DigSilent_Synchronous Generator Tech Doc

DIgSILENT Technical Documentation

Synchronous Generator

DIgSILENT GmbH Heinrich-Hertz-Strasse 9 D-72810 Gomaringen Tel.: +49 7072 9168 - 0 Fax: +49 7072 9168- 88 http://www.digsilent.de e-mail: [email protected]

Synchronous Generator Published by DIgSILENT GmbH, Germany Copyright 2007. All rights reserved. Unauthorised copying or publishing of this or any part of this document is prohibited. TechRef ElmSym V4 Build 331 18.10.2007


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Table of Contents

Table of Contents 1 General Description .............................................................................................................................................. 4 1.1 Mathematical Description ............................................................................................................................................... 5 1.1.1 Equations with stator and rotor flux state variables in stator-side p.u.-system .............................................................. 5 1.1.2 Mechanics ................................................................................................................................................................ 7 1.1.3 Equations with stator currents and rotor flux variables as used in the PowerFactory model ........................................... 7 1.1.4 Saturation ................................................................................................................................................................ 9 1.1.5 Simplifications for RMS-Simulation ........................................................................................................................... 10 1.2 Input Parameter Conversion ......................................................................................................................................... 10 1.2.1 Reactances, Resistances and Time Constants ........................................................................................................... 10 1.2.2 Saturation .............................................................................................................................................................. 13 1.3 Input-, Output and State-Variables of the PowerFactory Model ....................................................................................... 14 1.4 Rotor Angle Definition .................................................................................................................................................. 15 2 Input/Output Definition of Dynamic Models ...................................................................................................... 17 2.1 Stability Model (RMS) ................................................................................................................................................... 17 2.2 EMT-Model .................................................................................................................................................................. 19 3 References .......................................................................................................................................................... 21


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General Description

1 General Description The correct modelling of synchronous generators is a very important issue in all kinds of studies of electrical power systems. PowerFactory provides highly accurate models which can be used for the whole range of different analyses, starting simplified models for load-flow and short-circuit calculations up to very complex models for transient simulations. Basically there are two different representations of the synchronous generator: •

The round rotor generator or turbo generator

•

The salient rotor generator

The generators with a round rotor are used when the shaft is rotating with or close to synchronous speed of 1500 min-1to 3000 min-1. These types are normally used in thermal or nuclear power plants. Slow rotating synchronous generators with speed of 60 min-1 to 750 min-1, which are for example applied in diesel or hydro power plants, are realized with salient rotors. A schematic diagram of both types of machines is shown in Figure 1 and Figure 2. These figures are also indicating the orientation d- and q-axis according to the theory of the synchronous machine developed in the next section.

Figure 1: Schematic diagram of a three-phase round rotor synchronous machine


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General Description

Figure 2: Schematic diagram of a three-phase salient rotor synchronous machine In the figures the three stator windings are shown as well as the rotor windings. The winding ‘e’ is the excitation winding fed by the excitation voltage ve supplied by the excitation system. Then one damper winding can be defined for the direct (d-) axis and up to two damper windings can be included into the quadrature (q-) axes. All these windings are shown in Figure 2. The rotor is rotating with its speed ω. Also the rotor angle is ϑ the angle between the d-axis and the stator field.

1.1 Mathematical Description To describe the generator equations it is common practise not to use instantaneous values leading to a threedimensional problem in the abc coordinate system, but to transform all value into a rotating reference frame. This transformation is called dq0 or Park’s Transformation [1].

1.1.1 Equations with stator and rotor flux state variables in stator-side p.u.-system Stator voltage equations (the stator current are shown in generator orientation):

u d = rs id +

1 dψ d − nψ q ωn dt

u q = rs iq +

1 dψ q + nψ d ωn dt

u0 = rs i0 +

1 dψ 0 ωn dt


(1)

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General Description

Rotor voltage equations, d-axis:

ue = reie +

dψ e ωn dt

dψ D 0 = rD iD + ωn dt

(2)

Rotor voltage equations, q-axis, round rotor:

0 = rx ix +

dψ x ωn dt

0 = rQ iQ +

dψ Q

ωn dt

(3)

Rotor voltage equations, q-axis, salient pole:

0 = rQ iQ +

dψ Q

ωn dt

(4)

The Flux linkages are calculated as follows: d-axis:

ψ d = ( xl + xmd )id + xmd ie + xmd iD ψ e = xmd id + ( xmd + xrl + xle )ie + ( xmd + xrl )iD ψ D = xmd id + ( xmd + xrl )ie + ( xmd + xrl + xlD )iD

