Current Sensor Fault Diagnosis and Tolerant Control for Nine-Phase PMSM Drives Based on Improved Axis Rotation

Abstract

Fault probability rises with an increase in the number of current sensors in multi-phase permanent synchronous motor (PMSM) drives. This paper proposes an improved axis rotation method for fault diagnosis and tolerant control to make the multi-phase PMSM drives against current sensor loss signal and gain faults. This method can effectively diagnose and distinguish faults without selecting a threshold value, and the degree of fault can be further estimated. The proposed method makes current sensor fault diagnosis and tolerant control become an integrated module. The validity and accuracy of the proposed method is verified by different fault diagnoses and tolerant control experiments of a 9 kw nine-phase PMSM drive.

1. Introduction

Multi-phase motor drives have strong application requirements in the high-reliability of multi-electric aircraft, ships and EVs due to their higher power density, lower torque ripple and superior fault tolerant capability [1,2,3]. However, the fault probability rises with an increase in the number of current sensors. In the operation process of multi-phase motor drives, over-voltage, over-current, aging and other harsh working environments may cause the current sensor output signal to be inaccurate [4,5]. The current sensor faults should be diagnosed and tolerated rapidly and effectively in applications with high-reliability requirements. The different current sensor faults will cause a different degree of distortion in the current feedback signals. It is necessary to accurately detect different types of faults in order to design corresponding fault tolerant control strategies. Therefore, it is significant to study the diagnosis and tolerant control of current sensor faults in multi-phase PMSM drives [6,7,8].

Current sensor faults have been effectively diagnosed and located in many existing literatures. The extended Kalman filter is suggested to detect the loss signal fault in [9]. The Kalman filter is merged with a system reconfiguration under the faults of the current sensor methods in [10] to diagnose the faults. A valid sliding mode observer (SMO) is adopted in [11] to diagnose the current sensor loss signal fault in PMSM sensorless control systems. The current error module is built based on the estimated signal error of the other health current sensors. Three reliable observers are suggested in [12] to observe the real-time state of three current sensors, therefore the loss signal fault can be located by the observers. Ref. [13] proposes an observer-based strategy relying on the adaptive threshold for fault location and tolerance. However, these observer models are easily affected by the motor parameters and operating environments. The signal-based methods need no complex calculation; only by signal processing can accurate sensor fault detection and tolerant control be obtained. Current signal analysis is combined with an SMO in [14], and the sensor loss signal fault is diagnosed through the space vector error-projection algorithm. The phase and line voltage signal deviations strategy are presented in [15]. The loss signal faults can be diagnosed and isolated rapidly in both rectifier and inverter mode.

The above methods are only for current sensor loss signal fault detection and tolerant control. In addition to loss signal faults, the gain faults should also be detected and tolerated effectively. Fortunately, many efforts have been devoted to the research of loss signal and gain fault diagnosis. The state-observer-based algorithm shows its positive performance in current sensor loss signal and gain fault detection in [16,17]. In [18], the current estimations from an improved open-loop observer are adopted to obtain the fault detection and tolerant control. An algorithm programmed into the FPGA as a general controller is considered in [19]. For the three-phase PMSM systems, the fault diagnosis range in [19] is extended to loss signal and gain faults. The signal-based detection method with normalization in [20,21] has been widely used to achieve fault detection and tolerant control. The AC component of current analysis error is extracted by an improved filter in [22], and gain fault of the linear motor system is distinguished with other faults by the different frequency components. A modified algorithm combining a delay function and the current space vector is addressed in [23], which can detect and isolate loss signal and gain faults. Axis rotation is adopted in [24] for deciding the correct estimated value of the fault current sensor, and it shows good characteristics in the field of electric vehicle sensor fault detection and fault tolerance.

Considering different current sensor fault diagnosis and tolerant control for multi-phase PMSM drives, this paper proposes a method to diagnose both loss signal faults and gain faults based on improved axis rotation. Different from the existing literature, this signal-based method can accurately diagnose and effectively distinguish the loss signal and gain faults without additional hardware. By eliminating the selection of threshold value, the adaptability of this method to different operating conditions is also enhanced. Moreover, the degree of gain faults can be estimated quantitatively and it can be used to further tolerant control. Different types of current sensor faults are matched with corresponding fault tolerant control strategies. The method makes current sensor fault diagnosis and tolerant control become an integrated module. This article is structured as follows: The topology of nine-phase PMSM drives is introduced in Section 2. The influence on currents of current sensor loss signal faults and gain faults are analyzed in Section 3. The proposed diagnosis and tolerant control method are illustrated in Section 4 and Section 5, respectively. The corresponding experiments of a 9 kw nine-phase PMSM are carried out to verify the effectiveness and accuracy of the proposed method in Section 6. A conclusion is drawn in Section 7.

