A novel equivalent flux slidingmode observer (SMO) is proposed for dual threephase interior permanent magnet synchronous motor (DTIPMSM) drive system in this paper. The DTIPMSM has two sets of Yconnected stator threephase windings spatially shifted by 30 electrical degrees. In this method, the sensorless drive system employs a flux SMO with soft phaselocked loop method for rotor speed and position estimation, not only are lowpass filter and phase compensation module eliminated, but also estimation accuracy is improved. Meanwhile, to get the regulator parameters of current control, the inner current loop is realized using a decoupling and diagonal internal model control algorithm. Experiment results of 2MWlevel DTIPMSM drives system show that the proposed method has good dynamic and static performances.
1. Introduction
Multiphase motor drive system has been widely used in many applications, especially for highpower applications for their advantages compared to the standard threephase realizations, such as lower torque pulsations, less dclink current harmonics, reduced rotor harmonic currents, higher power per ampere ration for the same machine volume, etc.
[1

4]
. Among different multiphase motor drive solutions, one of the most widely discussed is the VSI fed dual threephase induction machine, having two sets of windings spatially shifted by 30 electrical degrees with isolated neutral points, and there are many literatures for induction, but relatively few research for multiphase permanent magnet synchronous motor (PMSM), especially for highpower PMSM.
In many high performance variable speed AC motor drives, field oriented or vector control is utilized, and rotorposition mechanical sensor is typically required in this method. However, the presence of mechanical sensor presents several drawbacks, such as increases the cost and size of motors, reduces system reliability, etc. Sensorless control technology can achieve the rotor position and speed estimation through exploiting the electrical information about the motor winding, and using a certain control algorithm, which represents the development direction of the AC motor drive system.
So far, several algorithms have been suggested in the recent literatures to estimate the rotor position and the speed of the motor. In the flux estimation methods
[5]
, the rotor flux is estimated by using the integral of the difference between the phase voltage and stator resistance voltage, but these methods are sensitive to machine parameter changes, especially the phased resistance. Drift and saturation problems may cause the controller to lose its synchronization ability consequently, especially at low speed. To get better performance, several improvement schemes to flux estimation have been undertaken in
[6

9]
. However, some methods designed for surface PMSM are not used for IPMSM, due to the differences between stator inductance on
d
axis and
q
axis.
Slidingmode observer (SMO) is an attractive solution compared with other algorithms due to several benefits, such as high state estimation accuracy, excellent dynamic properties, robustness to parameter variations, and the ability to handle nonlinear system like the IPMSM very well
[10

13]
. In this paper, a novel equivalent flux estimation based on SMO technology is proposed for DTIPMSM in a stationary reference frame, which also can be applied to both threephase surface PMSM and IPMSM. To simplify the structure of the controller, an equivalent flux linkage concept is employed in this method, and the conventional switch sign function is replaced by the sigmoid function to reduce system chattering; Moreover, a minimum four current regulators are presented to obtain better control performance. Simulation and experimental results will be presented to demonstrate the feasibility of the proposed control method.
2. Mathematical Mode of DTIPMSM
Using the vector space decomposition technique
[14]
, the machine model can be decoupled into three orthogonal subspaces, which are denoted as (
α
,
β
), (
z
_{1}
,
z
_{2}
) and (
o
_{1}
,
o
_{2}
). For machines with distributed windings, only (
α
,
β
) components contributed to the useful electromechanical energy conversion, whereas (
z
_{1}
,
z
_{2}
) and (
o
_{1}
,
o
_{2}
) components only produce losses. An amplitude invariant decoupling transformation is used as
Transformation (1) is the Clark’s matrix for DTIPMSM motor, (
o
_{1}
,
o
_{2}
) components are omitted from the consideration since the machine has two isolated neutral points. A rotational transformation is applied next to transform the (
α
,
β
) components into a synchronously rotating reference frame (
d
,
q
), which is suitable for field oriented vector control, i.e.,
The circuit equation of DTIPMSM on the
d

q
rotating coordinate and (
z_{1}
,
z_{2}
) coordinate are given respectively by
where,

[uduq]Tvoltage on thedqrotating frame;

[idiq]Tcurrent on thedqrotating frame;

[iz1iz2]Tcurrent on thez1z2frame;

[LdLq]Tstator inductance on thedqrotating frame;

Lzstator leakage inductance;

