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Regenerative energy saving in multi-axis servo-motor-drives
Article · September 2011
DOI: 10.1109/ECCE.2011.6064235
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Brian Hadley
Franklin Control Systems
Florida International University
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Regenerative Energy Saving in
Multi-Axis Servo-Motor-Drives
Ali K. Kaviani, Student Member, IEEE, and Brian
Hadley, Student Member, IEEE
Department of Electrical and Computer Engineering,
Florida International University, Miami, FL
Behrooz Mirafzal, Senior Member, IEEE
Department of Electrical and Computer Engineering,
Kansas State University, Manhattan, KS
Abstract- A method for the regenerative energy saving in
multi-axis servo-motor-drives is presented in this paper.
This energy management is achieved through a proper timecoordination between the speed commands of multi-axis
drives. Moreover, the proposed approach significantly limits
the peak value of the ac input current in these systems. It is
mathematically proved that a time-delay can significantly
increase the amount of utilized regenerative energy and
consequently decreases the amount of dissipated energy. In
this paper, a set of closed-form formulas is developed, where
the motor losses are neglected. The findings of this
investigation were experimentally verified using a two-axis
permanent-magnet (PM) motor-drive system, and the results
are presented in this paper. The experimental results are in
remarkable agreement with the developed closed-form
formulas.
I.
INTRODUCTION
The manufacturing industry utilizes different types of
electric motors for various applications and processes.
Some of these motors are either operating in a continuous
duty, where they are at constant speed for extended periods
of time, or they are operated in intermittent duty, where
they accelerate and decelerate (start and stop) a few times
per day. In many automated production and assembly lines
as well as manufacturing processes, such as packaging and
food industries, motors operate intermittently throughout
the day. Multi-axis servo-motor-drives are broadly
employed in parallel (and identical) manufacturing
processes, see Fig.1. In these applications, the servomotors usually run intermittent loads with trapezoidalshaped speed profiles. This means they have to accelerate
and decelerate within each operating cycle [1-4]. These
motor-drive systems typically consist of a diode bridge
rectifier, a dc-bus capacitor bank, and a PWM based
inverter, which controls the motor speed and consequently
the rotor’s position. Induction motors (IM) as well as
permanent magnet (PM) motors can both be utilized in
these systems. However, because of their advantages for
position control applications, PM motors are usually
preferred. Regardless of the type of the motors, the
regenerative energy during the deceleration time is a
common concern in these systems. During the acceleration
period, when the motor speed is increasing, the drive
draws a large amount of power, which results in a voltage
drop at the dc-bus. On the contrary, when the motor speed
is reducing during a deceleration period, the kinetic energy
of the mechanical system, which contains load and rotor
inertia, converts back to an electrical power, known as
regenerative power. A portion of this regenerative energy
is dissipated as heat in the motor and feeder cables, while
the rest is transferred back to the drive’s dc-bus and causes
U.S. Government work not protected by U.S. copyright
a voltage swell [5-6]. This overvoltage can cause failures
in the dc-bus capacitor and/or IGBT module.
Several common solutions to overcome the overvoltage
problem caused by the motor’s deceleration have been
addressed in [5-6]. A group of servo-drive designs
includes a speed regulator, which increases the drive’s
output frequency during a regenerative period to prevent
an overvoltage at the dc-bus. These drives are not capable
of supplying braking torque. Another group of servodrives makes use of resistive brakes to dissipate the excess
energy. Another solution is to use a bi-directional PWMbased module instead of a bridge rectifier. This type of
drive, the so called active-front-end drive, is expensive.
This bidirectional converter allows the regenerative energy
to be directed back to the ac power lines at the proper
voltage, frequency, and phase. Thus, the regenerative
energy will be available to all other loads connected to the
distribution system.
In addition to the overvoltage suppression, utilization of
the regenerative energy, in order to improve the overall
efficiency of the system, is another objective of the studies.
