ITEC-AP.2017.8080898

2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific)
A Comparison of Discrete-time Complex Vector
Current Regulators at Low Frequency Ratio
Zhen Dong
Yong Yu
School of Electrical Engineering and Automation
Harbin Institute of Technology
School of Electrical Engineering and Automation
Harbin Institute of Technology
Harbin, China
[email protected]
[email protected]
Harbin, China
Abstract?This paper focuses on the performance and
stability of discrete-time domain complex vector PI (CVPI)
current regulator operating at low carrier-wave frequency ratio.
A novel and accurate discrete-time induction motor model using
Pole-zero Matching method is proposed with consideration of
control delay and PWM delay. On the foundation of this, an
explicit comparison of discrete-time CVPI using Front-Euler
method, Back-Euler method, Tustin method and Direct Design
method is made to investigate the limit frequency ratio for
different discrete-time CVPI topologies. The simulation results
are provided with the same conditions except the adapted current
regulator for a relatively objective comparison.
Keywords?discrete-time domain; Pole-zero Matching Method;
complex vector current regulator ; induction motor
design of current regulator [9]. In this paper, an accurate
discrete-time induction motor model based on Pole-zero
Matching method is proposed. On the foundation of this, the
performance and stability of different discrete-time CVPI
current regulators are compared and analyzed in the condition
of low carrier-wave frequency ratio.
II. DISCRETE CVPI CURRENT REGULATOR DESIGN
A. CVPI Current Regulator Design
Considering that the Back_EMF is a relative tardy variant,
if the Back_EMF is compensated well with negligible effect on
current-loop, the IM model could be simplified as RL load in
stationary reference frame. And the transfer function is (1).
I. INTRODUCTION
Since the current loop, the inner loop, is the important part
for the performance of the vector control system, different
kinds of current loop topologies have been proposed in last
decades, like hysteresis-band control(HBC) [1], synchronous
reference frame PI(SFPI), feed-forward PI(FFPI) [2],feedback
PI(FBPI) and predict control[3]. Some of them with simple
topologies are wildly used by industrial applications but could
not realize totally decoupling of d_axis and q_axis current [2],
some of them with high control performance but the algorithm
is much more complicated [3][4]. Meanwhile, complex vector
PI (CVPI) current regulator proposed on the basis of SFPI has
been deeply discussed because of its comparatively feasible
structure and superior performance especially at low carrierwave frequency ratio or high speed terms [5-9].
In terms of current regulator discretization, generally the
Back-Euler method and the Tustin method could satisfy most
of the discrete requirement. However, since the switching
frequency is severely limited in some specially applications
like high-power or high-speed occasions because of the
intrinsic frequency of gate turn-off thyristors [8]. In order to
solve the problem caused by low frequency ratio, the
compensation of PWM delay and control delay caused by
frame rotation is considered [10]. Active damping is used to
eliminate the current oscillatory [8]. An improved Back-Euler
method is proposed in senseless control for a tradeoff between
the high-speed requirements and observer complexity [11].
Also, the step respond method is applied for direct discrete
G ps ( s ) =
(1)
Where: R =Rs + L2m L2r Rr , L=? Ls , Rs Stator resistance,
Rr Rotor resistance, Lm Mutual inductance, Lr Rotor selfinductance, Ls Stator self-inductance, ? Total leakage factor
( ? = 1 ? L2m Ls Lr ).
The transform between stationary reference frame and
synchronous reference frame is expressed as (2)(3):
f dqs = f dqe e j?et
j?e t
(2)
j?e t
sf = s( f e ) = [( s + j?e ) f ]e
(3)
Thus, the transfer function in synchronous frame is (4):
s
dq
e
dq
e
dq
i e ( s)
1L
=
(4)
e
u ( s ) s + j?e + R L
It is known that the current loop could be equivalent as a
first-order inertia loop, thus, the current loop is (5)
G pe ( s ) =
e
GCC
( s) =
i e ( s)
?cb?
=
i
( s ) s + ?cb?
e _ ref
(5)
Where ?cb? the expected bandwidth.
Project 51377032 supported by National Natural Science Foundation of
China; National science and technology support program, 2014BAF08B05.
978-1-5386-2894-2/17/$31.00 �17 IEEE
978-1-5386-2894-2/17/$31.00 �17 IEEE
i s ( s)
1L
=
s
u (s) s + R L
e
e
GCO
( s ) = GCR
( s ) * G pe =
?cb?
s
(6)
2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific)
TABLE I.
