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Stability and Oscillations of Electrical Machines of Alternating Current G. A. Leonov ∗ S. M. Seledzhi ∗ E. P. Solovyeva ∗ A. M. Zaretskiy ∗ ∗ Saint-Petersburg State University, Saint-Petersburg, Russia (e-mail: [email protected]). Abstract: An induction machine with the squirrel cage rotor and a synchronous machine with the four-pole rotor are considered and the new mathematical models of these machines are developed. The analysis of steady-state and dynamic stability of the systems, describing the electrical machines, is performed. The limit load problem is discussed and analytical and numerical estimations of the limit load for considered electrical machines are obtained by the equal-area method and the non-local reduction method. In the case of second method improved estimates of the limit load in comparison to the estimates, obtained by the equal-area criterion, are found. Keywords: Synchronous motors, induction motors, limit load problem, equal-area criterion, non-local reduction method. 1. INTRODUCTION Stability is an important qualitative characteristic of an electrical machine of alternating current, providing the reliability of its work. In practice of operation of electrical machines sudden disturbances occur, for example line fault, changes of the external load, changes of supply voltage, etc. In this case damage of the electrical machine or even its failure requiring overhaul can arise. By this reason the investigation of stability is one of the major scientific and technological problems in the design of electrical machines. By stability of an electrical machine we mean the ability of the machine to re-establish a steady-state mode after disturbances of the initial mode. The process of pull into synchronism after asynchronous start-up is also a property of stability of the machine. In 1931-1933 years Italian mathematician F. Tricomi (1931, 1933) first applied the strict mathematical methods for analysis of electrical machines. Equation of F. Tricomi describes the dynamics of a synchronous machine in the simplest two-dimensional idealization. A very effective method for investigation of stability of electrical machines became the second method of Lyapunov. This method was first used for stability analysis of synchronous motors by A.A. Yanko-Trinitskii (1958). The stability problems may be divided into two kinds: steady-state, or static stability and transient, or dynamic stability. The problem of dynamic stability of an electrical machune consists of not only checking whether the machine maintain synchronism after given dynamic disturbances, but also finding the limit permissible disturbance, corresponding to the boundary of dynamic stability. Therefore, the problem of dynamic stability is closely related to the limit load problem. The urgency of the limit load problem has increased significantly due to the large number of outages in the modern world. Numerical solution of the limit load problem for particular values of the parameters is given in works of W. V. Lyon, H. E. Edgerton (1930), as well as in monograph of J.J. Stoker (1950). In the engineering practice to determine the limit load it is used so called the equal-area method. Mathematical setting of the limit load problem for electrical machines and the methods of its solution are considered in J.M. Bryant, E.W. Johnson (1935); A. Pen-tung Sah (1946); F. A. Annett (1950); G. C. Blalock (1950); A.A. Yanko-Trinitskii (1958); U.D. Caprio (1986); E. A. Barbashin, V.A. Tabueva (1969); H.-Ch. Chang, M.H. Wang (1992); R. H. Miller, J. H. Malinowski (1994); S. A. Nasar, F. C. Trutt (1999); G.A. Leonov, N.V. Kondrat’eva, F.F. Rodyukov, A.I. Shepeljavyi (2001); J. C. Das (2002); N. Bianchi (2005); G.A. Leonov (2006); C. L. Wadhwa (2006); R.R. Lawrence (2007]); J.D. Glover, M.S. Sarma, T.J. Overbye (2008), where a mathematical justification of ”equal-area method” is given and the estimates of limit loads are obtained. In these works the different mathematical models of electrical machines and the Lyapunov function of the type: ”quadratic form plus an integral of nonlinearity” are used. The complexity of constructing Lyapunov functions for multi-dimensional models of dynamical