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Three-Phase Induction Motors
August 10, 2021Load Modeling Update
WECC MVS Meeting Online
Jamie Weber, Ph.D.Director of Software Development
1
Summary
β’ Provide a software history of 3-phase induction motor models
β’ CIM5, CIM6, CIMW, MOTOR1 before 1960β The 3-phase induction motor model
β’ Missing steady-state frequency-dependence of induction motor model CIM5β Problem noticed in 2019 by Kannan at ISO-NE
β’ Side Track: MOTORW β 1997β Important step in creating load componentsβ Double-cage model problem will be discussed
2
3-phase induction motor model.The real model
β’ Stator containing armature windingsβ’ Rotor windings are shortedβ’ Typically have 2 rotor windings (βdouble
cageβ)β βCageβ ultimately means a winding
β’ The mathematical theory of a 3-phase induction motor is the same as a 3-phase synchronous machineβ Synchronous machine is simplified because it
always operates very close to synchronous speed
3
Induction Motor Model: CIM5Whatβs different?
β’ Some sign differences because current conventions (motor vs. generator convention)
β’ Terms for βSLIPβ create a feedback between the flux dynamic statesβ Note: SLIP is not the normal definition of slip, but similar
β’ There is no input field voltage β Rotor windings are shorted
β’ Otherwise, these models are very similarβ 2 windings each have a complex flux, so thatβs 4 states
variables β same concept as GENROUβ Speed state is modeled, with mechanical torque a function
of rotor speed optionallyβ GENROU had a rotor angle which isnβt a state variable for
induction machine
4
Induction Motor and Synchronous Machine are very similar!
Double Cage Induction Motor GENROU
SLIP Terms do not exist for synchronous machineNo Efd on Induction Motor
5
Network Interfaces
CIM5, MOTOR1Double Cage Induction MotorEquation in Network Reference
GENROUBoth a dq rotor referenceAnd a network reference
Rotor Angle provides relationship between dq and network reference frame
6
3-phase Induction Motor Model History
β’ CIM5 is the load motor model from textbooks from well before the 1960sβ This is a PSS/E model and has variations named
CIM6 and CIMW which make modifications to the mechanical load connects to it
β 7 Circuit parametersβ’ Ra, Xa, Xm, R1, X1, R2, X2
β’ MOTOR1 is a generator model in PSLF which uses the same set of equationsβ 7 Dynamic parameters
β’ Ls, Lp, Ra, Tpo, Lpp, Tppo, Ll
7
Circuit Parametersvs Dynamic Parameters
β’ 7 parametersβ Algebraic relationship between parameter setsβ Doesnβt matter which one you use, but I prefer
circuit parameters
8
Mechanical Torque Equation
β’ CIM5:β’ CIM6:β’ Composite load model uses same as CIM5
β Parameter we use is normally βEtrqβ for βexponent of torque equationβ
β’ Mechanical Torque Equation impacts whether a motor will restart itself after voltage recoversβ Constant Torque (Etrq = 0) makes it harder to
restartβ Higher exponent (Etrq = 2) makes is easier for
motor to restart
9
Adding Frequency Dependence to Algebraic Models
β’ Algebraic Models that add frequency dependence
β’ LDFR β old PSS/E modelβ Provides a very simple algebraic model
which represents the frequency dependence
β Applies to constant power and current only
β’ WSCC Model β old PSLF Model
β’ IEEL Model β old PSS/E Model
10
3-phase Induction Motor, Stator Frequency Effect
β’ In 2019, Dmitry Kosterev of the Bonneville Power Administration (BPA) was working with Kannan Sreenivasachar from ISO-New England (ISONE)
β Kannan was noticing that as he decreased the frequency in the system, the steady state electric power of induction motor load was not impacted by frequency changes
β’ As a new steady-state is reached, a constant torque mechanical load results in a constant power electrical load
β Thus electrical frequency drop does not impact electrical power at steady state.
