 Energy and Power Engineering, 2013, 5, 667-669 doi:10.4236/epe.2013.54B129 Published Online July 2013 (http://www.scirp.org/journal/epe) Research of Network Transient Performance at Small Perturbations T. A. Makhkamov, A. M. Mirzabaev National Research University: Moscow Power Engineering Institute, Moscow, Russia, Tashkent State Technical University, Tashkent, Uzbekistan Email: temur.ma@gmail.com, solarmir@mail.ru Received February, 2013 ABSTRACT The advisability of the use of matrix methods of equations rearrangement for the investigated system which allows writing a secular equation is considered in this article. This approach greatly simplifies the analysis of performance of transient response in complicated multi-coupled electrical system at small perturbations. Keywords: Stability; Process Control Performance Factors; Transient Response; Matrix Analysis; Coefficient Matrix; Secular Equation 1. Introduction The stability is necessary but an insufficient condition of automatic control systems operability . The control system stability means only that there is a decaying of the transient response in the system under the influence of external control or perturbation action. Upon that, a process decaying time, maximum deviation of controlla-ble value and number of oscillations in the system are not defined, however, these values are very important proc-ess control performance factors. Process control performance factors can be defined by means of various methods. First of all, they comprise transient response design by the set closed-loop transfer functions, definition of performance factors by a disposi-tion of zeroes and poles, integral performance criteria, frequency-domain performance estimation and fre-quency-domain methods of transient response design [1-3]. 2. Analysis In case of electrical power system, the differential equa-tion system describing processes in such system are lin-ear (linearized) and look like in the matrix form : xAx, (1) where 11 12121 22212...........................................nnnn nnaa aaa aaa a (2) and х = [x1, x2, …, xn] – a column matrix comprising de-viations of required parameters of electrical power sys-tem condition. The problems listed above become insuperable in cas-es of the systems described by equations with high de-grees, i.e. complicated or multi-coupled systems. A pri-mal problem is deriving of a secular equation for the in-vestigated system. In this connection it is expedient to use matrix methods of rearrangement of the equations for the investigated system, then to receive a coefficient ma-trix for the differential equations A and further to set a secular equation under known algorithms of its setting. It is rather effective to apply Boher formulas  which generate factors of a secular equation of the investigated system. Let’s consider an algorithm of generation of se-cular equation coefficients by a known matrix of coeffi-cients of a matrix A. Product of characteristic numbers of a square matrix A is equal to a determinant of this matrix. Let’s notice that in case of equality to zero of any characteristic number the matrix A is singular [5, 7]. The sum of diagonal elements of a square matrix is equal to the sum of its characteristic numbers. Consider-ing importance of this property, the special title, namely, a matrix trace is appropriated to the sum of diagonal elements of such matrix. This property is used to form coefficients of a secular equation for the investigated system. Having designated the trace Aк (of the matrix A multi-plied k times by itself) by Tк it is possible to write the useful recurrence formula expressing coefficients of a secular equation in terms of various Tк, so: Copyright © 2013 SciRes. EPE T. A. MAKHKAMOV, A. M. MIRZABAEV 668 a1 = - Т1, а2 = - (1/2) (a1 T1 + T2), a3 = - (1/3) (a2Tl + alT2 + T3), (3) ап = - (1/n) (an-1 T1 + ап-2Т2 +... + а1 Tn-1 + Тn). This formula makes possible to define a secular equa-tion diversely. It is obvious that it is rather effective for algorithmization of determination of a secular equation’s coefficients. Let’s apply algorithm (3) to model of an electric sys-tem (electrical power system) which has n = 6 in the case of availability of a strong excitation on synchronous ge-nerator with automatic excitation control; the matrix of differential equation coefficients for the system looks like: 1221 22233233 344451 52535561 63660 00000000000 0000000aaa aaaaAaaa aaaa00ai (4) where aij - matrix elements which depend on condition and system parameters . The solution (1) looks like: 111111()2( ) ()(())kkinnttkkntHi ix tB eBeSintMD e  (5) where МН (i) and D'(i)) – polynomials defining zeroes (numerator) and poles (denominator) of a transfer func-tion of the investigated system . For the investigated system under initial conditions: U = 1, Рd = 3, xc = 0.3, xd = 2.3, j = 7 s, d0 = 2 s, e = 1 s, I = U = 0.1 s, k0U = 10, k1U = 30, k0 = 10, k1 = 10 and δ=700 elements of a coefficient matrix (4) are equal to: a12 =1, a21 =-19.0431, a22 =-134.5714, a23 =-18.3263, a32 = 2.6624, a33 =-1.9167, a34 = 1.9167, a44 =-1, a45 = 1, a52 = 75.1048, a55 =-10, a61 = 2.5035, a63 =-0.9014, a66=-10. Application of algorithm (3) to (4) gives the following secular equation: D (i) = a06 + a15 + a24 + a33 + a42 + a51 + a6 = 0 (6) with coefficients: a0=1, a1=157.5, a2 =3312, a3 =23373, a4 =62155, a5 =126430, a6=50859. Thereby roots of the secular equation: 1 =-10,00, 2 =-134,06, 3=-10,78, ,5=-1.07j 2.41, 6=-0.51. As may be inferred from structure of the roots, dominating roots have the real and complex values. It means that in the case if a system condition became heavier there are probable both non- periodic loss of stability and self-oscillation. Transient responses and their performances can be checked up with a traditional method applying a unit step excitation and delta function to an input as a result of which action it is possible to receive transient and weight characteristics of the investigated system [1, 4]. For this purpose it is necessary to uncover a right member (6) taking into account automatic excitation control parame-ters and roots of a secular equation and its derivative [8-10]. Figure 1 represents the transient characteristic in the case of excitation control by a deviation and the first- order derivative with respect to an angle. Figure 2 shows the pulse-response characteristic of the investigated electrical power system in the case of application of a delta function. For the given automatic Figure 1. The transient characteristic of the elementary electric al power sy stem in the case of availability of a strong excitation to synchronous generator with automatic excita-tion control: k= 50, k = 10, kU = 10, kU = 30. Figure 2. Pulse-response characteristic of the investigated model of an elementary electrical p o wer syst em in the case of availability of a strong excitation to synchronous generator with automatic excitation control: k= 150, k = 10, kU = 10, kU = 30. Copyright © 2013 SciRes. EPE T. A. MAKHKAMOV, A. M. MIRZABAEV Copyright © 2013 SciRes. EPE 669ntrol parameters the system is self-oscillat- 3. Conclusions ws, realization of the given technigement ofREFERENCES  N. N. IvaschenTheory and Systemanical Processes inexcitation coing. The analysis shows that self-oscillation is removed if to select (to optimize) amplification coefficients for the first-order derivative with respect to an angle and gen-erator voltage, provided that efficiency of the channel of a derivative with respect to an angle is essentially higher, than to the voltage channel. As the analysis shoque  K. R. Allaev and A. M. Mirzabaev, “Matrix Methods of Research of a Static Stability of Electric Systems,” Arti-cle in the Issue of the Magazine. for transient response research in case of complicated electrical power system is most effective, as the secular equation is formed on the basis of a coefficient matrix for the investigated system, and the higher degree of the dif-ferential equations, the more advantage as regards to formalization and computing time consumption. 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