Transcript Vector Control of Induction Machines

Vector Control of Induction
Machines
us
dq
ab
q
2
3
IM
3
2
ab
dq
q
is
Introduction
• The traditional way to control the speed of
induction motors is the V/Hz-control
• Low dynamic performance
• In applications like servo drives and rolling
mills quick torque response is required.
• Desire to replace dc drives led to vector
control
• Braunschweig, Leonhard, Blaschke, Hasse,
late 70-ies
What is vector control?
• Vector control implies that an ac motor is
forced to behave dynamically as a dc
motor by the use of feedback control.
• Always consider the stator frequency to
be a variable quantity.
• Think in synchronous coordinates.
Basic blocks of a vector controlled
drive
us
dq
ab
q
2
3
IM
3
2
ab
dq
q
is
Addition of a block for calculation of
the transformation angle
us
dq
ab
2
3
3
IM
qr
q
Transformation
angle
calculation
2
ab
dq
q
is
The current is controlled in the d- and
q-directions
ref
s
i
i
ref
sd
 ji
ref
sq
magnetization
torque production
Vector controller
isref
+
-
Current
controller
us
dq
ab
2
3
3
IM
qr
q
Transformation
angle
calculation
2
ab
dq
q
is
Stator and rotor of an induction machine
Magnetization current from the stator
The flux
The rotation
1
r
View from the rotor
2
Induced voltage and current
e  v  B dl
B
2
v
v
Torque production
F
2
Ampere-turn balance
2
Rotor flux orientation
• Difficult to find the transformation angle since
the direction of the flux must be known
• Flux measurement is required
• Flux sensors (and fitting) are expensive and
unreliable
• Rotor position measurement does not tell the
flux position
• The solution is flux estimation
Rotor flux orientation using measured
flux
Original method suggested by
Blaschke
•Requires flux sensors
•Flux coordinates: aligned with the
rotor flux linkage
  rb 
  arctan 

  ra 
Rotor flux orientation
q
b
d

r
y e
f
a
 j

y 


s
*
s
r
s
r
 s
 y


From Chapter 4
iss R s
Lsl

ims
s
s
u
d ss
 uss  R s  iss
dt
L rl i s
r
R r
Lm

j r  rs
(stator)
d rs
 jr  rs  R r  irs
dt
(rotor)
Transformation to flux coordinates
d
j
f
j
f
j
f
j
 e  j   s  e  us  e  R s  is  e
dt
f
s
d rf j 
 e  j   rf  e j   jr  rf  e j   R r  irf  e j 
dt
d sf
 usf  j1 sf  R s  isf
dt
d
f
f
  j2  r  R r  ir
dt
f
r
2  1  r
The flux coordinate system is ”synchronous” only at steadystate. During transients the speed of the rotor flux and the
stator voltage may differ considerably.
The rotor equation (5.9)
d
f
f
  j2  r  R r  ir
dt
1 f Lm f
f
ir   r 
is
Lr
Lr
f
r
d
R r f Lm R r f
f
  j2  r   r 
is
dt
Lr
Lr
f
r
Split into real and imaginary parts
d
 0
f
rq
f
rq
dt
0
d
R r f Lm R r f
   rd 
isd
dt
Lr
Lr
f
rd
Lm R r f
0  2  
isq
Lr
f
rd
Rotor flux dynamics are slow
Lr
Tr 
Rr
ψ
f
rd0
L i
f
m sd0
Torque control

3 Lm
f *
f
T  p
 Im   r   is
2 Lr
3 Lm
f
f
T  p
 rd  isq
2 Lr
ref
sd
i


ref
r
Lm

Rotor flux orientation using estimated
flux
• The rotor flux vector cannot be measured,
only the airgap flux.
• Flux sensors reduce the reliability
• Flux sensors increase the cost
• Therefore, it is better to estimate the rotor
flux.
The "current model" in the stator
reference frame
(Direct Field Orientation)
d
 jr  rs  R r  irs
dt
1 s Lm s
s
ir   r 
is
Lr
Lr
s
r
s
ˆ
 s Lm s
d r  1
    jr  ˆ r 
is
d t  Tr
Tr

The current model
T ref
 rdref
Current
control
usf
s
ˆ
isf
s
f
f
ˆ
IM
drive
uss
Current
model
r
iss
The "current model" in synchronous
coordinates (Indirect Field Orientation)
f
sq
f
rd
Lm Rr f
Lm i
0  2  
isq  2 

Lr
Tr 
f
rd

f
rd0
L i
f
m sd0

1 isq
2  
Tr isd
Transformation angle
q   1 d t
1 isq
1  r  2   r  
Tr isd
Remarks on indirect field orientation
• Does not directly involve flux estimation
(superscript f dropped)
• Not ”flux coordinates” but ”synchronous
coordinates”
• Since the slip relation is used instead of flux
estimation, the method is called indirect field
orientation
Indirect field orientation based on the
current model
T ref
 rdref
Current
control
usf
IM
drive
uss
sy
s
q
f
s
r
i
slip
relation
q
sy
s
iss
Feedforward rotor flux orientation
ref
sq
ref
sd
1 i
1  r  
Tr i
•Significantly reduced noise in the
transformation angle
•Fast current control is assumed
(ref.value=measured value)
•No state feedback => completely linear
The voltage model
•The current model needs accurate values of
the rotor time constant and rotor speed
•The trend is to remove sensors for cost and
reliability reasons
•Simulate the stator voltage equation instead of
the rotor voltage equation
dˆ
s
s
 us  Rs  is
dt
s
s
 s  Ls  is  Lm  ir
Solve for the rotor current and insert in
 r  Lr  ir  Lm  is
Lr
s
s
s
   s  Ls  is   Lm  is
Lm
s
r
Multiplication by
yields
Lm /Lr
2

