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D I g S I L E N T T e c h n i c a lD o c u m e n t a t i o n

Induction Machine



I n d u c t i o n M a c h i n e

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]

Induction Machine

Published by

DIgSILENT GmbH, Germany

Copyright 2005. All rights

reserved. Unauthorised copying

or publishing of this or any part

of this document is prohibited.

doc. TR-001, build 220

27 Februar 2007



T a b l e o f C o n t e n t s


Table of Contents

1 General Description........................................................................................................................... 4

1.1 Input Data ...............................................................................................................................................5 1.1.1 Equivalent Rotor Impedance.................................................................................................................8

1.2 Load Flow Analysis.................................................................................................................................. 10

1.3 Short Circuit Analysis ..............................................................................................................................10

1.4 Harmonic Analysis................................................................................................................................... 11 1.5 Stability/Electromagnetic Transients (RMS- and EMT-Simulation)............................................................... 12 1.5.1 EMT-Model ........................................................................................................................................14 1.5.2 Stability Analysis (RMS-Simulation) ..................................................................................................... 15 1.5.3 Mechanical Equations......................................................................................................................... 15 1.5.4 Mechanical Load ................................................................................................................................ 16 1.5.5 Initialization....................................................................................................................................... 17

2 Input/Output Definitions of Dynamic Models................................................................................. 18

3 Input Parameter Definitions ........................................................................................................... 20

3.1 Induction Machine Type (TypAsmo)......................................................................................................... 20

3.2 Induction Machine Element (ElmAsm)...................................................................................................... 21




1 General Description

The general induction machine model of DIgSILENT PowerFactory is the so-called Type 2 Asynchronous Machine

model that is available since version 12.0.

The model is basically a classical induction machine model including a frequency (or slip) dependent rotor

impedance (Figure 1).

Stator voltages and currents in these equivalent circuit diagrams are represented as instantaneous phasors in a

steady reference frame. Rotor voltages and currents are represented in a reference frame that rotates with

mechanical frequency. Hence, all quantities in these equivalent circuits are represented in their “natural”

reference frame. The machine model is supposed to be unearthed why no equation for the zero sequencecomponents is given. The rotor impedance is referred to the stator side, why the “rotating transformer” in Figure

1 does not show any winding ratio.

The winding resistance R s, the stator leakage reactance Xs, the magnetizing reactance Xm and the rotor

impedance Zrot characterize the model.

As already mentioned, Zrot can be frequency dependent and allows for modelling squirrel cage induction machines

over a wide speed or slip range. Zrot can be approximated by parallel R-L elements (index A1 and A2, see Figure

3).

Double cage induction machines are modelled by one additional R-L branch (index B, Figure 4) that is in parallel

to the described rotor impedance of cage A. Altogether, frequency dependence of the rotor impedance can be

approximated by up to three parallel R-L branches.

Main flux saturation (saturation of Xm) will follow in version 13.1 of DIgSILENT PowerFactory .

Rs Xs

Xm ZrotU

t jr e

ω :1

Figure 1: General Induction Machine Model




RrA

XrA

Ur'

Figure 2: Rotor Impedance of the Single Cage Rotor

RrA1

XrA1

Ur'

RrA2

XrA2

RrA0 XrA0

Figure 3: Rotor Impedance of Squirrel Cage Machines (with Current Displacement Effect)

RrB

XrB

Ur'

Xrm

RrA1

XrA1

RrA2

XrA2

RrA0 XrA0

Figure 4: Double-Cage Rotor

1.1 Input Data

Data can be entered either by directly specifying the resistances and reactances of the equivalent circuit diagrams

(electrical parameters) or by specifying characteristic points on the slip-torque and slip-current characteristic of

the machine.

If the input mode is set to “slip-torque/current characteristic”, the parameters of the equivalent circuit diagram

are automatically calculated from the nominal operation point and the maximum torque (torque at stalling point)

plus starting current and starting torque, if the model type is set to double cage machine or a squirrel cage rotor

is modelled.