Front-Euler
Back-Euler
z ?1
Ts
s=
DIESCRET METHODS
s=
z ?1
zTs
? (k )
Ts
0.5Ts
Tustin
s=
i (k )
u (k )
2( z ? 1)
Ts ( z + 1)
? ( k + 1)
i( k +1)
u ( k + 1)
u (k )
output
is ,ref +
Kp
?
is
K iTs
j?eTs
+
+
+
1
z ?1 +
us
Sampling and
Update Piont
Fig. 1. Block diagram of CVPI using Front-Euler Approximation.
Carrier Wave
And the open loop is (6), hence, the complex vector current
regulator in continuous domain is (7):
e
( s) =
GCR
K p Ki + jK p?e
u( s ) ?cb? s + j?e + R L
*
=K p +
(7)
=
1L
?i ( s )
s
s
Where K p = L?cb? ?Ki = R / L .
The discrete methods commonly used are Front-Euler,
Back-Euler and Tustin Approximation, as shown in Tab. 1.
Hence, we get the CVPI with Front-Euler method
expression (8), and Fig1 is the block diagram:
us ( z ) K p z + K p KiTs + jK p?eTs ? K p
=
(8)
?is ( z )
z ?1
Similarly, we obtain CVPI with Back-Euler method (9) and
Tustin method (10):
F
( z) =
GCR
B
GCR
( z) =
T
GCR
( z) =
=
us ( z ) ( K p + K p Ki Ts + jK p?eTs ) z ? K p
(9)
=
?is ( z )
z ?1
us ( z )
?is ( z )
(2 K p + K p KiTs + jK p?eTs ) z + ( K p Ki Ts + jK p?eTs ? 2 K p )
(10)
2( z ? 1)
Using the Pole-zero cancellation, the expression of direct
discrete-time CVPI current regulation in [9] is (11):
us ( z ) K ( ze j?eTs ? e ? RTs / L )
=
?is ( z )
z ?1
Where K is the coefficient related to the bandwidth.
D
GCR
( z) =
Computation
(11)
The expression (11) is derived from the discrete-time IM
model, and the explicit derivation is in the next part. We can
infer from (11) that since the exponential function is added in
the current regulator structure, theoretically, it should
correspond with that in continuous domain better while the
realized complexity on chip is neglected.
B. Time Delay Analysis in Digital Control System
In digital motor control system, time delay is inevitable
mainly because of program execution, PWM update and PWM
output while the sampling time, turn-on and turn-off delay of
IGBT are short enough to be neglectful. For double-samplingdouble-update, sampling point and PWM update happen in
Ideal PWM Signal
Ideal Modulating Wave
Output PWM Signal
Output Modulating Wave
Fig. 2. Time delay in digital control system with double-sampling-doubleupdate PWM.
is ,ref +
Kp
?
is
K iTs
j?eTs
+
+
+
z +1
+
2( z ? 1)
j?e
1.5 TTsj?
ee0.75
s
e
us
Fig. 3. Block diagram of CVPI using Tustin Method with phase advance
time delay compensation.
each Ts , during which program execution for uoutput is
finished. Then, in the next Ts , PWM output happens when the
output modulating wave intersects with the carrier wave, which
in average needs 0.5Ts . Hence, in sum, there is 1.5Ts delay for
control system. In stationary frame, control delay could be
?T s
equivalent to e ?1.5Ts s ( e p in accurate, if all the time delays are
included.). With Taylor Approximation, always it can be
written as 1?1.5Ts s + 1?. However, in synchronous frame,
? sample is related to both uoutput and isample , if ?e is thought to
be constant during the delay period, there would be an
0.75Ts?e error for frame rotation. Hence, the total time delay
above can be expressed as e ?1.5Ts ( s + j?e ) .Transforming to the
discrete-time domain, it could be written as z ?1e1.5Ts j?e .Thus, in
considering of time delay compensation, e1.5Ts j?e need be added
to the expression from (8) to (10). Meanwhile, since the direct
discrete-time current regulator has taken the PWM delay into
consideration as a zero-order hold, only e0.5Ts j?e is needed for
time delay compensation.
