Fault Tolerant Capability of Symmetrical Multiphase Machines
under One Open-Circuit Fault
W.N.W.A. Munim *†, Hang Seng Che *, Wooi Ping Hew *
* UMPEDAC, University of Malaya,
Kuala Lumpur, Malaysia
hsche@um.edu.my wphew@um.edu.my
Faculty of Electrical Engineering,
Universiti Teknologi MARA (UiTM), Malaysia
aishahmunim@salam.uitm.edu.my
†
Keywords: Multi-phase drives, fault-tolerance, current
optimization, symmetrical components.
references for different multiphase machines, based on
different optimization targets.
Open-circuit faults for dedicated phase such as five-phase
machine has been reported by Ryu et al. [14] highlighting
both minimum loss and maximum torque whilst Khalik et al.
[15] has proposed the calculation of derating factor to avoid
overheating of the machine after fault occurrence and
maximum torque in order to maintain equal phase currents.
On the other hand, different approach using Lagrange
multiplier method has been investigated in [16] for six-phase
focusing on minimum loss only and in [17] for seven-phase
machine solving minimum loss and maximum torque in
transient and steady-state operating conditions. Nevertheless,
the overall comparison of post-fault performance for faulted
n-phase indicating minimum loss, maximum torque as well as
derating factor still not been stated yet in literature.
To obtain a better overall view on the post-fault
performance for machines with different phase number, some
researchers have perform analysis on a range of multiphase
machines. Fu et al. [18] presented post-fault current analysis
for three- to seven-phase induction machines with the aim of
equalising amplitude of all remaining currents after fault
whereas in [19], Zheng et al. uses the symmetrical component
transformation based on Fortescue to analyse the stator
copper loss minimization and the inverter peak current
minimization for three- to nine-phase machines in steadystate operating conditions. In addition, Baudart et al. [16]
addressed the minimum loss of three- to eight-phase machines
operating under open-phase conditions. However, the paper
has not presented the derating factor which is the restriction
of the maximum post-fault current does not exceed the rated
phase current to maintain the reliability of the system.
Based on the literature reviewed, three optimizations
objectives are found to be common: equalising current
amplitudes after fault, minimising copper losses and
maximising torque/power under a given current limit. Among
these, minimum loss and maximum torque (hereafter referred
to as ML and MT respectively) are of particular importance.
In this paper, the fault tolerant capability of symmetrical fivephase (S5), six-phase (S6), seven-phase (S7), and nine-phase
(S9) in single isolated neutral under one open-circuit fault
(OCF) is investigated. The optimized post-fault currents
under ML and MT modes are first obtained, and the
performances of the machines in terms of derating factor (a)
and stator copper loss under both modes are evaluated. The
Abstract
This paper presents a fault-tolerant investigation of
symmetrical multi-phase induction machines (five-phase, sixphase, seven-phase and nine-phase) in terms of minimum loss
and maximum torque with single isolated neutral point under
open-circuit fault at one of the phases. Minimum
reconfiguration of the controller attainable for the post-fault
control by using normal decoupling transformation. The postfault performance is carried out based on derating factor,
minimum stator copper loss and minimum peak current. The
simulation results confirm the validity of the theoretical postfault current limits for symmetrical multi-phase induction
machines under one open-phase fault.
1 Introduction
Since beginning of the 20th century, the use of multi-phase
(more than three-phase) machines have been explored [1] for
their various advantages over conventional three-phase
machines, such as reducing the current per phase without
increasing the voltage per phase, lowering dc harmonics and
increasing reliability [2]. Even though multiphase grid does
not exist by default, advanced applications of power
electronic these days enables the use of multiphase electric
drives with literally no restriction on the number of phases.
This has opened the possibility for the utilisation of
multiphase machines on applications such as electric vehicles
[3], “more electric’ ships and aircraft, offshore wind farms
generators, high speed elevators and etc. In the light of this,
various research works have been devoted to the development
and study of multiphase machines and drives, particularly in
terms of fault tolerant design [4]–[8], modelling [9], [10] and
control [11]–[13].
One of the most acclaimed features of multiphase machine
is its ability to tolerate open-circuit faults in its phase
windings. To do this, a suitable set of new (post-fault) current
references is necessary to avoid negative sequence component
in the fundamental magneto-motive force (MMF) of the
machine, which will otherwise manifest as undesirable torque
and speed oscillations. Over the years, many works have been
devoted to the optimization of such post-fault current
1
results provide a guideline on the choice of multiphase
machine and mode of operation for a given fault tolerance
requirement.
This paper is organized in the following manner: Section 2
outlines the concepts of current optimization in post-fault
mode, while Section 3 explains the indicators used to evaluate
fault tolerant capability of a given machine. The theoretical
current waveforms and post-fault performances of the
machines are simulated and shown in Section 4, before
conclusions are given in Section 5.
2 Current Optimization in Post-Fault Mode
This section describes the healthy and faulted operation of
S5, S6, S7 and S9 drives and the optimization technique to
achieve disturbance-free fault-tolerant operation.
2.1 Overview of multi-phase drives
Symmetrical multi-phase induction machine is controlled
using vector space decomposition (VSD) model approach and
the generalized Clarke transformation matrix [2] as shown in
Equation (1).
Tn 
2
...
n












