This paper deals with faulttolerant control of fivephase induction motor (IM) drives under singlephase open. By exploiting a decoupled model for fivephase IM under fault, the indirect fieldoriented control ensures that electromagnetic torque oscillations are reduced by particular magnitude ratio currents. The control techniques are developed by the third harmonic current injection, in order to improve electromagnetic torque density. Furthermore, Proportional Resonant (PR) regulator is adopted to realize excellent current tracking performance in the phase frame, compared with Proportional Integral (PI) and hysteresis regulators. The analysis and experimental results confirm the validity of faulttolerant control under singlephase open.
1. Introduction
Multiphase drives are proper for high power application, due to their advantages over conventional threephase drives
[1

3]
. With the addition of extra phases, the current per phase is reduced for given phase voltage and total power rating, which makes it possible to use the standard power switches on a single device
[4

6]
. Increasing the number of phases is also a solution for torque pulsation reduction
[7

8]
. More vector planes in the multiphase motor can be manipulated to optimize airgap flux distribution and improve the torque density
[9

11]
. Another distinguished advantage is improved reliability and continuous system operation even in the fault condition
[12

13]
. This characteristic is crucial for all the applications requiring a high degree of reliability, such as Electric Vehicle (EV) or marine.
Many papers have focused on the model and control for multiphase motor under open phase condition. For the dualthree phase induction machine, a unified approach to represent the model with singlephase open has been presented by the fieldoriented control strategy
[13

15]
. The resilient current control of fivephase motor with sinusoidal and nonsinusoidal current excitations under fault condition is illustrated in
[16

18]
. Faulttolerant control techniques for fivephase PM motors with trapezoidal back electromotive forces have been discussed in
[19]
. The control scheme in
[20]
ensures the operation of a sevenphase drive under opencircuit. However, few works focus on the models of multiphase IM with concentrated winding structure under open phases, and the optimal harmonic current injection technique is rarely discussed.
Current regulator plays a key role in the high performance, and the primary goal is to track the reference current accurately during steady state and transient condition. The precise matched comparative analysis of different current regulation strategies is discussed in
[21]
, which is in the application of threephase IM under healthy condition. A theoretical connection between PR and synchronous frame PI regulators is established in
[22]
. Hysteresis current control is widely used for its simplicity, but it results in much more harmonic current
[17

18
,
21

22]
. The simple PI regulator realizes the excellent dc current control. However, its performance is unsatisfactory for the ac current control
[23]
. Fortunately, the PR regulator can achieve virtually the same steadystate and transient performance as synchronous reference frame PI regulator under healthy condition
[24]
. A further significant superiority of PR controller is its application to current control in the phase frame, where synchronous reference frame control is more difficult to realize under asymmetrical condition.
The decoupled model is established for the fivephase IM under open fault, and it contains inductance parameters derivation. The proposed method is based on the simple computation of the currents in the
αβ
subspaces, and electromagnetic torque oscillations are reduced by particular magnitude ratio currents. Then, a set of stator currents with third harmonic injection is obtained to improve the electromagnetic torque density. Third harmonic current is also utilized to decrease the amplitude of stator current, without exceeding the rate power of inverter. Furthermore, the PR current regulator is adopted to realize excellent current tracking in the phase frame. Finally, a fivephase induction motor drive with low DC link is constructed, and experimental results verify the feasibility of the faulttolerant control strategy under singlephase open.
2. Motor Model for Fivephase IM Under Singlephase Open
 2.1 Voltage equations in theαβsubspace
This paper presents a fourpole fivephase induction motor with identical quasiconcentrated phase windings, and each phase winding is constituted by two adjacent concentrated winding in series. The basic equations of fivephase IM under balanced condition are discussed in
[8

9]
. Given fivephase IM with phase “a” open, the stator and rotor voltage equations can be written as
where the current and voltage vectors are
In Eq. (2),
R
_{s}
,
L
_{ss}
,
R
_{r}
,
L
_{rr}
,
L
_{sr}
and
L
_{rs}
are parameter matrices described in
[12