Rstator resistance;

p=d/dtdifferential operator

ωeangular velocity at electrical angle;

ψfPM flux linkage
Transforming (3) into stationary reference frame
α

β
axis, (5) is derived
where, [
u_{α} u_{β}
]
^{T}
is the stator voltage on the
α

β
axes, [
i_{α} i_{β}
]
^{T}
is the stator current on the
α

β
axes, and with

Lα=L0+L1cos2θe,Lβ=L0L1cos2θe,

Lαβ=L1sin 2θe,L0= (Ld+Lq) / 2,L1= (LdLq) / 2.
Eq. (5) contains 2
θ_{e}
term, which is not easy for mathematical process. To eliminate the 2
θ_{e}
term, term, the impedance matrix is rewritten symmetrically like
where,
is defined as equivalent flux linkage.
The circuit equation on the on the
αβ
axis can be derived as (7)
where,
θ_{e}
is the rotor position in electrical radians, the PM flux linkage
ψ_{α,β}
projected onto the
αβ
axis can be represented as
From the new model (7)(8), the DTIPMSM can be described by a linear state equation as (9). Here, the state variables are stator current
i
and PM flux linkage
ψ
. Assuming that the electrical system’s time constant is smaller enough than the mechanical one, i.e.,
the velocity
ω_{e}
is regarded as a constant parameter.
The
W
_{2}
term in (9) is linearization error, this term appears only when
i_{d}
or
i_{q}
is changing. However, under the velocity control this happens in a very short time because of the high response of the current control loop. Besides, the proposed SMO has an embedded lowpass filter which can cut off the effect of
W
_{2}
.
3. Flux Slidingmode Observer Design
 3.1 Design of the observer
To achieve the flux linkage
ψ
, the proposed observer as (10) is designed based on the stator current model (9).
where, “^” denotes the estimated quantities, sgn(⋅) is the sign function,
K
is the designed parameter, and the (10) is the conventional SMO. To reduce the chattering phenomenon, the sign function is replaced by a continuous function, i.e., the sigmoid function, which is defined as
Here,
a
is a positive constant that can be adjusted the slope of the sigmoid function. And then, the SMO can be rewritten as
Assuming that the motor parameters
R, L_{d}
and
L_{q}
exist parameter errors, defined as follows
where, “^” denotes the estimated quantities, Δ
R
, Δ
L_{d}
and Δ
L_{q}
are the stator resistance error, the
d

q
inductance errors, respectively.
Considering the parameter variations of the motor, the observer (12) can be equivalent to
where,
W
_{1}
is the parameter error input matrix, defined as follows
with
and
K
is the designed constant parameter, satisfies
The sliding hyperplane is defined upon the stator current, i.e.,
s
= [
s_{α} s_{β}
]
^{T}
=

i
= 0. So the stator current estimation error dynamic function can be obtained from (9) and (14) as follows
where, ‘~’ denotes the estimated error, such as
 3.2 Lyapunov stability analysis
In order to prove the stability of the designed observer, the following Lyapunov function candidate is considered.
Differentiating (18) with respect to time and substituting (17) into it, then the following equation is obtained
Assuming that max(
W
_{1α}
, 
W
_{1β}
 ) <
k
′, the equation (19) can be given as
According to the Lyapunov stability theory, (20) must be obeyed to guarantee that the observer is stable, i.e.,
< 0 , the parameter
k
_{1}
can be chosen as
Hence,
V
decays to zero, then
and
are equal to zero. After slidingmode motion occurs, i.e.,
, the followings equation can be obtained from (17).
The flux linkage estimation error dynamic function can be obtained from (9) and (14) as follows
Substituting (22) into (23), then the following equation is obtained
Hence, if only the parameter
k
_{2}
is a positive gain, the (24) ensures the errors converge to zero, and the convergence rate of error dynamic is determined by
k
_{2}
. However, when
ω_{e}
is close to zero, the computed parameter may become illconditioned. To avoid this undesirable effect, we choose the observer poles as
k
_{2}
=
γ

