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  • 8/19/2019 Francis Turbine Oscillatory Problems

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    IEEE

    Transactions

    on Energy

    Conversion,

    Vol.

    12,

    No.

    4, December 1997

    419

    INVESTIGATION

    OF

    OSCILLATORY PROBLEMS OF HYDRAULIC

    GENERATING UNITS EQUIPPED WITH FRANCIS TURBINES

    D.N. Konidaris, Member,

    IEEE

    Public Power Corporation, Greece

    A bs t rac f i l n t he p resent paper a method

    Is

    presented

    for t he s tudy o f osc i l l a to ry p rob lems o f hydrau l i c

    generat lng uni ts equipp ed wi th Franc is turb ines. The

    acc ura t e equat lons o f t he tu rb ine a re imp lemented

    for the s imulat ion, tak ing into accoun t water hamm er

    e f fec t . Par t ia l der i va ti ves o f t u rb ine f l ow ra te and

    torque w i th respec t t o head and ga te open ing a re

    dedu ced from rea l measurements fo r t he who le

    load ing range. Speed governor dynamics a re f u l l y

    s imulated, and the inf luence of tors ion al v ibrat ions is

    s tudied. The v ibrat ion response of the uni t to draf t

    tube surges Is invest igated and the r isk of excess ive

    osc i l la t ions

    Is

    eva lua ted . The approach was de-

    ve loped fo r d iagnos is o f excess iv e v ib rat ions o f

    Kas t r ak i hydro po wer s ta tion in Greec e.

    K e y w o r d

    :

    ydraulic units, Francis turbines, Dra 7 tube

    su ges, Vi rations, Diagnosis

    LIST OF SYMBOLS

    6, , 6

    differential angular position of turbine, generator

    6 transient speed droop coefficient

    J permanent speed droop coefficient

    w differential angular velocity

    a velocity of water hammer waves in penstock

    T pilot valve time constant

    T dashpot time constant

    T water starting time constant

    T distributor valve time constant

    T penstock elastic time constant

    K

    torsional stiffness of shaft

    h differential head

    m differential mechanical torque of turbine

    G s) turbine transfer function (torque to speed relation)

    fh penstock characteristic frequency

    g drfferential gate opening

    J,

    ,

    J inertia constant

    of

    turbine, generator

    p,

    s

    derivative and Laplace operator

    PE-575-EC-0-02-1997

    A paper recommended and approved by the

    IEEE E nergy Development and Power G eneration Committee of the

    IEEE Power Engineering Society for publication in the IEEE

    Transactions

    on

    Energy Conversion. Manuscript submitted August 27,

    1996; made available for printing February 10, 1997.

    J.A. Tegopoulos, Life Fellow,

    IEEE

    National Technical University of Athens, Greece

    I. INTRODUCTION

    Vibrational problems of hydraulic generating units have

    long been recognized

    as

    a cause of decreased availability

    and increased production cost. Sometimes vibration level is

    so high that the operator has to reduce the power output

    below optimum. Large Francis turbines have been excep-

    tionally subject to vibrational difficulties due to flow induced

    structurdl vibrations, [I]. When operated at partial load

    Francis turbines may experience severe vibrational prob-

    lems due to the presence of intense pressure surges in the

    draft tube, [Z]. Electromechanicaloscillations may also arise

    leading to power swing, vibrations and noise.

    A hydroelectric plarit may be considered as a system

    consisting of several subsystems

    :

    the penstock including

    any surge chamber, the hydraulic machine, the speed

    governor, the tailrace and, finally, the generator and electri-

    cal network. Excitation of vibrations may arise from various

    sources, such as flow variation in the penstock, part load

    vortex in the draft tuble, load rejection, etc. The present

    state of knowledge perrnits to provide remedies to vibration

    problems in the field, after commissioning of the plant.

    Nevertheless the main cause is to predict vibration in the

    design stage.

    Simulation of hydro turbines has traditionally been

    performed neglecting water column elasticity, [3]. Although

    such models have been1 extensively used for system stability

    studies, they are not suitable for vibration analysis of

    hydroelectric plants, especially when long penstocks are

    implemented. The water column elasticity principle, [4], has

    recently been used for the identification

    of

    higher order

    hydraulic turbine models, [ 5 ] ,as well as for design purposes,

    Speed governor dynamics greatly influences hydro plant

    dynamic response in the significant frequency range of up to

    about 100 rls . Detailed hydro governor models are neces-

    sary to account for its filtering action on high frequencies of

    the penstock -turbine transfer function, [3]

    &

    [8].

