tema 1. introducción y modelado de máquinas eléctricas

1/34

UPC

PMSM Control (FOC & DTC) for WECS & Drives

• Introduction. Classification. Motor Types• Permanent Magnet Synchronous Machine (PMSM) Modeling• Control Strategies

–Field Oriented Control (FOC)•Current Loop•Speed Loop•PI tuning

–Direct Torque Control (DTC)•Flux Loop•Torque Loop•Hysteresis Comparators

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UPCClassification

• Types of electrical machines– DC – Universal – AC

• Single-phase AC Induction Motors• Single-phase AC Synchronous Motors• Three-phase AC Induction Motors• Three-phase AC Synchronous Motors

– Stepper– Permanent Magnet

• PMDC – Brushless • PMAC – SMPM, IPM

– Linear– Nano

Most important and used

3/34

UPCPermanent Magnet Machines

• Magnetic material to establish the rotor flux.• Most common magnetic material are samarium-cobalt (SmCo) and

neodymium-iron-boron (NdFeB) introduced in 1983 having superior magnetic characteristics at room temperature.

• Advantages:• No rotor currents => no rotor losses. • Higher efficiency => energy saving capability. Attractive for Wind

applications.• Smaller rotor diameters, higher power density and lower rotor

inertia.• Higher torque per ampere constant.• Weight and volume less than other type of machine for the same

power. Attractive for aerospace applications such as aircraft actuators.

• Other applications: machine tools, position servomotors (replacing the DC motors).

• Inconvenience: • synchronous machines => need for rotor position.• Price

4/34

UPCPM Motor Types

• Considering the shape of the back EMF• PMDC – brushless - trapezoidal• PMAC – sinusoidal

– Surface Mount PM– Interior Mount PM

SMPM IPM

5/34

UPCPMSM Dynamic Equations

• Space vector transformation– combines the individual phase quantities in to a single

vector in the complex plane

– Similar transformation are applied to• Stator voltages • Stator flux linkage

⎟⎟⎠

⎞⎜⎜⎝

⎛++= 3

43

2

)()()(32 ππ

αβ

j

c

j

ba etietitii

( ))()(23;)( titiitii cba −== βα

svs

ψ

a

b

c

i

αβ

ib

ic

ψb

ψc

ψa

β

α

i

ava +-

vc+

-

-

+vb

6/34


• Basic equation for phase windings voltages

• Total flux linkage

flux produced by the rotor magnet

Leakage inductance

Magnetising inductance

Self inductance

Mutual inductance

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

c

b

a

c

b

a

s

c

b

a

dtd

iii

rvvv

ψψψ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

)cos()cos(

)cos(

343

2

π

π

θθθ

r

r

r

m

c

b

a

ccbca

bcbba

acaba

c

b

a

ψiii

LMMMLMMML

ψψψ

mψ

lLmL

mlcba LLLLL +===

2m

cabcabLMMM −=== baab MM; = cbbc MM; = acca MM; =

a

b

c

i

ib

ic

ψb

ψc

ψa

ava +-

vc+

-

-

+vb

ψm

θ r

7/34


• voltage vector equation in the stationary α-β frame • Replacing the inductances values and applying the space vector transformation

⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡)cos(

)cos(23

2π

β

α

β

α

β

α

θθ

r

rmmls dt

dψii

dtdLL

ii

rvv

i

ψ

v +-

ψm

θ r

αα

α

iψ

v

+

-

β

ββ

a

b

c

i

ib

ic

ψb

ψc

ψa

ava +-

vc+

-

-

+vb

ψm

θ r

Clarketransformation

8/34


• Saliency– Variation of the stator phase inductance as function of the rotor position.