ω_{e}
; hence the observer gain is calculated as
 3.3 Estimation of speed and position
Conventionally, the rotor position can be estimated by using arcangent function
However, the existence of noise and harmonics may influence the accuracy of the position estimation, especially at very lowspeed, the obvious estimation error may occur using the arcangent function. To improve the position estimation for mitigation of the adverse influence, a phaselocked loop (PLL) method is employed. This scheme can be comparatively represented as a simple linearized closedloop system shown in
Fig. 1
.
Scheme of position estimation through PLL
As shown in
Fig. 1
, after the normalization of the flux linkage, the equivalent position error signal can be expressed as
The estimation of the electrical angular speed of the rotor is obtained using PI controller, i.e.,
where, the nonnegative gains
k_{p}
and
k_{i}
are selected as
[15]
where,
a
is the design parameter.
4. Current Control of DTIPMSM
From the model of the DTIPMSM, it seems that the two current loop control techniques of the threephase motors can be easily extended to the sixphase drives, as depicted in
Fig. 2
. The phase currents are applied to the transformation matrix (1) to obtain the stator current components in the stationary (
α
,
β
) reference frame. The (
d
,
q
) current components in the synchronous reference frame are obtained by using a rotor position from the flux SMO. The outputs of the PI current regulators, after an inverse Park transformation, are the stator voltage reference components in stationary reference frame (
α
,
β
) to be applied to the SVPWM modulator.
The conventional two current loop controls for DTIPMSM
The two current loop control strategy depicted in
Fig. 2
is very simple, but it is not able to compensate for the inherent asymmetries of the drive. Due to the small asymmetries in the stator windings and supply voltages, the two sets of threephase stator currents have rather different amplitudes depending on the operational conditions, and the harmonic currents in (
z
_{1}
,
z
_{2}
) subsystem is not eliminated effectively. To obtain better controller performance, a current control technology with four current loop regulators in (
d
,
q
) and (
z
_{1}
,
z
_{2}
) subsystems are adopted in this paper. Moreover, in order to overcome the current coupling terms on (
d
,
q
) subsystem, the decoupling and diagonal internal model control (DIMC)
[8
,
16]
structure for current control in the drive. The command voltage are now given by
where,
β
is the desired closedup bandwidth as determined by the specified rise time of the current controller. The DIMC involves only a single parameter, so tuning of the controller to give specified performance is easier. The speed controller outputs the
q
axis current reference
i_{q}
^{*}
, and the
z
_{1}
axis and
z
_{2}
axis current references
i
_{z1}
^{*}
and
i
_{z2}
^{*}
are set to zero, so the overall block diagram of the DTIPMSM control scheme can be shown in
Fig. 3
.
To obtain better results and implement simply, it is suggested in this paper the dual threephase SVPWM technique as shown in
Fig. 4
is used for modulation in this paper, the detailed discussion can be seen in
[17]
. It is has many advantageous, such as existing algorithms and tested threephase modulation methods can be effectively utilized, which can save time and trouble. It also makes the method computationally efficient since years of extensive study and wide usage have made space vector modulation a very simple task.
Block diagram of the sensorless flux SMO control drive scheme
The dual threephase space vector classification PWM technique
5. Simulation and Experiment Results
To check the feasibility of the proposed rotor position and speed estimation schemes, the simulation and experimental studies are carried out with a reference to a 2 MW DTIPMSM drives system. The block diagram of the sensorless flux observer control drives is presented in
Fig. 5
, and the machine parameters are given in
Table 1
. The same controller parameters are used both in simulation and experimental results. A stator current controller bandwidth
β
of 30rad/s, and the PLL system bandwidth
a
of 3rad/s are chosen. The parameters of flux observer are chosen as followings:
k
_{1}
+
k
′ = 300,
L_{d}
⋅
γ
= 50.
the parameters of DTIPMSM
the parameters of DTIPMSM
 5.1 Simulation results
Figs. 5
and
6
show the two sets of simulation waveforms when the reference speed is a step signal. In the simulation, the reference speed is changed from 5 to 17 r/min, and the load torque is 5.6 × 10
^{5}
N.m.
The simulation waveforms obtained by the conventional SMO method using sign function:(a) Actual and estimated speeds; (b) Actual rotor position, estimated rotor position, and estimated error; (c) Estimated flux ψ_{α,β}.
The Simulation waveforms obtained by the conventional method using sign function: (a) Actual and estimated speeds; (b) Actual rotor position, estimated rotor position, and estimated error; (c) Estimated flux ψ_{α,β}.
Fig. 5
displays the simulation waveform obtained by the conventional SMO method using a sign function.
Fig. 6
shows the simulation waveform obtained by the method proposed in this paper. It can be seen from