systems has led to the necessity for the development of various generalizations of the second Lyapunov method. In G.A. Leonov (2001) and in the monograph V.A. Yakubovich, G.A. Leonov, A.Kh. Gelig (2004) for investigating the stability of electric motors, in addition to typical functions of Lyapunov, the functions, involving the information on solutions of equation of comparison, namely Tricomi equation, are used. These Lyapunov-type functions constitute the essence of tne non-local reduction method. Further the method of non-local reduction has been developed in G.A. Leonov (2006); G.A. Leonov, N.V. Kondrat’eva (2009), where the dynamic stability and oscillations of differential equations of electrical machines are studied. 2. MATHEMATICAL MODELLING OF ELECTRICAL MACHINES Electrical machines obey the inversability principle and, therefore, they can operate as either generator or motor. The principle of inversability allows one to conclude that mathematical models of electrical machines, operating in the mode of electric energy generation, preserve the same structure as electric motors. In what follows we consider synchronous and induction machines operating as a motor. It is reasonable that any mathematical model of an electrical machine is a certain idealization. In the present paper the basic assumption, according to which the dynamics of an electrical motor is determined by the dynamics of the rotor, is that electromagnetic processes in windings of the rotor do not influence on the parameters of the rotating magnetic field, i.e. the magnetic field vector is constant in magnitude and rotates with constant speed. Such rotating magnetic field generated by alternate current in stationary windings of the stator of an electrical machine was invented by N. Tesla and G. Ferraris in 1888. Till now this phenomenon is a base of construction of alternating current machines: the synchronous and asynchronous ones. All the equations derived here are obtained by means of the general approach: introducing the system of coordinates rigidly connected to rotating magnetic field and considering the motion of electromechanical models of electrical motors in this system of coordinates. 2.1 Mathematical Model of Induction Motor with the Cage Rotor Let us study the cage rotor of induction motors consisting of n bars and two end rings (Fig. 1). The whole structure looks like a squirrel cage, so the rotor is also called squirrel cage rotor. dik 2kπ + R ik = l0 lB cos(θ + )θ̇, k = 1..n, dt n n X 2kπ )ik (t) − M, J θ̈(t) = l0 lB cos(θ(t) + n L (1) k=1 where ik – current in k bar; R – resistance of the bar; L – inductance of the bar; l, l0 – radius and length of the squirrel cage respectively; θ – angle between the radius vector of the bar with current in and the magnetic field vector B; J – the moment of inertia of rotor; M – the moment of resistance (so called load torque). Let transform (1) to more convenient form for further study. Using a nonsingular change of coordinates θ 7→ −θ, s = θ̇, x= n 2L X 2kπ )ik , sin(θ − nl0 lB n k=1 n 2L X 2kπ y= )ik , cos(θ − nl0 lB n k=1 zk = m X π i(k+j) mod n − cot( )ik , n j=−m k = 2..n − 1. system (1) takes the following form where θ̇ = s, ṡ = ay + γ, ẋ = −cx + ys, ẏ = −cy − xs − s, żk = −czk , k = 2..n − 1. a= n(l0 lB)2 , 2JL γ= M , J c= (2) R . L In system (2) the variables x, y, zk determine electrical quantities in the rotor bars, the variable s is slip speed. Note that the equations żk = −czk can be easily integrated and equations (2) except the first do not depend on θ, hence, it is sufficient to consider the system ṡ = ay + γ, ẋ = −cx + ys, (3) ẏ = −cy − xs − s. On the basis of the developed mathematical model steadystate and dynamic stability of an induction motor with the cage rotor are studied. 