The rated mechanical power, the rated power factor, the efficiency at nominal operation and the nominal speed

of the machine specify the nominal operation point.

Figure 5: Basic Data Page of Induction Machine Type

Pressing the “Calculate” button starts the conversion to equivalent circuit parameters. If the conversion fails due

to inconsistent input parameters, a corresponding error message appears:

• “No convergence in iteration, parameter estimation used”This message means that the input data could not be fully matched during the parameter estimation

iteration. An estimate approximating the entered data is used instead. By analyzing the speed-torque

and speed-current characteristic, the user can verify how close the estimated parameters could match

the entered characteristics.

• “Estimated parameter inconsistent. Check nominal operating point”

Here, no solution, even not an approximate solution could be found. The user should first of all check

the data entered on the basic data page. For achieving convergence, the user should first try to find a

solution using the single cage model. Only if motor start-ups are calculated, it is important to reproduce

the speed-torque characteristic over the full range. Therefore, for many applications, the single cage




representation will be good enough. Otherwise, we recommend to reduce the starting current, because

measured starting currents are very often higher due to saturation of leakage reactance, which is not

represented in the model.

Figure 6: Load Flow page of the induction machine type




Figure 7: Speed-Torque and Speed-Current characteristic for different voltages

The curves showing the speed-torque or the speed-current characteristic (Figure 6 and Figure 7) are always

calculated from the steady state equations of the equivalent circuit. Hence, they truly represent the machine’s

characteristics. These graphical diagrams are also available when the parameters of the equivalent circuit

diagram are directly entered.

1.1.1 Equivalent Rotor Impedance

Sometimes, neither the equivalent circuit parameters nor the speed-torque characteristic is given but values of

the equivalent rotor impedance according to Figure 1 for different frequencies (or slip-values).




The general formula that relates the equivalent rotor impedance to equivalent circuit parameters of a rotor circuit

approximated by two ladder circuits according to Figure 3 is:

( )( ) ( )

( )( ) ( )221

22

21

2121

2

1

2

221

21

22

21

2

12

2

21

2

2121

)(

)()(

A A A A

A A A A A A A A

A A A A

A A A A A A A Ar

X X s R R

X X X X s X R X Rs Xr

X X s R R

X R X Rs R R R Rs R

+++

+++=

+++

+++=

(1)

The rotor leakage impedance was assumed to be zero in this case.

The values at stand-still (s=1) and synchronous speed (s=0) are:

( )

( )221

1

2

221

21

21

)0(

)0(

A A

A A A A

A A

A Ar

R R

X R X R Xr

R R

R R R

+

+=

+= (2)

( )( ) ( )

( )( ) ( )221

2

21

21211

2

221

21

2

21

212

2212121

)1(

)()1(

A A A A

A A A A A A A A

A A A A

A A A A A A A Ar

X X R R

X X X X X R X R Xr

X X R R

X R X R R R R R R

+++

+++=

+++

+++=

(3)

This set of non-linear equations can be solved by an iterative procedure, e.g. a Newton-Raphson iteration.

The iteration is highly simplified using the following substitution:

( )

x A

x A A

r A

r A A

X X

X X X

R R

R R R

−=

−=

1

12

1

12

)0(

)0(

(4)

The auxiliary variable x X can directly be calculated from the given values for rot Z and is defined by:

( ))1()0(

)0()1()1(2

r r

r r r x

X X

R R X X

−

−−= (5)

Reasonable starting values are:

21

21

2

5

5

1

A A

A A

X X

R R

=

=

(6)




1.2 Load Flow Analysis

For representing induction machines in load flow analysis, the user has the choice between two representations:

• Slip Iteration (AS)

• Constant P-Q model (PQ)

The “slip iteration” representation is the more accurate representation and is based on the equivalent circuit

diagrams according to Figure 1 to Figure 4. Here, the model equations are evaluated in steady state. The user

defines only the (electrical) active power of the machine. During the load flow iteration, the corresponding slip is

calculated from the steady state model equations and the reactive power (Q) is resulting.

