Mathematical and Computer Modeling of the Stability of Complex Electric Power Systems Maksat Kalimoldayev Institute of Information and Computational Technologies CS MES RK, Pushkin str.125, 050010 Almaty, Kazakhstan. mnk@ipic.kz Maksat Akhmetzhanov Institute of Information and Computational Technologies CS MES RK, Pushkin str.125, 050010 Almaty, Kazakhstan. maks714@mail.ru Assel Abdildayeva Institute of Information and Computational Technologies CS MES RK, Pushkin str.125, 050010 Almaty, Kazakhstan. abass 81@mail.ru Fariza Galiyeva Institute of Information and Computational Technologies CS MES RK, Pushkin str.125, 050010 Almaty, Kazakhstan. galyeva.faryza@gmail.com Abstract This article discusses the development and study of mathematical model of complex power systems for the global asymptotic stability problems. The conditions have been obtained for the global asymp- totic stability of nonlinear control systems. Control actions that ensure stabilization of complex electric power systems have been found. The software package of dynamic study of complex electric power systems has been developed in Visual Studio using the programming language C-Sharp. 1 Introduction The industrial development of modern society leads to a constant increase in electricity consumption. Complex electric power systems are created to meet these growing needs. Copyright ⃝ c by the paper’s authors. Copying permitted for private and academic purposes. In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org 273 Under current conditions of Kazakhstan’s power generating industry development is at a high level of wearing out and insufficiently high rates of equipment modernization, this question is even more acute [Jacobson & Ycesan, 2008]. Besides Kazakhstan’s power industry modernization necessity, it is required to estimate effectiveness of modern equipment use and conditions of its normal operation. By their normal functioning electric power systems (EPS) provide operation of industry, transport, popu- lation household - the entire life of the country. Development of power systems is moving along the path of creating large energy associations, in some cases covering entire continents, such systems include large num- ber of generators, the emerging tendency of transferring generators to the reactive power consumption mode in order to normalize voltage levels in the network has complicated the problem of ensuring the EPS stabil- ity [Gelig et al., 1978]-[Kalman, 1963]. All these negative factors caused frequent violations of the vibrational stability of EPS, emphasizing the relevance and practical importance of solving this problem. These circum- stances required the improvement of research methods and in-depth study of EPS’s dynamic properties. At this stage, the development of highly efficient methods for numerical solution of global asymptotic stability problems, T-controllability, optimality, and carrying out wide-ranging computational studies were of great importance. The importance of the problem ensuring stable operation of electric power systems is confirmed by a large number of developments, both domestic and foreign scientists. We can note the works of such scientists as P. Kalman, P. Anderson, A. Fouad [Anderson & Fouad, 2002], L.E. Jones, J. Klir, S. Arimota , etc., and also scientists from near abroad L.Y. Anapolsky, A.A. Yanko-Tryniski, I.I. Blekhman, A.H. Gelig [Gelig et al., 1978] etc. and from Kazakhstan S.A. Aisagaliyev [Aisagaliev & Kalimoldayev, 2013], S.A. Aipanov, M.N. Kalimol- dayev [Kalimoldayev et al., 2014], M.T. Jenaliev, G.A. Leonov, V.M. Matrosov, V.A. Yakubovich, K. Tokty- bakiev, V.A. Korotkov, L. Kopbosyn, etc. 2 Formulation of the Problem The system of nonlinear differential equations: dx = A (t) x + D (t) u + f (t, u, x) , t ∈ [t0 , t1 ] , x (t0 ) = x0 dt     0 Si 0∗ni 0 A =  0 −Ki c∗i  , D =  0  0ni bi Ai qi     δi 0 x =  Si  , f (x) =  −fi (δi ) − ψi (δ)  xi 0ni Consider the general mathematical model for electric power systems: dδi =Si , dt dSi = (Wi −Ki Si −fi (δi )) −ψi (δi∗ ) , wi =Ci∗ xi , (1) dt dxi =Ai xi +qi Si +bi ui +Ri (Si , xi ) , i=1,l, (2) dt where function ∑ l ψi (δi∗ ) = Pik (δik ) , δik =δi −δk (3) k= 1 k ̸= i where δi – the angular coordinate; Si – angular velocity; xi , ni – the state vector of the regulator; wi - the control action of the regulator; Ki > 