x( n  4)/2 


y( n  4)/2 


0 
0 




x1
y1
x2
y2
...
1
cos 
cos 2
cos 3
...
0
sin 
sin 2
sin 3
...
1
cos 2
cos 4
cos 6
...
0
sin 2
sin 4
sin 6
...
1
cos 3
cos 6
cos 9
...
0
sin 3
sin 6
sin 9
...
...
...
n2
n2
n2
1 cos 
  cos 2 
  cos 3 
  ...
 2 
 2 
 2 
n2
n2
n2
0 sin 
  sin 2 
  sin 3 
  ...
 2 
 2 
 2 
1
1
1
1
...
2
2
2
2
1
1
1
1


...
2
2
2
2
where α = 2/n.
cos  n  1 



cos 2  n  1  

sin 2  n  1  
cos 3  n  1  

sin 3  n  1  


n2 
cos  n  1 

 
 2  
n2 
sin  n  1 
 
 2  

1

2


1


2

sin  n  1 
(1)
Decoupling transformation [Tn] will be used to transform
the n-phase variables to the n decoupled variables namely α-β
components which related to flux and torque, x-y currents ((n4)/2 for n=even or (n-3)/2 for n=odd) and zero sequence
components 0+0- which are not involved in the energy
conversion process and only contribute to loss. For odd phase
numbers, 0- is omitted as it is only exist in even number of
phases. Fault-tolerant control targeting to preserve the prefault torque with no additional torque ripple as in [11] is
assumed here. In order to do that, new optimized post-fault
current references need to be defined, based on the
Minimum loss (ML) and maximum torque (MT) optimization
objectives commonly used in literature.
2.2 Minimum Loss (ML)
When the phase A current is restricted to zero, the
objective of ML is to minimize the stator copper losses
defined by the cost function 𝐽𝑀𝐿 :
2
2
𝐽𝑀𝐿 = min{√(𝑖𝛼2 + 𝑖𝛽2 + 𝑖𝑥2 + 𝑖𝑦2 + 𝑖0+
+ 𝑖0−
)}
(2)
However, ML will results in unequal phase currents eg:
S5 produces two sets of equal phase currents with different
amplitude, S6 results in two sets of equal phase currents and
one phase current with different amplitude. This mode also
leads to the reduced of maximum achievable torque.
2.3 Maximum Torque (MT)
Since inverter is usually designed to operate with current
limited to the nominal value, this current limit should be
obeyed even during fault to avoid damaging the drive. Hence,
in MT post-fault operation, the aim is to minimize the
maximum phase current of the remaining healthy phases. This
is equivalent to the inverter peak current minimization used in
[19]. In this case, the cost function 𝐽𝑀𝑇 targets to maximize
the torque, which in turn indicates maximizing the amplitude
of the α-β phasor:
𝐽𝑀𝑇 = max⁡(|𝐼𝛼𝛽 |)
(3)
Subject⁡to:
 𝐼𝑓𝑎𝑢𝑙𝑡 = 0 ∈ {Faulted⁡phases}
 𝑖0+ = 0
 min⁡the⁡maximum⁡phase⁡current ∈
{Healthy⁡phases}
(4)
2.4 Optimization
There are various ways to optimization post-fault currents,
and the approach in [11], i.e. based on decoupled variables,
have been adopted here in this paper. Coefficients “K” are
used to relate the non-energy-converting currents with the α-β
references. The number of optimized currents depends on the
phase number n, where there are (n-4)/2 or (n-3)/2 x-y
currents in addition to 2 or 1 zero sequence currents for even
or odd phase machines respectively. Equations (5) shows the
coefficients used in S9:
∗
𝑖𝑥1