    Pressure surges in the draft tube have been extensively

    studied, and a great deal of knowledge has been gained and

    published, [I],

    [9]

    & [ IO] . Information is nowadays available

    about the frequency, intension and inception of vortex

    phenomenon.

    Technical literature lacks papers making use of the recent

    contributions to the study of the dynamic response of the

    integrated hydro plant, [ Il l, [I21

    &

    [13].

    In the present paper a transfer function approach is used

    for vibration analysis of hydro plants with Francis turbines.

    Full equations are implemented for the accurate simulation

    of the turbine - penstock subsystem, taking into consider-

    atiin water column elasticity. Torsionals of the shaft system

    are also t aken into consideration, while speed governor is

    161

    & [71.

    0885-8969/97/$10.00 997 IEEE

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    420

    simulated by the full representation model. The approach is

    demonstrated on diagnosis of vibrational problems of

    Kastraki hydro power plant in Greece.

    II. SIMULATION MODEL

    A

    Penstock - urbineModel

    According to the water hammer theory, which has been

    developed by Allievi in the early 19OO's, pressure waves

    may arise in long penstocks propagating with water hammer

    velocity a. Water hammer theory takes into consideration

    the water column elasticity, [4],and in terms of transfer fun-

    ctions can be translated for the frictionless case of a uniform

    conduit as followes

    :

    hlq (s)= - (TWIT,) tanh(T,s)

    (1)

    where

    :

    h(s) is the differential head at the turbine inlet (P.u.),

    q(s)

    is

    the differential flow rate (P.u.),

    T is the water starting time constant (s),

    T

    is the penstock elastic time constant (s).

    It is worth noting that the relationship between the turbine

    rate of flow and the turbine head depends only on the

    penstock, being independent of turbine's characteristics. If

    friction losses are to be taken into account, a second term

    4,

    has

    to

    be subtracted from the right hand side of

    (I).

    he

    p.u. value o f this term is in the order of I x I O - ~ proportional

    to the steady state flow rate, and is usually neglected in

    frequency response analysis,

    [5]

    & [Ill.

    For small variations around an equilibrium point the

    turbine can be represented by the following linearized

    equations

    :

    4 = all

    + a12 Wt a13

    g

    m = a2, h

    +

    aZ2u +

    a23

    g

    where

    :

    wt s ) is the differential angular speed o f the turbine (P.u.),

    g(s) is the differential gate opening (P.u.),

    m(s) is the differential mechanical torque (P.u.).

    Parameters a,J are the partial derivatives of q

    &

    m with

    respect to h, wt and g respectively, and they remain con-

    stant for variations near the equilibrium point (qo, mo). The

    values

    of

    these parameters depend upon the initial steady

    state point of the machine, and they have to be measured

    accurately in the field, or taken from model tests. The

    influence of these coefficients on the model accuracy is

    critical.

    From (1)

    & (2)

    the following equation may be deducted:

    m

    = G,g

    +

    Gw

    (3)

    where the transfer functions

    G S) & G,(s)

    relate the mech-

    anical torque with the gate opening and speed respectively

    G s)

    = a23[1+A(T,/T,)tanh(T,s)]/[l+a~l(Tw/Te)tanh(Tes)]

    . ,

    B = a11 - a21a12/a22

    As it can be seen from (4) the gain and phase of these

    transfer functions depend upon the operating point, by

    means of parameters a,]. Water starting time constant

    T

    depends upon the length

    L

    and cross section area S of the

    penstock :

    T=

    L

    Qb/g

    s

    Hb

    5)

    where Q and H are the base values of flow rate and head

    respectively. Penstock elastic time constant T is the time

    taken for the pressure wave to travel the length of the

    penstock at velocity a

    :

    1

    = Lla (6)

    Apart of the length of the penstock, reflection time depends

    also upon water hammer velocity a in the conduit. Computa-

    tion of this velocity is not so straight forward as usually

    presented in bibliography, since it varies not only according

    to the geometric dimensions of the pipe but depends upon

    the type of its fixings too,

    [4].

    A

    close examination of G,(s) reveals that this transfer

    function gain varies between two limit values at the odd and

    even harmonics

    of

    its characteristic hydraulic frequency f

    = 1/4 T respectively

    :

    7 )

    (8)

    limit1 G, s) = a23 AI /a,,

    limit2

    Gg s) =

    a23

    It has to be noted here that this characteristic frequency f

    in the order of about 1 Hz, does not vary with load and

    depends only upon the dimensions of the penstock.