The result of an investigation on optimal torque profiles
for induction motors has been reported in [7], in which an
optimal torque command, to the employed vector control
scheme, guarantees the maximum exploit of regenerative
energy during the motor deceleration period. The
regenerative energy is stored and injected to the motor
during acceleration process. For this purpose, a capacitor
and a bidirectional dc-dc converter are incorporated into
the dc link. In [8], the effect of regenerative energy, from
an induction motor, on the pulse-load capability of a
shipboard power system has been studied. The results
demonstrate that the regenerated energy can be used to
increase the pulse-load capability from 1.49pu to 1.67pu.
Furthermore, proper load management (load coordination)
can increase this value to 2.09pu.
The objective of this study is to obtain an efficient
technique to exploit the regenerative energy, caused by
deceleration of a motor, via optimal coordination of the
time schedule of two (although in general it can be more)
process lines that perform multi tasks in parallel. They
may resemble, for instance, a pair of production or
assembly lines employed in a food or packaging industry,
or some sort of material handling line, see Fig.1. The
presented technique may reduce or eliminate the use of the
above-mentioned line regeneration modules as well as the
bidirectional converters and the storage systems through a
simultaneous use of the regenerated power in multi-axis
motor drive systems. It should be emphasized that the
3450
Unprocessed
products
identical, and all of the system losses, including the energy
losses (in mechanical and electrical systems), as well as
the time delays (in sensing, controlling and switching
processes), are neglected. Meanwhile, the motor loads are
assumed to be inertial loads, and the speed profiles follow
a trapezoidal-shaped command as shown in Fig. 3. In
general, the dc-bus voltage increases when the stored
energy in the motor-load inertia is fed back into the
electrolytic capacitors connected to the common dc-bus
during a regenerative time-interval. In this situation, if the
dc-bus voltage exceeds a certain limit, the resistive brake
dissipates the extra energy in order to maintain the dc-bus
voltage within its nominal range. Also, when the dc-bus
voltage exceeds a certain level, the rectifier’s diodes
become reverse-biased until the dc-bus voltage drops again
and the system begins to absorb energy from the grid.
Motor 1
Line 1
Drive 1
Common
AC bus
dc bus
Grid rectifier
Drive 2
Processed
products
III. OPTIMUM TIME DELAY FOR SYMMETRICAL
TRAPEZOIDAL-SHAPED SPEED COMMANDS
Line 2
Motor 2
Figure 1. Two parallel manufacturing lines with a two-axis drive system
coordinated motors cannot be in the same process line due
to the sequential operation of these processes. This concept
can best be demonstrated in Fig. 1.
It should be noted that this study focuses on trapezoidalshaped speed commands in two-axis servo motor systems.
A trapezoidal-shaped speed profile is a typical speed
trajectory in position control drives. However, the
proposed methodology can be applied to the general
polynomial and trigonometric s-curve motion profiles [4].
Furthermore, the developed method can be extended to
multi-axis drive systems. The procedure for extension of
the proposed methodology to multi-axis drive systems,
with more than two motors, will be discussed in Section
VI. A comprehensive version of this paper, which includes
the formulation of non-symmetrical trapezoidal-shaped
speed commands, has been presented in [10].
Besides the introduction, this paper contains six
additional sections. In Section II, the case study system is
described. In Section III, the optimum time-delay between
two identical-symmetrical trapezoidal-shaped speedcommands of a two-axis drive is mathematically
formulated. In Section III, the formulated time-delays are
verified using experimentally obtained data. Finally,
Section V and Section VI are devoted to the discussion and
conclusion, respectively.
II.
As mentioned in the introduction, the objective of this
research is to utilize the regenerative energy, occurring
during the deceleration periods. One can intuitively think
of a simple time-coordination between two motors, such
that the acceleration period of one of the motors is just
overlapped with the deceleration period of the other one.