2. Influence of Current Sensor Faults on Drive Systems

The nine-phase PMSM in this article is composed of three sets of three-phase open-end windings which are spatially shifted π/9 degrees. Each phase is equipped with an independent current sensor. An H-bridge inverter circuit drives nine-phase effectively with more control, reliability, and independence. Figure 1a shows the nine-phase PMSM windings structure, and Figure 1b shows the nine-phase H-bridge drive topology.

The probability of faults increases with the number of current sensors required for the nine-phase PMSM drives. The fault diagnosis and tolerant control of nine-phase PMSM drives are shown in Figure 2.

3. Influence of Current Sensor Faults on Motor Drives

Based on the frequency of faults and the degree of negative impact on the motor drives, the typical current sensor faults distinguished by output characteristics can be classified as gain fault and loss signal fault. Taking phase-A₁ current sensor faults as an example, the measured phase currents of each sensor can be expressed as:

$$\begin{array}{l}{i}_{\mathrm{A}1}={\lambda i}^{\ast }{}_{\mathrm{A}1}={\lambda I}_{\mathrm{am}}\cos (\omega t+{\theta }_{\mathrm{ori}})\\ i_{\mathrm{A}2}=i^{\ast }{}_{\mathrm{A}2}\\ \hspace{1em}⋮\\ i_{C3}=i^{\ast }{}_{C3}\end{array}$$

(1)

where, i^*_A1, i^*_A2 … i^*_C3 are the real phase currents, i_A1, i_A2 … i_C3 are the measured values by nine-phase current sensors, λ is the gain coefficient of fault current sensor, I_am is the phase current amplitude, θ_ori is the initial phase angle and ω is the electrical angular frequency respectively.

In normal conditions, λ = 1; when loss signal faults occur, λ = 0; when gain faults occur, λ > 1. The expression of i_d1 and i_q1 in the d-q coordinate is in Equation (2).

$$\left[\begin{array}{c}{i}_{d1}\\ i_{q1}\end{array}\right]=\frac{2}{9}\ast \left[\begin{array}{ccc}\cos {\theta }_e& \dots & \cos {(\theta }_e-14\pi /9)\\ -\sin {\theta }_e& \dots & -\sin {(\theta }_e-14\pi /9)\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{A}1}\\ \dots \\ i_{\mathrm{C}3}\end{array}\right]$$

(2)

Assuming that the fault of the phase-A₁ current sensor in the multi-phase motor drives has little effect on the normal phase current amplitude, Equation (1) is substituted into Equation (2), then the measured value of i_{q1_m} can be expressed.

$$\begin{array}{c}{i}_{q1}{}_{\_m}=i_{q1}-\frac{2}{9}\ast (\lambda -1){\ast i}_{\mathrm{A}1}\ast \sin {\theta }_e-\frac{2}{9}\ast \sin {\theta }_e\\ =i_{q1}\underset{\mathrm{A}\mathrm{C}}{\underbrace{-\frac{(\lambda -1)I_{am}\sin (2{\omega t+\theta }_{ori})}{9}}}\underset{\mathrm{D}\mathrm{C}}{\underbrace{+\frac{(\lambda -1)I_{am}\sin ({\theta }_{ori})}{9}}}\end{array}$$

(3)

• In normal conditions, λ = 1, DC component is zero, so i_q1 is equal to i_{q1_m} with no bias.
• When loss signal faults occur, the real phase currents are not measured by current sensors. The AC component are zero-mean pulsations; DC component is a bias less than zero due to λ = 0.
• When gain faults occur, the AC component is still zero-mean pulsations, but the DC component is a bias greater than zero due to λ > 1.

Therefore, the i_{q1_m} impacted by the DC component can be used as the characteristic quantity to distinguish normal conditions, loss signal faults and gain faults. In addition to the fault type, the location of the fault point is also essential, so it is necessary to further propose a perfect diagnosis strategy combined with i_{q1_m}.

