On Strongly Nonlinear Phenomena in Electrical Machines - wseas

12th WSEAS International Conference on SYSTEMS, Heraklion, Greece, July 22-24, 2008

On Strongly Nonlinear Phenomena in Electrical Machines BOHUMIL SKALA1, JOSEF HRUSAK2, DANIEL MAYER3, MILAN STORK2 Department of Power Electronics1 Department of Applied Electronics and Telecommunications2 Theory of Electrical Engineering3 University of West Bohemia P.O. Box 314, 30614 Plzen Czech Republic

Abstract: - The paper deals with some general aspects of strongly nonlinear real-world systems which behavior can be adequately described by means of finite number of internal interactions. New concept of abstract state space energy has been introduced and developed as a tool for strongly nonlinear system analysis, synthesis and design. The basic idea is motivated by real-world experiments inspired by dynamo effect as mechanism of the Earth`s geomagnetic field generation including its chaotic reversals. Theoretical concepts are illustrated by real data gained from a simple AC power system and tested by simulation including that of chaotic phenomena. Key-Words: Strongly nonlinear, chaotic, power, system, state space, abstract energy, geodynamo nonlinear 3-phase AC power system manifesting several pathological features is presented. The next paragraph §3 is devoted to theoretical analysis of gained experimental data and starts to interpret them from a system-theoretic point of view. In §4 a new fundamental concept of the abstract state space energy is introduced and discussed from its structural consequences point of view. The last §5 demonstrates the applicability of the proposed energy oriented approach. As an example, the nontrivial problem of Earth’s geomagnetic field chaotic reversals has been successfully solved. The obtained theoretical results are compared with computer simulations. In contrary to the standard two-disc Rikitake dynamo model, which can be considered as „stationary“ special case, ignoring any Earth’s moving effects, the derived strongly nonlinear system representation includes Coriolis effects both of the moving subsystems automatically.

1 Introduction Even though asynchronous motors have been used for more than a hundred years, there are some problems which are still open. For instance effective methods of high quality control of these machines is still an active field of current research. It is well known that induction machines have very good qualities, such as high reliability, ruggedness, relatively low cost etc., which are of crucial importance for industrial applications. On the other hand, from the control synthesis and practical design of controller’s point of view, the situation can in many regards be significantly different. At first, every adequate mathematical model of a system containing the asynchronous machines is strongly nonlinear. Some state variables, particularly magnetic fluxes, or equivalently the rotor currents cannot easily be measured and thus nonlinear filters or state reconstruction systems can sometimes be required for high performance system operation. In many situations in addition to nonlinearities of internal system interactions some parameters, e.g. rotor resistances or load torques, etc. can be unknown and/or strongly timevarying. Hence some principles of robust or adaptive control system design are needed in high demanding control applications. It should be expected that high quality controllers will be more complicated, more expensive and with regard to the growing complexity can be less reliable. As a result, nonlinearly generated phenomena, such as nonlinear instability, relaxation effects and chaotic-like behavior especially in systems of “sensorless control” should be predicted and eliminated. In this paper just such questions are addressed. In §2 an experimental study of a simple real-world strongly

ISBN: 978-960-6766-83-1

2 Motivation example: Nonlinear Phenomena in a power system For motivation and better understanding of the proposed general theoretical constructions a strongly nonlinear but relatively simple electromechanical system consisting of two asynchronous machines was used. Both the machines shown in the Fig.1. are 4-pole machines. The smaller one of a power 11 kW is supplied by the regulated soft voltage source and works as a motor. The second one has the power of 12 kW. In order to work as an asynchronous self-excited generator a destabilizing linear feedback is introduced by means of the set of

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data, i.e. from the speed and the input power of the machine 11 kW. By a proper choice of the load parametrization instead of an expected “stationary working point” a sort of nonlinearly generated chaoticlike oscillations can be observed as it is demonstrated in the Fig.2.

capacitors. The generated power is changed to heat in loading resistors which can be considered as an active part of the load. The second set of capacitors is introduced in order to represent a reactive part of the linear load. 3x 400V

3x 170V

3xU 3xI PC

AM 11 kW

n

Fig. 3. Frequency oscillations of the generator 12 kW AM

3xU 3xI

12 kW

R 3x15Ω

C1 3x150 uF

Fig. 4. The speed, output power, voltage and current oscillations of the asynchronous generator 12 kW.