    The transfer function G,(s) has a similar behaviour, and

    the same equations hold for its maxima and minima, except

    that a23& A are replaced by a22& B respectively.

    In Fig.(l) the gain of the transfer function G,(s) is shown,

    for a typical unit, at full load. Standard values for parameters

    a,, for an ideal lossless turbine at full load are :

    a,, =

    0.5,

    aI2 = 0.0, a13 = 1

    0,

    a21=

    1.5,

    a22= 0.0, a23 = 1.0

    where the dependence upon speed has been neglected. In

    fact partial derivatives can be extracted by differentiation of

    the following equations representing the laws of similitude

    :

    Q

    = G

    M

    =

    Q

    H

    (n/n,)/n

    (9)

    where capital letters have been used for the absolute p.u.

    values and n = noin, is the ratio of efficiencies at the current

    operating point over that of base values. For deviations

    around rated speed and head, the partial derivatives versus

  • 8/19/2019 Francis Turbine Oscillatory Problems

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    421

    load are deduced:

    I

    all = 0.5G0, a13= 1.0,

    a21= 1.5 n Go, a23

    =

    n

    where Go is the p.u. piston displacement at the current

    operating point “0. ariation of flow rate with speed, a,2, is

    usually considered to be negligible, [14], in which case the

    transfer function G

    (s)

    depends mainly on the

    so

    called

    turbine gain a23. Tur%ine gain is a critical parameter for an

    accurate approximation of unit dynamics, and has to be

    measured precisely in the field. The deviation of mechanical

    torque with speed, a22, known as turbine’s self regulation, is

    negative with an absolute value usually near unity. In case

    that this dependency of mechanical torque deviation upon

    speed is neglected, the transfer function

    GJs)

    disappears.

    It is worth noting that more accurate values of turbine

    parameters, with the exception of

    ~123,

    an be deduced from

    the characteristic curves of the turbine, drawn from model

    tests, [15].

    B. Speed

    Governor

    Model

    In the steady state the shaft speed signal is compared to

    the speed reference, modified by the permanent speed

    droop times gate position. When gate position is changing,

    a transient droop setting is developed to oppose fast

    changes in gate position, [8]. The transfer function of a

    mechanical-hydraulic governor, Fig.2, is shown in Fig. 3

    :

    It can be readily seen that the speed governor operates as

    a low pass filter, when combined with the penstock

    -

    turbine

    transfer function. Fig.4 shows the gain of the transfer

    function G (s)”G (s), i.e. the mechanical torque to speed

    relation, inl he s byane.

    C. Shah+ orsionals

    The hydro turbine generator set may, under specific

    circumstances, oscillate due to the torsional elasticity of the

    interconnecting shaft. The application of Newton’s second

    law of motion to the rotating inertias of turbine and genera-

    tor is translated into the following matrix equation

    :

    J p2d

    K d =

    (11)

    where

    :

    J

    =

    diag

    [Jj, Jg]

    is the diagonal inertia matrix,

    K i s he stlffness matrix, and

    I=T -T,JT is the column matrix of excitation torques.

    The eigenfrequency ft of this subsystem can be easily

    computed from the

    charaderistio

    equation :

    0.01 0,l

    1

    FREcaJwcy W)

    Fig.1 Tran sfer functicln

    G

    (s) of an ideal turbine

    at

    full

    load

    (T, = 0.384 s,T = 1.637 s)

    RATE POSITW

    LIMITS LlMIT j

    j

    vawE

    OISTRI UTOR

    vncE

    Y AN0 GATE

    SERVOMOTOR SERVOMOTOR

    TRANSIENT DRO O P

    COMPENSATION

    PERMANENT D K W

    COMPENSATION

    Fig.2 Speed governor

    50

    40

    30

    20

    z.

