Damping torsional oscillations due to network faults using the dynamic flywheel damper

W.-C.Tsai C. Chyn T. -P. Tsa o

Indexing terms: Turbine generator, Flywheel damper, Mechanical equivalent of heat

Abstract: The technical feasibility of using a dynamic flywheel damper to damp shaft torsional oscillations in large turbine generators is investigated. Torsional damping is achieved using the slide friction force of the flywheel damper based on generator speed. A system is defined and the shaft torques following various electrical system disturbances are evaluated and analysed. It is shown that substantial damping can be implemented with a well designed flywheel damper, thus reducing the risk of significant shaft damage due to network disturbances. The equivalent models and designs for the flywheel dampers fitted into the shaft couplings of turbine mechanical systems are also described. This method can effectively damp the torsional oscillations due to the high reliability of the damper.

1 Introduction

Recently, much research attention has been directed toward suppressing the torsional damage to the shafts of large turbine generators caused by electrical system disturbances. Numerous countermeasures to the tor- sional oscillation problem such as an excitation con- troller [ 1, 21, a dynamically controlled three-phase resistor bank [3], static VAR compensators (SVC) [4- 91, superconducting magnetic energy storage units [lo], thyristor-controlled series compensation (TCSC) [ 1 11 and many others [ 13-1 71 have been developed since the first two shaft failures occurred at the Mohave station in 1970 and 1971 [17]. The dynamic flywheel damper which can be installed on the shaft coupling of the effected synchronous generator to depress unstable tor- sional oscillations will be proposed in this paper.

Results from these studies indicate that certain elec- trical system disturbances can significantly reduce the life expectancy of turbine shafts. The shaft damage that

0 IEE, 1997 IEE Proceedings online no. 19971264 Paper first received 9th August 1996 and in final revised form 26th February 1997 W.-C. Tsai and T.-P. Tsao are with the Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, 80424 Taiwan, Republic of China C. Chyn is with the Department of Electrical Engineering, Nan-Tai College, Yungkung, Tainan, Taiwan, Republic of China

would result from a particular disturbance depends on factors such as the initial magnitude of the induced tor- sional oscillations and the damping associated with the torsional modes. If the induced oscillations arc large and the damping associated with the torsional modes is small, a large number of high amplitude stress cycles can occur, contributing to a significant loss of life [3]. Furthermore, the damping associated with the torsional modes can indirectly influence the peak magnitude of the induced oscillations. If supplementary mechanical damping is added to the torsional modes of the turbine generator, the impact of electrical system disturbances on the turbine shafts would be considerably reduced.

A method of damping the shaft torsional oscillations using a dynamic flywheel damper is examined. The two ABB flywheel couplings with a diameter of 2170" and a net weight of 10 x 103kg have been installed to the LPB-ROTOR GEN shaft sections to solve the existed torsional oscillations and prevent stress damage cracks on the no. 1 and no. 2 turbine generator shafts of the no. 3 nuclear power plant (951 MW turbine gen- erator) in Taiwan. The design work was completed in December 1989 and the manufacture and installation of the flywheel couplings was completed by the end of 1990 [14].

2 Description of the system

A schematic diagram of the electromechanical system considered in this paper is shown in Fig. 1. The electri- cal components of the three-phase system consist of the synchronous generator, exciter, AVR control, trans- former, contact breakers, transmission lines and infinite bus.

The electromagnetic torque is given by [14] W

T - - {$cl ( i b - i c ) +$b ( ic - ia) +$Ji, - i b ) } (1) "-3A Three phases of the transmission line are modelled as separate resistances and reactances [ 141.

Fig. 2 shows the turbine-generator lumped parameter model. The mechanical data arc given in the Appendix (Section 8).