0 - damping factor; ci , qi , bi – constant ni – dimensional vectors; Ai – constant (ni × ni ) matrix; ui – control of the feedback type. The symbol (*) denotes transposition. Second order differential equations (1) describe the processes in the object of control, and the vector differential equation (2) 274 determines the state of the regulator of the i-th isolated subsystem, in which the phase trajectories tend to the particular stable equilibrium position. Consider the global asymptotic stability of the coupled system with many angular coordinates in the case when Ri (Si , xi ) = ei ϕi (σi ) , σi = gi∗ xi + γi Si , i = 1, l, (4) where gi ei - constants, ni - vectors, γi - scalar constant. (4) change shows the Euclidean space and consist of x and S. The system (1)-(3) takes the form: dδi dSi = Si , = wi − Ki Si − fi (δi ) − ψi (δi∗ ) , wi = Ci∗ xi , (5) dt dt dxi = Ai xi + qi Si + bi ui + ei ϕi (σi ) , i = 1, l (6) dt or in vector-matrix form dδ dS = S, = w − KS − f (δ) − ψ (δ∗ ) , w = C ∗ x, (7) dt dt dx = Ax + qS + bu + eϕ (σ) , σ = g ∗ x + γS, (8) dt where e = colon (e1 , . . . , el ) , γ = colon (γ1 , . . . , γl ) , g = diag {g1 , . . . , gl }. Characteristics of nonlinear elements ϕii (σi ) are continuous functions satisfying the conditions 0 ≤ ϕii (σi ) σi ≤ ρi σi2 , ϕi (0) = 0, i = 1, l ( ) ∀σi ∈ (0, +∞) , ∀σi ∈ Ri1 (9) If there are inequalities for the function ϕii (σi ): ρ1i σi2 ≤ ϕii (σi ) σi ≤ ρ2i σi2 , ( ) ρ1i σi2 , ρ2i σi2 ∈ (−∞, +∞) , ∀σi ∈ Ri1 , after substitution ϕi = ϕi − ρ1i σi it leads to considered case; where ρi = ρ2i − ρ1i . The differential equation (6) will be rewritten in the form dxi = Ai xi + qi Si + bi ui + ei ϕi (σi ) , i = 1, l, dt where Ai = Ai + ρii ei gi∗ , qi = qi + ρ1i γi , and nonlinearity ϕi (σi ) satisfies the condition 0 ≤ ϕi (σi ) σi ≤ (ρ2i − ρ1i ) σi2 , ( ) ρ2i − ρ1i = ρi ∈ (0, +∞) , ∀σi ∈ Ri1 . Constraint (9) is equivalent to the inequality ( ) ϕi (σi ) σi − ρ−1 i ϕi (σi ) ≥ 0, (10) ( ) ρi ∈ (0, +∞) , ∀σi ∈ Ri1 , i = 1, l. The function ∫ σi ψi (σi ) = ϕi (λ) dλ, i = 1, l (11) 0 is positive semi-definite function. Theorem 1. Let there exist scalars Di , τi > 0 such that is globally asymptotically stable 275 1. The phase system of the second order (1) is globally asymptotically stable (i.e. Di > (Di )kp ). 2. A should be Hurwitz matrix Ãi ei , Gi ) – completely observable pair. 3. (A ei , Qi ) – completely controllable pair . 4. (A [ ] ( ) e −1 hi χi ̸= 0 i = 1, l . Γi > 0, det 2Γi − χ∗i h∗i D i e i > 0. 5. Γi + Re Wi (jω) ≥ 0 (∀ω ∈ (−∞, +∞)) , Li ≥ 0, D Then control ∑ ui = a∗i xi + θi Si + εi ?i (δi ) + Sxi ψ∗ H i (δi∗ ) at zi ∈ i −1 i i bi ∑ (12) ui ̸= −(b∗i Hi bi ) (b∗i Hi Ai xi + b∗i Hi qi Si + b∗i Hi ei ϕi (σi )) at zi ∈ / i , i = 1, l ensures global asymptotic stability of the system (8), (9). 3 Numerical Example We will consider motion stabilization of electric power system when x(t1 ) = x1 (two synchronous generators). The equations (5) and (6) describe the operation of the system, and the equations (7) and (8) describe the state of the controller dδ1 dt = S1 , (13) dt = c1 x1 − K1 S1 − f1 (δ1 ) − P12 , dS1 dδ2 dt = S2 , (14) dt = c2 x2 − K2 S2 − f2 (δ2 ) − P12 , dS2 dx1 = A1 x1 + u1 , (15) dt dx2 = A2 x2 + u2 , (16) dt where x1 , x2 – phase variables; c1 , c2 – scalars; u1 , u2 – controls initial conditions: δ10 = 1.34; δ20 = 0.84 S10 = 0.0; S20 = 0.0; x10 = 0.001; x20 = 0.001; All conditions of the theorem 1 have been checked. dδ dS = S, = w − KS − f (δ) − ψ (δ∗ ) , w = C ∗ x, (17) dt dt dx = Ax + qS + bu + eϕ (σ) , σ = g ∗ x + γS, (18) dt We use the 4-th order Euler and Runge-Kutta methods to obtain the numerical solutions of equation systems (13)-(16). The results of the numerical solution are shown in figures (1)-(4). We use the 4-th order Adams-Bashford, Adams-Moulton and Runge-Kutta methods for more accurate results. Adams-Bashford method: h yn+4 = yn+3 + (55f (tn+3 , yn+3 ) − 59f (tn+2 , yn+2 )+ 24 251 5 +37f (tn+1 , yn+1 ) − 9f (tn , yn )) , h (η). 720 276 Adams–Moulton method: h yn+4 = yn+3 + (9f (tn+4 , yn+4 ) + 19f (tn+3 , yn+3 )− 24 19 5 −5f (tn+2 , yn+2 ) + f (tn+1 , yn+1 )) , − h (η). 