– For example

)2cos(

)2cos(

)2cos(

32

34

π

π

θΔ

θΔ

θΔ

−−+=

−−+=

−+=

rmmla

rmmlb

rmmla

LLLL

LLLL

LLLL

)2cos(2

)2cos(2

)2cos(2

34

32

π

π

θΔ

θΔ

θΔ

−−−=

−−=

−−−=

rmm

ca

rmm

bc

rmm

ab

LLM

LLM

LLM

β

α

ψm

d

q

mml

rmmla

LLL

)θ2cos(LLLL

Δ-+=

=Δ-+=

β

α

ψm

θ r

dq

ml

rmmla

LL

)θ2cos(LLLL

+=

=Δ-+=

β

α

ψm

θ r

d

q

mml

rmmla

LLL

)θ2cos(LLLL

Δ++=

=Δ-+=

9/34


• voltage vector equation in the stationary α-β frame considering saliency

• Where

• If there is no saliency and it is obtained the previous equation.

⎥⎦

⎤⎢⎣

⎡−

+

⎥⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡+

−+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

)cos()cos(

)2cos()2sin()2sin()2cos(

2π

β

α

β

α

β

α

θθ

θΔθΔθΔθΔ

r

rm

rssrs

rsrsss

dtdψ

ii

LLLLLL

dtd

ii

rvv

ms

mls

LL

LLL

ΔΔ23

23

=

+=

0=mLΔ

10/34


• voltage vector equation in the synchronous reference d/q frame fixed on the rotor – Angle chosen equal to the PM position

differential operator ; direct axis and quadrature axis inductancespssd LLL Δ= - ssq LLL Δ+=

ψm

θ r

α

d

β

id+

-vd

ψ

vq

+

-

iq

qψ

Parktransformation

i

ψ

v +-

ψm

θ r

αα

α

iψ

v

+

-

β

ββ

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡ −+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡10

rmq

d

qrd

rqd

q

ds

q

d ψii

pLLLpL

ii

rvv

ωω

ω

11/34


• expression for the instantaneous torque for the PM synchronous machine

• first term, usually called as magnet torque, is directly proportional to and independent of .

• second term, or reluctance torque, is only present in salient machines where and is proportional to the current product .

– Motion equation: • J rotor inertia • D friction

{ })(2

3qdqdqme LLiiiψ PT −+=

qi

di

0≠− qd LLqdii

12/34

UPCTypes of Controllers

13/34

UPC

14/34

UPCField Oriented Control (FOC) for PMSM

• Instantaneous torque for the PM synchronous machine

• id is kept to zero, for not demagnetizing the PM machine. Therefore, the reluctance torque will be zero.

• Electromagnetic torque will be regulated with iq .

{ })(2

3qdqdqme LLiiiψ PT −+=

β

α

ψm

θ r

dq

Iq>0

β

α

ψm

θ r

dq

Iq<0

Te > 0 Te < 0

15/34

UPCFOC in PMSM

Four quadrant Motor (Q1 & Q3) Generator (Q2 & Q4)

16/34

UPCFOC Scheme

• 3 PI control loops– 2 identical inner (and faster) current loops, d & q axis.– 1 outer (and slower) speed loop.

• Typical dynamic values (for a PMSM 3.8kW)– Current PI loop band width: 100 µs – 10kHz– Speed PI loop band width: 5ms – 200Hz

17/34

UPCCurrent PI Controller loop

• Inner faster loop.• D and Q current loops are closed by identical PI.• From PMSM model

• If d-q coupling terms are eliminated a first order system is obtained.

s

dq

s

dq

s

sdqdq

dq

RL

sRL

RRsLsv

si=

+=

+= τ;

1

11

)()(

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡ −+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡10

emq

d

qed

eqd

q

ds

q

d ψii

pLLLpL

ii

Rvv

ωω

ω

18/34


dqs

dqs

dqs

dq

t

sdq

vRtit

vRtit

vRtit

RLveRti

⋅==

⋅==

⋅==

=−=−

993.01)(;5

95.01)(;3

632.01)(;

;)1(1)(

τ

τ

τ

ττ

Model: 142UMC30

Number of poles: 6

Rated speed: 3000 (rpm)

