Lecture 3: Vector-Controlled Induction Motor DriveELEC-E8402 Control of Electric Drives and Power Converters (5 ECTS)
Marko Hinkkanen
Spring 2017
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Learning Outcomes
After this lecture and exercises you will be able to:I Explain the principle of rotor-flux orientationI Derive the rotor-flux orientation equations (torque, flux dynamics, slip relation)
using the inverse-Γ modelI Draw block diagrams for the most typical control schemes and explain themI Derive the current model and explain its properties
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Vector Control Methods
I Based on the dynamic motor modelI Rotor-flux-oriented vector control,
direct torque control (DTC)I Torque can be controlledI High accuracy and fast dynamicsI Speed measurement can be
replaced with speed estimation inmost applications
M
Vectorcontroller
TM,refωM,ref
Speedcontroller
ωM
Udc
However, it will be omitted in the followingblock diagrams (or constant Udc is assumed).
DC-link voltage is typically measured.
ia, ib, ic
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Outline
State-Space Representation
Principle of Rotor-Flux Orientation
Flux Estimation With the Current Model
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Review: Model in Synchronous Coordinates
I Voltage equations
us = Rsis +dψsdt
+ jωsψs
uR = RRiR +dψRdt
+ jωrψR = 0
I Flux linkages
ψs = Lσ is + ψR
ψR = LM(is + iR)
I Steady state: d/dt = 0
dψR
dt
iR
LMdψ
s
dt
Lσ
iM
is
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State-Space Representation
I Stator current is and rotor flux ψR are selected as state variablesI Derivation: rotor current iR and stator flux ψs are eliminated from the voltage
equations by means of the flux equations
Lσdisdt
= us − (Rs + RR + jωsLσ)is +
(RR
LM− jωm
)ψR
dψRdt
= RRis −(
RR
LM+ jωr
)ψR
I Dynamics of the stator current are governed by current controlI Dynamics of the rotor flux are taken into account by rotor-flux orientation
Go through the derivation of these equations (see the compendium).6 / 28
Outline
State-Space Representation
Principle of Rotor-Flux Orientation
Flux Estimation With the Current Model
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Rotor-Flux Dynamics
I Fast closed-loop stator-current controller is usedI Stator current is the input from the point of view of the rotor-flux dynamicsI Rotor equations in synchronous coordinates rotating at ωs
dψRdt
= −RRiR − j (ωs − ωm)︸ ︷︷ ︸ωr
ψR
ψR = LM(is + iR) ⇒ iR = ψR/LM − is
I Rotor current can be eliminated
dψRdt
= −(
RR
LM+ jωr
)ψR + RRis
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Rotor-Flux Orientation
I d-axis of coordinate system is fixed to the rotor flux
ψR = ψRd + jψRq = ψR + j · 0, is = id + jiq
I Real and imaginary parts of the rotor-flux dynamics
dψR
dt= −RR
LMψR + RRid (in the steady state ψR = LMid)
0 = −ωrψR + RRiq
I Rotor-flux magnitude ψR follows id slowly,
ψR(s) =LM
1 + sτrid(s) (in the Laplace domain)
due to the rotor time constant τr = LM/RR (typically 0.1. . . 1.5 s)
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Rotor-Flux Orientation
I d axis of coordinate system is fixed to the rotor flux:
ψR = ψR + j · 0, is = id + jiq
I Electromagnetic torque
TM =3p2
Im{
isψ∗R
}=
3p2ψRiq
I If ψR is constant, the torque can be controlled using iq (without delays)
The coordinate system could be fixed to the stator flux ψs
instead of the rotor flux. This stator-flux orientation would simplify the field weakening,but other parts of the control system would become more complicated.