720 Runge-Kutta method: h yn+1 = yn + (k1 + 2k2 + 2k3 + k4 ) , 6 Comparison of the used methods is shown below: Figure 1: methods of Adams-Bashford (line - - -), Adams–Moulton (line ...) of the 4-th order a) time change of S ; b) time change of δ. Figure 2: methods of Adams-Bashford (line- - -), Adams–Moulton (line ...) and Euler (line -) of the 4-th order a) time change of S ; b) time change of δ The main purpose of using different methods for solving this problem is to decrease the amplitude over the same period of time. Since large values can affect the operation of the system and the life of the equipment. The fourth-order Euler method was used for comparison with other methods. As can be seen in Figure 2, the methods Adams-Bashford and Adams-Moulton show a smaller amplitude than the Euler method. Figure 1 shows the methods Adams-Bashford and Adams-Moulton, it follows that the method Adams-Bashford gives a good result. If compare the time for compilation, then the method Adams-Bashford also shows a better result than the rest. This is due to the fact that method Adams-Moulton is implicit. Figures 3 shows a comparison of Euler and Runge-Kutta. Figure 4 shows a comparison of the methods Adams-Bashford and Adams-Moulton with the Rung-Kutta 277 Figure 3: methods of Runge-Kutta (line - - -) and Euler (line -) of the 4-th order a) time change of S ; b) time change of δ Figure 4: methods of Adams-Bashford (line - - -), Adams–Moulton (line ...) and Runge-Kutta (line-) of the 4-th order a) time change of S ; b) time change of δ . According to the obtained results it is clear that the increase of more than 4 is not necessary, as they equally converge to zero. For this task the Adams-Bashford and Runge-Kutta methods converge to zero faster than when using the method of Adams-Moulton. It allows to reduce time and speed up the process of determining emergency situation. Since the Adams-Moulton method is implicit and requires the solution of the “historical” values, which takes computation time. The software package of dynamic study of complex electric power systems has been developed in Visual Studio using the programming language C-Sharp. A detailed analysis of the problem with definition and the identification of input and output information has been conducted. Two types of testing have been carried out: functional and structural. In functional testing the programs are verified the compliance behavior of the program to its external specifi- cation. The logic of the program is verified when a structural test. The software products were initially created for the individual tasks of complex electrical power systems. Next, they were merged into a single set of programs. The 4-th order Adams-Bashford, Runge-Kutta and Adams-Moulton methods have been used when creating a software product. The figures (5-8) have shown the interface of software package. 278 Figure 5: Figure 6: Main menu. 4 Conclusion In this paper we solve the problem of global asymptotic stability of phase systems. According to the results obtained, it is clear that an increase in the order of more than four times is not necessary, since they converge to zero identically. For this problem, the Adams-Bashford and Runge-Kutte methods converge to zero faster than using the Adams-Moulton method. This allows you to reduce the time and speed up the process of determining the emergency situation. The software package of dynamic study of complex electric power systems has been developed in Visual Studio using the programming language C-Sharp. 279 Figure 7: Numeric data entry on the selected method. Figure 8: Illustration of graphics on the screen. Acknowledgements This work is partially supported by Grant No.3314/GF4 of a Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan. References [Jacobson & Ycesan, 2008] Jacobson, S. H., & Ycesan, E. (2008). Development and implementation of an information-computing system for the study of the dynamic stability of electric power systems. Com- putational technologies., 59-68. [Gelig et al., 1978] Gelig A.H., Leonov G.A., V.A. Yakubovich (1978) Stability of nonlinear systems with a nonunique equilibrium state. Moskow: Nauka. [Kalman, 1963] Kalman R.E.(1963) Lyapunov Functions for the problem of Lur’e in Automatic Control, Pro- ceedings of the National Academy of Science of USA, 1963, 2, 201-205. [Anderson & Fouad, 2002] Anderson P. M., Fouad A. A. (2002) Power System Control and Stability, Second Edition. Moskow: Nauka. 280 [Aisagaliev & Kalimoldayev, 2013] S.A. Aisagaliev, M.N. 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