Rated torque: 12.2 (Nm)

Rated power 3.82 (kW)

Kt: 1.6 (Nm/Arms)

Ke: 98.0 (Vrms/krpm):

Inertia: 20.5 (kgcm²)

R (ph-ph): 0.94 (Ohms)

L (ph-ph): 8.3 (mH)

Continuous stall: 15.3 (Nm)

Peak: 45.9 (Nm)

Manufacturer’s data for the PM machine from Control Techniques under the commercial name UNIMOTOR.

sdqdq

dq

rsL1

)s(v)s(i

+=

Step input time open loop response

)(83.847.01015.4

1·47.01015.4

47.01

47.01015.41

)()(

3

33

ms

sssvsi

dq

dq

=⋅=

+⋅=

+⋅=

−

−−

τ

19/34


• Root locus and step response with a P controller

• Slow dynamics• E0. Position Error

-400 -350 -300 -250 -200 -150 -100 -50 0-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Root Locus Editor (C)

Real Axis

Imag

Axi

s

Step Response

Time (sec)

Am

plitu

de

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

dqs

Lr-s =

1=PK

• Solution: add a PI controller

ps

s

s

p KRR

RKE

+=

+=

1

10

20/34


-1600 -1400 -1200 -1000 -800 -600 -400 -200 0-800

-600

-400

-200

0

200

400

600


Real Axis

Imag

Axi

s

Step Response

Time (sec)

Am

plitu

de

0 1 2 3 4 5 6 7 8x 10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Because of the KI E0=02nd order system and response

800-KK5,5;pK

p

I ==

PI

KKszero

spole-:0:

=

=

s

sKK

Ks

KsKs

KKsPI I

P

IIPI

P

1)(

+=

+=+=

Damping factor equal to 0,707 The overshoot is too large !

21/34


2nd order system and response depending on the damping factor value

http://upload.wikimedia.org/wikipedia/commons/9/94/2nd_Order_Damping_Ratios.svg

22/34


• Second order transfer function:

• The closed loop transfer function gives an unwanted zero

• A pre filter is used to get rid of the unwanted zero

22

2

2 nn

n

ss ωξωω

+⋅+

( )

LK

LK

LRss

sKK

LK

GHGsW

IPs

I

PI

+⎟⎠⎞

⎜⎝⎛ ++

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=+

=2

1

1

23/34


⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⋅⎥⎦

⎤⎢⎣

⎡ −+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡10

emq

d

qed

eqd

q

ds

q

d ψii

pLLLpL

ii

Rvv

ωω

ω• Feed forward crossed terms

id*

Lq·we·iq

vd*

iq*

Ld·we·id + PM·we

vq*

24/34


• Implementing PI in a DSP– From S to Z domain

– Ts=100us

Step Response

Time (sec)

Am

plitu

de

0 1 2 3 4 5 6 7 8

x 10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1-z)Ts-1(-z

K)z(PIs

sK)s(PI P

IP

IK

K

P

KK

P =→+

=

1-z,920-z

5,5)z(PIs800s

5,5)s(PI =→+

=

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


Real Axis

Imag

Axi

s

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Root Locus Editor (C)

Real Axis

Imag

Axi

s

25/34


// start iq PI controlleriq_error = iq_ref-iq;

vq_ref = vq_ref_last+c_Kp*(iq_error-c_Ki_Kp*iq_error_last);iq_error_last = iq_error;

if (vq_ref > VPI_MAX) vq_ref = VPI_MAX;if (vq_ref < -VPI_MAX) vq_ref = -VPI_MAX;vq_ref_last = vq_ref;// end iq PI controller

• Implementing a PI Controller in a DSP– From Z to discrete time domain

– C code for the TI DSP 6711

1-z)Ts-1(-z

K)z(error_iq

)z(ref_vq)z(PI P

IK

K

P==

))]Ts-1(-z(K)[z(error_iq)1-z)(z(ref_vq PI

KK

P=

))]K_K_c(z-1(K)[z(error_iq)z-1)(z(ref_vq pi-1

P-1 =

))K_K_c(last_error_iq-error_iq(Klast_ref_vqref_vq piP+=

26/34

UPCSpeed PI Controller loop

• The plant can be simplified as follows• First order with one pole • Mechanical time constant might be 50 times slower than the

electrical one. Current loop is neglected.• Typical sampling time 5ms.

