Taking advantage of the high reliability, multiphase permanent magnet synchronous motors (PMSMs), such as five-phase PMSM and six-phase PMSM, are widely used in fault-tolerant control applications. And one of the important fault-tolerant control problems is fault diagnosis. In most existing literatures, the fault diagnosis problem focuses on the three-phase PMSM. In this paper, compared to the most existing fault diagnosis approaches, a fault diagnosis method for Interturn short circuit (ITSC) fault of five-phase PMSM based on the trust region algorithm is presented. This paper has two contributions. (1) Analyzing the physical parameters of the motor, such as resistances and inductances, a novel mathematic model for ITSC fault of five-phase PMSM is established. (2) Introducing an object function related to the Interturn short circuit ratio, the fault parameters identification problem is reformulated as the extreme seeking problem. A trust region algorithm based parameter estimation method is proposed for tracking the actual Interturn short circuit ratio. The simulation and experimental results have validated the effectiveness of the proposed parameter estimation method.
1. Introduction
Owing to high torque-to-current ratio, large power-to-weight ratio, high efficiency, high-power factor, high fault tolerance, robustness, and so forth, multiphase PMSMs have been paid more attention in high-power and high-reliability applications [1–3]. Compared with the traditional three-phase PMSM, with the added phase number, the fault tolerance of the multiphase PMSM is enhanced, and thus the reliability of the multiphase PMSM is improved. Therefore, multiphase PMSMs are widely used in fault-tolerant control systems [4, 5].
Fault diagnosis is the foundation of the fault-tolerant control of the electrical machines. In PMSMs, the usual faults include electrical faults, mechanical faults, and magnetic faults [6]. In electrical faults, short circuit faults form 21% of the faults occurring in electrical machines. The stator winding ITSC fault is the commonest short circuit fault in PMSMs. It always occurs due to insulation failures but develops into more serious faults very quickly [7]. So it is meaningful to research the effective fault diagnosis methods of stator winding interturn short circuit for PMSMs.
The current existing detection and diagnosis methods of ITSC fault can be commonly divided into off-line methods and on-line methods [8]. Compared to the off-line methods, in on-line methods, the PMSMs do not have to be taken out of service and predicting health condition and detecting faults at an incipient stage are made easier [9]. In recent years, with the application of neural network, fuzzy logic and particle swarm optimization (PSO), the artificial intelligence (AI) on-line fault detection, and diagnosis methods have drawn the attention of many authors [10]. The AI methods improve the robustness and efficiency of the fault diagnosis and have no need to interpret the collected data in relation to the occurring fault.
In some AI fault detection and diagnosis methods, such as literature [11], in order to detect and diagnose the severity of the stator winding interturn short circuit fault of PMSM, a mathematical model that can describe both healthy and fault conditions is needed first. Literature [12] built power losses model of five-phase PMSM with ITSC fault and analyzed the changes in power losses due to faults occurrence by finite elements simulations. However, this fault model is not suitable for AI fault diagnosis based on parameter optimization. Literature [13] and literature [14] proposed two mathematical models of PMSM with ITSC fault for fault diagnosis. Unfortunately, these models are all about three-phase PMSM and relatively complex. If the fault model of five-phase PMSM was built by the way shown in literature [13] and literature [14], the model would be more complex, and the calculation for the subsequent fault diagnosis based on parameter optimization would increase greatly. Thus, the efficiency of fault diagnosis would be affected. Therefore, it is meaningful to establish a relatively simple five-phase PMSM model with ITSC fault for fault diagnosis.