(5)

q-axis, full-rotor:

ψ q = (xl + xmq )iq + xmq ix + xmq iQ

ψ x = xmqiq + (xmq + xrl + xlx )ix + (xmq + xrl )iQ

ψ Q = xmq iq + (xmq + xrl )ix + (xmq + xrl + xlQ )iQ

(6)

q-axis, salient rotor:

ψ q = (xl + x mq )iq + x mq iQ

ψ Q = x mq iq + (x mq + x rl + xlQ )iQ


(7)

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General Description

Electrical torque te in [p.u.]:

te = ψ d iq −ψ q id

(8)

1.1.2 Mechanics The accelerating torque is the difference between the input torque (mechanical torque) tm and the out put torque (electromechanic torque) te of the generator. The inertia of the generator-shaft system is then accelerated or decelerated, when an unbalance in the torques occurs. The equations of motion of the generator can then be expressed as

Jω n2 dn dn = Ta = tm + te 2 p z Pr dt dt dϑ = ωn n dt

(9)

The inertia of the generator and the turbine can then be expressed in a normalized per unit form as the inertia time constant H in [s], with

H=

1 Jω02 2 p z2 Pr

(10)

where pz is the number of pole pairs of the machine. The inertia time constant H can be given based on the rated apparent generator power, as shown in the equation above, or based on the rated active generator power. The mechanical starting time or acceleration time constant TA in [s] is then

Ta = 2 ⋅ H

(11)

Both H and TA can be entered in PowerFactory based on Sr or Pr.

1.1.3 Equations with stator currents and rotor flux variables as used in the PowerFactory model Subtransient Flux:

ψ d'' = keψ e + k Dψ D ψ q'' = k xψ x + kQψ Q


(12)

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General Description

with

ke =

xmd xlD xd 2

kD =

xmd xle xd 2

kx =

xmq xlQ

kQ =

xmq xlx

(13)

xq2 xq2

with

xd 2 = xle xlD + ( xmd + xrl )( xle + xlD ) xq2 = xlx xlQ + (xmq + xrl )(xlx + xlQ )

(14)

Using:

ψ d = xd'' id + ψ d''

(15)

ψ q = xq'' iq + ψ q'' and

u d'' =

1 dψ d'' − nψ q'' ωn dt

'' 1 dψ q u = + nψ d'' ωn dt

(16)

'' q

Stator equations with stator currents and subtransient voltages:

ud = rs id + uq = rs iq + u0 = rs i0 +

xd'' did − nxq'' iq + ud'' ωn dt xq'' diq

ωn dt

+ nxd'' id + uq''

(17)

x0 di0 ωn dt


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General Description

1.1.4 Saturation So far saturation effects where not included in the description of the equivalent circuits. The exact representation of saturation is very complex, but normally not necessary to obtain good results from simulations. Therefore in most cases saturation is represented by the saturation of the mutual reactances xmd and xmq only. Consideration of saturation of magnetizing reactance in d- and q-axis:

x md = k satd x md 0

(18)

x mq = k satq x mq 0 Saturation depending on magnitude of magnetizing flux:

ψ m = (ψ d + xl id )2 + (ψ q + xl iq )2

(19)

The saturation of the mutual reactance xmq in the q- axis can not be measured. Thus the characteristic is assumed to be similar to the one of the d-axis. For the round rotor machine the saturation is equal in d- and qaxis. In the salient rotor machine the characteristic is weighted by the ratio xq/xd. If

ψ m 〉 Ag : Bg (ψ m − Ag )

2

csat =

ψm

(20)

else:

csat = 0

(21)

The saturation coefficient ksat in d- and q-axis are calculated as follows:

k satd =

1 1 + csat

k satq =

1 1+

xmq 0 xmd 0

(22)

csat

Saturated magnetizing reactances applied to all formulas (5),(6),(7) and (12),(13),(14). Saturation in subtransient reactances is not considered.


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General Description

The saturation of the leakage reactance is not included in the model. This saturation is a current saturation, i.e. high currents after short-circuits will lead to a saturation effect of the leakage reactance xl. Here it is common practice to use unsaturated values only. Although to neglect this type of saturation may lead to an underestimation of the short-circuit currents. Hence there is a way to model this effect explicitly. This saturation is an effect, which influences the SC current only in the first milliseconds, i.e. it can be assumed to be a subtransient effect. For the definition of the input parameter in the PowerFactory model please refer to section 1.2.2.