= 𝐾1 · 𝑖𝛼∗ + 𝐾2 · 𝑖𝛽∗
∗
𝑖𝑦1 = 𝐾3 · 𝑖𝛼∗ + 𝐾4 · 𝑖𝛽∗
∗
𝑖𝑥2
= 𝐾5 · 𝑖𝛼∗ + 𝐾6 · 𝑖𝛽∗
∗
𝑖𝑦2 = 𝐾7 · 𝑖𝛼∗ + 𝐾8 · 𝑖𝛽∗
∗
𝑖𝑥3
= 𝐾9 · 𝑖𝛼∗ + 𝐾10 · 𝑖𝛽∗
∗
𝑖𝑦3
= 𝐾11 · 𝑖𝛼∗ + 𝐾12 · 𝑖𝛽∗
(5)
In the case of S9, the x1y1, x2y2, and x3y3, currents
references for the corresponding post-fault modes can be
obtained by optimizing the coefficients K1, K2,…, K12, based
on different optimization objective (ML or MT). Zero
sequence 0+ is set to be zero. Meanwhile, for S6, only one set
of x-y currents and two zero sequence currents 0+0- need to
be optimized, as shown in Equation (6).
𝑖𝑥∗ = 𝐾1 · 𝑖𝛼∗ + 𝐾2 · 𝑖𝛽∗
⁡⁡⁡𝑖𝑦∗ = 𝐾3 · 𝑖𝛼∗ + 𝐾4 · 𝑖𝛽∗
∗
⁡⁡⁡𝑖0+
= 𝐾5 · 𝑖𝛼∗ + 𝐾6 · 𝑖𝛽∗
∗
⁡⁡⁡𝑖0−
= 𝐾7 · 𝑖𝛼∗ + 𝐾8 · 𝑖𝛽∗
(6)
This paper using a non-linear optimization method, i.e. the
generalized reduced gradient (GRG) approach, provided by
“Solver”, an add-on in MS Office Excel. The optimization
objectives for ML and MT modes based on (2) and (3)
respectively and are subjected to (4). The coefficients will be
varied at each iteration and the subsequent phase currents
amplitudes are obtained by applying [Tn]-1 onto the VSD
currents.
Derating factor, a (%)
100
𝑎=
|𝐼𝛼𝛽|
(7)
𝑅𝑎𝑡𝑒𝑑
The mode of operation factor (kM): enhancement
of the derating factor a with maximum torque (MT)
compared to minimum loss (ML) criterion:
𝑘𝑀 =
𝑎𝑀𝑇 −𝑎𝑀𝐿
𝑎𝑀𝐿
x100
(8)
While ML mode improves efficiency, it gives less
post-fault torque/power than MT mode. The value of
kM hence indicates the gain in torque/power if MT
mode is chosen over ML mode.
Phase
S5Ø
a
0.6813
S6Ø
0.6882
S7Ø
0.7043
S9Ø
0.7403
Minimum Loss, ML (p.u)
Ploss
Coefficients, 𝑲
1.4999
𝑲𝟏 = -1
𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
1.3333
𝑲𝟏 = -0.6667 𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
𝑲𝟓 = 0
𝑲𝟔 =
𝑲𝟕 = -0.4714 𝑲𝟖 =
1.2499
𝑲𝟏 = -0.5
𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
𝑲𝟓 = -0.5
𝑲𝟔 =
𝑲𝟕 = 0
𝑲𝟖 =
1.1666
𝑲𝟏 = -0.3333 𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
𝑲𝟓 = -0.3333 𝑲𝟔 =
𝑲𝟕 = 0
𝑲𝟖 =
𝑲𝟗 = -0.3333 𝑲𝟏𝟎 =
𝑲𝟏𝟏 = 0
𝑲𝟏𝟐 =
50
40
30
20
10
4
5
6
7
Number of phases
Maximum Torque
8
9
10
Current Limit
Figure 1. Comparison of derating factor for S5, S6, S7 and
S9.
16.56
Derating factor can be used to evaluate the post-fault
torque available for given faulted machine without
violating nominal current limit. A higher derating
factor indicates higher maximum torque can be
obtained for a given current limit.

60
Minimum Loss
km %
15
The derating factor (a): per unit value of the postfault α-β current phasor modulus, with limitation that
the maximum post-fault phase current does not
exceed the rated phase current [11].
𝑃𝑜𝑠𝑡−𝑓𝑎𝑢𝑙𝑡
70
20
Several performance indicators have been selected to
examine the performance of the machines after fault.
|𝐼𝛼𝛽|
80
0
3
3 Post-Fault Performance Indicator

90
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
15.28
12.03
10
6.21
5
0
3
4
5
6
7
Number of phases
8
9
10
Figure 2. Mode of operation factor (kM) versus number of
phases for 1 OCF.

