ABOUT DYNAMIC STABILITY OF HIGH POWER SYNCHRONOUS MACHINE. A REVIEW

Rev. Roum. Sci. Techn. Électrotechn. et Énerg. Vol. 62, 1, pp. 8 13, Bucarest, 217 Dedicated to the memory of Academician Toma Dordea ABOUT DYNAMIC STABILITY OF HIGH POWER SYNCHRONOUS MACHINE. A REVIEW AUREL CAMPEANU 1, RADU MUNTEANU 2, VASILE IANCU 2 Key words: Synchronous machine, Dynamic stability, Simulation. There is analyzed in details the complicated process on which the dynamic stability of a high power synchronous machine (HPSM) depends. There is pointed out the necessity to use dynamic angular characteristics in the stability analysis and the fact that the speed of rotating magnetic field can be widely variable, beyond the operation in synchronism. It is shown that there is an infinity of dynamic angular characteristics and that mathematical models are an efficient and fast way to respond if a given disturbance of a steady state synchronous operation leads or not to loosing synchronism. 1. INTRODUCTION Heavy electromagnetic systems and their interconnecting make possible using HPSM in modern complex driving systems frequently present in industry of cement, petroleum etc. High investments of intelligence and materials have been carried out for analyzing the dynamic behaviour of HPSM. Advanced mathematical models have been conceived [1 11] which take into consideration the fundamental processes inside the machine, as processes conditioned by saturation and magnetic asymmetry. In [1 9] there is pointed out in details that establishing mathematical models must take into account the constructive solution and the power of the synchronous machine. On this basis, modelling is able to pre-establish, with a precision acceptable in practice, the dynamic evolution, in all respect, of HPSM and of the entire system in which it is incorporated. This paper is addressed to the detailed analysis, with mathematical models specific to salient pole HPSM, of the factors conditioning dynamic stability; there are evoked the effects caused by electromagnetic and mechanic inertia, generally important. 2. MATHEMATICAL MODEL The mathematical model of the synchronous machine accounting for the main flux saturation is used in the form [1, 3, 5, 6] and others. dx A + BX = [ u] [] u = [ ud uq ue ]T. (1) dt State variables may be chosen currents only, flux linkages only or a combination of currents and flux linkages, leading to different form of vector X and the matrices A, B. In the following a hybrid mathematical model, valid for a high power synchronous machine with magnetic asymmetry, is used. It is considered that is, ie, ψ are state variables. m To the state vector X = [ i i i ψ ψ ] T (2) d q E md corespond the matrices (see [9]) mq Lsσ A = LDσ Rs ωlsσ B = RD Lsσ LQσ ωlsσ Rs RQ RE RD LEσ LDσ ω RD Lmd 1 1 L 1+ Dσ Lmdt LQσ 1+ Lmq ω RQ Lmq. 1 The motion equation is added J dω 3 m M r =, m = p( ψdiq ψqid ). (4) p dt 2 (3) In (1) u d, u q, u E stator windings voltages in Park coordinates and excitation winding voltage, E; in (2) i d, i q ψ md, ψ mq components of stator current vector i s and of the magnetizing flux vector ψ by d, q axes; i E current of m the excitation winding; in (3) R s and R D, R Q are the stator and rotor resistances; L md, L mq are the stator d and q axis cyclic magnetizing inductances, L mdt is the stator d axe differential inductance; index σ is attached to leakage inductances; ω is the electrical speed of the rotor; in (4) m, M r, J, p are electromagnetic torque, load torque, inertia moment and number of poles pairs. All the notations are widely used [1 2, 4, 6 7] and others. Equations (1 4) fully describe the dynamic behaviour of the synchronous machine. We point out that in particular cases, mathematical models of synchronous machine can be based on general equations of induction machine [12, 13, 15]. In [14] there are justified research efforts for using efficiently electrical machines, especially high power machines. 1 University of Craiova, 17 Decebal Street, Craiova, 244, E-mail: acampeanu@em.ucv.ro 2 Technical University of Cluj Napoca