β’ Dmitry emailed me to ask why this was occurringβ My response initially was βthatβs what is supposed to happenβ
β’ I had noticed this in the equations coded for a three-phase induction motor 10 years ago, and just assumed it was fine
β BPA, ISONE, and EPRI have experience testing hardware, and to all of them this did not seem right
β They tested in 4 different software tools and all showed the same response
β Time to revisit induction machine theory!
11
Revisit network interfaces Induction and Synchronous Machines
Double Cage Induction MotorEquation in Network Reference
GENROUBoth a dq rotor referenceAnd a network reference
Multiply by frequency was not there!This is what is being fixed
12
Network Interface for Models
β’ Differential equations give output of fluxesβ’ The network interface equation is a voltage
relationship, so we should multiply by the stator electrical frequency in per unit ππππππππ
β’ This term was missing in traditional software implementation
πππππ π πππ π π π π π = ππππππππ πΈπΈπππππ π + πππΈπΈππππππ
13
Reminder: Why do we multiply flux by ππ to get voltage
β’ Remember these things are all phasors which represent time-varying waveform at a frequency
β’ Ο = Οππππππ cos ππππ β ππ + ππ π π π π π π ππππ β ππ
β’ Ο = Οππππππππππ ππππβππ
β’ ππ = ππΟππππ
= Οππππππ ππππ ππππβππ ππππβ’ ππ = ππππ Οβ’ Voltage = Flux multiplied by ππ and also apply a
90 degree phase shift (multiply by j)
14
This fixes our problem
β’ With multiplication added, a constant torque load will no longer behave as a constant electric power load at the network level
β’ PowerWorld Simulator has added a new model named InductionMotor3P_A
β’ https://www.powerworld.com/WebHelp/#TransientModels_HTML/Load%20Characteristic%20InductionMotor3P_A.htm
15
InductionMotor3P_A
Flag parameterindicates whether to include this multiplication
16
Similar to Synchronous Machine
β’ Same concept is in the synchronous machine models too
β’ GENROU and related models have multiplied by rotor speed instead
β’ ππ = rotor speed deviation in per unit
β’ Should we be using the stator electrical frequency for a synchronous machine too?β We donβt need to because πππ π πππ π in synchronous machine
this approximation is fine for synchronous machine
ππππ + ππππππ = βΟππβ²β² + ππΟππβ²β² 1 + ππ
17
Comparing Induction Machine and Synchronous Machine
SynchronousMachine
InductionMachine
Transient Time Frame Small deviations in rotor speed during simulation
Large deviations in rotor speed during simulation
Steady State Rotor speed and stator electric speed are equal at steady state
Rotor speed is not equal to electric speed at steady state (always a slip)
Is using Rotor Speed instead of stator electric speed a good approximation?
YES, Good approximation
NO,Must use stator frequency in per unit
18
Side Track: MOTORW
β’ The PSLF model MOTORW is important in the history of including induction motor modelsβ Was adding after 1996 blackout in WECC
Oscillations in real life
Did NOT match numeric simulations
19
Side Track: MOTORW
β’ Provided a first step in modeling a βpartialβ load characteristic
β’ MOTORW: had a parameter indicating a fractionβ βLoad is 20% 3-phase induction motorββ New concept which started us down the path of the
composite load modelβ’ MOTORW was called the βinterim load modelβ from
about 1997 β 2016, so it had a long run!β’ For a single-cage motor [(Tppo = 0) or (Lpp=Lp)]
MOTORW and CIM5 simply to same equationsβ Historically MOTORW models had input data that
represented single-cage motors onlyβ Old WECC MOTORW model always used
Tppo = 0 and Lpp=Lp=0.17
20
Side Track: MOTORW Double Cage Model
β’ Howeverβ¦. MOTORW double-cage equations are not the same as CIM5
β’ 6 parameters: Ls, Lp, Ra, Tpo, Lpp, Tppoβ It is missing the leakage reactance (Ll)β Model however is just different
β’ It is not simply assuming something about leakage reactance
21
CIM5 will never match MOTORW for Double-Cage Motor
Work in 2014 by Dr. Tom Overbye at University of IllinoisNo assumed value of Llgives a matchItβs a different model!
