tema 1. introducción y modelado de máquinas eléctricas
TRANSCRIPT
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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
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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
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UPCPM Motor Types
• Considering the shape of the back EMF• PMDC – brushless - trapezoidal• PMAC – sinusoidal
– Surface Mount PM– Interior Mount PM
SMPM IPM
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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
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UPCPMSM Dynamic Equations
• 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
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UPCPMSM Dynamic Equations
• 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
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UPCPMSM Dynamic Equations
• 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-+=
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UPCPMSM Dynamic Equations
• 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
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UPCPMSM Dynamic Equations
• 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
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UPCPMSM Dynamic Equations
• 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
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UPCTypes of Controllers
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UPC
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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
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UPCFOC in PMSM
Four quadrant Motor (Q1 & Q3) Generator (Q2 & Q4)
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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
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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
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UPCCurrent PI Controller loop
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
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UPCCurrent PI Controller loop
• 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
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UPCCurrent PI Controller loop
-1600 -1400 -1200 -1000 -800 -600 -400 -200 0-800
-600
-400
-200
0
200
400
600
800Root Locus Editor (C)
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 !
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UPCCurrent PI Controller loop
2nd order system and response depending on the damping factor value
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UPCCurrent PI Controller loop
• 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
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UPCCurrent PI Controller loop
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡ −+⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡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*
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UPCCurrent PI Controller loop
• 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
1Root Locus Editor (C)
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
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UPCCurrent PI Controller loop
// 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+=
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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
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UPCSpeed PI Controller loop
s1
• Windup phenomenon– Limits of the real plant (currents and voltages)– Might end up with instability
• Anti Windup PI
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UPC
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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
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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.
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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
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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
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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
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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