2 Dynamic stability of high power synchronous machine 9 3. SIMULATION RESULTS The simulations aim at pointing out explicitly the factors conditioning the dynamic stability of a 8 kw synchronous motor having the parameters mentioned in the annex. With specific mathematical model (1 4) the following dynamic processes are analyzed. The motor is started and synchronized with a loading torque M r =, when as a rule, a synchronous operation is reached. Finally after resynchronization (after the dynamic process determined by excitation current is stabilized), the motor in loaded with a given M r. In this paper there are analyzed only dynamic processes (dynamic stability) caused by the simultaneous presence, M r, i E. We denote by a and b the intervals before and after connection of excitation winding to dc source, and by 1 and 2 the begin and respectively the end of dynamic process determined by this connection. In the following the dynamic processes from the zones a, b are supposed known [16]; point 2 corresponds to a synchronism operation and the time t 1, to applying M r. In zone c for t > t 1, the dynamic stability of the synchronous machine is analyzed; if the synchronism is finally reached, the end of the dynamic regime is 3. 3.1. Let us consider, U = U n = 5 V, M r = 4 1 4 Nm, u E = 2.1 V, t 1 = 23 s. Figure 1a presents the dynamic angular characteristic in the zone c corresponding to given i E, M r. There results a tight connection between the rotor oscillations and mechanical stresses and, very important, the fact that the dynamic process l is finalized in the point 3. In the figure there also is plotted the static angular characteristics m(θ) for considered u E. It is noticed that the point 3 of the dynamic characteristic is plotted very precisely on the static characteristic. This comes to confirm mutually the two characteristics and to emphasize the necessity to consider a covering time for defining correctly the point 3. In Fig. 1b, by using the same static characteristic, there have been plotted the dynamic characteristics corresponding to some torque shocks from zero to 2 1 4 Nm and from 2 1 4 Nm to 4 1 4 Nm; they are obviously different from the static characteristic. It is noticed that the final points 3 are also placed precisely on the static characteristic. It results that the dynamic stability of the motor is ensured if M r = 4 1 4 Nm is applied. 3.2. Let us consider M r = 5 1 4 Nm. The other initial conditions are from 3.1. The beginning of the zone c records the electromechanical shock and a synchronous operation until t 7 s is apparently established; beyond this, an asynchronous operation occurs, with major electromagnetic and mechanic oscillations and with rotor speed around the synchronism speed. Fig. 1a M r = 4 1 4 Nm. Fig. 2a Detail zone c. Fig. 1b M r = ( 2 1 4 ) Nm; (2 1 4 Nm); (4 1 4 Nm). Figs. 1 Dynamic and static characteristics m(θ), M r = 4 1 4 Nm. Fig. 2b Detail zone c d. Figs. 2 Characteristics ω(t), ω ψ (t), M r = 5 1 4 Nm.

1 Aurel Campeanu, Radu Munteanu, Vasile Iancu 3 The processes develop cyclically and there can be defined close curves with two branches (limit cycles), which point out that the machine is excited. It results that the dynamic stability, is not ensured for u E = 2.1 V. At t 2 = 114 s, u E = 3V, is applied and after other large electromechanical oscillations, the synchronization is reached. The zone d is considered corresponding to t > 114 s and there the points from t 2 = 114 s, respectively from the synchronization final are noted with 4, 5. In Fig. 2 there are plotted together the characteristics ω(t), ω ψ (t) for the zones c, d. Figure 2a details the zone c, with over and sub-synchronous oscillations of ω, ω ψ (electrical speed of the main rotating magnetic field), immediately subsequent to t 1 = 23 s; in the figure there follow the relatively large interval of practically synchronous operation and 3 identical complex oscillations, also over and sub-synchronous. Each of these complex oscillations has two branches of important amplitudes, especially for ω, which correspond to a quasi steady-state asynchronous operation (to a consecutive operation on limit cycle). In Fig. 2b the interference zone c, d is detailed, resulting a tight interdependence of the curves ω, ω ψ and passing to a synchronous operation. It also results that the time t 2 = 114 s is unfavourable because the synchronization process is carried out with difficulty, with large oscillations of ω, ω ψ. Fig. 3c m(ω) detail zone d. Fig. 3d m(θ) detail limit cycle. Fig. 3a m(t) detail zone c. Fig. 3e m(θ) detail zone d. Figs. 3 Characteristics m(t), m(ω), m(θ), M r = 5 1 4 Nm. Fig. 3b m(ω) detail limit cycle. In Fig. 3 there are plotted the characteristics m(t), m(ω), m(θ) in the zones c, d. The characteristic m(t) from the zone c (Fig. 3a) is fully justified by Figure 2a. In Fig. 3b there is plotted the characteristic m(ω) on the limit cycle with two branches, corresponding to asynchronous operation; there are determined the variation limits of the electromagnetic torque, the motor speed and the correspondence of the maximum values of torque with the values from Fig. 3a.