    3 0

    z

    < O

    ?lo

    -20

    -30

    40

    0,Ol 0.1 1

    10

    FREQUENCY

    Hz)

    0

    -20

    -40

    z

    -120

    -140

    -160

    180

    Fig.3 Transfer ,function G,(s) of speed governor

    (.--- gain,

    ---

    phase)

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    422

    0

    20

    n

    U

    0

    2

    -23

    -40

    o m 01

    1

    FRE (W

    I -I

    Fig.4 Transfer function of turbine with speed governor, at full load

    (torque to - speed relation)

    det [H - K

    -

    (2n ft

    )*

    I] = 0

    so ft = [K (Jt + Jg)/Jt

    J ]

    121-1

    (12)

    (13)

    D. Network interaction

    Generator rotor is coupled to the electric network via the

    electrical torque. The differential electrical torque acting on

    generator rotor can be written in the following form, making

    use of the well known synchronizing and damping torque

    coefficients

    :

    T = K

    6

    + K wg

    (14)

    where the torque is expressed in terms of two components,

    the synchronizing and damping torque. The synchronizing

    and damping torque coefficients K and K, are variable in s,

    depending upon the automatic voltage regulator, system

    configuration, operating point of generator and parameter

    values, [16]. Under the ovlersimplified assumption of a

    generator without voltage regulator, connected to an infinite

    bus with nominal voltage, the synchronizing coefficient

    equals tan6,, where 6o s the angle between generator bus

    and the infinite system. This interconnection of generator

    rotor to the network produces the electromechanical oscilla-

    tion frequency :

    feI= (1/2n)[U, K,/2JJ

    (15)

    E. Draff tube surges

    Surge phenomena in the draft tube are the most frequent

    source of trouble for Francis turbines. During partial load

    operation the instability of flow leads to

    the

    well known

    vortex rope, which rotates iin the same direction as the

    runner, [I]. The oscillations

    of

    vortex rope produce pressure

    fluctuations that alter the net head acting on the turbine and

    lead to generator power pulsations.

    The frequency of these pressure surges is variable with

    load, depending upon the dimensions of draft tube and flow

    conditions. By experience as well as by theoretical analysis,

    [IO]

    it

    is known that this frequency varies between 0.25 and

    0.5 of the frequency corresponding to synchronous speed.

    The relative amplitude of pressure surges in the draft tube

    has been studied theoretically by Hosoi, [IO], and was found

    to depend upon the geometric dimensions of the draft tube

    and runner as well as upon Thoma's coefficient of cavita-

    tion. Usual relative amplitude of draft tube surges varies up

    to 5% of rated head, and even more, under the most

    unfavourable conditions.

    Ill.

    KASTRAKI HPP CASE STUDY

    Kastraki Hydro Power Plant consists of four units of 80

    MW each, equipped with vertical Francis turbines, operating

    at a head of 74.5 m at 166.67 rpm. There are four metallic

    outdoor penstocks which are different in length. The power

    station experienced strong vibrations from the commission-

    ing date in 1969. These vibrations are so high that unit

    I

    is

    only used for full load operation.

    In order to simulate unit performance, a model was

    constructed according to section

    II.

    A synopsis of main data

    s shown in Table I.

    Table I

    aka

    for

    d

    units

    nominal net head 74.5 m

    nominal flow 122 m3/s

    rated generator output 80000 k W

    penstock cross section 26.421 in2

    penstock wall thickness

    16

    m m

    generator inertia constant 3.23 s

    turbine inertia constant

    0

    108 s

    shaft stiffness 6.752 p.u.

    water hammer wave velocity 712 m/s

    ~ = 0 , 0 2 , 6 = 0 . 3 8 7 , T , = 5 . 0 , T , = 0 . 0 4 , T , = 0 . 2

    SQ.&ecl YeLw-data

    Main data for simulation

    Water hammer wave velocity a depends not only on the

    dimensions of the penstock but also on the type of fixing

    used for the different penstock sections,

    [4].

    In other words

    the ability of the pipe to move in the longitudinal direction

    influences the wave velocity by as much as 10%.

    Turbine gain aZ aries

    with load and

    has

    been measured

    in the field, since

    it

    is critical in order to obtain close agree-

    ment between the model and the unit. Fig.5 contains

    information about the values of turbine characteristic

    parameters a,, in the whole loading range. Since turbine gain

    aZ3gets its maximum value of about 1.2 p.u. in the range of

    0.4 to 0.6 p.u. loading, t is evident that the unit responds the

    strongest while partially loaded.

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    QO

    l2

    a3

    (14 a5 0 6

    a7 Q

    Q9 10 11

    [email protected]

    Fig 5 Turbine parameters

    a,,

    vs. power output (at rated

    head)

    A

    Frequency control oop

    An examination of turbine transfer function of Kastraki

    HPP

    units reveals that this is similar with the one of an ideal

    turbine shown in Fig.1 only at full load. In fact at full load

    Gg(s) has its maximum equal to “limitl”, (7), t the odd

    harmonics of f,,, while at the even harmonics the minimum

    occurs, equal to “limit2”, (8). This is a consequence of the

    relative values of “limitl”

    &

    “limit2” arising from the variation

    of turbine characteristics versus load. At partial load, “limitl”

    is less than “limit2”, so the turbine tranfer function exhibits its

    minimum

    &

    maximum at the odd and even harmonics off,,

    respectively.