Figs. 5
that the sign function can cause to chattering phenomenon, the low pass filter and phase compensation part must be used, and therefore, the rotor position estimation accuracy is not high. However, it can be seen from
Fig. 6
that the chattering phenomenon of the estimated rotor position and speed is reduced, and the accuracy of rotor position estimation is improved to some extent.
Fig. 7
shows the estimation performance of the proposed method when the parameters of DTIPMSM vary. In the simulation, the reference speed is 17 r/min, and the load torque is 5.6 × 10
^{5}
N m. It can be seen from
Fig. 7
that, when the resistance or the inductance of the motor changes, the estimated speed can still converge to the actual value, which verifies the robustness of the proposed approach.
Simulation waveforms when the parameters of DTIPMSM are changed: (a) Waveforms when the resistance is changed; (b) Waveforms when the inductance is changed.
 5.2 Experiment results
The effectiveness of the proposed sensorless control scheme for 2 MWlevel DTIPMSM drive is tested using the experiment setup shown in
Fig. 8
. In the experiment setup, a highpower backtoback converter system is used to feed the drive system, and the controller and machine parameters are same with the simulation, and the sampling period of the control system is set as 50
^{μs}
, the deadtime is set as 10
^{μs}
, the switch frequency is set as 1kHz. The overall system control algorithm is developed in Matlab/ Simulink, followed by implementation on an OPAL RTLab (Realtime Digital Simulator) controller board. The motor parameters are given in
Table 1
.
The overall experiment setup
Figs. 9
and
10
show the control performance when the DTIPMSM is running with the reference speed steps up from 2 to 10 r/min.
Fig. 9
shows the waveforms when the conventional method based on the sign function and lowpass filter is adopted. As can be seen from
Fig. 9
, due to the use of the sign function, the chattering of the estimated rotor position and speed obtained by the traditional method is significant.
Operating waveforms obtained by the conventional control method using a sign function: (a) From top to bottom are the estimated rotor speed, rotor error and estimated fluxlinkage, respectively; (b) From top to bottom are the estimated rotor position, and estimated error, respectively.
Operating waveforms obtained by the proposed method using a sigmoid function: (a) From top to bottom are the estimated rotor speed, rotor error and estimated fluxlinkage, respectively; (b) From top to bottom are the estimated rotor position, and estimated error, respectively.
When the proposed flux SMO is employed,
Fig. 10
presents a good dynamic performance of estimated speed, the estimated speed follows the reference speed very well, and the speed estimated error is very small. Especially, there is a small ripple (±0.1r/min) when DTIPMSM drive runs in a stable speed range. And the chattering is reduced when the sign function is replaced by the sigmoid function, and the waveforms of the estimated rotor position and speed obtained by the proposed flux SMO are smooth.
6. Conclusion
In this paper, a novel flux linkage slidingmode observer for DTIPMSM sensorless drives has been proposed. To simply the machine model, an equivalent flux linkage concept is employed. The sign function is replaced by the sigmoid function to reduce the chattering, and the conventional SMO is improved. From the design process we can see that the presented observer has a simple structure with less control parameters. Meanwhile, the details of the observer parameters and the rotor position and speed estimators are given. The feasibility of the proposed scheme is verified and confirmed through simulation and extensive experiments.
Acknowledgements
The work described in this paper was fully supported by a grant from the Major State Basic Research Development Program of China (No.2013CB035601), and the Project supported by the Program for New Century Excellent Talents in University of Ministry of Education of China (No.NCET110871).
BIO
JianQing Shen received the Ph.D. degree from Naval University of Engineering (NUE) in 2007, and is presently a research fellow at NUE. His areas of research are control method design of new type of electrical machine, power electronics and its applications in industry and power system, and power system integration technique applications
Lei Yuan received his B.S degree from Naval University of Engineering (NUE) in 2010, and is currently working toward the Ph.D. degree at NUE. His current interests include power electronics and the control method design of new type of electrical machine.
MingLiang Chen received the Ph.D. degree from Naval University of Engineering (NUE) in 2008, and is presently an associate research fellow at NUE. His areas of research are control method design of new type of electrical machine, power electronics and its applications in industry and power system.
Zhen Xie received the B.S and Ph.D. degrees from Naval University of Engineering (NUE) in 2009 and 2013. His current interests include highpower power electronics technique applications.
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