4 Dynamic stability of high power synchronous machine 11 Figure 3c details leaving the limit cycle m(ω) in the point 4 and passing to synchronous operation (curve d). Figures 3d, 3e repeat the characteristics 3b, 3c in m(θ) coordinates. There are re-confirmed the two branches of the limit cycle and the extreme values of the dynamic electro-magnetic torque. Figure 3e, as well as Fig. 3c shows a difficult synchronization (curve d which has the initial point in 4 and the final point in 5 on the static characteristic corresponding to u E = 3 V. The results from Figs. 3c, 3e confirm the comments regarding the unfavourable timer t 2 = 114 s. Observation. If the excitation u E = 3 V is applied (as for M r = 4 1 4 Nm) previously to the time t 1 = 23 s, the synchronization process is carried out directly according to the dynamic and static characteristics m(θ) from Fig. 4 intersected in the point 3. 3.3. Let us consider M r = 6 1 4 Nm. The other initial conditions are those from 3.1. In conditions u E = 2.1 V when M r = 6 1 4 Nm is applied, there is established again, subsequently to the shock, apparently, a synchronous operation which finally, after a shorter time than previous analysis, turns into an asynchronous operation marked as above, but with sub-synchronous oscillation of ω. Figure 5 plots together the characteristics ω, ω ψ in the zone c where the observations presented before are valid regarding the sub-synchronous oscillations of ω. Fig. 6a m(ω) details zone a, b, c. Fig. 4 Dynamic and static characteristics m(θ), M r = 5 1 4 Nm. Fig. 6b m(ω) detail limit cycle 1, limit cycle 2. Synchronization is not possible anymore even for an excitation voltage u E = 8V sensitively higher than in 3.2; the zone d is characterized by an oscillating asynchronous operation, with more important electromagnetic and mechanical oscillations. In this analysis the zone (23 5 s) is the zone c; beyond 5 s there is the zone d. Fig. 6c m(θ) detail limit cycle 1, limit cycle 2. Figs. 6 Characteristics m(ω), m(θ), M r = 6 1 4 Nm. Fig. 5 Characteristics ω(t), ω ψ (t), M r = 6 1 4 Nm detail zone c. Figure 6 plots the characteristics m(ω), m(θ). Figure 6a also emphasizes the adjacent zones a, b: the final of the zone a, in the point 1; the zone b, insignificant, corresponding to an excitation of u E = 2.1 V, finalized in the point 2 practically

12 Aurel Campeanu, Radu Munteanu, Vasile Iancu 5 overlapped to 1; the zone c, in which after oscillations around the synchronism speed, the permanent oscillating asynchronous operation is reached (limit cycle 1, u E = 2.1 V); the zone d which passes, through oscillations, towards a new permanent oscillating operation (limit cycle 2, u E = 8 V). In Figs. 6b, 6c there are detailed, for clarity, the two limit cycles in coordinated m(ω), m(θ) and passing from a cycle to another; in Fig. 6c there is plotted the dynamic process starting from the point 2. Observations. If a slightly increased excitation u E = 4 V is applied, as before, previously to the time t 1 = 23 s, the synchronization process is carried out directly according to the static and dynamic characteristics m(θ) from Figure 7 intersected in the point 3.3.4. Considering magnetic stresses. In all the dynamic processes analyzed before, the magnetic stresses vary between large limits. For exemplification in Figs. 8a, 8b there are plotted the characteristics of the inductances of L md (t), L mdt (t) for all the zones from the analyzed dynamic process which end with M r = 5 1 4 Nm. It results that for valuable results, the mathematical models must take these major evolutions into consideration. 4. CONCLUSIONS Fig. 7 Dynamic and static characteristics m(θ), M r = 6 1 4 Nm. Fig. 8a L md (t). The main goal of the paper is the analysis of dynamic stability of synchronous machine, particularly of high power synchronous machine. From those presented before it clearly results that in analysis of dynamic stability there must be necessarily taken into account the afferent dynamic angular characteristics, which are generally structurally different, from static angular characteristics. Passing from an oscillating synchronous operation to a synchronous operation by increasing the excitation winding voltage, is only possible if the asynchronous oscillations are around the