The equations in the mechanical axes for the inline shaft and blade system with the flywheel coupling and damper installed into the shaft section between LP2R and the generator are given by [ 151

P 4 j = w j (2)

P d B j = W B j ( 3 )

IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 5, September 1997 495

transmission tine R t , X t infinite b + b

fault

@ “&x+ tlr..‘] Fig. 1 Schematic diagram of electromechanical system of951 M W turbinegenerator

Fig.2 Lumped parameter model of 951 M W mechanical system

P W , = {TI”, - Tout3 - (D, + D,-l,J + DJ,J+l)LJJ + DJ-l, , * WJ-1 + D,,,+l * LJ3fl

- ( G I , , + KJ,3+l + KBJM,

combined, as the hollow flywheel, to be mounted on the shaft couping by the bolts. The coupling and the flywheel coupling can therefore be disassembled or fit- ted in the same way due to their similar designs.

+ KJ-LJ * 43-1 + Km+l * 43+1

+ K B ~ * @Bg)/IJ (4)

P W B ~ = ( T ~ B ~ - ToutB3 + K B ~ * 4, - KBj * @BJ - DB, * W B j ) / I B j (5)

and for the flywheel damper and coupling system by

P@CP = W C P (6)

P 4 F W = WFW (7) PLJCP

= {-(DCP + DLZ-CP + DCP-G + DCP-FW)UCP + DLZ-CP * W L ~ + DCP-G * WG + DCF * W F W

~ (KL2--CP + ICCP-G)4CP + K L S - C P * 4 L 2

+ KCP-G * $G}/ICP ( 8 )

- DFW * W F W ) / r F W (9)

PWFW = (DcP-FW * WCP + DCF * WFW

where 4, w, D, K and T are the deviation angle, speed, per unit damping coefficient, per unit shaft stiffness and per unit torque, respectively. The schematic dia- gram of an ABB flywheel coupling is shown in Fig. 3. Two sets of same semicylinders (stainless steel) are

I

Fig. 3 Schematic diagram of ABBj’lywheel coupling

3 Slide friction principles

In the cylindrical-shaped slide surface, the slide friction coefficient is given as

(10) 7 . 2 7 ~ . N 2 ~ r 7 . N ~ . _ _ - - F 7 . v -

’ ”=@==- A . p . 6 A . 6 p

where F (kg) = friction force, W (kg) = vertical weight, q (cp) = absolute viscosity of lube, 6 (cm) = the mean gap between shaft and bearing, U (cmis) = the mean linear velocity of the rim, N (r.p.m.) = the shaft speed, A (cm2) = cross-sectional area of rim, p (kg/cm2) =

496 IEE ProcCener. Transm. Distrib., Vol. 144, No. 5, September 1997

weightlper unit area, and r (m) is the radius of gyration of the rim. The previous term r (cm) can be considered as the constant if the shaft is the same. Also, q(Nlp) is defined as the characteristic of the shaft bearing. If the shaft has a large diameter and is heavy, it is necessary to use strongly viscous oils as the gear oil including the no. 7 and no. 8 pressurised additives. The dynamic vis- cosity of these oils are 25-30 and 3 0 - 4 0 ~ ~ ~ respectively, under 98.9"C. The greases with the various additives have a different friction effect.

The calculation of the friction occurring at the shaft bearing can be obtained using

kW (11) E ~ . W . V p u w - v H = - = hp E 75 75 100

where E (kg-mls) is the friction energy, H and Q are the friction horsepower and quantity of heat, respec- tively, U (mls) = sliding velocity, and J (= 429.6kg-mi kcal) is the mechanical equivalent of heat. If the fly- wheel is hollow, then the weight, inertia and heat energy can be calculated by eqns. 13-15, respectively

w = ~ ( R $ - R ; ) - p . h (13)

(14) 1 1 2 2

I = - W . R; + - W . R;

El, -I ,w 1 2 2

4 Design of flywheel damper

Applying the slide friction theory to two current tur- bine generators, if the designed flywheel damper depicted in Fig. 3 is fitted between the LP2R low-pres- sure steam turbine and generator, the material and size are given as: p = 7.9 x lo3 kg/m3 (stainless steel); RI = 0.2 m; R2 = 1.4 m; R3 = 1.6 m; and h = 0.8 m. Then, the flywheel weight is equal to: W = n(R3, - RZ) . p . h = 11.9 x lo3 kg and the inertia is approximately equal to I = 1/2(W. RZ) + 112(W. R;) = 26.9 x 103kg-m2. Since the generator is four-pole and the system fre- quency is 60Hz, the angular velocity and kinetic energy are given as

27T x 2f = 188.5rad/s

P 1 2

EI, = - I . w2 = 4.78 x lo8 J/s

For a 951MW generator set, the base of the kinetic energy is defined as EKbase = 951 x 108Jls. Thus, the fly- wheel inertia is inertia = 0.503MW - slMVA. The boundary slide velocity between the flywheel and shaft coupling is U = o . R2 = 263.9mls.