13]
.
T
_{5}
is transformation matrices under healthy condition contained in
[9]
, and the stationary transformation matrices
T
_{4(1)}
and
T
_{4(3)}
decomposed into
α
_{1}
−
β
_{1}
and
α
_{3}
−
β
_{3}
planes are given as
where
c
_{0}
= 0.25, and
γ
= 2
π
/5 . The electromechanical energy conversion takes place in
α
_{1}
β
_{1}
and
α
_{3}
β
_{3}
subspace. In (4) and (5), all the row vectors can’t realize the same norm simultaneously, while keeping the orthogonal property to each one. Additionally, the third row
z
_{1(3)}
is one degree of freedom to realize the same magnitude of stator currents
[12]
, and the fourth row of the matrices satisfies zero sequence current equation. Then, the correlative inductance parameter sunder singlephase open can be derived, as shown in
Table 1
.
Correlative inductance parameters under single phase open
Correlative inductance parameters under single phase open
 2.2 Electromagnetic torque ripple analysis
Reference
[13]
demonstrates that the electromechanical energy conversion takes place only in the
α
_{1}
−
β
_{1}
plane for dualthree phase IM. The fivephase IM with concentrated winding structure has two energy conversion planes, so the currents for the dual planes can be defined as
I
_{α1}
,
I
_{β 1}
,
I
_{α 3}
,
I
_{β3}
are the magnitude in the
α
_{1}
−
β
_{1}
and
α
_{3}
−
β
_{3}
planes, respectively. The steady state is contained in the paper, and the Eq. (6) is the reference stator current. At the fault occurrence, the magnitude of
I_{α}
and
I_{β}
are changed according to the load torque
[15]
.
The electromagnetic torque derivation for
α
_{1}
−
β
_{1}
planes has been discussed in the reference
[17]
. Similarly, the electromagnetic torque for is derived as
Eq. (8) exhibits clearly that the electromagnetic torque can be decomposed into four components: constant part
and oscillation component
for
α
_{1}
−
β
_{1}
plane; constant part
and oscillation component
for
α
_{3}
−
β
_{3}
plane.
3. Indirect Fieldoriented Control
In order to eliminate the torque oscillations, the current magnitude ratio is given as
To maintain the same amplitude of stator current, the currents in the z plane are identified as
i
_{z1}
= 0.236
i
_{sβ1}
,
i
_{z3}
= 0.236
i
_{sβ3}
[12
,
15]
. The criterion (9) can be applied in the indirect fieldoriented control scheme where the
dq
currents are used to control electromagnetic torque. The synchronous rotating transformation matrices in
d
_{1}
−
q
_{1}
and
d
_{3}
−
q
_{3}
planes can be expressed as
where
, and
. It is assumed that the fundamental and third harmonic rotor fluxes are attached to
d
axis, and no component exists on
q
axis. The expressions of magnetizing currents are
To maintain a quasitrapezoidal airgap flux density distribution, the corresponding magnitude currents
i
_{sd3}
is given by
where
m
_{3}
=1/ 6 , as described in
[10

11]
. The slip angular speed are derived as
Similarly, the substitution of the current from (1213) into the torque expression (8), the electromagnetic torque can be derived as
The torque current component in the
d
_{3}
−
q
_{3}
plane can be expressed as
According to the analysis mentioned above, the diagram of indirect fieldoriented control for fivephase IM under singlephase open is shown in
Fig. 1
. The current regulator in the phase frame is elaborated in the next section.
Indirect fieldoriented control for fivephase IM under singlephase open fault
4. Current Regulator Design in the Phase Frame
This section depicts the principles of three methods for current regulator
[22]
, and the diagram is shown in
Fig. 2
.
Diagram of current control model in the phase frame
Since Electromotive Force (EMF) is not random disturbance, its effect on the fundamental tracking error can be substantially reduced by incorporating feed forward injection of an estimated compensation. Then, the motor transfer function is described as
where
T_{p}
=
L_{σs}
/
R
_{s}
. Furthermore, the current control transfer function
G_{c} (s)
is demonstrated in the following.
 4.1 Hysteresis Current Control (Method I)
Hysteresis control operates by comparison of the reference current against the feedback current. Switching state of inverter is set to high or low, depending on its error moving outside of fixed hysteresis width
[21]
, according to
This control strategy achieves a fast current response by simple programming, though it has the limitations of variable switching frequency. Another negative effect is much more harmonic current, which increases the copper loss and torque pulsation for the system drive.
 4.2 PI Current Regulator (Method II)
The forward transfer block
G_{c} (s)
for a simple PI controller is defined by
The magnitude of
G_{c}(s)
is essentially infinite at the dc because of the 1/(
sk_{i}
) term. However, this method can’t realize zero ac current error in the phase frame because of the limited gain in the regulator.
 4.3 PR Current Regulator (Method Ⅲ)
In this situation, the only remaining alternative is to increase the gain of the forward controller, but this would reduce the system of stability. Fortunately, PR regulator can realize zero ac current error in the phase frame, because the resonant term has large gain at the target frequency, which can achieve good response in the fault occurrence
[21