However, it will be mathematically proved herein that a
time-delay can significantly increase the amount of utilized
regenerative energy, and consequently decreases the
amount of dissipated energy in the resistive brake.
In any rotational electric machine, the equation of
motion can be written as the following [9]:
(1)
and
are respectively the electromagnetic
where,
and load torques in (N.m),
is the motor inertia in
(kg.m2), is the motor friction factor in (N.m.sec) and
is the mechanical speed of the rotor in (rad/sec). Here, the
load is merely considered as an inertia-load, i.e.
(
⁄ ) and the friction is neglected. Consequently
(1) can be rewritten as follows:
(
where,
)
(2)
is the total load-motor inertia, i.e.
.
Now, let us assume that the speed command profiles of
the two motors intersect, as one can see in Fig. 3. The time
Inverter
SYSTEM DESCRIPTION
As can be seen in Fig. 2, the case study system consists
of a full bridge rectifier, a resistive brake, and two
permanent magnet (PM) synchronous motors fed by two
inverters, which are energized from a common dc-bus.
This configuration is defined herein as a two-axis motordrive system. The resistive brake dissipates the
regenerative energy and maintains the dc-bus voltage
within a reasonable range. The objective is to minimize the
dissipated energy in the resistive brake, which
subsequently leads to a reduction in the amount of the
energy provided from the grid. For simplicity, the two-axis
(including; inverters, motors and loads) are assumed
1(
)
2(
)
PM
Resistive
Brake
Inverter
Rectifier
Common
dc-bus
PM
Figure 2. A two-axis motor-drive system fed through a common dc-bus
3451
( )
∆
( )
( )
1(
)
2
( )
2
( )
0
∆
0
∆
Figure 3. The speed-command profiles of a two-axis motor-drive system
Figure 5. Power and energy flow in the system for 0
difference between the acceleration and deceleration
periods ( and ), ∆ can be either positive or negative,
i.e.
∆
. The two possible cases, 0 ∆
and
∆
0, for symmetrical trapezoidal-shaped
speed commands,
, are investigated in the
following subsections.
deceleration of Motor-1. The hashed area, i.e. , indicates
the recovered energy which is directly transferred from the
decelerating motor to the accelerating one. This energy is
neither stored in the capacitors nor dissipated in the brake,
but travels from the rotating mass of the decelerating
motor-load to the rotating mass of the accelerating set. The
maximization of
, i.e. the direct flow of energy from
Motor-1 to Motor-2, could be considered as a brute
solution to efficiently exploit the regenerative energy in a
two-axis drive system. In general (regardless of any
simplifying assumptions), the optimal solution should
result in minimizing the energy purchased from the grid.
However, since maximizing
results in minimizing the
energy need from the capacitors and the grid, it implicitly
leads to the optimal coordination of the system. In this
particular case, i.e. 0 ∆
, the transferred energy,
, can be obtained from the calculation of the hashed area
shown in Fig. 5 as follows:
Case (I): 0
∆
In Fig. 4, the speed command profiles of the two motors
are depicted during the regeneration period where 0
∆
. During the deceleration period of Motor-1, the
motor speed, ( ), linearly decreases from
at
0
( ) can be expressed as
to zero at
, therefore,
follows:
( )
(
),
0
(3)
It should be noted that the developed equations for the
regenerated and consumed powers have zero values out of
⁄ is equal to
the indicated time intervals, where
( ) and
( ) for a positive ∆
zero. The profiles of
are depicted in Fig. 5. The hashed area, i.e. , represents
the transferred energy from the decelerating motor directly
to the one which is accelerating, neglecting the system
power losses. Three areas, , , and , are denoted in
Fig. 5, where
represents the regenerated
energy during deceleration of Motor-1 and
is the consumed energy by Drive-2 for accelerating Motor2. During this process, the first energy, , will be either
stored in the dc-bus capacitors, which results in an overvoltage, or dissipated in the resistive brake. The second
energy, , will be provided by the capacitors and the grid.