After injecting the appropriately viscous grease, the friction coefficient yields ,U = 2.0. Therefore, the fric- tion power and heat quantity of the generator set may be given by

p . W . V N 62473.2 kW

100 H =

E p . W * v 'v 14712.6 kcal/s

'= J = 426.9 By using the generator rating as MVA base, the damp- ing can be obtained as: damping = 0.0657MW-s/ MVA-rad if the coupling inertia = 0.15 (MW - s) l MVA.

The mechanical model of the flywheel damper (see Fig. 5 ) can be set up from the mechanical constructions shown in Fig. 4. It is assumed that the flywheel damper is installed at the LP2R shaft section between the low- pressure steam turbine and generator. The additional damping is located between two inertias representing the shaft coupling itself and the outer flywheel. This mechanical model can be simulated for the damping effect of shaft torsional oscillations. The mechanical model includes the flywheel, damper and coupling. It is more convenient to analyse this phenomenon using the electromechanical analogy method. By the conversion of the impedance analogy, the electromechanical anal- ogy model of the mechanical system is illustrated in Fig. 6. When the generator set is under the normal operation without any disturbances, the operating fre- quency relative to the system frequency can be zero. Clearly, the capacitances have open-circuited and the inductances short-circuited according to electrical cir- cuit theory. Therefore, the additional mechanical damping is bypassed by a small amount of damping so that power energy would not be lost under normal operation. System disturbances cause the large current to form a voltage on two terminals of the inductance. The inductance is analogous to the flywheel inertia by electromechanical analogy. This is the reason why non- system frequency current cannot pass the damper and dissipate energy on it.

flvwheel ollow il gland 'scous oil

I 1 I! I H --I I

Fig. 4 Schematic diugram of designed flywheel damper

.t DFW

f lywheel

k P - F W rlnmnnr

T

DL2-CP I %P-G

Lf DCP

7777777 Fig. 5 Mechamicul model of designed flywheel damper

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 5, September 1997 491

DFW I F W DCP I C P

KCP-G L- KLZ-CP DCP-FW

Fig. 6 Electromechanical model oj designed flywheel damper

For the case of a 150MW generator set, a flywheel damper made of stainless steel can be designed as below. The size is: p = 7.9 x 103kg/m3 (stainless steel); RI = 0.2m; R2 = 0.8m; R3 = 1.0m; and h = 0.6m. The flywheel weight then is equal to W x(R# - R3) . p . h = 5.361 x 103kg, and the inertia is I = 1/2(W. R:) + 1/2(W. R;) = 3.615 x 103kg-m2.

With the two-pole generator, the system frequency is 60Hz and thus the angular velocity and kinetic energy are given as

2i7 x 2f &J=- = 377rad/s

P 1 2

For the 750MW generator set, the base of the kinetic energy is defined as

Ek = - I . w 2 12.57 x 10' J/s

1 2

E k b a s e = - I . w2 = 7.5 x 10* J/s

Thus, the flywheel inertia M W - s

MVA inertia N 0.343

The boundary slide velocity between the flywheel and shaft coupling is U = w . R2 = 301.6mls. Spraying the appropriately viscous grease, the friction coefficient yields ,U = 2.0. Therefore, the friction power and heat quantity of the generator set may be given by

p , w . u Y 32101.1 kW

100 H =

E ~ + W * W N 7560 kcal/s ' 7 = 426.9

In terms of the generator rating as MVA base, the damping can be obtained as: damping = 0.0428MW-s/ MVA-rad if the coupling inertia = 0.15MW - s1MVA. Comparing with the given generator damping, it can be seen that the mechanical damping in this system has increased significantly.

