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Probabilistic eigenvalue analysis method of power system using central moment and cumulative quantity

classification number: TM 712 document identification code: a

article number: (2000) a hybrid algorithm using moment and cumulative

for power system probabilistic eigenvalue analysis Ke Wen, Zhong Zhi Yong, TSE c t, Tsang K m

(Department of electrical engineering, the Hong Kong Polytechnic University, Hong Kong, China) abstract:based o the current promising eigenvalue studies, this paper presents a hybrid algorithm using moments and quantities of random variables Uncertain considered are system multi- ④ operating conditions derived from operating curves of nodal loads and generations By means of the first and second order eigenvalue sensitivity representation with different approximation, moments and cumulants of eigenvalues are determined from the statistical nature of nodal voltages and nodal injections. Probabilistic distributions and stability probabilities of critical eigenvalues are calculated from the Gram-Charlier series. In the proposed hybrid algorithm, random variables can have arbitrary distribution. Dependencies among random variables, MIT aerospace engineer designed carbon nanotube "needle" interaction between expectations and variables, as well as the correction of covariance to expectation are all considered

KEY WORDS：dynamic stability; eigenvalue; probability; Cumulant1 introduction the main purpose of probabilistic stability analysis of power system is to determine the probability distribution of critical characteristic root and the probability of dynamic stability of the whole power system. The probabilistic method was first introduced into the dynamic stability analysis of power system in [1]. Under the premise of normal distribution, the mean value and variance of the real part of the characteristic root are calculated from the probability characteristics of some system parameters by using the linearized system model, and the stability probability of the whole system is calculated by using the concept of joint normal distribution. Later, the author extended the algorithm to include random variables with arbitrary distribution by using the method of higher-order origin moments. For this problem, the following methods are adopted in paper [3] to improve the calculation accuracy: the second-order sensitivity method, the polynomial curve fitting method and the neural network learning method. The algorithms introduced in [1 ~ 3] are implemented on an 11 node two machine system. The random factors considered are the uncertainty of generator rotation angle and mechanical damping. In order to consider the influence of node power fluctuation on the static stability of the system, paper [4] uses the method of accumulation and gram series to calculate the simple static stability criterion dpi/d δ Probability characteristic of I 0. The uncertainty considered in [5] is the system multi operation mode based on the node load operation curve. Under the assumption of normal distribution, the initial operating state of the system and the mean and covariance of node voltage are determined by probabilistic power flow calculation; Then the distribution probability of the critical characteristic root is obtained by using the linearized system model [6]. The concept of probabilistic dynamic stability has also been used to analyze the probabilistic characteristics of dynamic stability boundary curves [7] and constitute the probabilistic dynamic model of induction motors [8]

according to the influence degree of different order moments of the characteristic root on its overall probability distribution, the calculation of each order moment of the characteristic root is treated with different degrees of approximation. Among them, the average calculation is the most accurate, so I won't refuse to take into account the influence of variance on the average; Variance takes the second place; The third order uses the central moment taking into account the correlation of variables; The fourth and fifth orders use cumulative quantities to form the mixed use of central moment and cumulative quantities. Finally, gram Charlie series is used to determine the probability density and stability probability of characteristic roots. 2 multi machine system model in this paper, the "general multi machine system representation" (GMR) technology [6] is used to form the state space equation of the system. This technology allows the generator and its related control system to be described in any detail, and is conducive to the expression of the sensitivity relationship between the characteristic root and the node voltage. The load model adopts the following exponential form: PL = pl0val, QL = ql0vbl (1) take the linearized coordinate transformation into the generator model, and all generators can be directly connected with the external power network, as shown in Figure 1. For a detailed description of GMR, see [6]. Fig. 1 multi machine system model

fig.1 multi machine system representation3 the main difficulty of probabilistic dynamic stability analysis of mixed arithmetic is still the contradiction between the calculation accuracy and the calculation amount, which is mainly reflected in: ① the restriction of random variable types (such as normal assumption) and the consideration of the correlation between variables; ② The simplification degree of the expression between the characteristic root and the original random disturbance variable, that is, whether to retain the nonlinear term; ③ The influence of high-order moments on low-order moments in random variable operations, and so on. The order moment method [2] allows random variables to be arbitrarily distributed and takes full account of the correlation between variables, but the calculation of high-order central moments is considerable. The accumulation method [4] has the advantages of simple formula and fast calculation speed, but it is based on the assumption that random variables are independent of each other. Retaining the nonlinear term in the eigenvalue sensitivity formula can improve the calculation accuracy, but the amount of calculation increases sharply

in addition, the numerical calculation of probability distribution shows that in gram Charlie series, the higher the accumulation, the smaller the influence on the whole distribution curve. In order to reduce the amount of calculation, the paper will

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