It should be mentioned that the grid contributes only when
the dc-bus voltage drops below a certain level in which the
diodes of the bridge rectifier are forward-biased. For a
lossless system, the amount of energy absorbed from the
grid for the acceleration of Motor-2, is equal to the amount
of dissipated energy in the resistive brake during
(
∆
∆ )
, 0
∆
(8)
This equation, after some mathematical manipulation, can
be simplified to:
(∆ )
(
∆ ) ,
0
∆
(9)
The transferred energy, (∆ ), has a minimum of zero at
⁄2 at ∆
∆
, and a maximum of (0)
0.
∆
Case (II):
0
Fig. 6 shows the same graph as Fig. 5 for a negative
time delay,
∆
0. In this case, the sum of
and
can represent the stored energy in the common dc-bus
capacitors and/or dissipated energy in the resistive brake,
i.e.
, and the sum of energies
and
is
supplied from the capacitors and the grid, i.e.
. In this case,
can be calculated through the
following equation:
(∆ )
( )
( )
(∆ )
∆
∆
∆
(0)
( )
∆
,
0
(10)
Now, (10) can be rewritten as:
(∆ )
0
∆
(
2 ∆
3∆ ),
∆
∆
Figure 4. Speed command profiles of the motors for 0
2
∆
0
(11)
Taking the derivative of the above equation with respect to
∆ , yields:
3452
The same approach as discussed in this section has been
also extended to calculate the optimal time delay for any
non-symmetrical trapezoidal-shaped speed profile. The
method is to maximize the utilization of regenerative
( ) and
( ), and
energy, i.e. overlapping
maximizing the intersection area, see reference [10] for
more details.
( )
( )
( )
2
IV.
0
∆
∆
∆
Figure 6. Power and energy flow in the system for
A test setup, similar to Fig. 2, was developed in the
laboratory to verify the validity of the analytical discussion
and mathematical formulation that are presented in this
paper. The setup consists of a full-wave bridge-diode
rectifier feeding a common dc-bus; two parallel motordrives (inverters) connected to the common dc-bus; two
inertial-loads; a resistive brake; and a dSPACE board
interfacing with MATLAB environment for the timecoordination control purpose.
0
(∆ )
⁄3
2
2
⁄3
∆
0
versus ∆
Figure 7. Variations of
∆
(∆ )
1
3
∆
(12)
⁄3 which
Equating (12) to zero, gives ∆
(
⁄3)
corresponds to a maximum of
2 ⁄3. Moreover, at the boundary points,
and 0; one
) 0. Using (8) and
⁄2 and (
can write (0)
(11), the variation of ET versus the time delay, ∆ , can be
illustrated as shown in Fig.7.
The preceding discussions indicate that, in order to have
maximum exploitation of the regenerative power in a
lossless two-axis drive system with a symmetric
trapezoidal-shaped speed command profile, the
⁄3 timeaccelerating motor should start to accelerate
units before the decelerating motor starts to decelerate.
This finding will be verified through experimental data in
Section IV, whereas system losses are inherently included.
ωref (rpm)
3000
ω1
2000
ω2
1000
P grid (kW)
Vdc (V)
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
Time (sec)
7
8
9
10
380
360
340
320
300
280
1.2
1
0.5
0
EXPERIMENTAL RESULTS
Figure 8. Profiles of the motor speed commands, dc-bus voltage, and
grid power for uncoordinated loads (
0.5 and ∆
1
)
when the resistive brake is out
The resistive brake controller and speed references were
developed in MATLAB and applied to the drives through
a dSPACE 1104 board. The resistance of the resistive
brake was 40 Ω, total inertia of each coupled motor-load
set was around 24×10-4 kg.m2 (
0.237 kW.sec at
3000 rpm). The PM motor-drives were two
identical Allen-Bradley 2 kW servo-motor-drives (model
number: 2098-DSD-020X). In this setup, the inverters of
the two-axis drive were fed through an external rectifier
and the total dc-bus capacitance was 3760 µF.