5

To illustrate the effectiveness of the present flywheel damper for damping torsional oscillations under the disturbance conditions, the digital computer simula- tions are performed. The dynamic behaviour of the sys- tem subjected to the L-L and L-L-E faults which occurred at 0.1s and cleared at 0.279s is now investi- gated.

The system response is shown in Figs. 7-10 and 15- 18 without the flywheel damper in service and in Figs. 11-14 and 19-22 with the dynamic flywheel damper in service. Comparison of the transient torques in Figs. 7- 10 and 15-18 to those in Figs. 11-14 and 19-22 shows that the damping action has significantly reduced the maximum shaft torques. Furthermore, since the tor-

Results of time domain simulation

498

sional oscillations shown in Figs. 11-14 and 19-22 have a much shorter lifetime, significantly less shaft damage would occur with the flywheel damper in service.

0 1 2 3 L time,s

Fig.7 951 M W twbine generator shaft torque response due to worst case L-L-E fault 0.1s fault applied and clearing at 0.277s, load angle 107.5", autoclosure sequence not successful LP2R-GEN shaft torque

-1.5 L I I I I 0 1 2 3 L

time,s Fig.8 As for Fig. 7 but for LPIR-LP2Fshaft torque

0.8 0.71 I I I

~~

-0.31 I 1 I I 0 1 2 3 L

time,s Fig.9 As for Fig. 7 but for HP-LPIFshaft torque

2 002 E- 001 y 00 6% -0 01 0 -002

-003 -0.OL I I I I I

0 1 2 3 L time,s

Asfor Fig. 7 but for GEN-RECshaft torque Fig. 10

The peak magnitudes of the shaft torques following an unsuccessful high speed autoreclosure sequeme depend on the instantaneous values of the shaft torques at the time of autoreclosure. In particular, the torques induced due to reclosing may subtract from or add to the existing torsional oscillations. Figs. 7-10 and 15-18 illustrate the worst case cenarios of L-L and L-L-E faults, respectively, where the induced torques due to unsuccessful reclosing completely reinforce the existing oscillations, thereby doubling the peak to peak magni- tude of the torsional oscillations initially induced by the faults.

IEE Proc.-Gene?. Transnz. Distrib., Vol. 144, No. 5, September 1997

499

- 0 L I I I I I I I

0 1 2 3 L time,s

As for Fig. 19, but for HP-LPIFshuft torque Fig.21

0.05 0.041 i ,ihr .

"."_ -0.051 I I I I J

0 1 2 3 1 time,s

As for Fig. 19, but for GEN-RECshufi torque Fig.22

5. I The torsional torques due to the network disturbances and mechanical responses oscillate in the material of the shaft system and use up the life span of the shaft. Though time domain simulations can give the dynamic behaviour of the system subjected to a disturbance, they are not so useful for studying the rate of fatigue of shafts caused by torsional oscillation components in the system. A torsional fatigue estimation which is capable of performing this work will be applied to demonstrate the validity of the proposed flywheel damper for reducing the torsional oscillations of system responses.

5. I . I Fatigue theories and models: Fatigue models for the most part are based on empirical data. This data usually takes the form of a stress-life curve.

Torsional fatigue life estimations

The rotor materials used by the turbine generator being studied were tested. NiMoV is a generator field mate- rial, NiCrMoV is a low-pressure rotor material, and CrMoV is a high-pressure rotor material within the scope of ASTM specifications A469, A470 and A471. The torsional stress-strain curves, strain-life and stress-life diagrams for three materials are presented in [18] as the plots of shear strain and stress amplitude against cycles-to-failure.