C2 3x80 uF

Fig. 1. Simplified structure of a real-world power system.

Fig. 5. The input power, speed, current and voltage oscillations of the asynchronous motor 11 kW. In order to investigate the mechanism of non-linear internal system interactions in detail, we restrict further analysis to the steady state behavior of the system in the next. Therefore, all the data gained by measurements in transient phase will be ignored. Typical courses of basic measured variables of the asynchronous generator are depicted in the Fig. 4.

Fig. 2. Motor output mechanical power fluctuations. The speed of the machine unit, as well as the input power of the motor and the corresponding output power of the generator, are measured. The torque is not measured directly. It is computed from the measured

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3 Further Aspects of the Nonlinear Power – Informational Interactions In experiments above the asynchronous motor was supplied by means of a booster. This type of source is relatively soft voltage source. It is easy to demonstrate that just this fact, together with the typical form of the asynchronous motor nonlinearity, is responsible for observed system oscillations. It follows from the Fig.2. that the speed of the asynchronous generator oscillates in the range of approx. 500-1500 rpm. As a consequence also the instantaneous frequency of the generated electrical power oscillates correspondingly (see the Fig.3.). It is worthwhile to notice that, from the frequency of generator point of view, the system behavior is strictly causal, non-stochastic, and with exception of a transient response it seems to be quasi-periodic and even almost sinusoidal. Similarly, typical courses of basic measured variables of the asynchronous motor are depicted in the Fig. 5. Recall that due to the reduced voltage, the torque characteristics of the motor are not nominal. The input voltage is changing during one period of oscillation thanks to the varying load of the booster. This generator changes the mechanical energy to electrical energy and this energy is finally changed in the resistors to the heat. When the mechanical power input to the generator is not sufficient and the loading torque of the generator 12 kW is higher than mechanical torque of the motor 11 kW, the speed is decreasing, as it can become clear from the Fig. 6.

Fig. 7. The detail analysis of internal interactions between the motor 11 kW and the generator 12 kW. Zone No. 1. The output power of the generator 12kW is very low and asynchronous motor 11 kW runs up. The speed is increasing, as well as the input power, because the input power factor is increasing, as the speed of the motor is increasing. The end of this zone is in the point when the motor input power reaches its maximum value. The electric power is changing to mechanical power and speed is increasing. AS a result the energy is stored in the form of kinetic energy of the rotating mass in correspondence with the total moment of inertia. At this “working point” the motor operates in a “labile part” of the torque characteristic and typical features of instability phenomena appear. Zone No. 2. The instantaneous speed of the system is approaching the synchronous speed, and consequently the motor input power, as well as the instantaneous value of the power factor of the motor is decreasing. The asynchronous generator starts to excite the capacitors and consequently also its output power is increasing. Because the (mechanical) output power of the motor is sufficient in this zone, the instantaneous speed reaches its maximum value and the motor starts to operate near the “stable part” of its torque characteristic. Zone No. 3. The output voltage of the generator is extremely increasing. Consequently also the active output power of the generator increases to its maximum value. In correspondence with the energy conservation principle the motor input power from the supply source grows up to its maximum value, too. The instantaneous speed is slightly decreasing. The motor works near the maximum of its torque characteristic and further moves to its “labile part”. Zone No. 4. The voltage supply source capacity is insufficient to satisfy the mechanical input power requirements of the generator. The voltage of the supply source falls down, the motor input power falls down, too, and the speed continues to decrease to its minimal value. The generator power output is during this time covered from the accumulated mechanical energy, i.e. from the kinetic energy of the system. Again the asynchronous

Fig. 6. The speed, torque of the motor 11 kW and torque of the asynchronous generator 12 kW. It is obvious that non-sinusoidal character all of the internal energy state variables is due to non-linear system interactions. To explain the course of the active input power, one period of oscillation will be divided into 4 zones (see the Fig.7), at first.

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It would be a relatively easy task to translate the ideas above into some specific physically motivated mathematical form. This standard approach will be discussed in detail elsewhere. In the next paragraph a more general approach based on a new concept of the abstract state space energy will be proposed.

motor operates in the “labile part” of its torque characteristic. This quasi-periodically repeating process is depicted as projections of chaotic-like state space trajectories into the speed-to-torque plane (see the Fig.8.).