The measurements in this experiment were performed
using a 600MHz LeCroy Waverunner 64XI oscilloscope
with one 100MHz CP031 current probe, one 50MHz
CP030 current probe, and one 100MHz ADP305
differential voltage probe.
Case (I): Symmetrical trapezoidal-shaped speed profile
The symmetrical trapezoidal-shaped speed commands,
the dc-bus voltage and the input power are shown in Fig. 8
for ∆
1,
0.5 sec, and
3000 rpm. In
this test, the resistive brake was disconnected. It should be
mentioned that the dc-bus voltage and power waveforms
have been digitally filtered with a cut-off frequency of 8
Hz for the demonstration purposes in this section. Also,
the scaling factor of the speed command is 300 rpm/V, and
the speed response satisfactorily follows the command
signal. As can be seen, if the resistive brake does not
operate, the regenerative energies result in voltage swells
up to ~380 V at the dc-bus, while its rated value is around
290 V. Notice that, this overvoltage is a function of
,
, , dc-bus capacitance, as well as the resistive and
frictional (or damping) coefficient of the system.
On the other hand, the performance of the system under
the same conditions, when the resistive brake was a part of
the circuit, is presented in Fig. 9. It should be mentioned
that, because of the limited number of oscilloscope
channels, the speed command of Motor-2 could not be
shown in Figs. 9, 11, 13, and 15. In these figures, a
channel was dedicated to measuring the resistive brake
current in order to calculate the dissipated power. In this
case, the dc-bus voltage is compared with a certain
reference voltage, here
300 V, and the result is the
input to a proportional controller with a gain of 0.02. Then,
3453
the output duty cycle is compared with a 2 kHz saw-tooth
waveform to generate switching signals for the brake
IGBT. The profile of dissipated power in the resistive
brake, P
, is shown in the bottom graph of Fig. 9.
Also, it can be observed that the resistive brake limits the
peak of the dc-bus voltage to 320 V.
0.14
0.12
Ebrake (kW.sec)
0.1
In Figs. 8 and 9, one can observe that a small
regeneration occurred right after the acceleration period,
while the motor speed was settling at
(e.g. around
3 sec in Fig. 8). These small generation episodes are
due to the oscillatory pattern of the motor speed.
In order to investigate the validity of the derived
equations in Section III, two independent experiments
were carried out for different time delays over a range of
∆
0.5, 0.5 sec, with the steps of 0.1 sec, and the
dissipated energy in the resistive brake during one
deceleration period was measured. The outcomes are
demonstrated in Fig. 10. As can be observed, the results of
the both experiments are quite consistent with each other,
and quantitatively, the correlation is more than 97%.
Moreover, the minimum energy dissipation can be
achieved at ∆
0.2 sec.
When considering the resolution of the experimental
data, this is the closest obtainable value to the theoretical
optimal time delay derived in Section III, i.e. ∆
⁄3
0.17 sec. A comparison between Fig. 10 and
Fig. 7 validates the proposed model and the developed
formulations. However, it is worth to mention that, since
TABLE I. SYSTEM PERFORMANCE UNDER DIFFERENT TIME DELAYS
∆
(
-0.5
(
)
.
0.094
)
( )
( )
0.986
278.0
323.6
(
0.924
)
-0.2
0.021
0.187
0.761
281.4
303.9
0.0
0.039
0.367
0.948
279.9
308.3
+0.2
0.074
0.683
0.955
278.5
317.1
+0.5
0.098
0.708
1.099
278.8
317.2
+1.0
0.131
0.834
0.954
279.5
317.9
st
1 run
-0.4
-0.3
-0.2
-0.1
0
0.1
Δt (sec)
0.2
0.3
0.4
0.5
Figure 10. Dissipated energy in the resistive brake during the
deceleration period, versus ∆ , for two independent runs when
0.5 sec
practically it is very difficult to measure the exact amount
of regenerative power transferred from one motor to the
other, in the current study, the reduction in dissipated
energy in the resistive brake has been adopted as a
measure for the utilization of the regenerative energy. The
reason is that the dissipated energy in the brake is
reciprocally related to the utilized regenerative energy.