After the establishment of the fault model, in order to diagnose fault severity of the fault motor, the parameters associated with fault severity need to be identified. However, for the complex distribution of the parameters in the fault model, the identification problem is extremely difficult for nonlinear identification techniques. To overcome this difficulty, the fault diagnosis problem is transformed into a corresponding optimization problem and then solved by intelligent algorithm [15]. In recent years, many authors focus on PSO parameter optimization to deal with this problem, such as that shown in literature [16] and literature [17]. PSO is an evolution computation technique based on swarm intelligent methodology. PSO is initialized as a swarm of arbitrary particles (arbitrary solution), and then the optimal solution is discovered by iteration. However, the PSO algorithm creates the problems of partial convergence and precocious convergence when the particles’ diversity is decreasing. Therefore, finding a better parameter optimization algorithm for five-phase PMSM fault diagnosis is essential.
In this paper, relatively simple mathematics models of the five-phase PMSM under both healthy and ITSC fault situations are established, respectively. Furthermore, a novel fault diagnosis method of ITSC based on the trust region algorithm is proposed for five-phase PMSM. With the aid of the trust region algorithm which is global convergence, the interturn short circuit ratio μ is estimated with a short time transient. The simulation and experimental results have validated both the correction of the established models and the effectiveness of the proposed parameter estimation method.
2. Model Analysis2.1. Five-Phase PMSM Healthy Model
In order to establish the healthy model of five-phase PMSM, without loss of generality, the following assumptions are as follows:
The magnetic circuit is linear. It is, in turn, that the magnetic circuit is not saturation.
The stator winding current is sinusoidal, symmetrical, and without harmonics. The air gap magneto motive force (MMF) is sinusoidal.
The rotor MMF is sinusoidal and the slot effect is neglected.
The five-phase PMSM is nonsalient pole structure.
Eddy currents and hysteresis losses are negligible.
With these assumptions, the five-phase PMSM model can be provided by(1)Us=RsIs+pψs,(2)ψs=LsIs+ψm,(3)Te=∂W∂θ=P12IsT∂Ls∂θIs+Is∂ψm∂θ.
Equation (1) is the voltage balance equation, (2) is the flux equation, and (3) is the torque equation, where the stator phase voltage vector Us=[uaubucudue]T; the stator phase current vector Is=[iaibicidie]T; the stator winding resistance Rs=rs×I5×5; the stator flux vector ψs=[ψaψbψcψdψe]T; the rotor flux vector ψm=ψmcosθcosθ-αcosθ-2αcosθ-3αcosθ-4αT (θ is the rotor electrical angle and α=72°); Ls is the stator inductance matrix; p=d/dt is the differential operator; P is the number of pole pairs; and Te is the electromagnetic torque.
Because of adding two-phase windings, compared to traditional PMSM, the stator inductance matrix Ls of five-phase PMSM is more complex and it can be represented by(4)Ls=LmMabMacMadMaeMabLmMbcMbdMbeMacMbcLmMcdMceMadMbdMcdLmMdeMaeMbeMceMdeLm,where Lm is the self-inductance of phase winding A (B, C, D, and E) and Mab(c,d,e) is the mutual- inductance between phase windings A and B (C, D, and E). Actually, the mutual-inductances can be expressed by Mab=Lmcosα, Mac=Lmcos2α, Mad=Lmcos3α, and Mae=Lmcos4α.
2.2. Five-Phase PMSM Fault Model
Without loss of generality, assume that phase winding A causes ITSC fault and the rest of the phase windings is in healthy state. The five-phase PMSM with ITSC is shown in Figure 1. Note that a short circuit loop current if, which gives birth to braking torque, is produced in phase winding A. And thus the braking torque affects the motor performance seriously. Besides, the effective turns number of phase winding A is reduced, and the values of the phase winding resistance, the self-inductance, the mutual-inductance, and the flux linkage are all changed accordingly.
The schematic of five-phase PMSM with interturn short fault.