4 State Space Energy Concept and its Structural Consequences Consider a continuous-time strictly causal system S given by a semi-linear representation:

R{S} : x ( t) = Ax( t ) + Bu( t ) , x( t0 ) = x0, y = Cx( t)

(1)

where x(t) represents the state vector with dim x(t)=n, u(t) represents the input signal vector, dim, u(t)=r and y(t) is the observed output signal vector with dim y(t)=p. A concept of physically correct state equivalent finite-dimensional representations is induced by the dissipation normal form as proposed in [4]. A crucial concept of abstract state space energy E(x) is defined as a measure of distance of the instantaneous state x(t) from an equilibrium state 1 (2) E ( x ) = ρ 2  x ( t ) , x*    2 and for the zero equilibrium state x* = 0 we get the expression 2 1 1 n (3) E ( x ) = x ( t ) = ∑ xi2 ( t ) 2 2 i =1 The corresponding system output signal power P(t) representing the instantaneous state energy dissipation by means of the output signal y(t) is natural to define as follows

Fig. 8. Projections of the chaotic-like state space trajectories As was already mentioned above, the whole system has one main source of energy – an external power supply source. Furthermore, from the active load point of view, there are at least 10 additional internal energy sources – energy accumulation elements (6 inductances, 3 capacitors, 1 moment of inertia). Thus, in principle the instantaneous energy flow to the loading resistors is generated not only from the mechanical power input of a generator, but even from the energy, which is accumulated in the rotary mass of the machine unit, as well as in the others internal energy accumulators of the system. Notice that the speed of asynchronous machines is varying very strong. The motor even exceeds the maximum torque point and starts to operate in the labile mode – labile part of the torque characteristic. Due to this fact the speed can very strongly fall down , in principle almost to zero. Consequently, not only the speed and current of the machine 11 kW, but also the input power and especially the instantaneous value of the power factor is very strongly time-varying, as it is shown in the Fig. 9.

P (t ) = y (t )

2

(5)

Considering the zero future input, an abstract form of state energy conservation principle can be expressed in differential form by the following output signal power balance relation

dE ( x , t ) = −P (t ) = − y (t ) dt

2

(6)

or equivalently in the following integral form

E (t0 ) =

∞

∫

y (t )

2

dt

(7)

t0

The structure of the dissipation normal form as defined in [4] can be interpreted as a chain of undamped linear or nonlinear oscillators with linear or nonlinear couplings with just one aggregated dissipation element. Such a structure has an important property – it is parametrically minimal. The dissipation of the whole system is concentrated in the dissipation parameter α11, expressing the aggregated dissipation in the direction of

Fig. 9. The course of the instantaneous power factor

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only one observed state component x1. In some applications multidirectional dissipation has proven to be more useful. Let us consider a chain of nonlinear oscillators which can be characterized by 2(n-1) nonlinear coupling parameters together with at most n nonlinear dissipation parameters αii given by the state dependent matrix A as follows.

5 Illustrative Example: Chaos in Earth’s Electromagnetic Field As an illustrative example, let us consider a 4-th order system consisting of two nonlinearly coupled nonlinear oscillators with linear damping, given by the following structure:

0 x1   0 x1   −α11 α2x4 0 x  −α x −k α 0 x2  β1 2 2 4 1 3   =   +  x3   0 −α3 −k2 −α4x1x3   0       0 α4x1 −α44 x4   0 x4   0

0 0 0 0  −α11 α2 −α −α 0 0 0  22 α3  2  0 −α3 −α33 % 0 0  A=   % % 0 αn−1 0  (8)  0  0 0 0 −αn−1 −αn−1,n−1 αn    0 0 0 −αn −αn,n  0

0

0  u1 



β2  u2  (9)



0

The nonlinear couplings between the two oscillators can be interpreted as power interactions. All the dissipation parameters are supposed to be constant and nonnegative. One of many physical motivations for this structure comes from a well known standard hypothesis concerning the geomagnetohydrodynamic explanation for chaotic changes of Earth’s geomagnetic field [1, 2]. The two-disc Rikitake dynamo [1, 3, 5, 6] represents a well-known and very simple electromechanical system characterized by chaotic reversal. Its behavior has been studied for many years in association with the geomagnetohydrodynamic phenomena occurring in the geodynamo, in order to find possible physical analogies. And even when it was shown that such a simple system cannot simulate extremely complicated processes in the surface layers of the liquid Earth core, it has still been recognized for its ability to demonstrate fundamental aspects of chaos. Clearly, the given matrix form of the system representations is equivalent to the following set of state equations represented by a set of four 1-st order nonlinear ordinary non-homogenous differential equations x1 = −α11 x1 + α 2 x2 x4