This means that the more energy dissipation in the resistive
brake is equivalent to the less regenerative energy
utilization, and vice versa. Figs. 7 and 10 demonstrate a
quite similar pattern of regenerative energy utilization over
a wide range of ∆ , which indeed verifies the validity of
the derived equations in which the system was assumed
lossless.
Several performance indices of the system operation
under different time delays are summarized in Table I. It
is evident that the optimal time coordination enhances the
performance of the system in terms of energy and power
losses, peak demand from the grid, and dc-bus voltage.
For instance, a comparison between the performance of
the two-axis system under the optimal coordination, i.e.
∆
0.2 sec, and a partially coordinated condition, i.e.
∆
0, results in the following: (1) the resistive brake
3000
ωref-1 (rpm)
0
1
2
3
4
5
6
7
8
9
2000
1000
0
10
Vdc (V)
380
360
340
320
300
280
0
1
2
3
4
5
6
7
8
1.2
1
9
P brake (kW)
0.5
0
1
2
3
4
5
6
Time (sec)
7
8
9
Figure 9. Profiles of the speed command of Motor-1, dc-bus voltage, as
well as grid and resistive brake powers for uncoordinated loads (
0.5 and ∆
1
)
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
Time (sec)
7
8
9
10
1.2
1
0.5
0
10
0
380
360
340
320
300
280
10
P grid (kW)
P brake (kW)
ωref-1 (rpm)
Vdc (V)
2 run
The average
0
-0.5
1000
P (kW)
nd
0.02
2000
0
0.06
0.04
3000
0
0.08
Figure 11. Profiles of the speed command of Motor-1, dc-bus voltage, as
well as resistive brake power for optimally coordinated loads (
0.5 and ∆
0.2
)
3454
0.12
0.14
0.11
0.12
0.1
0.1
Ebrake (kW.sec)
Ebrake (kW.sec)
0.09
0.08
0.07
0.06
0.06
0.05
0.04
0.04
-0.2
-0.1
0
0.1
0.2
Δt (sec)
0.3
0.4
0.5
0.02
-1
losses reduce from 0.039 to 0.021 kW.sec, (a 46%
reduction), (2) peak of the power dissipated in the resistive
brake reduces from 0.367 to 0.187 kW (a 49% reduction in
the resistive brake capacity). This means that a less
expensive and smaller resistive brake can be utilized, (3)
peak of the power demand from the grid reduces from
0.948 to 0.761 kW (an 20% reduction in the dc-bus size),
and (4) the range of variation in the dc voltage reduces
from [279.9, 308.3] to [281.4, 303.9] volts, which is a 21%
reduction in the voltage variation across the dc-bus.
Notice that such a comparison between the optimally
coordinated and uncoordinated (i.e. ∆
1
)
conditions, results in a 84% reduction in the brake losses, a
78% reduction in the peak power of the resistive brake, a
20% reduction in the peak demand from the grid, and a
41% reduction in the dc-bus voltage variation, which
constitutes a significant improvement. The performance of
the system under the optimal time delay, i.e. ∆
0.2
sec is presented in Fig. 11. A comparison between Fig.11
and Fig. 9 also confirms that the optimal coordination
significantly enhances the performance indices of the
system with respect to an uncoordinated system. From Fig.
11, it may be concluded that this system does not
necessarily need a resistive brake while it is optimally
coordinated. The rationale behind this stipulates that under
Case (II): Non-symmetrical trapezoidal-shaped speed
profile
The same set of tests was conducted for the two cases of
non-symmetrical trapezoidal-shaped speed profiles, i.e.
when the deceleration period is longer than the
acceleration period (
( ⁄ ) 1), and when the
acceleration period is longer than the deceleration period
(
1).