2014: PowerWorld added an option implementing MOTORW using the PSLF block diagram
Different Model βnow it matches
22
Side Track: MOTORW Double Cage Model
β’ Present Software Statusβ MOTORW is a PSLF model. The CMPLDW model in PSLF uses this
as component for 3-phase motorsβ PowerWorld Simulator and PowerTech TSAT have implemented
the ability to use these MOTORW equations only to match the results our customers who use PSLF see
β Siemens PSS/E has not added implementation these special MOTORW equations
β’ Summaryβ Move away from MOTORWβ PSLF has a modular load component model named _cmp_mot3
which is based on MOTORWβ PSLF needs to make a new one that is based on the correct 3-
phase motor model instead and users can transition to that (maybe name it _cmp_cim5? and use circuit parameter input parameters?)
23
Summary
β’ 3-phase induction motors are 3-phase machinesβ Very similar to 3-phase synchronous machines
β’ Need to add new models that include frequency-dependence
β’ Transition away from using MOTORW
24
25
Following Slides:Frequency Dependence
β’ Following slides cover more theory on frequency dependence terms of induction motor
β’ Explains why they should be modeled
26
Proving this with Math:Krause Book
β’ Another bookAnalysis of Electric Machine and Drive Systems, Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarekβ First Edition published in 2003β Third Edition is 2013
27
Proof that Stator Electric Frequency should be used
β’ Start with Krause et. All equation 6.5-22 and 6.5-23 on page 227 of the book
β’ These equations are on an arbitraryreference frame (ππ has not been chosen)
π£π£ππππ = πππππ π ππππ + ππππππππππππ + 1
ππππ
ππππππππππππ
6.5-22
π£π£ππππ = πππππ π ππππ βππππππππππππ + 1
ππππ
ππππππππππππ
6.5-23
28
Synchronous Machine ignoring stator transients
β’ For a synchronous machine we used the rotor reference frame and thus the choice of ππ = πππ π is made
β’ We end up writing the following by assuming derivatives go quickly to zero in a new pseudo steady state
β’ This is a valid approximation for a synchronous machine in the rotor reference frame β This feels like itβs always correct, but it depends on the reference
frame and the type of machine
π£π£ππππ = πππππ π ππππ + ππππππππππππππ + 1
ππππ
ππππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ + 1
ππππ
ππππππππππππ
π£π£ππππ = πππππ π ππππ + ππππππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ
29
How about induction motor on Synchronous Reference Frame
β’ Induction Motor uses synchronous reference frame, so this choice is ππ = ππππ
β’ The induction motor models in all our software tools then did same approximation to ignore stator transients
β’ Mistake was just made! β This is the source of our troubleβ Thatβs why Bernieβs simulations are differentβ Can NOT just assume derivatives go to zero
π£π£ππππ = πππππ π ππππ + ππππππππππππππ + 1
ππππ
ππππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ + 1
ππππ
ππππππππππππ
π£π£ππππ = πππππ π ππππ + ππππππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ
30
Do ππππππππ/ππππ and ππππππππ/ππππ always quickly go to zero? NO!
β’ Depends on the reference frame and the machine typeβ Magnitude reaches new steady state quickly with ππππππ
ππππ0
β ππππππ and ππππππ reach a steady state spinning around a circle with a frequency of πππ π β ππ
β ππππππππ
ππππand ππππππππ
ππππderivatives become
sinusoids and do NOT go to zeroβ’ In rotor reference frame ππ = πππ π ,
so that new steady state has the fluxes states spinning at πππ π β πππ π β In a synchronous machine, the new
steady state will have πππ π = πππ π , thus that means the red dot becomes stationary
β For synchronous machines it is correct that ππππππππ
ππππand ππππππππ
ππππwould
reach a new steady state and would go toward 0 quicklyβ This is NOT true for all machines and references
ππππππ
ππππππ
31
Synchronous MachineGraphical Explanation
ππππππ
ππππππ
Synchronous Machine at new steady state in rotor reference frame ππ = πππ π
Rotates at πππ π β πππ π = 0
Synchronous machine, so we assume πππ π = πππ π is achieved very quickly
This is what βignoring stator transientsβ means!