4. Diagnosis for Current Sensors Faults

4.1. Improved Axis Rotation

From the transformation matrix of the semi-symmetric nine-phase PMSM natural coordinate system to the synchronous stationary coordinate system (Clark), it can be seen that in the standard transformation shown in Figure 3, A₁-axis and α₁-axis are in the same direction. The component of phase current i_A1 on β₁-axis is zero and the component on α₁-axis is 100% i_A1.

Therefore, from the i_α1 and i_β1 in Equation (4), only the current sensor fault of phase-A₁ has the least influence on i_β1 and the largest influence on i_α1 among all nine phases.

$$\left[\begin{array}{c}{i}_{\alpha 1\_1}\\ i_{\beta 1\_1}\end{array}\right]=\frac{2}{9}\ast \left[\begin{array}{cccc}1& \cos \frac{\pi }{9}& \dots & \cos \frac{14\pi }{9}\\ 0& \sin \frac{\pi }{9}& \dots & \sin \frac{14\pi }{9}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{A}1}\\ i_{\mathrm{A}2}\\ \dots \\ i_{\mathrm{C}3}\end{array}\right]$$

(4)

where, i_{α1_1} and i_{β1_1} are fundamental wave currents of α and β when phase-A₁ is used as reference axis.

In general, regarding the other phases, when different phase-N is selected as the reference axis, the Clark matrix will shift different angles θ_N, and current sensor faults of reference phase-N has the least influence on the corresponding i_{β1_N}, and the largest influence on i_{α1_N}. The general formula of i_{α1_N} and i_{β1_N} can be obtained in Equation (5).

$$\left[\begin{array}{c}{i}_{\alpha 1-N}\\ i_{\beta 1-N}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}\cos (0-{\theta }_N)& \cos (\frac{\pi }{9}-{\theta }_N)& \dots & \cos (\frac{14\pi }{9}-{\theta }_N)\end{array}\\ \begin{array}{cccc}\sin (0-{\theta }_N)& \sin (\frac{\pi }{9}-{\theta }_N)& \dots & \sin (\frac{14\pi }{9}-{\theta }_N)\end{array}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{A}1}\\ i_{\mathrm{A}2}\\ \dots \\ i_{\mathrm{C}3}\end{array}\right]$$

(5)

where, the θ_N of stator windings of semi-symmetric nine-phase motor system can be expressed as:

$${\theta }_N\in [\begin{array}{ccccccccc}0& \frac{2\pi }{9}& \frac{4\pi }{9}& \frac{6\pi }{9}& \frac{7\pi }{9}& \frac{8\pi }{9}& \frac{12\pi }{9}& \frac{13\pi }{9}& \frac{14\pi }{9}\end{array}]$$

(6)

When different phases are selected as reference axes, the expressions of i_{α1_N} and i_{β1_N} can be obtained in Equation (7).

$$\begin{gathered} \left[\begin{array}{c}{i}_{\alpha 1-1}\\ i_{\alpha 1-2}\\ \dots \\ i_{\alpha 1-9}\end{array}\right]=\left[\begin{array}{c}{i}_{\mathrm{A}1}\ast 1+i_{\mathrm{A}2}\ast \cos \pi /9+\dots +i_{C3}\ast \cos 14\pi /9\\ i_{\mathrm{A}1}\ast \cos \pi /9+i_{\mathrm{A}2}\ast 1+\dots +i_{\mathrm{C}3}\ast \cos 13\pi /9\\ \dots \\ i_{\mathrm{A}1}\ast \cos 14\pi /9+i_{\mathrm{A}2}\ast \cos 13\pi /9+\dots +i_{\mathrm{C}3}\ast 1\end{array}\right] \\ \left[\begin{array}{c}{i}_{\beta 1-1}\\ i_{\beta 1-2}\\ \dots \\ i_{\beta 1-9}\end{array}\right]=\left[\begin{array}{c}{i}_{\mathrm{A}1}\ast 0+i_{\mathrm{A}2}\ast \sin \pi /9+\dots +i_{\mathrm{C}3}\ast \sin 14\pi /9\\ -i_{\mathrm{A}1}\ast \sin \pi /9+i_{\mathrm{A}2}\ast 0+\dots +i_{\mathrm{C}3}\ast \sin 13\pi /9\\ \dots \\ -i_{\mathrm{A}1}\ast \sin 14\pi /9-i_{\mathrm{A}2}\ast \sin 13\pi /9-\dots -i_{\mathrm{C}3}\ast 0\end{array}\right] \end{gathered}$$