In the ITSC fault model of five-phase PMSM, one of the most important parameters is the interturn short circuit ratio μ, which is defined as the ratio of the shorted turns number to the total turns number. When the ITSC fault occurs in phase winding A, depending on the physical relationship of the windings, the equivalent circuit of five-phase PMSM with ITSC fault is shown in Figure 2, where Ra(b,c,d,e)_af is the resistance of phase winding A (B, C, D, and E); Rsa_af is the short circuit winding resistance; La(b,c,d,e)_af is the self-inductance of phase winding A (B, C, D, and E); Lsa_af is the self-inductance of the short circuit winding; Lm is the self-inductance of phase winding under normal conditions. Actually, the resistance and the inductances can be expressed by(5)Ra_af=1-μRs,Rb_af=Rc_af=Rd_af=Re_af=Rs,Rsa_af=μRs,La_af=1-μ2Lm,Lsa_af=μ2Lm,Lb_af=Lc_af=Ld_af=Le_af=Lm.
The equivalent circuit of five-phase PMSM with interturn short circuit fault.
Besides, there are also three kinds of mutual-inductances existing. The first is the mutual- inductance Mab(c,d,e)_af between the remaining normal winding of phase A and other phase (B, C, D, and E) windings, where Mab(c,d,e)_af=1-μMab(c,d,e). The second is the mutual-inductance Masa_af between the remaining normal winding of phase A and the short circuit winding of phase A, where Masa_af=(1-μ)μLm. The third is the mutual-inductance Msab(c,d,e)_af between the short circuit winding of phase A and other phase (B, C, D, and E) windings, where Msab(c,d,e)_af=μMab(c,d,e). Since the phase windings B, C, D, and E are healthy, the mutual-inductances of the phase windings B, C, D, and E remain the values of the healthy model.
When phase winding A causes ITSC fault, the fluxes between the windings and the rotor can be derived as(6)ψa_af=1-μψmcosθ,ψb_af=ψmcosθ-α,ψc_af=ψmcosθ-2α,ψd_af=ψmcosθ-3α,ψe_af=ψmcosθ-4α,ψsa_af=μψmcosθ.According to the equivalent circuit and the analysis above, the voltage balance equation renders as(7)Va=Ra_afia+La_afdiadt+Mab_afdibdt+Mac_afdicdt+Mad_afdiddt+Mae_afdiedt+Masa_afdia-ifdt+dψa_afdt,Vb=Rb_afib+Lb_afdibdt+Mab_afdiadt+Mbc_afdicdt+Mbd_afdiddt+Mbe_afdiedt+Mbsa_afdia-ifdt+dψb_afdt,Vc=Rc_afic+Lc_afdicdt+Mac_afdiadt+Mbc_afdibdt+Mcd_afdiddt+Mce_afdiedt+Mcsa_afdia-ifdt+dψc_afdt,Vd=Rd_afid+Ld_afdiddt+Mad_afdiadt+Mbd_afdibdt+Mcd_afdicdt+Mde_afdiedt+Mdsa_afdia-ifdt+dψd_afdt,Ve=Re_afie+Le_afdiedt+Mae_afdiadt+Mbe_afdibdt+Mce_afdicdt+Mde_afdiddt+Mesa_afdia-ifdt+dψe_afdt,0=Rsa_afia-if+Lsa_afdia-ifdt+Masa_afdiadt+Msab_afdibdt+Msac_afdicdt+Msad_afdiddt+Msae_afdiedt+dψsa_afdt.
The voltage balance equation (7) can be rewritten as(8)Vaf=RafIaf+LafdIafdt+dψafdt,where(9)Vaf=VaVbVcVdVe0T,Iaf=iaibicidieifT,Raf=Rs0000-μRs0Rs000000Rs000000Rs000000Rs0μRs0000-μRs,Laf=LmMabMacMadMae-μLmMabLmMabMacMad-μMabMacMabLmMabMac-μMacMadMacMabLmMab-μMadMaeMadMacMabLm-μMaeμLmμMabμMacμMadμMae-μ2Lm,ψaf=ψmcosθcosθ-αcosθ-2αcosθ-3αcosθ-4αμcosθT.
The electromagnetic torque equation of five-phase PMSM is(10)Te=∂W∂θ=p12IafT∂Laf∂θIaf+Iaf∂ψaf∂θ=pIaf∂ψaf∂θ.