Steady State is reached with a fixed red dot that is stationary
32
Compare to Induction MachineGraphical Explanation
ππππππ
ππππππ
ππππππ
ππππππ
Synchronous Machine at new steady state in rotor reference frame ππ = πππ π
Induction Machine at new steady state in synchronous frame ππ = ππππ
Rotates at πππ π β πππ π = 0
Rotates at πππ π β ππππ
33
Algebra: Need to go back to the ABC phase phasors
ABC Phase Quantitiesππππππ = 2ππππ πππππ π πππ π ππππππ = 2ππππ πππππ π πππ π β
2ππ3
πππ π ππ = 2ππππ πππππ π πππ π + 2ππ3
πππ π = ππππππππππ
After DQ transformation in an Arbitrary Reference Frameππππππ = + 2ππππ cos πππ π β ππππππππ = β 2ππππ sin πππ π β πππππ π ππ = 0ππ = ππππ
ππππ
ππππ =ππππππ2 +ππππππ
2
2
34
Now Take Derivatives
ππππππππππππ
= ππππππ
2ππππ πππππ π πππ π β ππππππππππππππ
= ππππππππππ
2 πππππ π πππ π β ππ β 2ππππ π π π π π π πππ π β ππ ππππππππππ
β ππππππππ
ππππππππππππ
= ππππππππππ
2ππππ π π π π ππ ππππβππππππ
+ β 2ππππ π π π π π π πππ π β ππ ππππππππππ
β ππππππππ
ππππππππππππ
= ππππππππππ
ππππππππππ
+ ππππππ πππ π β ππ
ππππππ = + 2ππππ cos πππ π β ππ
ππππππ = β 2ππππ sin πππ π β ππ
ππππππππππππ
= ππππππ
β 2ππππ π π π π π π πππ π β ππππππππππππππ
= ππππππππππ
β 2ππππ π π π π π π πππ π β ππ β 2ππππ πππππ π πππ π β ππ ππππππππππ
β ππππππππ
ππππππππππππ
= ππππππππππ
β 2ππππ πππππ π ππππβππππππ
+ β 2ππππ πππππ π πππ π β ππ ππππππππππ
β ππππππππ
ππππππππππππ
= ππππππππππ
ππππππππππβ ππππππ πππ π β ππ
35
Apply this to our equations in the Arbitrary Reference Frame
ππππππππππππ
= ππππππππππ
ππππππππππ
+ ππππππ πππ π β ππππππππππππππ
= ππππππππππ
ππππππππππ
β ππππππ πππ π β ππ
π£π£ππππ = πππππ π ππππ + ππππππππππππ + 1
ππππ
ππππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππ + 1
ππππ
ππππππππππππ
π£π£ππππ = πππππ π ππππ + ππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
+ ππππππ πππ π β ππ
π£π£ππππ = πππππ π ππππ βππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
β ππππππ πππ π β ππ
π£π£ππππ = πππππ π ππππ + ππππππππππππ + ππππ
ππππππππππ β
ππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππ β
ππππππππππππππ + ππ
ππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
π£π£ππππ = πππππ π ππππ + ππππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
Substitute in our derivative calculation
Expand Terms
Cancel Terms
36
Ignore Stator Transients in any Reference Frame or Machine
β’ Approximation to ignore stator transients is
β’ Result for an arbitrary reference frame
β Use this for our induction machine!β Could also use this for a synchronous machine
β’ Also valid to use ππππππππ
for a synchronous machine though, so we do that because itβs easier (πππ π is a state)
π£π£ππππ = πππππ π ππππ + ππππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ + 1
ππππ
ππππππππππ
ππππππππππ
π£π£ππππ = πππππ π ππππ + ππππππππππππππ
π£π£ππππ = πππππ π ππππ βππππππππππππππ
Terms quickly go to zero regardless of reference frame choice and machine type ππππ is magnitude of flux phasor
ππππππππππ
β 0