Because fatigue damage is assessed on a closed hys- teresis loop basis, the rotor deformation history must be analysed to identify the reversals that form closed hysteresis loops or cycles. A cycle-counting technique, currently considered to be the best available and known as the rain-flow cycle counting method, uses closed loops in the stress-strain plane. The Palmgren- Miner rule states that the damage associated with each closed hysteresis loop found in a complex load history may be added together to obtain the cumulative dam- age. This is known as linear damage cumulation theory and can be

where ni is and Nfi is

expressed mathematically as

$ = fraction of life lost (16) i

the number of cycles at the ith stress level, the number of cvcles to failure at the ith

stress level. When the fraction of life expended equals one, the linear damage cumulation predicts failure of the material [18].

The maximum shear stress on the shaft is the shear stress at the outer radius [ 181

where Ro is the radius of a uniform shaft and z is the net torque. The influence of the safety factor, as altered by the radius, on the cumulative damage of the shaft system is also investigated in the paper.

5.1.2 Results of fatigue prediction: The results in Table 1 show torsional fatigue damage at 951MW

Table 1: Torsional fatigue due to worst-case L-L fault (Fault applied and cleared at 0.1 s and 0.279s, machine malsynchronisation, auroreclosure sequence not successful)

Time (as %)

0.1-1 .I s 0.1-3.1 s 0.1-5.1s 0.1-7.1 s Safety factor

1.75 0.805885 5.626513

2.0 0.064358 0.440665

2.25 0.007268 0.049579

Shaft

HP-LPIF (clearing at 0.279s, load angle 112.5")

2.5 0.001 104 0.006966

3.0 17.1 51 160 44.132060

3.5 0.969947 2.497299 0.081212 0.210813

4.5 0.009425 0.024401

HPIR-LP2F (clearing at 0.279s, load angle 112.5") 4.0

3.0 2.885353 6.643706 3.5 0.167660 0.386734

0.014576 0.033470

4.5 0.001618 0.003764

HP2R-Gen. (clearing at 0.279s, load angle 112.5") 4'0

2.0 6.791221 6.792237

2.25 0.461196 0.461219 2.5 0.034652 0.034652

3.0 0.000160 0.000160

Gen.-Rec. (clearing at 0.331 s, load angle 150")

6.505370

0.509897

0.057223

0.008078

49.6221 60

2.81 5381 0.237434 0.027424 7.667989 0.445393

0.038561

0.004334

6.792237

0.461 21 9

0.034652 0.0001 60

6.564077

0.514468

0.008 145

0.008 145

49.622310

2.815389 0.237434 0.027424 7.6681 27 0.44540 1

0.038561

0.004334

6.792237

0.461 219

0.034652 0.000160

500 IEE Proc.-Geneu. Tuansm. Distrib., Vol. 144, No. 5, September 1997

Table 2: Torsional fatigue due to worst-case L-L fault with 0.164MW-s/MVA-rad flywheel damper installed (0.1 s fault applied, machine malsynchronisation, autoreclosure sequence not successful)

Shaft

Time (as %)

Safety 0.1-1.1s 0.1-3.1s factor

0.428916 HP-LPIF (clearing at 0.279s, load angle 112.5")

HPIR-LP2F (clearing at 0.279s, load angle 112.5")

F LYW-G e n . (clearing at 0.279s. load angle 112.5")

Gen.-Rec. (clearing at 0.331 s, load angle 150")

1.75

2.0

2.25

2.5

3.0

3.5

4.0

4.5

3.0

3.5

4.0

4.5

2.0

2.25

2.5

3.0

0.42871 1

0.035585

0.00401 1

0.000615

0.028283

0.001650

0.000148

0.000016

0.000562

0.000035

0.000003

0.000000

1.616591

0.097783

0.006478

0.000016

0.035602

0.00401 1

0.000615

0.028283

0.001 650

0.000148

0.000016

0.000562

0.000035

0.000003

0.000000

1.61 6591

0.097783

0.006478

0.000016

0.1-5.1s

0.428916

0.035602

0.00401 1

0.000615

0.028283

0.001650

0.000148

0.000016

0.000562

0.000035

0.000003

0.000000

1.616591

0.097783

0.006478

0.00001 6

0.1-7.1 s

0.428916

0.035602

0.00401 1

0.00061 5

0.028283

0.001650

0.000148

0.000016

0.000562

0.000035

0.000003

0.000000

1.616591

0.097783

0.006478

0.00001 6

turbine generator shafts without the flywheel damper in service following a line-to-line fault which was applied and cleared at 0.1 s and 0.279s, respectively. The influ- ence of the 0.164MW-s/MVA-rad flywheel damper installed and various safety factors selected at the design stage to the accumulative fatigue life expendi- ture on turbine shafts are shown in Table 2. The fatigue damage is evaluated at 1.0s, 3.0s, 5.0s and 7.0s. The effect of autoreclosure on the fatigue life expendi- ture of shafts is not considered.