    In order to investigate the stability of the frequency control

    loop one has to express turbine’s transfer function Gg(s) n

    a more convenient form. Using Taylor series expansion the

    fourth order approximation gives :

    where

    :

    N,(s)

    = s4 + A,s3

    +

    A2s2

    +

    A,s

    +

    A

    Dg(s)= s4

    +

    B,s3

    +

    B2s2

    + B,s + Bo

    A = 4ATw/Te2,A

    =

    12/Te2,A,

    =

    24ATW/Te4, = 241Te4

    B

    = 4a,,T,/Te2, 8 = A

    6

    24allT,/Te2, Bo= A

    It can be clearly seen that the coefficients Ai

    &

    4 depend

    upon the loading of the machine through A

    &

    a,,. Since all

    B;s are positive, the turbine transfer function is always

    stable. An examination of the poles of Gg(s)vs. load for unit

    I

    reveals that they form two complex conjugate pairs with

    corresponding angular frequencies at

    0.97

    &

    1.86 r/s. The

    damping constants of the poles are variable with load, Fig.

    6. Due to the variability of a the poles approximate the jw

    axis for high loading, resulting in lower damping. Since B,’s

    are inversely proportional to the penstock length

    L,

    the

    poles of the turbine transfer function of unit IV are found to

    have higher angular frequencies, namely 1.1

    &

    2.1 rls,

    and

    423

    Q2 Q 4 0 6 0.8 1 12

    l-cm

    @.U

    Fig.6 Damping constant

    of

    poles ( A ) & zeros (.) of the turbine transfer

    function (Kastraki HPP, unit I)

    damping constants. This is an interesting conclusion stating

    that for a hydro power station equipped with more than one

    identical Francis units, the one with the shortest penstock

    has the smallest egenfrequencies with the smallest damping

    constants.

    The open loop zeros of the frequency control

    loop,

    originating from the turbine transfer function (14), are the

    roots of the numerator N,(s). Due to the variability of

    parameter A versus load the coefficients A

    &

    A, become

    negative for loadings, higher than about

    0.5

    P.u.. In this

    loading range the zeros of the turbine transfer function

    appear with positive damping constants, corresponding to

    instability. In Fig.6 the variation of the damping constants of

    the two zeros is also shown for unit

    I.

    It is evident that for

    huh values of the gain the closed loop will become unstable.

    It is worth noting here that the preceding analysis did not

    take into consideration the friction losses in the penstock.

    The inclusion of these friction losses in the transfer function

    helps in increasing the damping constants.

    B. Vibration anawsis

    In order to investigate the problem, vibration measure-

    ments were undertaken in all four units for the whole

    operating zone, Table II. It can be clearly seen that vibration

    level

    is

    higher in unit I, while unit

    IV

    seems to have best

    vibration performance. Frequency analysis of vibration

    signal showed that the vortex component increased cons-

    iderably in the loading range where vibrations were maxi-

    mum. Vortex frequency was found to vary from 0.7 Hz, at

    0.5

    p.u. output, to almost 1Hz, at

    0.75

    p.u. These values

    agree with measurements conducted in other Francis units,

    as well as with theoretical predictions, [IO] Fig.7 shows

    unit’s I shaft vibration spectrum at 50 MW. Main frequency

    components lie at vortex frequency, synchronous frequency

    as well as at 1.7 and :3.8 z.

    Given a value of about -0.8 p.u. for the turbine’s self

    regulation it is concluded that the maximum gain of the

    transfer function

    GJs)

    in the same loading range equals

    about 65% of G,(s). This means that speed variations

    i n f l u e n c e o n

    mechanical

    orque is substantially

    lower than

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    0 2

    M 6 8

    10

    Fig.7 Unit's I shaft vibration spectrum at 0.6 p.u. (X direction)

    Table

    II

    unit power output (P.u.)

    0.25

    0.5

    Shaft

    vibrations near generator guide bearing (pm p-p)

    that of gate position.

    Due

    to

    the increasing penstock length for units

    I

    to IV, the

    Characteristic frequency f varies from 0.687 Hz, for the first

    unit, to 0.755 Hz, for the fourth. Head variations due to

    vortex pulsations at this frequency cause maximum flow

    rate variations, according

    to

    (1).