1.1.5 Simplifications for RMS-Simulation Stator voltage equations (see Eq.(17)): Neglecting stator flux transients:

ud = rs id − xq'' iq + ud''

(23)

uq = rs iq + xd'' id + uq'' with:

ud'' = − nψ q''

(24)

uq'' = nψ d'' Assumption that magnetizing voltage is approx. equal to magnetizing flux (for saturation):

ψ m ≈ um =

(u

+ rs id − xl iq ) + (uq + rs iq + xl id ) 2

d

2

(25)

1.2 Input Parameter Conversion 1.2.1 Reactances, Resistances and Time Constants The set of input parameters is specified as follows: d-axis:

xd'' , xd' , xd , xl , xrl , Td'' , Td'


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General Description

q-axis, round rotor:

xq'' , xq' , xq , xl , xrl , Tq'' , Tq' q-axis, salient pole:

xq'' , xq , xl , xrl , Tq'' The internal model parameters are: d-axis:

x d'' , xl , x rl , xle , xlD , re , rD q-axis, round-rotor:

xq'' , xl , xrl , xlx , xlQ , rx , rQ q-axis, salient pole:

xq'' , xl , xrl , xlQ , rQ Auxiliary variables:

x1 = x d − xl − x rl x 2 = x1 −

( x d − x l )2 xd

x1 x d'' x2 − xd x3 = x d'' 1− xd

x d ' ⎛ x d x d ⎞ '' Td + ⎜⎜1 − ' + '' ⎟⎟Td x d' ⎝ xd xd ⎠ T2 = Td' + Td''

(26)

T1 =

(27)

T3 = Td' Td''


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General Description

a=

x2T1 − x1T2 x1 − x2

(28)

x3 b= T2T3 x3 − x2 −a a2 + −b 2 4

Tle =

(29)

−a a2 TlD = − −b 2 4 Calculation of internal model parameter:

Tle − TlD T1 − T2 TlD + x1 − x2 x3

xle =

xlD =

TlD − Tle T1 − T2 Tle + x1 − x2 x3

re =

xle ωnTle

rD =

xlD ωnTlD

(30)

q-axis, round rotor machine: - analoguous to d-axis parameter q-axis, salient pole machine:

xlQ = rQ =

(x

q

(

− xl ) xq'' − xl xq − x

'' q

xq'' xq − xl − xlQ xq

) (31)

ωnTq''


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General Description

1.2.2 Saturation Figure 3 shows the definition of the saturation curve of the mutual reactance. The linear line represents the airgap line indicating the excitation current required overcoming the reluctance of the air-gap. The degree of saturation is the deviation of the open loop characteristic from the air-gap line.

Figure 3: Open loop saturation The characteristic is given by specifying the excitation current I1.0pu and I1.2pu needed to obtain 1 p.u respectively 1.2 p.u. of the rated generator voltage under no-load conditions. With these values the parameters sg1.0 (=csat(1.0pu) ) and sg1.2 (=csat(1.2pu) ) can be calculated. Calculation of internal coefficients based on

s g1.0 = s g1.2

ie (1.0 p.u ) −1 i0

i (1.2 p.u ) = e −1 1.2i0

(32)

For quadratic saturation function

1.2 − 1.2 Ag = 1 − 1.2 Bg =

s g1.2 s g1.0

s g1.2 s g1.0

(33)

s g1.0

(1 − A )

2

g


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General Description

1.3 Input-, Output and State-Variables of the PowerFactory Model Per-unit system of rotor-flux and rotor currents: Rotor currents:

~ ie = xmd 0 ie ~ iD = xmd 0 iD ~ ix = x mq 0 i x ~ iQ = xmq 0 iQ

(34)

Rotor-flux:

ψ~e =

xmd 0 ψe xe 0

ψ~D =

xmd 0 ψD xD 0

x ψ~x = mq 0 ψ x xx 0

ψ~Q =

xmq 0 xQ 0

(35)

ψQ

With

xe 0 = xmd 0 + xlr + xle xD 0 = xmd 0 + xlr + xlD x x 0 = xmq 0 + xlr + xlx

(36)

xQ 0 = xmq 0 + xlr + xlQ Rotor voltage equations, d-axis:

dψ~e ~ u~e = ie + Te 0 dt dψ~D ~ 0 = iD + TD 0 dt


(37)

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General Description

Rotor voltage equations, q-axis, round rotor:

dψ~x ~ 0 = ix + Tx 0 dt dψ Q ~ 0 = iQ + TQ 0 dt

(38)