6 Conclusions

The technical feasibility of using a controlled flywheel damper to damp shaft torsional oscillations in a large- scale turbine generator has been considered. From the results in Section 5 and 5.1 it can be seen that almost all of the torsional fatigue damage at the generator shafts occurs in the first three seconds immediately fol- lowing the worst-case fault clearance, however, the tor- sional fatigue only has an effect on the shaft life expenditure in first second with the flywheel damper in service.

The torsional shaft torques following worst-case fault clearance are significantly reduced as the flywheel damper activates in service. Therefore, more fatigue life is expended by this flywheel damper at the LP2R-GEN shaft coupling.

The damping of shaft torques due to electrical distur- bances is achieved by dissipating power using the fly- wheel damper installed between the LP2R low-pressure turbine and generator. The operation of the flywheel damper during severe electrical disturbances has been illustrated. Even though the disturbances studied herein

response can be particularly severe unless supplemen- tary damping is added. The flywheel damper would be equally effective in damping torsional oscillations

may be infrequent for most aystems, thc torsional

IEE pro^ -Gener TrunJm Dirtrib, Vol 144, No 5, September 1997

induced by less severe but more common disturbances, e.g. line-ground faults, line switching etc. Here, the benefits of the added damping would be less pro- nounced for any individual event, however, since they occur more frequently, the benefits of the added damp- ing may accumulate during the operating life of the turbine generator.

Ideally, since the mechanical structure of the flywheel damper is simple and solid, its life expectancy should be same as the design life of the turbine generator set.

Under normal operating conditions, there are no tor- sional vibrations and relative motions between the shaft coupling and the hollow flywheel. The friction and heat energy in the flywheel damper cannot be pro- duced. Therefore, the viscous oil should deteriorate quite slowly. It is only necessary to monitor the quality and quantity of the viscous oil between the coupling and hollow flywheel daily. When the oil has deterio- rated, it may be renewed during the annual down- machine maintenance period.

Since the tooth-shaped parts of the coupling and hol- low flywheel are embedded as the flywheel damper, it is required further to design its mechanical structure to solve the installation problem. One of the feasible methods is that two sets of semicylinders (stainless steel) are combined as the hollow flywheel by using the bolts. With such a design, maintenance staff may read- ily disassemble, fit and maintain the flywheel damper.

Since using the mechanical method to depress the vibrations caused by fault or condition disturbances is simple and reliable, this paper proposes that a flywheel damper is installed at the pivot position of the shaft coupling to increase the mechanical damping in vibra-

under normal operation. Furthermore, the slide friction principle is applied to two current generator sets for the accurate designs and models.

tion Goniponcnts without dissipating ally power energy

501

~

7

1

2

3

4

5

6

7

8

9

References

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WASYNCZUK, 0.: ‘Damping subsynchronous resonance using reactive power control’, IEEE Trans., 1981, PAS-100, pp. 1096- 1104 HAMMAD, A.E., and EL-SADEK, M.: ‘Application of a thyris- tor-controlled VAR compensator for damping subsynchronous oscillations in power systems’, IEEE Trans., 1984, PAS-103, pp. 198-2 12 BALDA, J.C., EITELBERG, E., and HARLEY, R.G.: ‘Optimal output feedback design of a shunt reactor controller for damping torsional oscillations’, Electr. Power Syst. Res., 1986, 10, pp. 25- 33 WANG, L., and HSU, Y.Y.: ‘Damping of subsynchronous reso- nance using excitation controllers and static VAR compensators - a comparative study’, IEEE Trans., 1988, EC-3, pp. 6-13 BYERLY, R.T., POZNANIAK, D.T., and TAYLOR, E.R.: ‘Static reactive compensation for power transmission systems’, IEEE Trans., 1982, PAS-101, pp. 39974005 XING, K., and KUSIC, G.L.: ‘Damping subsynchronous reso- nance bv Dhase shifters’. IEEE Trans.. 1989. E C 4 . DD. 344-350