The “P-Q” representation corresponds to the classical way of representing induction machines in load flow

programs. By assuming that the machine operates at a certain power factor, independent of the bus bar voltage,

the machine can be approximated by a standard P-Q load model.

The “slip iteration” is of course the more precise method of representing induction machines in load flow

programs. Since this model is consistent with dynamic models it should always be used when the load flow is

used for initializing a transient analysis. However, it requires the full machine characteristics why it is sometimes

more suitable to use the simple P-Q approach, especially in load flow planning studies, when no transients have

be calculated or when no concrete data are available.

1.3

Short Circuit Analysis

Figure 8: Short-Circuit Model

For short circuit analysis, a voltage source behind the subtransient impedance (rs+jx’’) generally represents

induction machines (see Figure 8).

The value of the subtransient impedance is either directly taken from the speed current characteristic (“Consider

Transient Parameter”) or it can be entered separately. This is sometimes the more accurate approach because

under short circuit conditions saturation effects of the leakage reactance that are not represented in the standard

model can occur.

''U

'' x sr




Figure 9: Short Circuit Input Dialogue Box

The relationship between the locked rotor current ratio and the subtransient impedance is the following:

''// Z Z I I nna = (7)

The actual value of the subtransient voltage depends on the short circuit method applied. Also, the model

according to Figure 8 is only able to represent the subtransient behaviour correctly. For calculating DC time

constants, transient or permanent short circuit currents, the rules defined in the individual short circuit standards

are applied.

1.4 Harmonic Analysis

The induction machine model for harmonics analysis can directly be derived from the equivalent circuits according

to Figure 1 to Figure 4.

The value of this impedance is either calculated from the equivalent circuits according to Figure 1 to Figure 4. For

higher frequencies, the induction machine impedance corresponds to the subtransient value. Only for frequencies

around fundamental frequency, the actual slip dependence is important. This accurate representation is especially

required for subsynchronous resonance studies or self-excitation studies of induction machines.

It is possible to neglect the effect of slip dependence by disabling the flag “consider transient parameters”.




1.5 Stability/Electromagnetic Transients (RMS- and EMT-

Simulation)

The dynamic models for RMS (stability) and EMT-simulations can be derived from the equivalent circuits

according to the Figures 1-4.

Possible state variables of a general induction machine model are either current or flux variables.

As long as no saturation is considered, the actual choice of state variables doesn’t have any influence to the

results, only the numerical behaviour of the solution algorithm will depend on it.

The PowerFactory model uses stator currents and rotor flux as state variables because this choice leads to the

best decomposition of time frames and has therefore the best numerical properties.

The voltage equations of an induction machine model with a number of n R-L rotor-loops are the following:

Rn

Rref

n

R R R

S n

ref

n

S

S S S

jdt

d

jdt

d ir u

ψ

ψ

iR 0ω

ω ω

ω

ψ ω

ω

ω

ψ

−++=

++=

(8)

The equations are expressed in a rotating reference frame common to the stator and the rotor equations. The

dimension of the rotor-flux vector and the rotor-current vector is equal to the number of rotor-loops.

The flux linkage equations are the following:

R RRS RS R

R

T

SRS SS S

i

i x

iXxψ

ix

+=

+=ψ

(9)

For formulating the induction machine equations with stator current and rotor flux as state variables, the flux

linkage equations must be solved for the non-state variables, which are stator flux and rotor currents:

R RRS RS R

R

T

SRS S

i

i x

ψXk i

ψk

1

''

−+−=

+=ψ (10)

The new coefficients are:

( )

RS RR RS

RR

T

SR

T

SR

RS RR

T

SRSS x x

xXk

Xxk

xXx

1

1

1''

−

−

−

=

=

−=

(11)




With these definitions, the stator-voltage equation results in:

dt

d j

dt

id xi x jr u

nn

ref S

n

S

n

ref

S S ω

ψ ψ

ω

ω

ω ω

ω ''''

'''' +++⎟⎟ ⎠

⎞⎜⎜⎝

⎛ += (12)

The subtransient flux is defined by:

R

T

SRψk =''

ψ (13)

Main Flux Saturation

Will be included in the version 13.1 model.