(7)

Therefore, the above formula shows that the i_{α1_1} changes the most in all i_{α1_N} when loss signal fault (i_A1 = 0) occurs in phase-A₁, which is equivalent to the i_{α1_1} losing 100% of i_A1 and i_{α1_1} has the lowest mean value in all i_{α1_N}. Also, i_{β1_1} has the smallest change in all i_{β1_N}, and the other eight i_{β1_N} have lost sinπ/9, sin2π/9 … sin14π/9 times of i_A1, respectively.

• In normal conditions, nine i_{α1_N} have the same mean value, and so do nine i_{β1_N}.
• When loss signal fault (i_A1 = 0) occurs in phase-A₁, the above formula shows that the i_{α1_1} changes the most in all i_{α1_N}, which is the equivalent to the i_{α1_1} losing 100% of i_A1, and i_{α1_1} has the lowest mean value in all i_{α1_N}. Also, i_{β1_1} has the smallest change in all i_{β1_N}, and the other eight i_{β1_N} have lost sinπ/9, sin2π/9...sin14π/9 times of i_{α1_1}, respectively.
• When gain fault (i_A1 =λi^*_A1) occurs in phase-A₁, i_{α1_1} has the largest change in all i_{α1_N}, equivalent to i_{α1_1} increased by 100% “(λ − 1)i_A1”, which has the largest mean of all i_{α1_N}. Additionally, i_{β1_1} has the smallest change in all i_{β1_N}. The other phases have gained sinπ/9, sin2π/9 … sin14π/9 times of (λ − 1)i_A1, respectively. i_{β1_1} has the smallest mean in all i_{β1_N}.

Therefore, if the mean value difference of each i_{α1_N} and i_{β1_N} is selected for comparison, the difference corresponding to the fault phase is the largest, which can be used as the characteristic quantity for locating the fault phase. Taking phase-A₁ faults as an example, in combination with Equation (7), the reference vectors of i_{α1_N} and i_{β1_N} in each axis rotation coordinate system are shown in Figure 4.

The proposed method is based on the principle of axis rotation; i_{α1_N} and i_{β1_N} always change in reverse direction after a fault, so the diagnosis variables can be designed by combining them.

4.2. Diagnosis Strategy

Based on the above derivation, this paper proposes a fault diagnosis method of current sensor based on improved axis rotation. The fault diagnosis of a multi-phase motor can be realized by analyzing the sampled current information of the system. Its diagnosis principle is shown in Figure 5.

Step 1: The rotation angle θ_N of different phases in the multi-phase motor should be determined, and the expressions of i_{α1_N} and i_{β1_N} with different reference phases as α₁-axis should be established according to θ_N. Nine-phase motors correspond to 2 × 9 expressions.

Step 2: i_{α1_N} and i_{β1_N} are respectively defined as i_{α1_N} = I_am1cos(ωt − θ_N), i_{β1_N} = I_am2sin(ωt − θ_N), where I_am1 and I_am2 are the current amplitude. M[|i_{α1_N}|] and M[|i_{β1_N}|] are respectively obtained by the absolute-mean processor in Equation (8).

$$\begin{array}{l}\mathrm{M}[|i_{\alpha 1\_N}|]=\frac{\omega }{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{\omega }}|I_{\mathrm{am}1}\cos (\omega t-{\theta }_N)|}\mathrm{d}\mathrm{t}=\frac{2}{\pi }{I}_{\mathrm{am}1}\\ \mathrm{M}[|i_{\beta 1\_N}|]=\frac{\omega }{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{\omega }}|I_{\mathrm{am}2}\cos (\omega t-{\theta }_N)|}\mathrm{d}\mathrm{t}=\frac{2}{\pi }{I}_{\mathrm{am}2}\end{array}$$

(8)

When current sensor fault occurs, the absolute-mean difference between i_{α1_N} and i_{β1_N} is the largest in the axis rotation coordinate system corresponding to the fault phase. Therefore, the difference of M[|i_{α1_N}|] and M[|i_{β1_N}|] can distinguish between fault and health phases.