synchronism speed. Otherwise, these oscillations are amplified, with all the consequences presented before; when mechanical oscillations occur, the speed of the rotating magnetic field ω ψ can vary between large limits too. As a mathematical tool, the mathematical model adequate to the considered synchronous machine must be taken into account. The mathematical model provides an immediate and detailed response for complex problems of dynamic stability. In case of HPSM, consequently to important electromagnetic and mechanical inertia, there must be considered enough times t i for each disturbance of the steady-state process. In general, mathematical models considering magnetic saturation and its dynamic variation, as well as the magnetic and electric asymmetry provide accurately valuable results, in a range of admissible errors and practically they are compulsory in the design stage as well as in anticipating the behaviour of a built machine, operating in a given dynamic process. APPENDIX Fig. 8b L mdt (t). Figs. 8 Characteristics L md (t), L mdt (t) for M r = 5 1 4 Nm. The motor rated values are U = 2,887/5, V, P = 8, kw, n = 1,5 r.p.m., f = 5 Hz. The motor parameters are: R s = 32.967 1 3 Ω, L sσ =.795 1 3 H, L Eσ = 1.823 1 3 H, L Dσ =.838 1 3 H, 3 Q σ =.921 1 H L, R E = 1.798 1 3 Ω, R D = 92.46 1 3 Ω, R Q = 115.5 1 3 Ω, J = 616 kg m 2. The saturation characteristic is imd f ( imd ) = 1.9 9.189 arctan. 823.867

6 Dynamic stability of high power synchronous machine 13 Received on November 9, 216 REFERENCES 1. E. Levi, Saturation modeling in D-Q axis models of salient pole synchronous machines, IEEE Trans. on Energy Conversion, 14, 1, pp. 44 5, 1999. 2. S.A. Tahan, I. Kamwa, A Two-factor saturation model for synchronous machines with multiple rotor circuits, IEEE Trans. on Energy Conversion, 1, 4, pp. 69 616, 1995. 3. I. Iglesias, L. Garcia Tabares, I. Tamaret, A., D-Q Model for the Self Comutated Synchronous Machine Considering the Effect of Magnetic Saturation, IEEE Trans. on Energy Conversion, 7, 4, pp. 768 776, 1992. 4. I. Boldea, S.A. Nasal, Unified Treatment of Losses and Saturation in the Ortogonal Axis Model of Electric Machines, IEEE Proceedings, 1, 6, pp. 134B, 1987. 5. T. Dordea, Beitrag zur Zweeiachsentheorie der Elektrischen Maaschinen, Archiv fur Elektrotechnik, 5, 6, pp. 362 371, 1966. 6. T. Dordea, R. Munteanu, A. Campeanu, Using the Circuit Theory to Derive Matematical Models of AC Machines. A Review, Proc. Advanced Topics in Electrical Enginering ATEE, Bucharest, 211. 7. A. Campeanu, I. Cautil., M. Nicolaescu, T.Cîmpeanu, The effect of magnetic asimmetri on the behavior of the synchronous machine in dynamic regime, Proc. International Conference on Electrical Machines ICEM, Vilamoura, 28. 8. I. Holz, The reprezentation of AC machines dynamics by complex signal flow graph, IEEE Trans. Ind. Electron., 42, pp. 263 271, 1995. 9. A. Campeanu, Nonlinear dynamical models of saturated salient pole synchronous machine, Rev. Roum. Sci. Techn. Electrotechn. et Energ., 46, 1, pp. 89 99, 21. 1. A. Campeanu, M. Stiebler, Modeling and Simulation of Dynamical Processes in High Power Salient Pole Synchronous Machines, Rev. Roum. Sci. Techn. Electrotechn. et Energ., 56, 2, pp. 179 188, 211. 11. M. Stiebler, A Campeaanu, Simulation of Saturation in Salient Pole Synchronous Machines, Proc. International Conference on Electrical Machines ICEM, Rome, 21. 12. M. Mihalache, Eqivalent circuit parameters and operating performances of three-phase asynchronous motors, Rev. Roum. Sci. Techn. Electrotechn. et Energ., 55, 1, pp. 32 41, 21. 13. N. Galan, G. Ciumbulea, Mathematical models and electrical equivalent schemes of the induction motor, Rev. Roum. Sci. Techn. Electrotechn. et Energ., 53, 2, pp. 151 162, 28. 14. T. Dordea, A. Campeanu, The Necessity for Increasing Electrical Machines Efficiency, WEC Regional Energy Forum Forem 24, Code S 5-2, Neptun. 15. A. Campeanu, Transient performance of the saturated induction machine, Electrical, Engineering (Archiv fur Elektrotechnik), 78, 4, pp. 241 247, 1995. 16. A. Campeanu, I. Vlad, S. Enache, G. Liuba, L. Augustinov, I. Cautil, Simulation of Asynchronous operation in High Power Salient Pole Synchronous Machines, Proc. International Conference on Electrical Machines ICEM, Marseille, 212.