Normalised Post-fault Stator Copper Loss (Ploss):
since different mode gives different phase currents,
the stator copper loss is expected to differ. The stator
copper loss, normalised to that of a healthy machine,
can be calculated based on:
𝑃𝑙𝑜𝑠𝑠 =
2 +𝑖 2 +𝑖 2 +𝑖 2 +𝑖 2 +𝑖 2 }
{𝑖𝛼
𝛽 𝑥 𝑦 0+ 0−
(9)
2
2
𝑖𝛼_𝑅𝑎𝑡𝑒𝑑
+𝑖𝛽_𝑅𝑎𝑡𝑒𝑑
Figure 2 shows the mode of operation factor (kM) versus
number of phases for 1 OCF. It shows that MT gives
approximately 6-17% of improvement in a compared to ML
mode, with the improvement being increasingly significant
for higher number of phases.
a
0.7236
0.7710
0.8119
0.8629
Maximum Torque, MT (p.u)
Ploss
Coefficients, 𝑲
1.5279
𝑲𝟏 = -1
𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
1.4013
𝑲𝟏 = -0.6484 𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
𝑲𝟓 = 0
𝑲𝟔 =
𝑲𝟕 = -0.4972 𝑲𝟖 =
1.3003
𝑲𝟏 = -0.5864 𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
𝑲𝟓 = -0.4136 𝑲𝟔 =
𝑲𝟕 = 0
𝑲𝟖 =
1.1937
𝑲𝟏 = -0.4250 𝑲𝟐 =
𝑲𝟑 = 0
𝑲𝟒 =
𝑲𝟓 = -0.3747 𝑲𝟔 =
𝑲𝟕 = 0
𝑲𝟖 =
𝑲𝟗 = -0.2003 𝑲𝟏𝟎 =
𝑲𝟏𝟏 = 0
𝑲𝟏𝟐 =
0
-0.2361
0
-0.3681
0
0
0
-0.1935
0
-0.2198
0
-0.1355
0
-0.0820
0
0.0347
Table 1: Reference x-y, 0+-0- Currents under Fault Tolerant ML and MT Modes of Operation for Symmetrical Five-, Six-,
Seven- and Nine-Phase Machine.
The overall derating factor performance with 1OCF
comparing S5, S6, S7 and S9 is illustrated in Fig. 1. Since the
post-fault current and torque production is proportional to a
and a2 respectively in induction machine [11], the derating
factor becomes significant as the number of phases reduced in
ML mode, with more improvement for higher number of
phases. However, with each additional phase added, the
improvement gained becomes increasing marginal where the
gain is seen to saturate from S7 to S9.
2
ia
1.5
ib
ic
id
The outcomes of the optimization results are given in
Table 1, including the optimized coefficients as well as the
performance indicators.
4 Simulation Results
To visualise and verify the optimization results in Table I,
simulation studies have been conducted using Matlab
Simulink. Phase currents for the studied machines pre- and
2
ie
Currents (p.u.)
Currents (p.u.)
0
-0.5
-1
0.005
0.01
0.015
0.02
0.025
Time (s)
id
ie
if
0.5
0
-0.5
-1
0.03
0.035
-2
0
0.04
0.005
0.01
0.015
(a)
2
0.02
Time (s)
0.025
0.03
0.035
0.04
(b)
ia
1.5
ib
ic
id
ie
if
2
ig
ia
1.5
ib
ic
id
ie
if
ig
ih
ii
1
Currents (p.u.)
1
Currents (p.u.)
ic
-1.5
-1.5
0.5
0
-0.5
-1
0.5
0
-0.5
-1
-1.5
-1.5
-2
0
ib
1
1
0.5
-2
0
ia
1.5
0.005
0.01
0.015
0.02
0.025
Time (s)
0.03
0.035
-2
0
0.04
0.005
0.01
0.015
0.02
0.025
Time (s)
0.03
0.035
0.04
(c)
(d)
Figure 3. Stator phase current waveforms for healthy and ML modes (with 1OCF) for
symmetrical multiphase machines a) S5, b) S6, c) S7 and d) S9.
2
ia
1.5
ib
ic
id
2
ie
0.5
0
-0.5
-1
-1.5
id
ie
if
0
-0.5
-1
-1.5
0.005
0.01
0.015
0.02
0.025
Time (s)
2
0.03
0.035
-2
0
0.04
0.005
0.01
0.015
0.02
Time (s)
0.025
0.03
ic
ie
0.035
0.04
(b)
ia
1.5
ib
ic
id
ie
if
2
ig
ia
1.5
1
ib
id
if
ig
ih
ii
1
Currents (p.u.)
Currents (p.u.)
ic
0.5
(a)
0.5
0
-0.5
-1
-1.5
-2
0
ib
1
Currents (p.u.)
Currents (p.u.)
1
-2
0
ia
1.5
0.5
0
-0.5
-1
-1.5
0.005
0.01
0.015
0.02
Time (s)
0.025
0.03
0.035
0.04
-2
0
0.005
0.01
0.015
0.02
Time (s)
0.025
0.03
(c)
(d)
Figure 4. Stator phase current waveforms for healthy and MT modes (with 1OCF) for
symmetrical multiphase machines a) S5, b) S6, c) S7 and d) S9.
0.035
0.04
2
2
Ploss S5
1.5
Ploss (p.u.)
Ploss (p.u.)
1.5
1
Healthy
MT
ML
1
Healthy
0.01
0.02
0.03
Time (s)
0.04
0.05
0
0
0.06
0.01
0.02
(a)
0.04
0.05
2
Ploss S7
0.06
Ploss S9
1.5
Ploss (p.u.)
1.5
Ploss (p.u.)
0.03
Time (s)
(b)
2
1
Healthy
MT
ML
1
Healthy