Rotor voltage equations, q-axis, salient pole:

dψ~Q ~ 0 = iQ + TQ 0 dt

(39)

With

Te 0 =

xe 0 reωn

TD 0 =

xD 0 rDωn

(40)

x Tx 0 = x 0 rxωn TQ 0 =

xQ 0 rQωn

1.4 Rotor Angle Definition The actual position of the rotor d-axis with respect to the network voltages is monitored and is important for the behaviour of the machine and for assessing its stability. It is expressed as the rotor angle. In PowerFactory the rotor angle is available with several reference angles. The angles available are: •

fipol / [deg]:

Rotor angle with reference to the local bus voltage of the generator (terminal voltage)

•

firot / [deg]:

Rotor angle with reference to the reference voltage of the network (slack bus voltage)

•

firel / [deg]:

Rotor angle with reference to the reference machine rotor angle (slack generator)

•

dfrot / [deg]:

identical to firel

•

phi / [rad]:

Rotor angle of the q-axis with reference to the reference voltage of the network (=firot-90°)

All rotor angles are shown in Figure 4.


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General Description

Additionally there is the variable ‘dfrotx’ available at each generator, which is indicating the maximum value of dfrot for all generators in the system. This variable can assist you to indicate, if a generator is falling out of step with respect to the reference machine angle.

Figure 4: Rotor Angle Definition


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Input/Output Definition of Dynamic Models

2 Input/Output Definition of Dynamic Models 2.1 Stability Model (RMS)

psie psiD psix psiQ xspeed phi ve

fref ut/utr/uti

pt

pgt ie

xmdm

pgt outofstep xme xmt cur1/cur1r/cur1i P1 Q1

Figure 5: Input/Output Definition of the synchronous machine model for stability analysis (RMSsimulation)


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Table 1: Input Definition of the RMS-Model Parameter

Description

Unit

ve

Excitation Voltage

p.u.

pt

Turbine Power

p.u.

xmdm

Torque Input

p.u.

Table 2: Output Definition of the RMS-Model Parameter

Description

Unit

psie

Excitation Flux

p.u.

psiD

Flux in Damper Winding, d-axis

p.u.

psix

Flux in x-Winding

p.u.

psieQ


p.u.

xspeed

Speed

p.u.

phi

Rotor Angle

rad

fref

Reference Frequency

p.u.

ut

Terminal Voltage

p.u.

pgt

Electrical Power

p.u.

outofstep

Out of step signal (=1 if generator out of step)

p.u.

xme

Electrical Torque

p.u.

xmt

Mechanical Torque

p.u.

cur1

Positive-sequence current

p.u.

cur1r


p.u.

cur1i


p.u.

P1

Positive-sequence active power

p.u.

Q1

Positive-sequence reactive power

p.u.

utr

Terminal Voltage, real part

p.u.

uti

Terminal Voltage, imaginary part

p.u.


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2.2 EMT-Model

psie psiD psix psiQ xspeed phi fref

ve

ut/utr/uti pt

pgt ie

xmdm

pgt outofstep xme xmt cur1/cur1r/cur1i P1 Q1

Figure 6: Input/Output Definition of the HVDC converter model for stability analysis (EMTsimulation)

Table 3: Input Definition of the EMT-Model Parameter

Description

Unit

ve

Excitation Voltage

p.u.

pt

Turbine Power

p.u.

xmdm

Torque Input

p.u.


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Table 4: Output Definition of the EMT-Model Parameter

Description

Unit

psie

Excitation Flux

p.u.

psiD


p.u.

psix

Flux in x-Winding

p.u.

psieQ


p.u.

xspeed

Speed

p.u.

phi

Rotor Angle

rad

fref

Reference Frequency

p.u.

ut

Terminal Voltage

p.u.

pgt

Electrical Power

p.u.

outofstep

Out of step signal (=1 if generator out of step)

p.u.

xme

Electrical Torque

p.u.

xmt

Mechanical Torque

p.u.

cur1


p.u.

cur1r


p.u.

cur1i


p.u.

P1

Positive-sequence active power

p.u.

Q1

Positive-sequence reactive power

p.u.

utr

Terminal Voltage, real part

p.u.

uti

Terminal Voltage, imaginary part

p.u.


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References

3 References [1] P. Kundur, Power System Stability and Control, McGraw-Hill, Inc., 1994.


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DigSilent_Synchronous Generator Tech Doc

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