The mechanical motion equation of five-phase PMSM is(11)Te-TL=Jdωdt+Bω,where Te is the electromagnetic torque; TL is the load torque; J is the rotational inertia; B is the viscous friction coefficient; ω is the mechanical angular velocity.
3. Fault Diagnosis3.1. Trust Region Algorithm
Trust region algorithm is a method for the extreme seeking. The method sets a trust region radius as the upper bound of the displacement length and, with the current iteration point as the center, determines a closed spherical region named trust region. By solving the optimal point of the quadratic approximation model to determine the candidate displacement, the nonlinear extreme problem is transformed into the extreme problem of solving the approximation quadratic model of the objective function within the trust region [18, 19]. If the candidate displacement enables the sufficient reduction to the objective function, the candidate displacement is adopted to be the new displacement and simultaneously maintains or expands the trust region radius for a new round of iteration. Otherwise, it indicates the approximate degree of the quadratic model and the objective function is unsatisfactory, the trust region radius should be reduced, and the extreme problem of the approximation quadratic model should be solved within the new trust region radius to obtain the new candidate displacement [20]. By the continuous iteration, the extreme optimization of nonlinear function is achieved.
The trust region methods can be understood by a typical unconstrained minimization problem:(12)minx∈Rnfx,where f(x) is objective function to be minimized.
Suppose that xk is the kth iteration, fk=f(xk), gk=∇f(xk), and Bk is the kth approximation of the Hesse Matrix ∇2f(xk). So the trust region subproblem is(13)minqkd=gkTd+12dTBkds.t.d≤Δk,where Δk is the trust region radius and · is the vector norm, usually ·2 or ·∞. Suppose that the optimal solution of (13) is dk; Δfk is the decreasing of the kth iteration, and Δfk=fk-f(xk+dk); Δqk is the predicted decreasing correspondingly, and Δqk=qk(0)-qk(dk). Define rk as(14)rk=ΔfkΔqk.Generally, Δqk>0. So if rk<0, Δfk<0, xk+dk will not be the next iteration point. To solve the subproblem, the trust region radius needs to be reduced. If the value of rk is closed to 1, it indicates that the quadratic model is a good approximation for the objective function within the trust region, xk+1≔xk+dk can be used as the new iteration point, and meantime the trust region radius can be increased in the next iteration. For other cases, the trust region radius remains unchanged.
3.2. Fault Parameter Identification
In order to diagnose the fault severity of the fault motor, the interturn short circuit ratio μ needs to be identified based on the fault model. Because of the complex distribution of the parameter in the fault model, the identification problem is extremely difficult for nonlinear identification techniques. To overcome the difficulty, the fault diagnosis problem is first transformed into a corresponding optimization problem and then solved using trust region algorithm. Because of its simplicity, global convergence, and computational efficiency, trust region algorithm has been used extensively to solve a broad range of optimization problems. Numerous applications have applied trust region algorithm for parameter tuning and identification. The applications based on trust region algorithm are not limited by model structures, as are many traditional identification algorithms. As long as the model performs differently with different parameters, which is almost always true, trust region algorithm will be able to identify the unknown parameters in the models.
The principle of fault parameter (the interturn short circuit ratio μ) identification for five-phase PMSM under ITSC fault is as shown in Figure 3.
The block diagram of fault parameter identification based on trust region algorithm.
To begin with, give an arbitrary constant μ^(0)∈0,1 as the initial estimation value of the interturn short circuit ratio for the five-phase fault PMSM model. Meanwhile, sample both the five-phase PMSM drive voltage V and the phase currents i from the actual PMSM. And then, a reference model of the five-phase PMSM with ITSC fault is established as (7), and its drive voltage is the same as the actual PMSM.