100, (7), pp. 3340-3349

8 parameters

8. I Electrical system data Synchronous-generator (Rating: 1057.5MVA, 23.75KV, 60Hz, 4 pole)

Appendix: 951 MW Turbine generator system

Xd 1 1.574 Xq = 1.490 x, = 0.190 X’j = 0.168 Xk,j = 0.110 Xkq = 0.490 R, 0.00359 R k d = 0.02571 Rkq 0.02571 Rf = 0.000698

Step-up transformer (rating: 1057.5MVA, 23.751 345 KV)

Rt = 0.00192 Transmission line (rating: 1057.5MVA, 345KV) X, = 0.14304

X ~ A = X i , XIc = 0.1088 R ~ A = R ~ B = R ~ c = 0.0073 rz X ~ B = X2c 0.1088 R ~ A = R ~ B R ~ c = 0.0073

Initial operating conditions PO = 0.90 Q, = 0.10 V, = 1.03

8.2 Mechanical system data Mass Inertia Damping Stiffness

10 LEE, Y:S:, and WU, C:J.: ‘Application of supercodducting mag- netic energy storage unit on damping of turbogenerator subsyn- chronous oscillation’, IEE Proc. C, Gener. Transm. Distrib., 1991,

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12 SHELTON, M.L., MITTELSTADT, W.A., WINKELMAN, P.F., and BELLERBY, W.J.: ‘Bonneville power administration 1400-MW braking resistor’, IEEE Trans., 1975, PAS-94, (2), pp. 602-61 1

13 YOSHIDA, K., FUKUNISHI, M., and AOI, J.: ‘Development of system-damping resistors for stabilizing bulk power transmis- sion’, Elect. Eng. Jpn., 1971, 91, (3), pp. 79-90

14 LIANG, C.-C.: ‘Torsional response of rotor system in steam tur- biiiegenerator’, Taipower Eng. J., 1993, 538, (6), pp. 35-52

15 TSAO, T.P., and CHYN, C.: ‘Restriction of turbine blade vibra- tions in turbogenerators’, IEE Proc. C, Gener. Transm. Distrib., 1990, 137, (5) , pp. 339-342

16 TSAO, T.P., CHYN, C., and NIEN, H.H.: ‘A large-scale mechanical filter application of turbogenerator flywheel coupling’, Electr. Power Syst. Res., 1992, 10, pp. 35-45

17 HALL, M.C., and HODGES, D.A.: ‘Experience with 500KV subsynchronous resonance and resulting turbine generator shaft damage at Mohave generating station’. IEEE, No. 76, New York, 1976, pp.22-29

18 CHYN, C., WU, R.C., and TSAO, T.P.: ‘Torsional fatigue of turbinegenerator shafts due to network faults’, IEE Proc. Gener. Transm. Distrib., 1996, 143, (5), pp. 479486

138, pp. 419-426

pp. 7-12

HP

LPlF

Blade LPlR

Blade LP2F

Blade LP2R

Blade Gen.

Rec.

Exc.

H (MW-s MVA

0.1787

0.6546

0.0134 0.6486

0.0134 0.6575

0.0134 0.6676

0.0134 1.1616

0.00334

0.00236

1 MW-S MW-s ( MVA-r ad K(MVA-rad

0.00180 144.15

0.000230

0.000004 0.000210

0.000004 0.00021 0

0.000004 0.00021 0

0.000004 0.00012

1595.0 36.2

24.069 36.2

1584.9 36.2

325.28 36.2

117.16 0.0

1.61 0.0

502 IEE Proc.-Gener. Transm. Distuib., Vol. 144, No. 5, September 1997