In the first test the acceleration and deceleration periods
were set to be 0.5 and 0.25 sec, respectively (i.e.
0.5,
0.25
0.5). The experimentally obtained
ωref-1 (rpm)
1000
3
4
5
6
7
8
9
2000
1000
10
0
10
380
360
340
320
300
280
1
2
3
4
5
6
7
8
9
P brake (kW)
1
0.5
0
3000
Vdc (V)
Vdc (V)
2
380
360
340
320
300
280
0
P brake (kW)
1
0
1
2
3
4
5
6
Time (sec)
0.5
this situation the amount of energy dissipation in the brake
is significantly low. Therefore, the dc-bus voltage will not
exceed its acceptable range even without use of the brake.
Hence, a new test was conducted without the resistive
brake in the circuit, while the time delay was set to be
equal to ∆
0.2 sec. It was observed that the dc-bus
voltage variation was bounded within a range of [292.3,
311.9] V, which is still much better than that for the
uncoordinated condition in the presence of the resistive
brake. Therefore, a time coordination may even result in
removal of the resistive brake. This will consequently lead
to a reduction in the cost and size of the overall system.
2000
0
0
Figure 14. Dissipated energy in the resistive brake during the
deceleration period, versus ∆ , for two independent runs when
0.5
and
1.0 sec
3000
0
-0.5
Δt (sec)
Figure 12. Dissipated energy in the resistive brake during the
deceleration period, versus ∆ , when
0.5 and
0.25 sec
ωref-1 (rpm)
0.08
7
8
9
10
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
Time (sec)
7
8
9
10
1
0.5
0
Figure 13. Profiles of the speed command of Motor-1, dc-bus voltage, as
well as resistive brake power for optimally coordinated loads (
0.5,
0.25 sec, and ∆
0)
0
Figure 15. Profiles of the speed command of Motor-1, dc-bus voltage, as
well as resistive brake power for optimally coordinated loads (
0.5,
1.0, and ∆
0.7 sec)
3455
data for dissipated energies versus the time delay are
demonstrated in Fig. 12. As can be observed, the minimum
energy dissipation is obtained at ∆
0 which is the
closest obtainable value to the theoretical optimal time
delay presented in [10], i.e. ∆
0.036 sec
for
0.5. Also, the performance of the system under
this situation is shown in Fig. 13.
For
1, the acceleration and deceleration periods
were set at 1 and 0.5 sec, respectively, (
0.5,
1
2), and the results are shown in Fig. 14. As can
be seen, the minimum energy dissipation is obtained at
∆
0.7 sec, which is fairly close to the theoretical
optimal time delay presented in [10], i.e. ∆
0.57
sec for
2. Also, the performance of the system under
this situation is shown in Fig. 15.
V.
DISCUSSION
The developed closed-form formulas for a two-axis
motor-drive system can be extended to a multi-axis motordrive system for parallel industrial processes. It should be
clarified that a multi-axis system is defined herein as a
multiple number of inverter-motors, which are fed from a
common dc-bus. Moreover, it should be noted that each
one of the motors are also part of a sequential control
process in each parallel line (e.g. assembly or packaging
lines). In other words, the proposed coordination must be
implemented between the motors that are in parallel
processes, while each process has its own required
sequential controller. For any number of the motors in
parallel processes, the developed closed-form formula can
be applied for each pair of parallel motors which ensures
that the system operates, at least, under some partial
optimal condition. For instance, in the case of four parallel
packaging processes with the symmetrical trapezoidalshaped speed profile, the programmable logic controller
(PLC) can optimally coordinate Motor-1 with Motor-2 and
Motor-3 with Motor-4. This approach results in saving
67% of the total regenerated energy of the system, see Fig.