Single Cage Model

The flux-linkage and the resistance matrices of the single cage model according to Figure 1 and Figure 2 can be

expressed as follows:

mS SS x x x += (14)

mSR x x = (15)

m RS x x = (16)

mrA RR x x x += (17)

rA R Rr = (18)

Squirrel Cage Rotor

The flux-linkage and the resistance matrices of the squirrel cage rotor model according to Figure 1 and Figure 3

are the following:

mS SS x x x +=

(19)

[ ]mm

T

SR x x=x (20)

⎥⎦

⎤⎢⎣

⎡=

m

m

RS x

xx (21)

⎥⎦

⎤⎢⎣

⎡

+++

+++=

mrArAmrA

mrAmrArA

RR x x x x x

x x x x x

020

001X (22)




⎥⎦

⎤⎢⎣

⎡

+

+=

200

010

rArArA

rArArA

R R R R

R R RR (23)

Double Cage Rotor

The flux-linkage and resistance matrices of the double cage model with three R-L-rotor loops according to Figure

1 and Figure 4 are the following:

mS SS x x x += (24)

[ ]mmm

T

SR x x x=x (25)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

m

m

m

RS

x

x x

x (26)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++++

++++++

++++++

=

mrmrBmrmmrm

mrmmrmrArAmrmrA

mrmmrmrAmrmrArA

RR

x x x x x x x

x x x x x x x x x

x x x x x x x x x

020

001

X (27)

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

+

+

=

rB

rArArA

rArArA

R

R

R R R

R R R

00

0

0

010

001

R (28)

1.5.1 EMT-Model

In the EMT simulation, PowerFactory uses a steady state reference frame for expressing the stator equations.

The stator-voltage equation in a steady state reference frame is:

t j

nn

ref S

n

S S S

ref edt

d j

dt

id xir u

ω

ω

ψ ψ

ω

ω

ω ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +++=

''

''''

(29)

In the EMT-model, the reference frame, in which the rotor equations are expressed, rotates with nominal

frequency, hence:

nref ω ω = (30)

The resulting stator-voltage equation of the EMT model is therefore:




t j

n

S

n

S S S ne

dt

d j

dt

id xir u

ω

ω

ψ ψ

ω ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +++=

''

''''

(31)

This equation corresponds exactly to the equivalent circuit according to Figure 8, with the following definition for

the subtransient voltage:

t j

n

nedt

d ju

ω

ω

ψ ψ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ +=

''

'''' (32)

1.5.2 Stability Analysis (RMS-Simulation)

For stability analysis, the induction machine model has to be reduced. In accordance with the steady state model

of the electrical network that is applied in stability analysis, the stator equations of the induction machine model

are reduced to steady state equations. The following voltage equation is resulting:

''''ψ

ω

ω

ω

ω

n

ref

S

n

ref

S S ji x jr u +⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ += (33)

This is a steady state representation of the equivalent circuit according to Figure 8. The subtransient voltage is

here defined as:

''''ψ

ω

ω

n

ref ju = (34)

In the stability model, the stator equations are expressed in a reference frame that rotates with the global system

reference that is usually fixed to the rotor of the reference generator (or an external network or a voltage source,

depending on the load flow reference).

Because stator transients are neglected, the choice of the reference frame has actually an influence to the stator

voltage equations. For avoiding any dependence on the actual choice of the reference machine, the influence of

the reference frequency is not considered in subtransient reactance of the PowerFactory stability model.