Afterwards, to diagnose the fault severity of the five-phase PMSM, or in other words, to identify the fault parameter (the interturn short circuit ratio μ) by trust region algorithm, a quadratic fitness function is introduced as(15)fμ^=∑n=15in-in∗2,where i and i∗ are the phase current of the actual PMSM and reference model, respectively. In fact, the objective function is related to the parameters the interturn short circuit ratio u and its estimation μ^. It can be proved that the objective function is equal to zero if and only if the actual interturn short circuit ratio is equal to its estimation.
At last, use the trust region algorithm to seek the interturn short circuit ratio estimation value μ^ such that the fitness function achieves its minimum value zero. The specific steps of the trust region algorithm parameter identification can be realized as follows. In the process of parameter optimization, the gradient and the approximation of the Hesse Matrix for the quadratic fitness function are calculated, respectively, as (16) and (17).
Step 0. Select the initial parameters, where 0≤η1<η2<1, 0≤τ1<1<τ2, and 0≤ε≪1. Consider μ^(0)∈0,1. The upper limit of the trust region radius is Δ-, Δ->0, and the initial trust region radius is Δ0, Δ0∈(0,Δ-). Set k≔0.
Step 1. Calculate gk as shown in (16); if gk≤ε, stop the iteration(16)gk=∇fμ^=∂fμ^∂i1∗∂fμ^∂i2∗∂fμ^∂i3∗∂fμ^∂i4∗∂fμ^∂i5∗T.
Step 2. Solve the trust region subproblem of the objective function (15), and the solution is dk.
Step 3. Calculate the value of rk as (14), and in the calculation of rk, qk is calculated as shown in (14), where Bk is the kth approximation of the Hesse Matrix, as shown in (17)Bk=∇2fμ^=∂2fμ^∂2i1∗∂2fμ^∂i1∗∂i2∗∂2fμ^∂i1∗∂i3∗∂2fμ^∂i1∗∂i4∗∂2fμ^∂i1∗∂i5∗∂2fμ^∂i2∗∂i1∗∂2fμ^∂2i2∗∂2fμ^∂i2∗∂i3∗∂2fμ^∂i2∗∂i4∗∂2fμ^∂i2∗∂i5∗∂2fμ^∂i3∗∂i1∗∂2fμ^∂i3∗∂i2∗∂2fμ^∂2i3∗∂2fμ^∂i3∗∂i4∗∂2fμ^∂i3∗∂i5∗∂2fμ^∂i4∗∂i1∗∂2fμ^∂i4∗∂i2∗∂2fμ^∂i4∗∂i3∗∂2fμ^∂2i4∗∂2fμ^∂i4∗∂i5∗∂2fμ^∂i5∗∂i1∗∂2fμ^∂i5∗∂i2∗∂2fμ^∂i5∗∂i3∗∂2fμ^∂i5∗∂i4∗∂2fμ^∂2i5∗.
Step 4. Correct trust region radius as follows:(18)Δk+1≔τ1Δk,if rk≤η1,Δk,if η1<rk≤η2,minτ2Δk,Δ-,if rk≥η2,dk=Δk.
Step 5. If rk>η1, set μ^k+1≔μ^k+dk, Bk≔Bk+1, k≔k+1, and go to Step 1; else set μ^k+1≔μ^k, k≔k+1, and go to Step 2.
4. Simulation Analysis
In order to verify the model of the five-phase PMSM under ITSC fault and the trust region algorithm in the process of the fault parameter identification, the simulation has been done by MATLAB/Simulink. The parameters of five-phase PMSM are shown in Table 1.
Parameters of five-phase PMSM used for simulation.
Stator winding resistance
Rs
17.4 (Ω)
Self-inductance of phase winding
Lm
4.5e-2 (H)
Flux amplitude
ψm
0.1827 Wb
Moment of inertia
J
0.8e-6 (kgm^{2})
Friction constant
B
1e-3 (Nms)
Number of pole pairs
p
4
Load torque
TL
0.4 (Nm)
Input sinusoidal voltage amplitude
V
28 (V)
Input voltage frequency
f
20 (Hz)
Figure 4 shows the phase current waveforms in healthy state. Figures 5 and 6 show the phase current waveforms in fault state. Notice that the current of phase A (the yellow line) in fault state is significantly larger than the healthy ones. And with the increase of the fault parameter u, the current of phase A in fault state also increases. And meanwhile, the current waveforms of other phases are also affected by the fault phase A.