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Steady-State Equivalent Circuit in Rotor-Flux Coordinates
is jωsLσ
RR
ωr/ωsus
Rs
jωsLM jωsψR
jiq
id
is
iR = −jiq
id
jiq
ψR
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Stator Coordinates (αβ)
I Vectors are rotating(in the steady state ϑs = ωst)
I Controlling the torque
TM =3p2
Im{
iss(ψs
R
)∗}=
3p2(iβψRα − iαψRβ
)would be difficult
α
β
iss
ψsR
jiβ
ϑs
iα ψRα
jψRβ
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Rotor-Flux Coordinates (dq)
I Variables are constantin the steady state
I Torque
TM =3p2
Im{
isψ∗R
}=
3p2ψRiq
easily controllable via iq
α
β
is
ψR
id
d
q
ϑs
jiq
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Rotor-Flux Coordinates (dq)
I Variables are constantin the steady state
I Torque
TM =3p2
Im{
isψ∗R
}=
3p2ψRiq
easily controllable via iq
is
d
q
ψR
id
jiq
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Rotor-Flux Coordinates (dq)
I Variables are constantin the steady state
I Torque
TM =3p2
Im{
isψ∗R
}=
3p2ψRiq
easily controllable via iq
is
q
ψR
d
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Rotor-Flux Coordinates (dq)
I Variables are constantin the steady state
I Torque
TM =3p2
Im{
isψ∗R
}=
3p2ψRiq
easily controllable via iq
is
d
q
ψR
jiq
id
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Example Measured Waveforms: 45-kW Induction Motor Drive
−0.5
0.5
1
−1
1
0 1 2 3 4 5
Spe
ed(p
.u.)
Cur
rent
(p.u
.)
Time (s)
iqid
0
0
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Rotor-Flux-Oriented Vector Control
M
Currentcontroller
Fluxestimator
ψR,ref
ϑ̂s
ωM,refCurrent
referenceSpeed
controller PWM
dqabc
ia, ib, ic
dqabc
is
ωM
p
us,refis,ref ua,ref,ub,ref,uc,ref
TM,ref
ωm
Very fastcurrent-controlloop
Stator coordinatesEstimated rotor-flux coordinates
Rotor-flux angleis estimated
Calculatedin estimatedrotor-flux coordinates
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Space-Vector and Coordinate Transformations
I Space-vector transformation (abc/αβ)
iss =23
(ia + ibej2π/3 + icej4π/3
)I Transformation to rotor coordinates (αβ/dq)
is = isse−jϑ̂s
I Combination of these two transformations isoften referred to as an abc/dq transformation
I Similarly, the inverse transformation isreferred to as a dq/abc transformation
Space-vector
issαβ
abc
dq
αβ isiaibic
ϑ̂s
is
transformationCoordinate
transformation
dq
abciaibic
ϑ̂s
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Current References
1. Flux-producing current reference
id,ref =ψR,ref
L̂M(where the hat refers to estimates)
I Integral term based on umax − |us,ref| can be used for field weakeningI Flux controller could be used for improving the flux dynamics in the case of
varying ψR,ref
2. Torque-producing current reference
iq,ref =2TM,ref
3pψR,ref
I Flux reference ψR,ref is often replaced with the estimate ψ̂R
For more information on field weakening and flux control, see: Z. Qu, M. Ranta, M. Hinkkanen, J. Luomi, “Loss-minimizing flux level control ofinduction motor drives,” IEEE Trans. Ind. Applicat., vol. 48, no. 3, 2012.
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Outline
State-Space Representation
Principle of Rotor-Flux Orientation
Flux Estimation With the Current Model
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Current-Model Flux Estimator in Stator CoordinatesI Current model is based on the rotor voltage equation
dψ̂sR
dt= −
(R̂R
L̂M− jωm
)ψ̂
sR + R̂Riss
I Corresponding forward Euler approximation
ψ̂sR(k + 1) = ψ̂
sR(k) + Ts
{−
[R̂R
L̂M− jωm(k)
]ψ̂
sR(k) + R̂Riss(k)
}
where Ts is the sampling period and k is the discrete-time indexI At each time step, the angle of the flux estimate ψ̂
sR = ψ̂Rα + jψ̂Rβ is
ϑ̂s = atan2(ψ̂Rβ, ψ̂Rα
)In practice, the forward Euler approximation should not be used in stator coordinates due to its poor accuracy and limited stability.