The stator voltage equations are therefore:

( ) ''''ψ ji jxr u

S S S ++= (35)

1.5.3 Mechanical Equations

The model is completed by the mechanical equation:




me R M M J −=ω & (36)

• J : Inertia

• e M : Electrical torque

• M M : Mechanical torque

• Rω : Angular velocity of the rotor (mechanical)

The mechanical equation can be rated to the nominal torque:

z

nn

mnn

ps

P M ω

)1( −= (37)

Resulting in the following, normalized mechanical equation:

meag

T

z

n

mn

z

nn

mmnT n pP

ps J

ag

−==

−

&&

4 4 34 4 21

ω

ω )1(

(38)

The following variables have been used in the normalized equations (38):

• nω : Nominal electrical frequency of the network

• :ns Nominal slip

• mnP : Rated mechanical power

• z p : Number of pole-pairs

• agT : Acceleration time constant

1.5.4 Mechanical Load

Mechanical loads can generally be defined by connecting a so-called mdm-model (motor-driven machine) to theinput xmdm (m m in Eq. (39)) of the induction machine. Such an mdm-model can either be defined by a DSL-

model or by one of the built-in models (MDM_1, MDM_3).

If no separate mdm model is defined, the induction machine uses the speed-torque characteristic of the built-in

mdm-model:

ex

pm nlm = (39)




The parameters used in this equation are:

1. l p : Proportional factor of the motor-driven machine (parameter mdmlp )

2.

ex : exponent of mdm-characteristic (parameter mdmex )

1.5.5 Initialization

All state variables of the model are initialized from a preceding load flow calculation so that a simulation starts

from a steady state condition.

If the default orientation of the induction machine is set to “motor”, the mechanical load torque xmdm is

initialized. In case of “generator” orientation, the turbine power pt is used for establishing the active power

balance of the model.

In case of a running machine, the proportional factor l p of the built-in mdm or analogous factors of separately

modelled motor-driven machines are calculated during the initialization process. In case of a disconnected

machine, e.g. if a motor start-up is simulated, the user-defined variable of the input dialogue is used instead.




2 Input/Output Defini tions of Dynamic Models

Figure 10: Input/Output Definition

The following per-unit systems are used

• Rated Apparent Power, Rated Voltage:

r

r br r

S

V Z V S

2

,, =

• Rated (Electrical) Active Power:

)cos( r r er S P ϕ =

• Rated Mechanical Power:

r er mr PP η =

r η : Rated efficiency

• Rated Mechanical Torque

z

nr

mr

rn

mr r

ps

PP M

ω ω )1( −

==

r s : Rated slip

nω : Nominal electrical angular velocity

z p : Number of pole pairs

Tabelle 1: Input Variables (signals)

Parameter Symbol / Equ. Description Unit

pt Turbine power, (rated tomechanical power) p.u

xmdm mm /(38) Mechanical Load Torque. (rated to mechanical

torque)

p.u

rradd Additional rotor resistance p.u.

pt

xmdm

rradd

xspeed

pgt




Tabelle 2: Output Variables (signals)Parameter Symbol / Equ. Description Unit

xspeed n /(38) Mechanical Speed p.u.

pgt Electrical Power. (rated to electrical active power) p.u

Tabelle 3: State Variables (signals)

Parameter Symbol / Equ. Description Unit

speed n/ (38) Mechanical Speed p.u.

phi Electrical Power rad

psiA1_r Rψ /(9) Flux of loop A1, real p.u.

psiA1_i R

ψ /(9) Flux of loop A1, imaginary p.u.

psiA2_r R

ψ /(9) Flux of loop A2, real p.u.

psiA2_i R

ψ /(9) Flux of loop A, imaginary p.u.

psiB_r R

ψ /(9) Flux of loop B, real p.u.

psiB_i R

ψ /(9) Flux of loop B, imaginary p.u

Tabelle 4: Additional Parameters and signals (calculation-parameter)