22]
.
The forward gain block
G_{c}(s)
of a practical PR control is described by the transfer function
From this expression, it can be seen how the term of
s
/(
s
^{2}
+
𝜔_{e}
^{2}
) creates infinite forward controller gain at the target frequency
𝜔_{e}
, which is utilized to achieve zero error for tracking ac current.
The phase angle of forward loop at the cross over frequency
𝜔_{c}
is given by
where
ø_{m}
is target phase margin, and the value is usually set to
π
/6 ⌷
π
/4. As described in the reference
[21]
, the maximum cross frequency can be derived
Then the maximum possible magnitude of
k_{p}
can be derived by setting the open loop gain to unity
[24]
, when gives
k_{i}
≈ 10/
𝜔
_{c(max)}
.
Appling the approach to the test system described for the controller gains (
k_{p}
= 0.3,
k_{i}
= 0.0025
s
), the bode plots of forward open loop for PI and PR regulator are shown in
Fig. 3
.
Bode plots of PR and PI control
5. Experimental Verification
In order to evaluate the capability of the indirect fieldoriented control under fault condition, experimental tests are executed for the steady state and transient response. The drive system is depicted as
Fig. 4
, and the nominal parameters of fivephase IM in the healthy condition are shown as
Table 2
.
Fivephase IM drive system
Nominal parameters of fivephase IM
Nominal parameters of fivephase IM
The performances of the drive system for three current regulators are extensively explored in the experimental tests, when the motor operates at 600 r/min with the 15N. m load. Experimental results contain the performances of steady state and transient response under phase “a” open. For the aspect of EMF compensation in the feed forward loop, the commanded ratio of
V/f
curve is set to 0.40.
The experimental results in
Fig. 5
exhibit the current responses of drive system controlled by hysteresis regulator (method ↑). There is relatively larger steady state and transient current errors, and the maximum amplitude is about 60A shown in
Fig. 5(b)
. Worse still, many low order harmonic currents are contained in hysteresis control, and the THD is about 21.5% by the algorithm of FFT.
Experimental results for method I
The performance of PI current regulator (method II) is demonstrated in the
Fig. 6
, and it is can be seen that the harmonic current is reduced significantly compared with that of hysteresis regulator. However, current pulse error at the transient response is still large.
Experimental results for method II
The results of PR current regulator (method Ⅲ) with EMF compensation are expressed in the
Fig. 7
. It can be clearly seen that the stator current is sinusoidal without low order harmonic, and the current error shown in
Fig. 7(b)
is minimized distinctively, especially for the transient response at fault occurrence. On the other hand, the experimental for PR regulator without EMF compensation is contained in the
Fig. 8
. Consequently, the current error is amplified without forward feed compensation, identifying the theoretical analysis in section 3.
Experimental results for method Ⅲ
Experimental results for error between reference and actual current of phase “b”
Fig. 9
depicts five stator currents and motor torque when one phase is open without any faulttolerant control. It can be seen that torque oscillations is significantly larger, and the maximum magnitude is reach to 5N.m. Additionally, the magnitude of phase “b” increases by 65%, and it means that the load should be restricted by the inverter with same capability.
Experimental results under no faulttolerant control
The torque ripple is reduced significantly with faulttolerant control, and the value is less than 1.5 N.m at the bottom of
Fig.10(b)
. To further verify the effectiveness of the third harmonic current injection, the experiments are executed. The maximum of stator current in the
Fig. 10 (a)
is decreased by 15.6%, compared with that of
Fig. 7 (a)
. Moreover, the electromagnetic torque is increases by 11%, as shown in
Fig. 10(b)
.
Experimental results under faulttolerant control with third harmonic current injection
Fig. 11 (a)
illustrates the experimental results for fundamental and third harmonic reference currents under singlephase open, which are mapped into
d
_{1}
−
q
_{1}
and
d
_{3}
−
q
_{3}
planes, respectively. It can be seen that