Step 3: The fault phase of the current sensor can be located by diagnosis variables “D_N = Max(|M[|i_{β1_N}|]-M[|i_{α1_N}|])” after screened by the maximum calculator. The proposed method eliminates the selection of threshold value, and the phase with the strictest maximum in the all D_N corresponding to the nine-phase currents is the fault phase, which greatly reduces the probability of misdiagnosis.

Step 4: The above algorithm realizes the location of the fault current sensor. However, to distinguish between gain faults and loss signal faults, the q₁-axis current i_{q1_m} is used. After processed by a mean processor and combined with Equation (3), the expression of M[i_{q1_m}] can be obtained:

$$\begin{array}{ll}\mathrm{M}[i_{q1}{}_{\_\mathrm{m}}]& =\frac{\omega }{2\pi }{\displaystyle {\int }_0^{\frac{2\pi }{\omega }}{i}_{q1}{}_{\_\mathrm{m}}}\mathrm{dt}\\ & =\mathrm{M}[i_{q1}]-\underset{\mathrm{M}[\mathrm{A}\mathrm{C}]}{\underbrace{\mathrm{M}[\frac{(\lambda -1)I_{\mathrm{am}}\sin (2\omega t+{\theta }_{\mathrm{ori}})}{9}]}}+\underset{\mathrm{M}[\mathrm{DC}]}{\underbrace{\mathrm{M}[\frac{(\lambda -1)I_{\mathrm{am}}\sin ({\theta }_{\mathrm{ori}})}{9}]}}\\ & =i_{q1}+\frac{(\lambda -1)I_{\mathrm{am}}\sin ({\theta }_{\mathrm{ori}})}{9}\end{array}$$

(9)

• In normal conditions, λ = 1, DC = 0, M[i_{q1_m}] = i_{q1_m}. There was no change in i_{q1_m}.
• When loss signal fault occurs, λ = 0, M[AC] is a zero-mean value. So, M[i_{q1_m}] = i_q1-I_amsin(θ_ori)/9.
• When gain faults occurs, λ > 1, M[AC] is still a zero-mean value, but M[i_{q1_m}] = i_q1 + (λ − 1)I_amsin(θ_ori)/9.

Therefore, M[i_{q1_m}] will be biased with different properties when the current sensor fault occurs, and the bias caused by loss signal faults and gain faults must be opposite. Therefore, the next key is to determine the DC component “I_amsin(θ_ori)/9”.

Since i_{q1_m} can be considered to be derived from the nine-phase current through Park transformation in Equation (5), M[i_{q1_m}] is equivalent to the mean action of the nine-phase currents, that is, M[i_{q1_m}] = M[cosθ_e × i_A1] + … + M[cos(θ_e − 14π/9) × i_C3]. Therefore, M[i_{q1_m}] will become “8/9” of its original value after a phase current sensor fault. The expression of M[i_{q1_m}] after faults can be obtained:

• In normal conditions, i_{q1_m} has no DC component, i_{q1_m} = i_q1.
• When loss signal fault occurs, the DC component causes i_{q1_m} to be 1/9 less than the i_{q1_m} in normal conditions, that is, I_amsin(θ_ori)/9 = i_q1/9.

  $$\mathrm{M}[i_q{{}_{1\_}}_{\mathrm{m}}]=i_q{}_1-\frac{I_{\mathrm{am}}\sin ({\theta }_{\mathrm{ori}})}{9}=\frac{8}{9}{i}_q{}_1<i_q{}_1$$

  (10)
• When gain fault occurs, the DC component causes i_{q1_m} more (λ − 1)I_amsin(θ_ori)/9 than i_{q1_m} in normal operation.

  $$\mathrm{M}[i_q{{}_{1\_}}_m]=i_q{}_1+\frac{(\lambda -1)I_{\mathrm{am}}\sin ({\theta }_{\mathrm{ori}})}{9}=i_q{}_1+\frac{(\lambda -1)i_q{}_1}{9}>i_q{}_1$$

  (11)

Therefore, the loss signal faults and gain faults can be distinguished by comparing the bias direction of M[i_{q1_m}] after the faults occur.