MT
ML
0.5
0.5
0
0
MT
ML
0.5
0.5
0
0
Ploss S6
0.01
0.02
0.03
Time (s)
0.04
0.05
0.06
0
0
0.01
0.02
0.03
Time (s)
0.04
0.05
0.06
(c)
(d)
Figure 5. Ploss under healthy, ML and MT modes (with 1OCF) for symmetrical multiphase machines a) S5, b) S6, c) S7 and d)
S9.
post-fault have been plotted based on the information in Table
I. The simulation has been set to maintain the same α-β
magnitude (hence torque/power) in pre- (𝑡 < 0.02𝑠) and post(𝑡 > 0.02𝑠) fault situations. Results for S5, S6, S7 and S9
under ML and MT mode in healthy and after faulty condition
are shown in Fig. 3 and Fig. 4 respectively.
Using ML criterion, the phase currents increases after
fault with the pre-fault current being 0.6813, 0.6882, 0.7043
and 0.7403 times the maximum post-fault currents for S5, S6
S7 and S9 respectively, in accordance with their respective
derating factor in Table 1.
If MT criterion used, the phase currents increases after
fault with the pre-fault current being 0.7236, 0.7710, 0.8119
and 0.8629 times the maximum post-fault currents for S5, S6
S7 and S9 respectively.
It is worth noting that ML will leads to unequal phase
currents. This can be seen from Fig. 3(a) whereas S5 produces
two sets of currents (ib and ie) and (ic and id) with 1.4678 and
1.2631 times of its rated value respectively. S6 however
results in two sets of equal phase currents (ib and if), (ic and
ie) and one phase current (id) with different amplitude. On the
other hand, MT mode gives rise to equal current amplitudes
in the remaining phases after fault, even though equalisation
of current amplitude was not specified as an optimization
objective.
Finally, Fig. 5 shows the comparison of normalised stator
copper losses Ploss for S5-S9 under healthy, ML and MT
modes respectively. The healthy state has been set to (𝑡 <
0.02𝑠), ML mode at (0.02𝑠 < 𝑡 < 0.04𝑠) and MT mode at
(0.04𝑠 < 𝑡 < 0.06𝑠), such that instantaneous power loss for
each mode over one fundamental cycle can be observed. It is
worth noting that for all studied machines, fault tolerant
operation under single OCF resulted in higher instantaneous
power loss (than healthy case) which is oscillating with twice
the fundamental frequency. As the number of phase increases,
the increase in average Ploss becomes less significant.
Furthermore, it can be seen that MT mode give more torque at
a marginal increase in stator copper losses. For S5, MT mode
gives 6.2% more torque increase at a cost of 1.9% additional
loss; while for S9, a 16.6% gain in torque with 2.3% increase
in loss is obtained when MT mode is chosen over ML mode.
This implies that generally, MT mode gives better
performance with marginal penalty in power loss. This is
more so as the number of phases increase.
5 Conclusion
In this paper, an analysis on the fault tolerant capability of
several popular symmetrical multiphase machines under one
open-circuit fault has been presented. The paper considers
comparison of minimum loss, maximum torque and derating
factor under fault scenario for S5, S6, S7 and S9.
Based on the theoretical and simulation study, it can be
concluded that MT mode is more favourable than ML mode,
since the gain in torque/power is generally more significant
than the increase in power losses. With every additional phase
increased, the gain in derating factor (hence torque/power)
become more marginal. It is hence not advantageous to opt
for high number of phases, as the gain in fault tolerance will
not justify the increase in complexity and cost. Five or sixphase machines are sufficient in terms of cost effectiveness.
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