III. PROPOSED DISCRETE-TIME IM MODEL
A. Discrete-time IM Model using Pole-zero Matching Method
In order to analyze the performance of discrete-time
complex vector current regulator theoretically, the accurate
discrete-time IM model under synchronic reference frame is
necessary to be built. The commonly known transition
approaches from continuous-time domain to discrete-time
domain includes Front-Euler, Back-Euler, Tustin transform,
Pulse Response Method, Step Response Method and Pole-zero
Matching Method. Though each approach has own advantages
in the appropriate occasions, none of them could transfer all the
information entirely. Since in the part 2, the complex vector
current regulator is designed using pole-zero cancellation,
2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific)
u
e ? RTs
e
L
+
1
2R
+
ze j?eTs +
?
e
j?eTs
z ?1
1
z ? e ? RTs
0.75
0.5 TTs sj?j?
ee
L
ee
ie
Fig. 4. Block diagram of IM model using Pole-zero Matching Method with time delay.
corresponding to the discrete method, the Pole-zero Matching
Method is most appropriate to ensure the pole-zero
consistency.
Considering of the time delay in part3, the accurate
discrete-time IM model with time delay z -1 ? e-0.5Ts j?e is obtained
(16):
Using Pole-zero Matching Method, which would be
expressed as follows:
i e ( z)
( z ? e j?eTs + 1)(1 ? e ? RTs L )
=
(16)
e
u ( z ) (2 R ? e j?eTs z ? 2 R ? e ? RTs L ) z ? e0.5Ts j?e
From (16), it is clear to see that the IM property is mainly
caused by varying poles related to ?e .
G( s) =
k ? ( s + zi )
sT
z =e
???
? G( z ) =
m
? (s + p )
k1 ? ( z -e ? ziT )
m
? ( z-e
j
n
? p jT
)
G pe _ td ( z ) =
( z + 1)n ? m
n
B. Comparison and Analysis of CVPI Current Regulator
In part2, CVPI with four different kinds of discrete
structure has been given without considering of time delay
compensation. In this part, both the discrete precision and the
time delay compensation would be taken into comparison.
(12)
Where zi : zero, p j : pole, m, n are the order of numerator
and denominator.
If m < n , it indicates that there is zero at s = ? ,
corresponding to z = ?1 .
And the gain k1 is derived by G ( s )
s =0
= G( z)
z =1
Fig.5 gives the eigenvalue migration with varying
frequency from 0Hz to 300Hz. In part II, we know that in
continuous domain, the excellent pole/zero cancellation can be
achieved with the CVPI structure. However, the zeros using
Front-Euler Approximation achieve the proper cancellation
under the low frequency. As the frequency going up, it
becomes improper, which means that the regulator is disabled.
Similarly, as shown in Fig.6, though the regulator zeros tend to
track the varying close-loop poles, the cancellation result is still
not proper especially in high frequency situation. However, it is
much better by using BE Approximation compared with the FE
Approximation. Therefore, to tradeoff the algorithm validity
.
The expression (1) transformed as discrete?time IM model
in stationary reference frame can be written in (13):
k1
( z + 1)
z ? e ? RTs L
G ps ( z ) =
Where k1 =
1 ? e ? RTs
2KR
(13)
L
.
2 Ri s ( k + 1) ? e ? j? ( k +1) ? e j?eTs ? u s (k + 1)(1 ? e ? RTs L ) ? e ? j? ( k +1) ? e j?eTs
Pole-Zero Map
= 2 K R e ? RTs Li s ( k ) ? e ? j? ( k ) + u s (k )(1 ? e ? RTs L ) ? e ? j? ( k )
1
0.6?/T
0.8
(14)
Then, the resulting equation can be transformed into the
difference equation in the synchronous frame (14), and finally
the discrete-time domain transfer function in synchronous
frame is obtained (15):
?
?
? j ? ( k )+?e ( k ) Ts ]
Where f dqe (k ) = f??s (k )e ? j? ( k ) , e ? j? ( k +1) = e [
0.5?/T
0.5
0.4?/T
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.1?/T
0.8
0.9
0.7?/T
0.6
0.8?/T
Imaginary Axis
0.4
0.2
0
-0.2
0.9?/T
0
1?/T
1?/T
0.9?/T
0.1?/T
-0.5
-0.4
0.8?/T
-0.6
0.2?/T
0.7?/T
-0.8
0.3?/T
0.6?/T
-1
-1
-0.8
-0.6
-0.4
-0.2
0.5?/T
0
-1
0.4?/T
0.2
0.4
0.6
0.8
1
0.6
0.7
0.8
0.9
1
1.1
Real Axis
G pe ( z ) =
j?eTs
? RTs L
e
? RTs L
i ( z ) z (1 ? e
)?e
+ (1 ? e
=
ue ( z)
2 R ? e j?eTs z ? 2 R ? e ? RTs L
)
Fig. 5. Eigenvalue migration of current close-loop with FE CVPI.