Different from the loss signal faults, if the gain coefficient λ can be obtained when the gain fault occurs, the fault tolerant operation can be realized only by adjusting the factor of current sampling. From Equation (11), it can be concluded that the gain fault causes the fault i_{q1_m} to be (λ − 1)i_q1/9 larger than normal i_{q1_m}. So λ can be obtained by Equation (12).

$$\lambda =\frac{9(\mathrm{M}[i_q{{}_{1\_}}_{\mathrm{m}}]-i_q{}_1)}{i_q{}_1}+1$$

(12)

Combining with the diagnosis variables D_N and M[i_{q1_m}], the current sensor gain and loss signal faults can be distinguished and located, the gain coefficient λ of fault degree is obtained, which is prepared for fault tolerant in the next stage. The diagnosis method proposed in this paper is shown in Figure 6.

5. Tolerant Control of Current Sensors Faults

According to Section 2, i_{q1_m} will produce different forms of pulsation and bias when the current sensor faults occur, thus affecting the stable operation of the control system. Therefore, the key to fault tolerant of current sensor depends on the correct estimation of feedback current i^*_d1 and i^*_q1. i_d1 and i_q1 are derived from i_{β1_N} and i_{α1_N}, so as long as the correct i^*_{β1_N} and i^*_{α1_N} is estimated by the appropriate algorithm, the fault tolerant operation of current sensors can be realized.

The fault diagnosis in Section 3 can locate the corresponding phase of the fault current sensor, and then determine the corresponding θ_A1 of axis rotation system. Taking phase-A₁ as a fault example, in its corresponding rotation system, the phase current i_{β1_1} remains unchanged before and after the fault, so i_{β1_1} can be used as the β₁-axis fault tolerant current i^*_β1 after the fault. The calculation of α₁-axis fault tolerant current i^*_α1 is discussed as follows. According to Equation (10), the M[i_{q1_m}] after loss signal fault and its value in normal conditions meet Equation (13):

$$i_q{}_1=\frac{9}{8}\mathrm{M}[i_q{{}_{1\_}}_{\mathrm{m}}]$$

(13)

So α₁-axis fault tolerant current i^*_α1 can be obtained by the anti-Park matrix in Equation (13).

$$i_{{{}_{\alpha }}_1}^{\ast }=\left[\begin{array}{cc}\cos ({\theta }_e-{\theta }_{\mathrm{A}1})& -\sin ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{A}1})\end{array}\right]\left[\begin{array}{c}\frac{9}{8}\mathrm{M}[i_q{{}_{1\_}}_{\mathrm{m}}]\\ i^{\ast }{}_{d1}\end{array}\right]$$

(14)

Among them, the rotation angle θ_A1 of the transformation matrix in Equation (14) should be selected in accordance with the θ_A1 determined by the fault diagnosis module.

After i^*_α1 and i^*_β1 are determined, the feedback current i^*_d1 and i^*_q1 can be obtained from the rotation matrix corresponding to θ_A1.

$$\left[\begin{array}{c}{i}_{d1}^{\ast }\\ i_{q1}^{\ast }\end{array}\right]=\left[\begin{array}{cc}\cos ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{A}1})& \sin ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{A}1})\\ -\sin ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{A}1})& \cos ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{A}1})\end{array}\right]\left[\begin{array}{c}{i}_{\alpha 1}^{\ast }\\ i_{\beta 1}^{\ast }\end{array}\right]$$

(15)

i^*_d1 and i^*_q1 are incorporated into the control system as fault tolerant feedback to realize fault tolerant operation of current sensor fault. Thus, i^*_α1, i^*_β1, i^*_d1 and i^*_q1 are obtained by rotation transformation with the same angle θ_A1. The θ_A1 corresponding to the fault can be determined only by the fault detection module, that is, fault diagnosis and fault tolerant operate in the same rotation transformation system, realizing the integration of diagnosis and fault tolerant. It is worth noting that the combination of fault diagnosis and fault-tolerant control module needs to consider the correspondence of θ_N. As a bridge connecting diagnosis and tolerance, the selection of θ_N determines the accuracy of fault tolerance current i^*_q1. Moreover, for gain faults, fault-tolerant control can be realized directly and accurately only by compensating a calculated gain coefficient λ (Equation (12)) for the fault phase current.