7. Accordingly, in the case of a three parallel processes,
Motor-1 and motor-2 are optimally coordinated together
which means Motor-2 has to start to accelerate ⁄3
before Motor-1 starts to decelerate. Then, Motor-3 can be
coordinated in such a way that it can absorb a large portion
of the regenerated power of Motor-1 which cannot be
transferred to Motor-2. One of the candidate solutions has
been shown in Fig. 16, in which Motor-3 starts to
accelerate 2 ⁄3 before Motor-1 starts to decelerate (The
reader should notice that, because of the overlap between
the operation of Motor-2 and Motor-3, this coordination
( )
2
( )
( )
( )
2⁄3
1⁄3
0
1⁄3
2⁄3
1
Alternatively one can use an intelligent or evolutionary
optimization technique, such as particle swarm
optimization (PSO) [11], in conjunction with the proposed
methodology for modeling the regenerative energy flow in
order to find the global optimum (or a near to the global
optimum) solution for the system. The overall procedure
can be summarized in three major steps:
i.
In each iteration (or generation) a set of potential
solutions (possible time delays ∆ ( , ), where
1, 2, … ,
1 for
number of motors, and
1, 2, … , for a swarm population of ) is generated.
ii.
The system is modeled for each time delay
∆ ( , ) according to the proposed methodology, and the
( ), and
instantaneous input powers to the motors,
( ), are derived.
regenerated powers,
iii.
Based on the intersection areas, the objective
function is evaluated for every potential solution of the
iteration and, based on the obtained objective values, the
algorithm generates an appropriate set of potential
solutions for the next iteration.
It should be noted that, in step ii, instead of applying the
proposed methodology, it is possible to simulate the
system for every different time delay. However, according
to our experience, dynamic simulations of a system
consisting of several power electronic converters (with
very fast switching times) and motors (with relatively long
time constants) over a long period of time (maybe some
tens of seconds) demands lots of computation time and
memory which results in either memory buffer overrun, or
several days of calculations for an algorithm with a large
number of population and iterations (or generations). On
the contrary, the proposed methodology does not suffer
from such drawbacks.
VI.
( )
( )
⁄
may result in larger voltage variations during acceleration
and deceleration periods of these two motors). As can be
seen in this figure,
8 ⁄9. This means that 16 ⁄9
can be saved during each operating cycle of the system,
which involves three accelerations and three decelerations.
Thus one may conclude that 16 ⁄9 out of a total
regenerated energy of 3
(i.e. 59.26 % of the total
regenerated energy) can be saved through this operating
strategy. Similar procedure can be extended to the systems
with five or more parallel motors. This approach
guarantees that the regenerative energies of some of the
motors are efficiently exploited. This method of operation
may not provide the global optimum condition; however, it
would be an adequate solution for a partial optimization.
⁄
Figure 16. Partial optimal operation of a three-axis motor-drive system
CONCLUSION
In this paper, an analytical approach for the time
coordination
between
multi-axis
(in
parallel
processes/tasks) servo-drives has been presented in order
to obtain the maximum exploitation of the regenerative
energy. Besides the energy saving, this will result in
significant reductions in the physical size and the peak
current rating of the common input converter (bridge
rectifier) and resistive brake circuit of multi-axis servodrives. The methodology is based on a fundamental
concept, which is to overlap the regeneration and
consumption power curves and maximize their intersection
3456
area. Experimental results verified the validity of the
proposed methodology for inertia-loads with trapezoidalshaped speed profiles. Extension of the proposed
methodology to other types of electric systems (e.g. the
time coordination between pulse-loads and flywheel
systems or pulse-loads and propulsion motors in shipboard
power systems) can be a matter of potential studies.
ACKNOWLEDGMENT
The authors wish to thank Mr. Tom Van Groll of
Rockwell Automation Inc. for providing two servo-PM
motor-drives along with associated cables and the control
software.
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