(15)
Pole-Zero Map
Pole-Zero Map
0.6?/T
0.8
0.5?/T
0.8?/T
Imaginary Axis
0.4
0.2
0
-0.2
0.9?/T
0.1?/T
-0.6
0.8?/T
-0.8
-0.6
-0.4
-0.2
0.5?/T
0
0
-0.2
0.9?/T
1?/T
1?/T
0.9?/T
0.1?/T
0.8?/T
0.2?/T
0.7?/T
-0.8
0.4?/T
0.2
0.2
-0.6
-0.8
0.3?/T
0.6?/T
0.6
0.8
1
0.3?/T
0.6?/T
-1
0.4
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.1?/T
0.8
0.9
0.8?/T
-0.4
0.2?/T
0.7?/T
0.4?/T
0.4
-0.4
0.9?/T
0.5?/T
0.7?/T
0.6
-0.2
1?/T
1?/T
-0.8
-1
-1
0.8
0
-0.4
-0.6
0.6?/T
0.2
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.8
0.1?/T
0.9
0.7?/T
0.6
1
0.4?/T
Imaginary Axis
1
0.6
0.7
0.8
0.9
1
-1
-1
-0.8
Real Axis
(a)
Fig. 6. Eigenvalue migration of current close-loop with BE CVPI including time delay compensation
-0.6
-0.4
-0.2
0.4?/T
0.5?/T
0
0.2
Real Axis
(b)
0.4
0.6
0.8
1
1.2
2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific)
Pole-Zero Map
0.5?/T
0.6?/T
0.8
0.6
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.8
0.1?/T
0.9
0.8?/T
Imaginary Axis
0.4
0.9?/T
0.2
Pole-Zero Map
-0.2
0.6?/T
0.8
0.2
-0.4
0.1?/T
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.1?/T
0.8
0.9
0.8?/T
0.9?/T
0.2
1?/T
1?/T
0
-0.2
0.9?/T
0.1?/T
-0.4
-0.4
0.8?/T
-0.6
0.7?/T
-0.8
-0.6
-0.4
0.5?/T
0
-0.2
-0.8
0.2?/T
0.7?/T
-0.8
0.3?/T
0.6?/T
0.4?/T
0.2
0.8?/T
-0.6
0.3?/T
0.6?/T
-1
-1
-0.6
0.2?/T
-0.8
0.4
0.6
0.8
1
0.6
0.8
-1
-1
1
-0.8
-0.6
-0.4
-0.2
Pole-Zero Map
0.6?/T
0.8
0.5?/T
0.6
0.8?/T
0
-0.2
0.9?/T
0.8
0.2
0.9?/T
-0.4
0.1?/T
-0.6
0.8?/T
-1
-1
-0.8
-0.6
-0.4
-0.2
0.5?/T
0
0.4?/T
0.6
0.8
1
0.2
0.4
0.6
0.8
0.4?/T
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.8
0.1?/T
0.9
0.8?/T
0
-0.2
0.9?/T
1?/T
1?/T
0.9?/T
0.1?/T
0.8?/T
0.2?/T
0.7?/T
-0.8
-1
Real Axis
0.2
-0.6
-0.8
0.3?/T
0.6?/T
0.4
-0.4
0.2?/T
0.7?/T
0.5?/T
0.4
0
1?/T
1?/T
-0.8
0.2
0.7?/T
0.6
-0.2
-0.4
-0.6
0.6?/T
0.4
0.1
0.3?/T
0.2
0.3
0.4
0.2?/T
0.5
0.6
0.7
0.8
0.1?/T
0.9
0.4
0.2
0.4?/T
Pole-Zero Map
1
0.4?/T
0.7?/T
Imaginary Axis
1
0.5?/T
0
Real Axis
Real Axis
Imaginary Axis
0.4?/T
0.4
-0.2
0.9?/T
0.5?/T
0.7?/T
0.6
0
1?/T
1?/T
0
1
0.4
0.4?/T
0.7?/T
Imaginary Axis
1
0.3?/T
0.6?/T
1
0.7
0.8
0.9
-1
-1
1
-0.8
-0.6
-0.4
-0.2
0.5?/T
0
0.4?/T
0.2
0.4
0.6
0.8
1
Real Axis
Fig. 7. Eigenvalue migration of current close-loop with Tustin CVPI and PZM CVPI including time delay compensation.