The proposed method used stator currents to diagnosis current sensor faults and obtained tolerant control by estimating and switching the i_q1. Therefore, is strongly adaptable to a different controller. Additionally, the discussion and verification of the proposed method in this paper are operated under a vector control system (Figure 7).

6. Experimental Verification

6.1. Experimental Setup

In order to verify the validity of the fault diagnosis and fault tolerant method proposed in this paper, an experimental platform of nine-phase PMSM was built. The platform consists of a DC power, a controller, a 9 kw nine-phase PMSM, a DC motor, a speed-torque measuring instrument, and a voltage source inverter (VSI). The 9 kw nine-phase PMSM is powered by VSI; H-bridge inverter circuits drive each phase respectively, and the load shaft is connected with a DC generator. The block diagram of the motor control principle is shown Figure 7. The parameters of nine-phase PMSM are shown in

Table 1. Nine-phase PMSM parameters.

Parameters	Value
Power	9 kw
Voltage	234 V
Current	4.6 A
Speed	900 rpm
Torque	95.5 Nm
d-axis inductance	41.2 mH
q-axis inductance	41.2 mH
Armature resistance	2.47 Ω
Magnet flux linkage	0.8524 Wb
pole-pairs	4
Rotational inertia/kg·m2	0.03128

.

6.2. Loss Signal Fault Diagnosis

Some literature on axis rotation uses the difference between i_{α1_1} and i^*_{α1_1} as diagnosis variables [24]. Thresholds are introduced to distinguish normal conditions from current sensor faults. Because this threshold is so similar in gain and loss signal faults (Figure 8), the algorithm presented in [24] cannot distinguish them effectively.

In this paper, the mean value difference of each i_{α1_N} and i_{β1_N} is selected for comparison, the “D_N = Max(|M[|i_{β1_N}|] − M[|i_{α1_N}|])” corresponding to the fault phase is the largest, which can be used to locate the fault without the selection of a threshold.

First, the motor operates stably with 25 Nm load. Then, the output of the phase-A₁ current sensor is set to zero through simulation in DSP. The corresponding i^*_{α1_N} and i^*_{β1_N} are shown in Figure 9.

When i_A1 = 0, the loss signal fault occurs in phase-A₁. According to the Equation (7), among the nine i_{α1_N} and i_{β1_N} obtained by the axis rotation system, i_{α1_1} is most affected and i_{β1_1} keeps the amplitude almost constant. Thus, it can be seen in the Figure 8 that i_{α1_1} becomes the current with the lowest amplitude, and conversely, i_{β1_1} becomes the one with the highest amplitude.

Figure 10 shows the M[|i_{α1_N}|] and M[|i_{β1_N}|] calculated by absolute-mean processors. Due to the linear relationship between amplitude and absolute-mean in Equation (8), the value of M[|i_{α1_1}|] is the smallest of the nine M[|i_{α1_N}|]. Similarly, M[|i_{β1_1}|] is the largest one of the nine M[|i_{β1_N}|].

Figure 11 shows the diagnosis variables D_N and M[i_{q1_m}]. In normal conditions, each D_N are almost equal and approximately zero. When a current sensor fault occurs, the difference between M[|i_{α1_N}|] and M[|i_{β1_N}|] of the fault phase is strictly maximum in Figure 10. Therefore, D₁ is the max value of nine D_N, so the fault phase can be located. Moreover, i_{q1_m} has a negative DC bias “−I_amsin(θ_ori)/9”, a zero-mean pulse. Thus, M[i_{q1_m}] in a loss signal fault is less than it is in normal conditions, so the loss signal fault of phase-A₁ current sensor can be diagnosed.

6.3. Gain Fault Diagnosis

This fault diagnosis experiment is to verify the validity of the proposed method for current sensor gain fault. When a 2.5 times gain fault of phase-A₁ current sensor is set through simulation in DSP. The corresponding i^*_{α1_N} and i^*_{β1_N} are shown in Figure 12.

When the loss signal fault occurs in phase-A₁ (i_A1 = 2.5i^*_A1), i_{α1_1} is most affected while i_{β1_1} keeps the constant amplitude. According to Equation (7), i_{α1_1} becomes the current with the largest amplitude and i_{β1_1} becomes the one with the smallest amplitude. The M[|i_{α1_N}|] and M[|i_{β1_N}|] calculated by absolute-mean processors are shown in Figure 12. Therefore, the value of M[|i_{α1_1}|] is the largest of the nine M[|i_{α1_N}|], and M[|i_{β1_1}|] is the smallest one of the nine M[|i_{β1_N}|].