The corresponding structure is depicted in the Fig. 10.

x2 = −α 2 x1 x4 − k1 x2 + α 3 x3 + β1u1 x3 = −α 3 x2 − k2 x3 − α 4 x1 x4 + β 2u2

x4 = α 4 x1 x3 + α 44 x4 It is easy to show that by a proper choice of the state variables a 4-th order abstract system based on the concept of abstract state space energy represents adequately the physical interactions of the Rikitake system [1], [3], [5] as displayed in the Fig. 11. In the case of geodynamo as a part of universe, an additional assumption of “absolutely hard” unique torque source seems quite reasonable and hence we put u2(t) = u1(t) = u(t) = const. (11) It is obviously equivalent to the assumption that any external random fluctuations are excluded as a cause of observed chaotic behavior in such a model. It means that if some chaotic phenomena occur then their causes must unavoidably be internal.

Fig. 10. Physically correct system structure. It is staightforward to prove that for any values of the coupling parameters αk the internal system interactions are neutral with respect to the state space energy dissipation. The most important consequence is that in such a case the state energy conservation principle holds structurally, i.e. without any respect to actual values of the physical system parameters [4].

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(10)

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A further important problem in studying strong nonlinear systems is to determine a set of parameters for which the system of a given structure behaves chaotically. Assume that only the others dissipation parameters k1 and k2 can be varied. The correlation of the state space energy should express a measure of inpredictability of the system behavior. Minimal correlation means maximal inpredictability. Notice that absolute minimum in our experiment (α3 = 0.01) has been reached for dissipation parameter value k=0.077245 (k2=k1=k). Result of simulations are illustrated by the Fig. 12., where time courses of all the state space variables in the chaotic mode are depicted.

Fig.11. Simplified structure of Rikitake dynamo From the fact that external inputs u1, and u2 are interpreted as mechanical torques it follows that the second and the third equation must be interpreted physically as the “torque balance relations” equivalent to the well known laws of Newton. Consequently the only correct interpretation of the corresponding state variables x2 and x3 is to consider them as normalized angular velocities. It implies that the first and fourth equation express the „voltage balance relations“ in form of the well known voltage Kircchhoff`s law. Consequently the only correct interpretation of the corresponding state variables x1 and x4 is to interpret them as normalized currents. With this knowledge in hand the physical interpretation of all individual terms, as well as all the structural parameters is very easy job. The physical meaning of terms k1x2 and k2x3 should be clear. They express torques of viscosious friction of the corresponding subsystems. Similarly, we can deduce that the rest dissipation parameters have the meaning of normalized resistances.

6 Conclusion New concept of abstract state space energy has been introduced and developed as a tool for strongly nonlinear system analysis, synthesis and design. The basic idea is motivated by real-world experiments and simulations.

Acknowledgment This work has been supported from Research Project Diagnostics of interactive phenomena in electrical engineering, MSM 4977751310 and GACR project No. 102/07/0147. References [1] T. Rikitake, Electromagnetism and the Earth`s Interior, Elsevier, Amsterdam, 1966. [2] P. Roberts and G. Glatzmaier, Geodynamo theory and simulations, Rev. of Modern Physics, Vol. 72, No.4, 2000. [3] R. Hide, Structural Instability of Rikitake Disc Dynamo, Geoph. Res. Lett. 22, 1057–1059, 1995. [4] D. Mayer and J. Hrusak, On correctness and asymptotic stability in causal systems theory, Proceedings of 7th World Multiconf. SCI, Orlando, FL.,Vol. XIII, pp. 355-360, 2003. [5] F. Plunian, P. Marty and A. Alemany, Chaotic Behavior of the Rikitake Dynamo with Symmetric Mechanical Friction and Azimuthal Currents, Proc.Roy.Soc.Lond.454,1835–1842, 1998. [6] D. Sisan, W. Shew and D. Lathrop, Lorenz force effects in magneto-turbulence, Physics of Earth and Planetary Interiors, 135, pp. 137-159, 2003.

Fig. 12. Time evolution of state space variables in the chaotic mode for k1= 0.077245

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