Parameter Description Unit

slip Slip p.u.

xme Electrical torque, based on rated mechanical torque p.u.

xmem Electrical torque (inverted sign), based on rated

mechanical torque

p.u.

xmt Mechanical Torque, based on rated mechanical

torque

p.u.

xradd Additional rotor reactance p.u.

addmt Additional mechanical torque, based on rated

mechanical toruqe

p.u.

ccomp Internal capacitance (for compensating reactive

power mismatch in case of PQ-load flow model)

p.u.

i_star i_star=1: Star Operation i_star=0: Delta Operation

(used for Star-Delta start-up)




3 Input Parameter Definit ions

3.1 Induction Machine Type (TypAsmo)

• All rotor resistances and reactances are expressed in p.u. referred to the stator side.

• Rotor impedances given in Ohm, referred to the stator side have to be divided by the base impedance of

the machine (Zbase=U2rated /Srated)


loc_name Name

ugn Rated Voltage kV

sgn Power Rating: Rated Apparent Power kVA

pgn Power Rating: Rated Mechanical Power kW

cosn Rated Power Factor

effic Efficiency at nominal Operation %

frequ Nominal Frequency Hz

anend Nominal Speed rpm

nppol No of Pole Pairs

nslty Connectioni_cage Rotor Model

aiazn Locked Rotor Current (Ilr/In) p.u.

amazn Locked Rotor Torque p.u.

rtox R/X Locked Rotor

amkzn Torque at Stalling Point p.u.

aslkp Slip at Stalling Point

amstl Torque at Saddle Point p.u.

asstl Slip at Saddle Point

rstr Stator Resistance Rs p.u.

xstr Stator Reactance Xs p.u.

xm Mag. Reactance Xm p.u.

xmrtr Rotor Leakage Reac. Xrm p.u.

i_cdisp Operating Cage/Rotor data: Consider Current

Displacement (Squirrel Cage Rotor)

rrtrA Operating Cage/Rotor data: Rotor Resistance RrA p.u.

xrtrA Operating Cage/Rotor data: Rotor Reactance XrA p.u.

rrtrA0 Operating Cage/Rotor data: Slip indep. Resistance

RrA0

xrtrA0 Operating Cage/Rotor data: Slip indep. Reactance

XrA0




r0 Operating Cage/Rotor data: Resistance RrA1

x0 Operating Cage/Rotor data: Reactance XrA1

r1 Operating Cage/Rotor data: Resistance RrA2

x1 Operating Cage/Rotor data: Reactance XrA2

rrtrB Starting Cage: Rotor Resistance RrB p.u.

xrtrB Starting Cage: Rotor Reactance XrB p.u.

i_trans Consider Transient Parameter

aiaznshc For Short-Circuit Analysis: Locked Rotor Current

(Ilr/In)

p.u.

iinrush Inrush Peak Current: Ratio Ip/In p.u.

Tinrush Inrush Peak Current: Max. Time s

Tcold Stall Time: Cold s

Thot Stall Time: Hot s

trans Consider Transient Parameter

xdssshc For Short-Circuit Analysis: Locked Rotor Reactance

rtoxshc For Short-Circuit Analysis: R/X Locked Rotor

xtorshc For Short-Circuit Analysis: X/R Locked Rotor

3.2 Induction Machine Element (ElmAsm)


loc_name Name

outserv Out of Service

ngnum Number of parallel Machines

i_mot Orientation: Generator/Motor

c_pmod Model

bustp Bus Type

pgini Active Power MW

qgini Reactive Power Mvar

i_rem Remote Control

p_cub Controlled Branch (Cubicle) (StaCubic*)

i_pset State Estimation: Estimate Active Power

iconfed Static converter-fed drive (short circuit analysiststart Starting Time (protection) s

mdmlp Mechanical Load: Proportional Factor p.u.

mdmex Mechanical Load: Exponent

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