Figure 13 shows the diagnosis variables D_N and M[i_{q1_m}]. Each D_N are approximately zero in normal conditions. When the fault occurs, the difference between M[|i_{α1_N}|] and M[|i_{β1_N}|] of the fault phase is strictly maximum in Figure 13. Therefore, D₁ is also the max value of D_N, so the fault phase-A₁ can be located. Moreover, i_{q1_m} has a distinct positive DC bias “1.5I_amsin(θ_ori)/9” and a zero-mean pulse. Thus, M[i_{q1_m}] in loss signal fault is greater than it is in normal conditions, which can be used to further diagnose tolerant control. So, the loss signal fault of phase-A₁ current sensor can be diagnosed.

The threshold selection is eliminated in the whole diagnosis process, and the loss signal faults can be located by selecting the phase with the maximum value of the diagnosis variables D_N. M[i_{q1_m}] has an obvious negative bias feature, which makes the distinction between loss signal and gain faults more accurate.

6.4. Loss Signal Fault Tolerant

In order to verify the effectiveness of the proposed fault tolerant method, the loss signal fault-tolerant control experiment is shown as follows: i_{α1_1} and i_{β1_1} are the actual currents, i^*_{α1_1} and i^*_{β1_1} are the fault tolerant currents. Since i_{β1_1} remains almost constant when fault occurs, i_{β1_1} can be directly referred to as i^*_{β1_1}. And the conversion between i_{q1_m} and fault tolerant current i^*_q1 is realized by the switch designed in DSP.

In Figure 14, i_{α1_1} and i_{β1_1} (i^*_{β1_1}) have similar amplitude in normal conditions. However, due to the existence of a correction factor of 9/8 in Equation (12), the amplitude of the fault tolerant current i^*_{α1_1} is slightly higher than i_{α1_1} and i^*_{β1_1}. When the loss signal fault occurs in A₁, i_{α1_1} is most affected and i^*_{β1_1} keeps almost constant but the tolerant current i^*_{α1_1} has the same amplitude as i_{α1_1} in normal condition. Before and after the fault-tolerant control, i^*_{α1_1} can maintain a stable correct amplitude, so that it can be connected to the system together with i^*_{β1_1} as the fault tolerant current.

Figure 15 shows that i_{q1_m} produces a large pulse due to the DC bias and AC pulsation when faults occur. At this time, the current i_Q1 connected to the feedback system is still i_{q1_m}. Then, the switch is given a fault tolerant signal and it selects i^*_q1 to be transferred to the output. Therefore, the feedback current i_Q1 is restored to i_{q1_m} in normal condition, and fault-tolerant control is realized.

6.5. Gain Fault Tolerant

The proposed method can estimate the gain coefficient λ, which can be used as the correction factor of the fault phase current sensor, and can achieve fault-tolerant control directly and accurately.

In Figure 16a, when a 2.5 times gain fault of a phase-A₁ current sensor is set, the estimated coefficient λ has good tracking effect in normal and fault conditions. Therefore, fault-tolerant control can be realized by compensating the gain coefficient of the fault current sensor in DSP. After compensation, the estimated λ and i_{q1_m} are both restored to normal values.

Different from the existing algorithms, the proposed method can estimate the gain coefficient λ, which can be used as the correction factor of the fault sensor. The fault-tolerant control achieves tolerant accuracy by applying this factor.

7. Conclusions

In this paper, a fault diagnosis and tolerant control method is proposed to different kinds of current sensor faults in multi-phase PMSM drives. It is desirable that the method utilizes the stator currents signal-processing, making it even simpler and less computationally demanding. It is also worth noting that the diagnosis variables have distinct characteristics which eliminate (to some extent) the margin of the design of the threshold. This method can distinguish between loss signal and gain faults effectively and lacks operating condition dependence. Moreover, the degree of current sensor gain fault can be quantitatively described by an estimated coefficient. Finally, fault diagnosis and tolerant control experiments are presented, which show the effectiveness and rapidity regarding the method. The optimization of diagnosis variables when multiple faults occur is the next focus.