tech ref induction machine
DESCRIPTION
digsilent technical referenceTRANSCRIPT
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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
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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
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T a b l e o f C o n t e n t s
I n d u c t i o n M a c h i n e
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
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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
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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.
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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
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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
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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).
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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)
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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
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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”.
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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)
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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)
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⎥⎦
⎤⎢⎣
⎡
+
+=
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:
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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:
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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)
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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.
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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
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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)
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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)
Parameter Description Unit
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
7/17/2019 Tech Ref Induction Machine
http://slidepdf.com/reader/full/tech-ref-induction-machine 21/21
I n d u c t i o n M a c h i n e
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)
Parameter Description Unit
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