Lecture 3: Vector-Controlled Induction Motor Drive · Vector Control Methods I Based on the dynamic...

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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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




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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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I Torque

TM =3p2

Im{

isψ∗R

}=

3p2ψRiq


is

d

q

ψR

id

jiq

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I Torque

TM =3p2

Im{

isψ∗R

}=

3p2ψRiq


is

q

ψR

d

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I Torque

TM =3p2

Im{

isψ∗R

}=

3p2ψRiq


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




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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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Lecture 3: Vector-Controlled Induction Motor Drive · Vector Control Methods I Based on the dynamic...

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