    It has already been men-

    tioned that the frequency

    of

    vortex pressure pulsations

    varies, for all units, from 0.68 Hz, at 0.5 p.ui. load, to almost

    1Hz at 0.75 p.u. load. So it is concluded that in the loading

    range where pressure surges due to vortex are maximum,

    the units are in hydraulic resonance, and that this phe-

    nomenon

    is more intense for the first wi t. In that case

    strong vibrations of the same frequency are induced in the

    shaft.

    The torsional eigenfrequency of the unit was computed

    according to (11) and was found to be 3.8 Hz. Small varia-

    tions in the electrical oading of the machine, always present

    in the steady state, may excite weak torsialnal vibrations of

    the inertias of turbine and generator. Turbine's speed

    variations at t h e frequency ft excrte, through variations in the

    flow rate and head, vibrations in the penstock itself. In fact

    the fiFth wave egenfrequency of the penstock cross section

    has been identified as 3 7 Hz. This frequency component

    was found in the spectrum o f shaft vibrations, due to the

    well known coupling between flexural and torsional vibra-

    tions, as well as in the penstock cross section vibrations.

    Although this component is rather weak in the shaft spec-

    trum, it is strongly amplified in the penstock due to reso-

    nance. For generator angles between

    20

    and

    60

    the elec-

    tromechanical oscillations frequency varies between

    1

    and

    1.7 Hz. At the high end of this range, elecromechanical

    oscillations are expected

    to

    influence penstock cross section

    vibrations by their second harmonic, in the same manner as

    torsional vibrations do. This is the reason why these vibra-

    tions are higher at higher unit loadings.

    IV. CONCLUSIONS

    A new method was presented for the investigation of

    oscillatory problems of hydro power plants equipped with

    Francis turbines. The approach of transfer functions was

    implemented, with the adoption of elastic water column

    principle. Turbine gain, as well as the partial derivatives of

    flow rate and torque with respect to head, were measured

    in the field. Full nonlinear equations were used for turbine

    and penstock modelling, while the full model of mechanical -

    hydraulic speed governor was implemented. Turbine

    -

    generator torsionals as well as hunting were taken into

    consideration and their influence on vibrational behaviour

    was investigated.

    The stability of the frequency control loop was studied and

    the influence of turbine characteristics as well as of conduit

    design was analyzed. It was found that for a hydro power

    station equipped with more than one identical Francis units,

    the one with the shortest penstock has the smallest eigen-

    frequencies with the smallest damping constants. For

    loadings higher than about

    50%

    it was found that the unit

    may experience instability, especially when high gain is

    present in the frequency control loop.

    Excessive vibrations in Kastraki hydro power plant, and

    mainly in unit I,were also studied. The main cause of these

    vibrations was found to be the near resonance of draft tube

    vortex pulsations with the characteristic frequency h in

    partial oading. The shaft vibrations spectrum was explained

    taking into account the torsional as well as the electrome-

    chanical oscillations behaviour of the unit. The excessive

    vibrations in the penstock cross section were also identified

    due to resonance of penstock's characteristic frequency

    with torsional and electromechanical oscillations at high

    loading.

    The inclusion of shaft torsionals in the unit model proved

    to

    be a critical parameter for the identification of penstock

    cross section Vibrations. The implementation of the full

    equations for the modelling of the unit, as well as the de-

    duction of turbine's parameters from real measurements

    proved to be necessary for an accurate simulation.

    V.

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    D.N.

    Konldar ls

    (M‘83) was born in loannina, Greece, on

    May 23, 1954. He received the Diploma and Ph.D. in

    Electrical Engineering in 1977 and 1996 respectively from the

    National Technical University of Athens. He joined the Public

    Power Corporation of Greece in 1983. From 1983-1988 he

    worked as an Operation Engineer in Kardia Power Station

    (4x300 MW). He

    is

    now engaged in electric measurements

    and tests. He is a member of IEEE and CIGRE. (e-mail

    :

    [email protected]).

    J.A. Tegopoulos (M156-SM’63-F’75)was born in Trikala,

    Greece, on September 30, 1924. He received the Diploma

    in Mechanical and Electrical Engineering in 1948 from the

    National Technical University of Athens and M.S. and Ph.D.

    degrees from Purdue University in 1954 and 1956. From

    1958-1966 he was with Westinghouse Electric Corp. and

    taught at the University of Pittsburgh. Now he is a professor

    of Electrical Engineering at the National Technical University

    of Athens.

    pp.1-18.

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