Lm s
Lm  s
s
 r   s   Ls 
  is
Lr
Lr 

2
m
L
Ls 
 Lσ
Lr
Solve for

s
r
Lr
s
s
ˆ 
 ˆ s  L  is 
Lm
s
r
Direct field orientation using the
voltage model
T ref
 rdref
Current
control
f
uss
s
ˆ
isf
s
usf
ˆ rdf
f
ˆ
Voltage
model
IM
drive
iss
Stator flux orientation
"Direct self-control" (DSC) schemes first suggested
by Depenbrock, Takahashi, and Noguchi in the
1980s.
dˆ
s
 us
dt
s
s
1 nominal
At low frequencies the current
model can be used together with:
Lm s
s
ˆ 
ˆ r  L  is
Lr
s
s
Field weakening
isdref
Maximum
torque range
 r max
Field weakening range
=> Reduced torque
 rref
Current control
i
u
s
+
s
R
L
+
e
s
s
di
s
s
s
L
 u  Ri e
dt
d
jq
jq
jq
jq
L i  e   u  e  R  i  e  e  e
dt
 d i jq
jq 
L   e + i  jω  e   u  e jq  R  i  e jq  e  e jq
 dt

di

L  + i  jω   u  R  i  e
 dt

di
L  u   R + j L   i  e
dt
d id
L
 ud  Rid   Liq  ed
dt
L
d iq
dt
 uq  R iq   Lid  eq
Transfer function and block diagram of
a three-phase load
G(s) 
1
 s  j  L  R
e
-
u
+
G(s)
i
Review of methods for current control
• Hysteresis control
• Stator frame PI control
• Synchronous frame PI control
Hysteresis control
(Tolerance band control)
• Measure each line current and subtract from
the reference. The result is fed to a
comparator with hysteresis.
• Pulse width modulation is achieved directly by
the current control
• The switching frequency is chosen by means
of the width of the tolerance band.
• No tuning is required.
• Very quick response
Drawbacks of hysteresis control
• The switching frequency is not constant.
• The actual tolerance band is twice the chosen
one.
• Sometimes a series of fast switchings occur.
• Suitable for analog implementation. Digital
implementation requires a very high sampling
frequency.
Stator frame PI control
• Two controllers: one for the real axis and one
for the imaginary axis
• Cannot achieve zero steady-state error
• Tracking a sinusoid means that steady-state is
never reached in a true sense
• Integral action is useless except at zero
frequency
Synchronous frame PI control
• In a synchronous reference frame the current
is a dc quantity at steady-state.
• Zero steay state error is possible.
• Coordinate transformations necessary
• Easily implemented on a DSP
• Usually the best choice!
Design of synchronous frame PI
controllers
di
L  u   R + j L   i  e
dt
Remove cross-coupling
u  u  j L i
di
L  u  R  i  e
dt
1
G( s ) 
sLR
e
iref
+
F (s)
-
u
G ( s )
+
G(s)
+
jL
i
Desired closed-loop system
Gc ( s ) 
a
s a
ki
F (s)  k p 
s
a tr  ln(9)
F ( s )G ( s )
Gc ( s ) 
1  F ( s )G ( s )
Choice of controller parameters
F (s) 
a
s
 G ( s )  
-1
a
s
 s L  R   a L+
kp  a L
ki  a R
aR
s
Speed control
• Applications: pumps and fans in the process
industry, paper and steel mills, robotics and
packaging, electric vehicles
• Very different dynamic requirements
• Most drives have low to medium high
requirements on dynamics. These drives are
considered here.
• Cascade control is sufficient
Block diagram of a speed-controlled
drive system
Inverter

mref


Speed
controller

i
ref
 I
i
Current
controller u ref
Electric
motor
m
The mechanical system
d m
Te  Tl  J
 b  m
dt
 Tl
Te 
1/  J s  b
m
The speed controller
• The task of the speed controller is to provide a
reference value for the torque (or current)
which makes the mechanical system respond
to the speed reference with a specified rise
time.
a  tr  ln(9)
Block diagram with speed controller
Speed
controller


ref
m
 
F  cT
Inner loop
ref
sq
i
1/ cT
 Tl
Te 
1
cT 
3 Lm ref
p   rd
2
Lr
1/  J s  b
m

ref
m
1
 m   F 
 m
J sb
1
F 
 Go
J sb
Go
a
Gc 

Go  1 s  a
Go 
a
s
Choice of controller parameters
1
a
F 

J sb s
F 
a
s
 J s  b   a  J 
P
a  b
s
I
Realistic choice of bandwidth
• Care must be taken that the bandwidth of the
speed controller is not unnecessarily high.
• In fact this should be decided during the first
steps in the design process of a drive system
• The bandwidth is directly connected to the
current rating of the inverter.
A change in the speed reference
iqp    a  J
How large steps should be foreseen?
 max  Cmax  mbase
With
iqp  I nom
and
   max
I nom  Cmax  mbase  a  J
I nom
a 
Cmax  mbase  J
Check if the current controller
is sufficiently fast.