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Preliminary study on classic rocking equation and chaotic phenomenon of generator

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The preliminary study of the classic rocking equation and chaotic phenomenon of the generator, the system does not have a chaotic phenomenon in strict sense. However, there is a time behavior similar to chaos, such as instantaneous chaos, and bifurcation and other phenomena. This indicates that the above model cannot be used to study the chaotic phenomenon in strict sense. The above conclusions are initially proved by the matrix eigenvalue analysis of the variation equation of the system equation.

0 Introduction In order to study chaos in the dynamic simulation of the actual power system, this paper begins with the simplest model, and gradually gives some results of the chaotic theoretical analysis or numerical simulation. It is believed that if the generator adopts a classic rocking equation model, the load is indicated by a constant passive impedance, and the system does not have a strict chaotic phenomenon, but there is a phenomenon such as instantaneous chaos.

Index and its significance for non-linear continuous autonomous systems, inter-line behavior is φ to guide the X-see for X, can be obtained: the decomposition I starting in the formula is a deflection number matrix of the unit array D, the following isamer.

Defining the variation equation of the equation (1), it is a characteristic value that describes φ) varied over time, the corresponding Lyapunov index is a feature root of the λ destination matrix), there is M T. That is, the LYAPUNOV index λ is equal to DF (X) eigenous real index and system time behavior of the system time behavior, the maximum Lyapunov index is λ 0 indicates that the system track shrink λ = 0 indicates that the system track is a cycle behavior. There is a λ 0, then its track is a chaotic behavior 2 eigenvalue analysis and a proof of chaotic demonstration 2. 1 DF (X) selector 3 of the classic swing equation containing 3 generators The selection generator 3 is a slack node. If the load is indicated by a constant impedance, the temporization network is read from the endpoint of the generator, and the network's gateway is I = Ye, where Y is the diagonal element of II, and the non-diagonal element is Y IJ. Then, the classic rocking equation of the other 2 generators is: the constant potential W for the transient reactance of each generator is constant.

To: wherein A = 2. 2DF (x) is analyzed for the n-step matrix M. The primary diagonal element is remembrenected as a NN, and its n feature value is λ [2]. For the above DF ( x), it is apparent that the following theorem.

Theorem 1λ Theorem 2λ = DF, is not equal to 0, that is, it is proved to first unfolded the remaining sub-estimation of DF, i.e., DF = AF BC. Then use the anti-counter method.

2. 3 The classic rocking equation containing 3 generators does not have a chaotic phenomenon in the constant load. 3 For systems containing 3 generators, if the generator uses a classic rocking equation model, the load is indicated by a constant passive impedance, then There is no chaotic phenomenon in strict sense.

It is proved that it is known from the variation equation (3) that at each time t, it is: it is a linear approximation of X = f (x) along its trajectory. In general, the feature value of the in each instant is O view, since the Lyapunov index λ is equal to the corresponding feature value of the DF (X) (this 1.2), the DF (X) is analyzed. The feature value, the type of time behavior of the system (6) (this article 3 and theorem 2) are known. That is: a. DF ≠ 0 means that each of the features of the DF is not equal to 0, and if each feature root is true, the time behavior of the equation (6) or converges to a balance point, or diverged.

b. Because DF is a real matrix, its feature root or is a real number, or is a paired plural. If a pair of pure imaginary numbers are included, the solution of equation (6) is periodic behavior. At this point, there is also DF ≠ 0.

c. Because the chaotic system must have a λ [1], the time behavior of the system (6) must contain many points to satisfy DF = 0.

It can be seen from theorem 2, and the DF is equal to 0 under certain limited specific conditions. Therefore, chaos phenomenon does not occur.

3 Some specific example LiU et al. Studied the system of 3 generators in the literature, and the generator adopts a classic rocking equation model, and the system has instantaneous chaotic phenomena and (similar to).

For the case in [4] 1: If the DF (x) in this article 2. 1, it is not equal to 0. Therefore, there is no real chaotic phenomenon in the system. From the results of numerical simulation, instant chaos is not true chaos.

After approximately 20 s ~ 30 s, each example has become a small cycle behavior. For example, the time behavior of the disturbance 1, P is shown in Fig. 1, and Fig. 2, respectively.

From the example of the above numerical simulation, it can be visually seen that there is no real chaotic phenomenon in the system.

For the case of literature [4] 2:? Academic papers? The preliminary study of the classic rocking equation and chaos phenomenon of Yang Zhengzi et al. The time behavior of the preliminary study of the chaotic phenomenon is 0, the system's damping coefficient is 0, that is, = 0, not satisfied with the necessary premise for the chaotic system λ 0, so it is not true Chaotic system, the system does not have a real chaotic phenomenon. From the results of numerical simulation, the time behavior of P is shown in Figure 3, and Fig. 4. About 200 S is previously similar to chaotic behavior, and bifurcated behavior after 200 s.

Time Behavior 4 For 4-machine or multi-machine system, the form of DF (X) is the same as DF (x) in this article 2. 1 in the form of DF (X) in this article, but the order is increased. It is not difficult to prove that DF (X) is equal to 0 for some specific points, so there is no chaotic in the true sense. Because there is a certain range of variables of the chaotic system, DF (x) does not appear to have a number of 0 values, which cannot meet some λ5 related issues of power systems have chaotic phenomena, and now there is no conclusion. Especially in experimental studies. Numerical simulation discovered chaos and bifurcation in a certain extent. This paper believes that there is no real chaotic phenomenon in a constant impedance load for a system containing 3 generators. The classic rocking equation is under constant impedance load. However, numerical simulations have found a phenomenon similar to chaos and bifurcation. Therefore, it is not ideal for studying chaos using a classic rocking square, preferably other generator models, or other load models. However, since the classic rocking equation is only suitable for 1 s, it is feasible to study instantaneous chaos.


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