2.2 Mathematical Model of Synchronous Motor with the Four-Pole Rotor Fig. 1. Cage rotor of induction motor We consider motion of the cage rotor in the rotating coordinate system, rigidly connected with the magnetic field vector B. By first and second Kirchhoffs laws for electrical circuit of the cage rotor and the equation of the moments of forces we obtain the following system of equations, which describes the behaviour of cage rotor induction motor The electromechanical model of a synchronous motor with the salient-pole rotor (Fig.2) is studied. We consider fourpole electromechanical model with damper winding. The damper winding is a short-circuited winding similar to the squirrel cage winding of an induction motor rotor. A scheme of four-pole model of rotor with damper winding is shown in Fig.3. This scheme consists of two orthogonal pairs of parallel turns of field winding with one excitation current i and damper winding with the currents ik , k = 1...n. θ̇ = s, ṡ = −µs − ax sin θ − b sin θ + γ, ẋ = −cx − ds sin θ. (5) Here √ √ √ R0 2 2n0 S0 B 2 2n0 S0 Be 2 2n0 S0 B ,c = ,d = , ,b = a= J JR0 L0 L0 µ= Fig. 2. Salient-pole rotor of synchronous motor n(SB)2 8JR . 3. THE STATEMENT OF LIMIT LOAD PROBLEM FOR ELECTRICAL MACHINES A typical situation for an electrical machine is as follows: the machine is started without load, then in transient process it is pulled in synchronism and only after that a load-on occurs. In this case the limit load problem is to find for what loads, after transient process, the electrical machine is pulled in a new synchronous operating mode. Fig. 3. Schematic of four-pole rotor. Field winding with current i and damper winding with currents ik are shown As before, we consider motion of windings in the rotating coordinate system rigidly connected to the magnetic field vector. We use the previous denotations of the squirrel cage winding for parameters of the damper windings. Let us introduce the following parameters of the field winding: L0 is inductance, R0 is resistance, S0 is area of separate turn of the field winding, m is the numbers of turns in the field winding. Parameters J, M have the same meaning as before. By first and second Kirchhoffs laws and the equation of the moments of forces we get the system of equations di L0 + R0 i = 2n0 S0 B(cos θ + sin θ)θ̇ + e, dt dik SB 2πk L + Rik = cos(θ + )θ̇, k = 1..n, dt 2 n (4) J θ̈ = 2n0 S0 B(cos θ + sin θ)i+ n 2πk SB X )ik − M. cos(θ + + 2 n In the work the limit load problem for induction motors with the cage rotor and synchronous motors with the quadrupole rotor is considered. Thus, the behaviour of these electrical motors may be described by the autonomous system of the form ż = f (z), z ∈ Rn . In the work global stability of electrical motors under noload condition is proved, so it is assumed that the motor operates in a synchronous mode. Let this operating mode correspond to the solution of the system z = z0 . Further at time t = τ the instantaneous load-on occurs. Thus, for t > τ the moment of the external load is not already zero. Hence, the operating mode of the motor changes, that is a new synchronous operating mode of the motor under load condition corresponds to the solution of the system z = z∗ . Thus, a mathematical setting of the limit load problem is the following: to find conditions, under which the solution z = z(t) with the initial data z = z0 is contained in the domain of attraction of the stationary solution z = z∗ . The latter means that the following relations lim z(t) = z∗ . t→+∞ must be satisfied. It follows that the limit load problem for an induction motor is reduced to the problem of finding the region of attraction of the stable equilibrium point. k=1 This system describes the behaviour of the synchronous motor with the four-pole rotor. 4. LIMIT LOAD ESTIMATIONS Let us neglect inductance of the damping winding, i.e. assume L = 0. Using nonsingular transformation of coordinates π θ 7→ −θ − , 4 e , x=i− R0 and taking into account π 2πk SB cos(θ + − )θ̇, k = 1..n, ik = 2R 4 n system (4) is reduced to the third order system Consider first the limit load problem for system (3) of equations of induction motor with the cage rotor. The equilibrium points of system (3) under condition γ > a2 are points: the asymptotically stable equilibrium point γ γs0 , y = − , s = s0 < c, x=− ac a which corresponds to an operating mode of induction motor, and the unstable equilibrium point γs0 γ x=− , y = − , s = s1 > c, ac a which represents a physically unrealizable mode. Here s0 , s1 are roots of the equation acs = γ. s2 + c2 Suppose, that to synchronous operating mode of induction motor under no-load condition corresponds an unique asymptotically stable equilibrium point of system (3) in the case γ = 0: s = 0, y = 0, x = 0. Further at time τ , the instantaneous load-on γ occurs. Thus, for solving the limit load problem we need to find conditions, under which the solution x(t), y(t), s(t) with the initial data s = 0, x = 0, y = 0 is contained in the domain of attraction of γ 0 the stationary solution x = − γs ac , y = − a , s = s0 < c: γs0 γ lim s(t) = s0 , lim x(t) = − , lim y(t) = − . t→+∞ t→+∞ ac t→+∞ a (6) The estimate of the limit load for induction motors with the cage rotor can be obtained by Theorem 1. If a quantity γ satisfies the inequalities a 0 < γ < min{ , 2c2 } (7) 2 and Zs1 γ 2 (− s2 + as − cγ) ds ≥ γ 2 , (8) c 0 then γ is the permissible load. Ideas of the equal-area method are used during the proof of Theorem 1. The estimate (8) can be improved by the nonlocal reduction method. Before this we transform system (3) to more convenient form for further study. The change of variables γ s ac transforms the system (3) into the form ṡ = η, η̇ = −cη + azs − ψ(s), (9) 1 γ ż = −cz − sη − η a ac and maps the initial data of system (3) into the initial data s = 0, η = γ, z = 0 of the new system. Here ψ(s) = − γc s2 + as − cγ. η = ay + γ, z = −x − Under the condition 0 < γ < a/2 the stationary set of system (9) consists of two points: the point s = s0 , η = 0, z = 0 is asymptotically stable and the point s = s1 , η = 0, z = 0 is unstable, where s0 and s1 are zeros of the function ψ(s) p p c a − a2 − 4γ 2 c a + a2 − 4γ 2 s0 = , s1 = . 2γ 2γ Theorem 2. Suppose that Zs1 (10) 2 ψ(s)ds + Γ2 s21 ≥ γ 2 , 0 is satisfied, where Γ = 2 max λ c − λ − λ∈(0,c) γ2 4c2 (c − λ) 1/2 . Then the solution of system (9) with the initial data s(0) = 0, η(0) = γ, z(0) = 0 satisfies the relations lim s(t) = s0 , lim η(t) = 0, lim z(t) = 0. (11) t→+∞ t→+∞ t→+∞ Proof. Consider the equation dF F = −ΓF − ψ(s), ds with initial data F (s1 ) = 0. In E. A. Barbashin, V.A. Tabueva (1969)it is shown that its solution F (s) in the point zero can be estimate by Zs1 1 F (0) > (2 ψ(s)ds + Γ2 s21 ) 2 . 0 Thus, if condition (10) is fulfilled, then the solution F (s) satisfies the condition F (0) > γ. (12) Let us consider the function a2 1 1 W (s, η, z) = z 2 + η 2 − F 2 (s). 2 2 2 For W (s, η, z) on the solutions of system (9) we have Ẇ (s, η, z) + 2λW (s, η, z) ≤ 0 (13) Then two cases are possible: either there exists a point s2 < 0 such that F (s2 ) = 0 and F (s) > 0 is fulfilled for all s ∈ (s2 , s1 ), or F (s) > 0 for all s ∈ (−∞, s1 ). In the first case let us define Ω as follows Ω = W (s, η, z) < 0, s ∈ [s2 , s1 ] . In virtue of (13) the set Ω is positively invariant. From the properties of F (s): F (s) > 0 for all s ∈ (s2 , s1 ) and F (s1 ) = F (s2 ) = 0 it follows the boundedness of the set Ω. In the second case the function W (s, η, z) induces a positive invariant cone Ω1 = {W (s, η, z) < 0, s ∈ (−∞, s1 ]}. Consider the function V (s, η, z) a2 1 V (s, η, z) = z 2 + η 2 + 2 2 Zs ψ(s)ds. s1 For V (s, η, z) on the solutions of system (9) we have aγ ηz − cη 2 ≤ 0. V̇ (s, η, z) = −ca2 z 2 − c Thus, the function V (s, η, z) also induces a positive invariant cone Ω2 = {V (s, η, z) < C, s ∈ [s3 , +∞)}. Here s3 is a solution of the equation Zs1 ψ(s)ds = C x < s1 x and constant C > 14 F 2 (0) is chosen so that s3 < 0. Consider the bounded set Ω∗ = Ω1 ∩ Ω2 . Since Ω1 and Ω2 are positively invariant, then Ω∗ is also the positively invariant set. Now we show that under the conditions of the theorem the positively invariant set Ω and Ω∗ contain the point (0, γ, 0) and (s0 , 0, 0). Under the condition (12) the following inequality 1 1 W (0, γ, 0) = γ 2 − F 2 (0) < 0 2 2 is fullfilled. It follows that (0, γ, 0)T ∈ Ω, (0, γ, 0)T ∈ Ω1 . Since s0 ∈ (0, s1 ) and F (s) > 0 for all s ∈ (s2 , s1 ), then γ = 0 corresponds to the synchronous operating mode of the synchronous motor under no-load condition. Further at time τ , the instantaneous load-on γ occurs. Thus, for solution of the limit load problem we need to obtain conditions, under which the solution θ(t), s(t), x(t) with the initial data θ = 0, s = 0, x = 0 is contained in the domain of attraction of the stationary solution θ = θ0 , s = 0, x = 0: lim θ(t) = θ0 , lim s(t) = 0, lim x(t) = 0. (14) t→+∞ t→+∞ t→+∞ The estimate of the limit load for synchronous