and complexity, BE Approximation is the common choice for
most under based speed occasion. Meanwhile, of another
particular interest is the poles in the dashed box, which would
migrate to or out of the unit circle as the synchronous
frequency increases. The system will become oscillated
without any compensation. Considering of the delay
compensation, phase advance method is used to compensate
the e ?1.5Ts j?e . Comparing with Fig.6 (a), the unstable poles can
be well cancelled in Fig6 (b). Hence, the whole system will be
in stability without any effects of the increasing synchronous
frequency.
Comparing the two latter methods in Fig.7: Tustin
Approximation and Direct Discrete-time Method. It is obvious
that both would satisfy most of the high-speed application
because of the excellent pole/zero cancellation. From the
enlarged part, nuanced matching still could be found. Thereby,
Direct Discrete-time Method is more precious than Tustin
Approximation especially when the varying frequency up to
200Hz. And the phase advance time delay compensation is
needed yet.
frequency is set to 400Hz(the frequency ratio is 8). Fig.9 shows
that BE Method tends to be unstable, which means that with a
TABLE II.
PARAMETERS OF THE INDUCTION MOTOR
Parameters
Value
Rated power Pe/kW
3.7
Rated voltage Ue/V
380
Rated current Ie/A
Rated frequency fe/Hz
8.9
2
Stator resistance Rs/?
1.142
Rotor resistance Rr/?
0.825
Mutual inductance Lm/mH
118.9
Stator, rotor inductance Ls,Lr/mH
124.4
Frequency Ratio 12
10
BE
DQ_axis Current
?A?
A given commend step speed from 0 to 50Hz, the
simulation results are in Fig.8, the first platform (0s-0.4s)
acceleration period, and the q-axis current is limited, and the
second platform (0.5s-0.8s) is the loading period. The
switching frequency is set to 600Hz(the frequency ratio is 12).
And the BF CVPI, Tustin CVPI and Pole-zero Matching are
included in terms of the operation stability. It can be clear seen
that all of them have a nice performance. While the switching
1440
Pole pair pn
IV. SIMULATION RESULTS
The simulation is based on the MATLAB Simulink
circumstance. The IM model is built by Simulink modules and
the control system is compiled by M files. The parameters
needed in simulation are in Tab.2.
50
Rated velocity ne/(r/min)
0
-10
10
Tustin
0
-10
10
Pole-zero
Matching 0
-10
0
0.5
isd ,ref
Time (s)
isq ,ref
1.0
isd
isq
Fig. 8. Performance of BE, Tustin and Pole-zero Matching under the
frequency ratio of 12.
2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific)
these above objectively, a novel and accurate discrete-time
inductive motor model using Pole-zero Matching Method is
proposed with consideration of control delay and PWM delay.
The eigenvalue migration is applied for theoretical analysis and
the simulation results prove the validity of the Pole-zero
Matching Method especially under the low frequency ratio.
Frequency Ratio 8
10
DQ_axis Current
?A?
BE
0
-10
10
Tustin
0
-10
Pole-zero
Matching
REFERENCES
10
[1]
0
-10
0
0.5
1.0
Time (s)
isd ,ref
isq ,ref
isd
isq
Fig. 9. Performance of BE, Tustin and Pole-zero Matching under the
frequency ratio of 8.
Frequency Ratio 6
50A
10
DQ_axis Current
?A?
BE
0
0.46s
-10
0.52s
50A
10
Tustin
0
0.46s
-10
0.52s
15A
10
Pole-zero
0
Matching
-10
0
0.5
isd ,ref
Time (s)
isq,ref
1.0
-10A
0.8s
isd
0.84s
isq
Fig. 10. Performance of BE, Tustin and Pole-zero Matching under the
frequency ratio of 6.
sudden reference, the response need an oscillated period to
follow the track. Meanwhile, the latter methods are still in
stability. Finally, the switching frequency is set to 300Hz (the
frequency ratio is 6). Both the BE Method and the Tustin
Method are out of control because of the too long control
period. However, the Pole-zero Matching is still effective
though the control performance is degraded in some certain.
V. CONCLUSION
This paper focuses on the performance and stability of
different discrete methods including FE Approximation, BE
Approximation, Tustin Approximation and Pole-zero Matching
Method based on CVPI structure. In order to compare all of
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