JD-s =

PITe*

DsJ1

e* TL

27/34

UPCSpeed PI Controller loop

s1

• Windup phenomenon– Limits of the real plant (currents and voltages)– Might end up with instability

• Anti Windup PI

28/34

UPC

29/34

UPC

• Direct Torque Control (DTC) was firstly introduced by Takahashi in 1986 [Ref 1]. It was an important new method becoming very popular.

• Depenbrock in 1988 [Ref 2] introduced a similar idea under the name Direct Self Control.

• However, just ABB company has got a popular commercial drive based on DTC, the ASC600.– Great number of applications such as: pumps, conveyors, lifts…from 2.2kW

until 630kW.– Thanks to its 40MHz Toshiba processor plus ASIC, ASC600/800 closes the

entire control loop every 25us.

Takahashi, I and Nogushi, T. “A New Quick-Response and High-Efficiency Control Strategy of an Induction Motor”, IEEE Trans. Industry Appl., Vol. 1A-22, pages 820-827, October 1986.

Depenbrock, M. "Direct Self Control of Inverter-Fed Induction Machines". IEEE Trans. on Power Electronics. vol: PE-3, no:4 October 1988. pp: 420-429.

Direct Torque Control

30/34

UPC

Classical DTC schematic

Features– Direct torque and stator flux control– Indirect control of stator currents and

voltages– Approximately sinusoidal stator

fluxes and currents

TableTorque

ref

Flux ref +-

+- VSI

stator fluxtorque

estimators

flux sector wm

DT

inductionmotor

Advantages– Absence of co-ordinate transform.– Absence of voltage modulator block, as well

as other controllers such as PID for flux and torque.

– Fast torque response.– Robustness for parameters variation.

31/34

UPC

Tangential component - torque Normal component - modulus flux

v1(100)

v4(011)

v3(010) v2(110)

v5(001) v6(101)

v3(FD,TI)

v2 (FI,TI)

1

65

4

32

v6(FI,TD)v5(FD,TD)

tg

n

DTC Principles

( )rss'r2

mrs

me sin

LLLLP

23t ρ−ρ⋅ψ×ψ

−=

Φ τ S1

TI V2

T= V0FI

TD V6

TI V3

T= V7FD

TD V5

tu ss Δ=Δψ

32/34

UPC

Classical Look up Table

Φ τ S1 S2 S3 S4 S5 S6

TI V2 V3 V4 V5 V6 V1

T= V0 V7 V0 V7 V0 V7FI

TD V6 V1 V2 V3 V4 V5

TI V3 V4 V5 V6 V1 V2

T= V7 V0 V7 V0 V7 V0FD

TD V5 V6 V1 V2 V3 V4

33/34

UPCConclusions

• Permanent Magnet Synchronous Machines– Advantages:

• No rotor currents => no rotor losses. • Higher efficiency => energy saving capability. • Smaller rotor diameters, higher power density and lower rotor

inertia.• Attractive for Wind applications, Aerospace applications such

as aircraft actuators, Machine tools, position servomotors (replacing the DC motors).

– Inconvenience: • Synchronous machines => need for rotor position. • Price

34/34

UPCConclusions

• PMSM dynamic Modelling• FOC for PMSM has been introduced

– Principles– Scheme– PI Controller design

• Classical DTC – Principles & features

– Scheme

• Research: Sensorless Control of PMSM

tema 1. introducción y modelado de máquinas eléctricas

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