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Current Model in Estimated Rotor Flux Coordinates
M
Currentcontroller
Fluxestimator
ψR,ref
ϑ̂s
ωM,refCurrent
referenceSpeed
controller PWM
dqabc
ia, ib, ic
dqabc
is
ωM
us,refis,ref ua,ref,ub,ref,uc,ref
TM,ref
ωmp
I Signals fed to the flux estimator are DC in the steady stateI Discrete-time implementation becomes easier
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Current-Model Flux Estimator in Estimated Rotor-Flux Coordinates
dψ̂Rdt
= −
(R̂R
L̂M+ jω̂r
)ψ̂R + R̂Ris ψ̂R = ψ̂R + j · 0
I Real and imaginary parts in estimated flux coordinates
dψ̂R
dt= − R̂R
L̂Mψ̂R + R̂Rid ω̂r =
R̂Riqψ̂R
I Flux-angle estimation
ϑ̂s =
∫ω̂sdt =
∫(ωm + ω̂r)dt
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Indirect Field Orientation (IFO)
M
Currentcontroller
Fluxestimator
ψR,ref
ϑ̂s
ωM,refCurrent
referenceSpeed
controller PWM
dqabc
ia, ib, ic
dqabc
is
ωM
us,refis,ref
ua,ref,ub,ref,uc,refTM,ref
ωmp
I Current reference is used as an input of the flux estimatorI Flux estimator is also simplified (see the following slide)
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IFO
I Flux-magnitude dynamics are omitted in the slip relation
ω̂r =RRiq,ref
ψR,ref
I Flux-angle estimation
ϑ̂s =
∫(ωm + ω̂r)dt
I Poor performance if the flux reference ψR,ref is not constantor if the current controller does not work as intended
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Properties of the Current Model and IFO
Disadvantages:I Rotor speed measurement is neededI Converges slowly (with the rotor time constant), which can be a problem if the
flux reference ψR,ref is variedI Inaccurate model parameters R̂R and L̂M cause errors in field orientation⇒ degraded control performance
Advantages:I SimplicityI Robustness
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Reasons for Parameter Detuning: Actual Motor Parameters Vary
I Resistances depend on:I Temperature (about 0.4%/K)I Frequency due to the skin effect
(especially the resistances of the rotor bars)I Inductances depend on the magnetic state
I Stator (magnetizing) inductance increases as the stator (rotor) flux decreases inthe field-weakening region
I Torque may also have an effect on the inductancesI Identification of the motor parameters is never perfectI Some phenomena are omitted in the model but exist in the actual machine
(e.g. core losses)
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Magnetic Saturation Characteristics: 2.2-kW Motor as an Example
0 0.4 0.8 1.20
0.4
0.8
1.2
0 0.4 0.8 1.20
1
2
ψs (p.u.) Ls (p.u.)
is (p.u.) ψs (p.u.)
Rated operating point ψs = ψsN
Field weakening at ψs = 0.5ψsN
No-load saturation characteristics Corresponding stator inductance
I Stator inductance Ls = Lσ + LM depends on the stator-flux magnitude ψs
I Effect should be taken into account in control, if field weakening is used26 / 28
Summary: Rotor-Flux Orientation
I Decoupled control of the flux and the torque, as in the DC machinesI d-axis of the coordinate system is fixed to the rotor flux vector
(or its estimate in practice)I Rotor-flux magnitude is controlled using the d-component of the stator currentI Torque is controlled using the q-component of the current
Similar control structure can also be used in sensorless methods,but a different flux (and speed) estimator is needed.
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Further Reading
ModelingI G. R. Slemon, “Modelling of induction machines for electric drives,” IEEE
Trans. Ind. Applicat., vol. 25, no. 6, 1989.Flux control, field weakening, loss minimization:
I Z. Qu, M. Ranta, M. Hinkkanen, J. Luomi, “Loss-minimizing flux level controlof induction motor drives,” IEEE Trans. Ind. Applicat., vol. 48, no. 3, 2012.
Sensorless methods:I M. Hinkkanen, L. Harnefors, J. Luomi, “Reduced-order flux observers with
stator-resistance adaptation for speed-sensorless induction motor drives,”IEEE Trans. Pow. Electron., vol. 25, no. 5, 2010.
I Z. Qu, M. Hinkkanen, L. Harnefors, “Gain scheduling of a full-order observerfor sensorless induction motor drives,” IEEE Trans. Ind. Applicat., vol. 50, no.6, 2014.
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