motors with (s0 , 0, 0)T ∈ Ω, (s0 , 0, 0)T ∈ Ω1 . the four-pole rotor can be obtained by It remains to show that the set Ω2 contain these points. Theorem 3. If a quantity γ satisfies the inequality Since s 1 Z Z0 1 2 1 2 1 2 1 2 V (0, γ, 0) = γ − ψ(s)ds < γ − F (0) < F (0) < C (sin θ − γ)dθ < 0, (15) 2 2 4 4 0 and V (s0 , 0, 0) = θ1 Zs0 s1 then (0, γ, 0)T ∈ Ω2 , =− Zs1 then γ is the permissible load. < C, s0 (s0 , 0, 0)T ∈ Ω2 . By the function V (s, η, z) it can be proved that system (9) is dichotomic. Thus, it follows that relations (11) are satisfied. From analytical estimates (7), (8) and (10), we obtain the numerical estimates of the limit load for an induction motor, presented in Fig. 4. Theorem 3 is a justification of the widely used in engineering practice equal-area criterion. The estimate (15) can be improved by the non-local reduction method. Theorem 4. If a quantity γ satisfies the inequality Zθ1 b sin θ − γ dθ ≥ −Cθ12 , (16) 2 0 where C= ( 2µ2 , 4a(µ − a), then γ is the permissible load. µ < 2a, µ ≥ 2a. Proof. Let us consider the equation p θ̈ + 2 λ(µ − λ)θ̇ + (sin θ − γ) = 0, where the parameter λ satisfies the inequalities 0 < λ < min(µ, a). (17) (18) Suppose that for solution of the equation (17) with the initial data θ = 0, θ̇ = 0, condition θ(t) ≤ θ1 , ∀t ≥ 0. (19) is satisfied. Fig. 4. Limit load estimates for induction motor. Area 1 is obtained by the equal-area criterion and area 1+2 is obtained by the non-local reduction method Consider further the limit load problem for the system (5) of equations of synchronous motor with the four-pole rotor. The equilibrium points of system (5) under condition γ < b are points: the asymptotically stable equilibrium points θ = θ0 + 2πk (k ∈ Z), s = 0, x = 0, which correspond to operating modes of synchronous motor, and the unstable equilibrium points θ = θ1 + 2πk (k ∈ Z), s = 0, x = 0, which represent physically unrealizable modes. Here θ0 = arcsin γb , θ1 = π − arcsin γb . Suppose, that the asymptotically stable equilibrium point θ(0) = 0, s(0) = 0, x(0) = 0 of system (5) in the case In E. A. Barbashin, V.A. Tabueva (1969) it is shown that in virtue of (19) equation (17) and p dF F (20) = −2 λ(µ − λ)F − (ϕ(θ) − γ), dθ are equivalent equations. Equation (20) has either solutions Fk (θ) such that Fk (θ2k+1 ) = 0, Fk (θ) 6= 0, ∀θ 6= θ2k+1 lim Fk (θ) = −∞, lim Fk (θ) = +∞, k ∈ Z (21) θ→+∞ θ→−∞ or solution F∗ (θ) defined on the interval (θ, θ1 ), θ < θ0 and such that F∗ (θ) = F∗ (θ1 ) = 0, F∗ (θ) > 0, ∀θ ∈ (θ, θ1 ). (22) Introduce the functions c 2 x + 2d c 2 x + W∗ (θ, s, x) = 2d Wk (θ, s, x) = 1 2 s − 2 1 2 s − 2 1 2 F (θ), 2 k 1 2 F (θ). 2 ∗ The relations and Ẇk (θ, s, x) − 2λWk (θ, s, x) ≤ 0 Ẇ∗ (θ, s, x) − 2λW∗ (θ, s, x) ≤ 0 imply a positive invariance of the sets Ωk = Wk (θ, s, x) ≤ 0 and Ω∗ = W∗ (θ, s, x) ≤ 0, θ ∈ (θ, θ1 ). Let us consider the set Ω = Ω0 ∩ Ω−1 , θ ∈ (θ1 − 2π, θ1 ). The sets Ω and Ω∗ are bounded and involve the points: the unique equilibrium point (θ0 , 0, 0) and (0, 0, 0). By the function Zθ c 2 1 2 (23) x + s + (sin(ζ) − γ)dζ. V (θ, s, x) = 2d 2 θ1 it can be proved that system (5) is dichotomic. Thus, it follows that relations (14) are satisfied. It is known from E. A. Barbashin, V.A. Tabueva (1969) that condition (19) is satisfied if p υ(λ) = λ(µ − λ) > 0 and Zθ1 2 4λ(µ − λ)θ1 + 2 (ϕ(θ) − γ)dθ ≥ 0. 0 The best estimate is achieved at the maximum of the function υ(λ). Taking into account (18), if µ < 2a the maximum of υ(λ) is achieved at λ = 12 µ, otherwise, at λ = a. From analytical estimates (15) and (16), we obtain the numerical estimates of the limit load for a synchronoud motor, presented in Fig. 5. Fig. 5. Limit load estimates for synchronous motor. Area 1 is obtained by the equal-area criterion and area 1+2 is obtained by the non-local reduction method REFERENCES A. Pen-tung Sah (1946). Fundamentals of alternatingcurrent machines. McGraw-Hill induction motor abook company, inc. A.A. Yanko-Trinitskii (1958). New method for analysis of operation of synchronous motor for jump-like loads. M.-L.: GEI. C. L. Wadhwa (2006). Electrical power systems. New Age International. E. A. Barbashin, V.A. Tabueva (1969). Dynamical systems with cylindric phase space. M.: Nauka. 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