = 21IL_ (4) «J "0,1,2 V ' Zij where zjj4,2 is the symmetrical component primitive impedance element between nodes i and j. For any kind of fault, three phase, three phase to ground, line to ground, line to line, line to line to ground faults, the corresponding fault impedance or admittance matrix is substituted into equations (2), (3) and (4) to obtain fault currents and voltages. In order to take into account of the variability of key parameters in short-circuit analysis, the probabilistic approach is developed. Probabilistic short-circuit analysis provides probability distributions of fault currents either in a region of interest or at a particular bus under study while the system experiences random parameter changes, component outages, operating decisions and load fluctuations as well as random fault conditions. This study present two methods for the development of the probability distribution of fault currents. The first one is analytical approach which uses the analytical expressions derived from probability theory. The second one is based on a Monte Carlo Simulation. Analytical method is only applicable to three phase faults, but cheap in terms of computer time. Monte Carlo approach is more genral but requires an extensive computations effort. Before going into these approaches, basic principles of probabilitiy theory random variables, Monte Carlo Simulation are given. Many simple power system problems can be analyzed with the help of elementary probability theory. Probabilistic theory helps the engineer to measure the uncertainly. In order to measure the uncertainly, first sample space and events are defined. Sample spaces can be classified as discrete or continuous. Whether a sample space is discrete or continuous, a collection of sample points contitutes an event. The probability of an event is best given by the relative frequency theory. If among N trials, the occurence of the event A is k then the probability of A can be given as xmP(A) = Hm£ (5) Then with the help of the set theory, the combinaiton of events, conditional theory and independence, and basic axiams and theorems of probability can be given. Random variable X is a function that assigngs a real number X(s) to each element s of a sample space S. The rule for describing the probability measures associated with the values of a random variable is called a probabiity distribution. For any randoim variable X, Fx(x) is defined as Fx(x) = P(X<x) (6) For discrete random variable X, its probability distribution may also be described in terms of probability mass function which is a function describing P(X=x) for all x. For a continuous random variable, an interval (a,b) on the reasl axis with a<b can be cosidered on the real axis and then the probability P(a<x<b) can be computed with the help of a probability density function f*x(x). Sometimes a random variable can be described with expected value or variance and standart deviation. If a function of a single variable is given like Y=g(x) (7) the problem is the find the probability distribution Y when the probability distribution of X is known. The general technique for solving this problem consists of three steps: 1. Express the event (Y<y) in terns of an event involving variable X. 2. FindF^y) 3. Differentiate FY(y) to give fy(y) and determine its range of definition. Monte Carlo simulation is based on selecting a set of values of system parameters and obtaining a solution of the system model for a selected set. By repeating the simulation process for different sets of parameters, different sample solutions are obtained. The sample solutions may be treated satisticafly. Every Monte Carlo study requires generation of the appropriate values of the random variables in accordence with the respective prescribed probability distribution. If u is a value of the standart uniform variate U, the corresponding values of X can be given with x = F;» (8) for cont,nuous random variables and with Fx(xi_1)<u<Fx(xi) (9) for discrete random variables. xrvAnalytical method involves the determination of the fault current magnitude at the bus of interest as a function of the fault location on given lines in the network. This function is used with the probability density distribution of fault locations on the lines to obtain the conditional probability density distribution of fault current magnitudes for faults on the various lines. By assuming the faults are uniformly distributed on the line, first the fault current magnitude probability density distribution for a simple single-line single-source system is derived. 1 V 1 V V fj(i) = --^4 - - - ^- (10) JLv Jit] JL VUo * Zj+1-j ^g In this equation, L is the length of the line, Vs is the phase-to-neutral system voltage, Zs is the positive sequence equivalent source impedance (ohm/phase), and Zj is the line impedance (ohm/phase per unit length). Then by considering successively more complicated systems, methods are developed for combining the fault current probability density distributions for faults on portions of the system to obtain the overall fault current probability density distribution, fI(i) = 2P(faultonline).fI.(i) (11) where n is the number of lines in the system. For multigenerator systems containing loops, since the function of fault current of fault location isn't one to one, it's hard to obtain the probability density function. In this case, Superposition Theorem which derives the analytical probability density functions for successive subsystem each containing one voltage source, all others being set to zero, and calculating the combined probability density function from a function of these partial probability functions may be used. Monte Carlo approach simulates the behavior of the power system under fault conditions by applying short-circuits at random times and random locations in the network. The fault times and locations are drawn randomly from probability density distributions for (i) the fault distribution of lines in the network, (ii) the locations of faults on the lines (in) the time of occurrence of faults. These distributions are taken from power system location fault data. At each simulation, fault location and fault type are specified by random number generators using these distributions. A new algorithm is developed to change the bus impedance matrix according to fault location. In this algorithm a new bus m is added at the fault location at line (ij) so a new bus impedance matrix Zb is formed which has one more line and column from the old bus impedance matrix Z. The elements of Zb is calculated as the function of elements of Z matrix, the parameter X which is a continuos random variable having uniform distribution and specifies the fault position and the line impedance zy, XVZbmv=(l~X)Zio+XZjo (12) Zbmm=(l-X)Z,+X2ZB+2X(l-X)Zji+X(l-X)zij (13) By using these expressions and the equations (3) and (4), bus voltages and line currents are calculated. After M simulations, the probability distributions of fault currents in the form of histograms which show the number of fault occurrences with certain values of the current being calculated can be given. In this study, because of the lack of statistical data, only the fault location is being changed randomly. For a sample system, a computer program is developed to obtain the probability distribution of fault currents for three-phase faults in the form of the histograms. XVI
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