TORSIONAL STRESSES IN TURBINE GENERATOR SHAFTS DUE TO ELECTRICAL DISTURBANCES

AN APPLICATION OF THE E.M.T.P. CODE

D.N. Konidaris N.D. Hatziargyriou Public Power Corporation of Greece National Technical University of Athens

Greece Greece ABSTRACT This report presents an application of the Electromagnetic Transients Program (E.M.T.P) for the evaluation of torsional torques that arise in turbine generator shafts due to electrical disturbances. It is part of the answer of the Greek National Committee of CIGRE to Questionnaire 84/2 of Working Group 11.01. Some comments are made as for the implementation of the EMTP code for the study of shaft torsionals, which could be interesting for other users too. GENERAL During the last two decades it became evident that turbine - generator rotor could no more be considered as a single inertia in transient studies. It was in 1970's that the first shaft failure occurred at Mohave Station due to the development of excessive alternating torsional torques. Today it is clear that turbine generator shaft is a complicated mechanical system with an infinite number of natural torsional frequencies. Under the condition that one or more of these coincide with a complement of a natural frequency of the electrical system, a disturbance may excite torsional vibrations, which may lead to failure of the shaft. In the aftermath of the above-mentioned shaft failure at Mohave the EMTP code was extended to include a new modeling feature for synchronous machines with shaft torsionals taken into account. APPLICATION Version M39 of EMTP was used for the evaluation of torsional alternating torques that arise at the coupling zones of a turbine generator subjected to symmetric three phase short circuit for different fault clearing times.

fig.1 CIGRE Benchmark Model for Computer Simulation of Torsional Vibrations

The turbine generator shaft train consists of four cylinders (one high pressure, one intermediate pressure and two low pressure cylinders for the turbine), generator and rotating exciter. In the following part comments will be made on the main application problems. Tolerance choice When using type - 59 synchronous machine source component one must pay attention to the correct choice of the tolerance values. Our first attempt to run the program for the data case of fig.1 led to "marginal convergence" of the iteration and to the printing of a very annoying and paper consuming warning "lack of convergence" for all steps. This was due to the fact that EPDGEL was equal to its very small default value of E-16. The final values for the set of tolerance parameters that were successfully used were: EPOMEG = 0.0001 EPDGEL = 0.00009 NIOMAX = 33 (default value) with a time step of 0.0001 s. Mechanical damping input EMTP uses a spring mass model (fig.2) for the representation of the mechanical system.

fig.2 Spring Mass Model of the Rotor. This model requires the input of mechanical damping in the form of dashpot damping for each mass and material damping for each shaft while only modal values can be measured accurately. A simple method to overcome this difficulty is to solve for the eigenvalues λi and eigenvectors ri of the matrix:

H-1

K where H is the inertia matrix and K is the stiffness matrix. We know that, since these eigenvalues are distinct, we have :

λi = ωmi

2

where ωmi are the angular eigenfrequencies of the mechanical system without mechanical damping. Since mechanical damping is very low, these are, with great accuracy the eigen-frequencies of the real system.

The transformation of the given logarithmic decrement damping values, log-deci, to mechanical damping values, Dmi, can be realized by means of the following equations:

Dmi = -2 σmi Hmi σmi = -(log-deci) fmi

Hm = diag (Hmi) = RT H R

where σmi is the decrement factor of mode i, Hmi is the modal inertia for mode i and R is the matrix of eigenvectors. Let us now consider the equation : Dm = RT D R where D is the damping matrix : D1+D12 -D12 0 0 0 0 -D12 D2+D12+D23 -D23 0 0 0 0 -D23 D3+D23+D34 -D34 0 0 D = 0 0 -D34 D4+D34+D45 -D45 0 0 0 0 -D45 D5+D45+D56 -D56 0 0 0 0 -D56 D6+D56 Although matrix H and K can be diagonized to Hm and Km respectively by a canonical transformation with matrix R, this is not true, in general, for matrix D. We can have a pseudo - diagonization of D if we neglect the off diagonal elements of R

T D R.

With this assumption the following relations are valid : Dm = diag (Dmi) = R

T D R

D = (RT)

-1 Dm (R)

-1

If we now assume that the dashpot damping terms, Di, are negligible then the mutual damping terms, Dij, can be evaluated and used as the mechanical damping input to EMTP. Synchronous machine representation A very important feature of EMTP code is that it allows for a more accurate rotor representation. Conventional models assume that the three per unit mutual inductances of the direct axis are all equal. However this theory gives good results only for the stator circuit quantities ([1], [2]). For the accurate modeling of the rotor it has been proven that it is necessary to introduce a new parameter in the equations to allow the mutual inductance between field and damper to differ from the other two. For the simulation of the system of fig.1 a model with 2+2 windings on the rotor was employed, suitable for the analysis of Subsynchronous Resonance problems ([3]). Results In Table I peak shaft torques at coupling zones 3 & 4 are presented for various fault clearing times (FCT).

Tclear (cycles) TLP1-LP2 (max) in p.u. TLP2-G (max) in p.u.