i
_{sd1}
and
i
_{sd3}
are constant, while
i
_{sq1}
and
i
_{sq3}
are approximately proportional to the electromagnetic torque illustrated in
Fig. 11(b)
. The flux producing current
i
_{sd1}
is controlled at 25A, and
i
_{sd3}
is controlled at about 9A.
Fig. 11 (b)
shows that the rotor speed has no disturbance under sudden load at 15 N. m, ensuring high performance for torque control under singlephase open.
Experimental results under sudden load
6. Conclusion
A modified model for fivephase IM under open fault has been exploited to realize high performance in this paper. The control scheme is suitable for the implementation of reducing the torque ripple, allowing disturbancefree operation with particular magnitude ratio currents. To assess the effectiveness of current control, three regulators have been analyzed in detail by comparison of current magnitude error for steadystate and transient response. The third harmonic current injection technique can be used to improve the iron utilization and torque density. The experimental results demonstrate that the proposed method is valid under singlephase open fault. Moreover, the method presented in this paper can be used for up to two open stator phases for fivephase IM.
Nomenclature
 Variable
V_{s} Stator voltage matrices V_{r} Rotor voltage matrices I_{s} Stator current matrices I_{r} Rotor current matrices R_{s} Stator resistance matrices R_{r} Rotor resistance matrices L_{ss} Stator selfinductance matrices L_{rr} Rotor selfinductance matrices L_{sr} Mutual inductance matrices M Mutual inductance under fault condition L Self inductance under fault condition P Number of poles ψ Rotor flux T_{em} Electromagnetic torque τ Rotor time constant ω_{e} Electrical angular speed θ Synchronous rotating angle ϕ_{0} Initial angle T_{d} Sampling and transfer delay time V_{DC} Voltage of DC link k_{p} Proportional gain k_{i} Integral gain T_{5} Transformation matrices under healthy condition T_{4} Transformation matrices under faulty condition
 Subscripts
1 Fundamental component 3 Third harmonic component s Stator component r Rotor component α, β Stationary frame component d, q Synchronous frame component
BIO
Wubin Kong He received a B.S. degree from the College of Electrical Engineering, Zhejiang University, Hangzhou, China, in 2009. He is currently working toward the Ph.D. degree. His research interests are in multiphase machines and drives.
Jin Huang He received the B.S. degree from the College of electrical engineering, Zhejiang University, Hangzhou, China, in 1982, and the Ph.D. degree in electrical engineering from the National Polytechnic Institute of Toulouse, Toulouse, France, in 1987, respectively. From 1987 to 1994, he was an associate Professor in College of Electrical Engineering, Zhejiang University, China. He is currently Professor of Zhejiang University. He is engaged in research on electrical machine, AC drives, multiphase machine and condition monitoring of electrical machines. Min Kang received the B.S. and Ph.D. degrees from the College of Electrical Engineering, Zhejiang University, Hangzhou, China, in 2003 and 2008, respective.
Min Kang received the B.S. and Ph.D. degrees from the College of Electrical Engineering, Zhejiang University, Hangzhou, China, in 2003 and 2008, respectively. From 2008 to 2011, he was a Lecturer in the College of Electrical Engineering, Zhejiang University of Science and Technology, China. He is currently associate Professor of Zhejiang of Science and Technology. His research interests are in multiphase bearingless machines and AC drives.
Bingnan Li He received the B.S. and M.S. degrees from the College of Electrical Engineering, Shenyang University of Technology, Shenyang, China, in 2006 and 2009, respectively. He is currently working toward the Ph.D. degree from the College of Electrical Engineering, Zhejiang University, Hangzhou, China. His research interests are in multiphase theory, multiphase bearingless machines and drives.
LiHang Zhao He received a B.S. degree from the College of Electrical Engineering, Zhejiang University, Hangzhou, China, in 2011, where he is currently working toward the Ph.D. degree. His research interests are in parameter identification of AC motor.
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