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 asymptotic 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.
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.
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, population 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 number 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 stability [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 circumstances 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. Kalimoldayev [Kalimoldayev et al., 2014], M.T. Jenaliev, G.A. Leonov, V.M. Matrosov, V.A. Yakubovich, K. Toktybakiev, V.A. Korotkov, L. Kopbosyn, etc. 2
The system of nonlinear differential equations: (1) (2) (3) dx dt = A (t) x + D (t) u + f (t, u, x) , t ∈ [t0, t1] , x (t0) = x0
A = = (Wi−KiSi−fi (δi)) −ψi (δi ) ,
wi=Ci xi, =Aixi+qiSi+biui+Ri (Si, xi) , i=1,l, δik=δi−δk 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)
where giei- 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: or in vector-matrix form dδi = Si, dt
dSi = wi − KiSi − fi (δi) − ψi (δi ) , wi = Ci xi, dt dxi = Aixi + qiSi + biui + eiϕi (σi) , i = 1, l dt dδ dt (ρ1iσi2, ρ2iσi2 ∈ (−∞, +∞) , ∀σi ∈ Ri1) ,
0 ≤ ϕi (σi) σi ≤ (ρ2i − ρ1i) σi2, (ρ2i − ρ1i = ρi ∈ (0, +∞) , ∀σi ∈ Ri1) .
ϕi (σi) (σi − ρi 1ϕi (σi)) ≥ 0, (ρi ∈ (0, +∞) , ∀σi ∈ Ri1) , i = 1, l. ψi (σi) = ϕi (λ) dλ,
i = 1, l ∫ i 0 (4) (5) (6) (7) (8) (9) (10) (11)
If there are inequalities for the function ϕii (σi): after substitution ϕi = ϕi − ρ1iσi it leads to considered case; where ρi = ρ2i − ρ1i. The differential equation (6) will be rewritten in the form dxi = Aixi + qiSi + biui + eiϕi (σi) , i = 1, l, dt where Ai = Ai + ρii eigi , qi = qi + ρ1iγi, and nonlinearity ϕi (σi) satisfies the condition
is positive semi-definite function.
Theorem 1. Let there exist scalars Di, τi > 0 such that is globally asymptotically stable 1. The phase system of the second order (1) is globally asymptotically stable (i.e. Di > (Di)kp).
Γi > 0, det [2Γi − χi hi Dei 1hiχi ̸= 0 (i = 1, l) .
] 5. Γi + Re Wi (jω) ≥ 0 (∀ω ∈ (−∞, +∞)) , Li ≥ 0, Dei > 0. Then control ui ̸= ui = ai xi + θiSi + εi?i (δi) + Sxii Hi(ibii ) at zi ∈ ∑i −(bi Hibi) 1(bi HiAixi + bi HiqiSi + bi Hieiϕi(σi)) at zi ∈/ ∑i, i = 1, l ensures global asymptotic stability of the system (8), (9). 3
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 (12) (13) (14) (15) (16) (17) dx2 = A2x2 + u2, dt where x1, x2 – phase variables; c1, c2 – scalars; u1, u2 – controls initial conditions: ddt1 = S1, ddSt1 = c1x1 − K1S1 − f1 (δ1) − P12, ddt2 = S2, ddSt2 = c2x2 − K2S2 − f2 (δ2) − P12, dx1 = A1x1 + u1, 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,
h 24 +37f (tn+1, yn+1) − 9f (tn, yn)) , yn+4 h 24 yn+4
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
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.
In functional testing the programs are verified the compliance behavior of the program to its external specification. 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.
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.
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.