3.0 1.191 1.135

3.5 2.085 2.105

4.0 2.452 2.585

4.5 2.437 2.746

5.0 2.185 2.605

5.5 2.007 2.162

6.0 1.378 1.593

6.5 1.659 2.123

7.0 2.453 2.741

7.5 2.735 2.960

8.0 2.282 2.608

8.5 1.781 2.187

9.0 1.391 1.586

9.5 1.438 1.842

10.0 1.935 2.213

Table I: Peak torques at coupling zones 3 and 4

From Table I it is evident that the maximum mechanical torque is highly dependent upon the exact FCT and has a semi - periodic nature. ([4]). The results are in rather good agreement with the results of other CIGRE members although different programs were used for the evaluation. To obtain the above results EMTP version M39 was used on a PRIME 2250 minicomputer with 1 Mb of main memory. For the simulation time of 1.5 s the execution time was 1139.395 CPU s (for FCT equal to 4 cycles with printer plottings of electrical torque and mechanical torques at couling zones 3 and 4). In addition, torsional torques on turbine generator shafts have been studied using ATP version on a IBM PC/AT compatible microcomputer under DOS 3.21. In Appendix A the respective Data file is shown and in Appendix B plottings of main quantities are presented. ACKNOWLEDGEMENTS The work presented here was carried out at the Electric Energy Systems Lab of Elecric Power Division of The National Technical University of Athens. The authors wish to thank the Director of the Laboratory Prof. B.C. Papadias. REFERENCES [1]. I.M. Canay, "Causes of Discrepancies on Calculation of Rotor Quantities and Exact Equivalent Diagrams of the Synchronous Machine", IEEE Trans. on PAS, Vol. PAS-88, July 1969.

[2]. B. Adkins, R.G. Harley, " The General Theory of Electrical Machines: Applications to Practical Problems", Chapman and Hall, London. [3]. IEEE Joint Working Group, "Current Usage and Suggested Practices in Power System Stability Simulations for Synchronous Machines", IEEE Trans. on EC, Vol. EC-1, nr. 1, March 1986. [4]. A. Abolins, D. Lambrecht, J.S. Joice, L.T. Rosenberg, "Effect of Clearing Short Circuits and Automatic Reclosing on Torsional Stress and Life Expenditure of Turbine Generator Shafts", IEEE Trans. on PAS, Vol. PAS-95, nr. 1, January 1976. APPENDIX: DATA Transformer: 21 V/380 kV, RT = 0.002 p.u., XT = 0.16 p.u. System 1: SCC1 = 10 GVA, R/X = 0.1 System 2: SCC2 = 10 GVA, R/X = 0.1 Generator: 850 MVA, 720 MW, 21 kV, 50 Hz, full loaded unit, Xs = 0.22, Rfd = 0.001, Xcd = 0.035, Xhd = 2.45, Xfd = 0.123 Rdd = 0.012, Xdd = 0.037, Ra = 0.0015 Xhq = 0.231, Xcq = 0.0, Xfq = 1.1, Rdq = 0.011, Xdq = 0.1 (p.u.) Mech. Data: H1 = 880, H2 = 2200, H3 = 13800, H4 = 13800, H5 = 11300, H6 = 520 in Kg m2, K12 = 1.6, K23 = 2.55, K34 = 2.4, K45 = 1.15, K56 = 0.9 in 106 Nm Percentage of load for each cylinder: 24, 27, 24.5, 24.5% Logarithmic decrement for modes 1,...5: 0.0030, 0.0025, 0.0015, 0.0010 & 0.0015 APPENDIX: DATA CASE BEGIN NEW DATA CASE POWER FREQUENCY 50. 0.0001 2.0 50. 50. 1000 1 1 1 GENTEATRANSA .00104.08301 GENTEBTRANSBGENTEATRANSA GENTECTRANSCGENTEATRANSA TRANSAINFINA .00439.04388 TRANSBINFINBTRANSAINFINA TRANSCINFINCTRANSAINFINA BLIN2AELIN2ATRANSAINFINA BLIN2BELIN2BTRANSAINFINA BLIN2CELIN2CTRANSAINFINA BLIN2ABLINBA .00001 BLIN2BBLINBB .00001 BLIN2CBLINBC .00001 BLANK CARD ENDING BRANCH CARDS TRANSABLIN2A-1.0 0.12 TRANSBBLIN2B-1.0 0.12 TRANSCBLIN2C-1.0 0.12 ELIN2AINFINA-1.0 0.12 ELIN2BINFINB-1.0 0.12 ELIN2CINFINC-1.0 0.12 BLINBA 10. BLINBB 10.

BLINBC 10. BLANK CARD ENDING SWITCH CARDS 14INFINA 1+17307.16250. 0.0 -1. 14INFINB 1+17307.16250. 0.0 -1. 14INFINC 1+17307.16250. 0.0 -1. 59GENTEA 17146.4282 50.0+0.9.773843 GENTEB GENTEC TOLERANCES 0.0001 0.00009 0605060002 1.1 850.0 21. 1900. 2.608