The phase current waveforms in normal state.
The phase current waveforms in fault state (u=0.2).
The phase current waveforms in fault state (u=0.5).
The parameters of five-phase PMSM for fault parameter identification based on trust region algorithm are shown in Table 2.
Parameters of trust region algorithm.
η1
0.1
η2
0.75
Δ-
2.0
τ1
0.5
τ2
2.0
ε
0.1
The Simulink simulation model of five-phase PMSM for fault parameter identification based on trust region algorithm is shown in Figure 7.
The simulation model of five-phase PMSM for fault parameter identification.
The results of five-phase PMSM for fault parameter identification based on trust region algorithm are shown in Table 3. The actual value of the interturn short circuit ratio μ is 0.2 (20% of the total turns occurring are shorted) and the initial estimation value of the interturn short circuit ratio u∗ is 0.3. After six steps of the trust region seeking calculation, the estimation value of interturn short circuit ratio u∗ converges to the actual value.
Result of fault parameter identification.
k
Δ0
rk
u∗
1
1.0
0.1799
0.2834
2
1.0
0.2072
0.2668
3
1.0
0.2462
0.2502
4
1.0
0.3061
0.2335
5
1.0
0.4050
0.2167
6
1.0
0.5298
0.2001
5. Experimental Results
To verify the model of the five-phase PMSM under ITSC fault and the trust region algorithm in the process of the fault parameter identification, the experiment has been done. The parameters of five-phase PMSM are the same as shown in Table 1. The experimental devices are as shown in Figure 8. Figure 9 is the phase current waveforms of the five-phase PMSM under 20% ITSC fault in phase A. Figure 10 is the fitness function curve in optimization process under 20% ITSC fault in phase A.
Experimental devices.
Phase current waveforms of the five-phase PMSM under 20% ITSC fault in phase A.
Fitness function curve in optimization process under 20% ITSC fault in phase A.
Comparing the experimental results with the simulation results, it can be seen that the actual phase current waveforms are almost the same as the simulation analysis from Figures 5 and 9. The peak current values of the nonfault phases are all 1 A approximately in both the simulation results and the experimental results. And the peak current values of the fault phase are all about 1.2 A in both the simulation waveform and the experimental waveform. It proves the correction of the mathematics models built for the five-phase PMSM under both healthy and ITSC fault situations. The only difference between simulation results and experimental results is that the experimental phase current waveforms are not so smooth as shown in the simulation results. The reason is that there may be interference during the actual current sampling process, which results in the current glitch.
Figure 10 is the fitness function curve in optimization process under 20% ITSC fault in phase A. It can be seen that the value of the fitness function is close to zero at the tenth optimization. And from Table 3, it can be seen that, in the simulation, after six steps of the trust region seeking calculation, the estimation value of interturn short circuit ratio u∗ converges to the actual value. Compared to simulation results, the optimization number of the trust region seeking in the experiment is larger than the simulation shown, for the reason that the actual phase voltage and current of the motor are sampled by the voltage sensor and current sensor, and there are errors during the sampling procedure. So the optimization number is added. However, it is obvious that both the simulation and the experiment are all successful implementations of the parameter identification and prove the correctness of the proposed parameter estimation method for the fault diagnosis.
6. Conclusions
In this paper, the mathematics models for the five-phase PMSM under both healthy and ITSC fault situations are established, respectively. Furthermore, a novel fault diagnosis method of ITSC based on the trust region algorithm is proposed for five-phase PMSM. With the aid of the trust region algorithm, the interturn short circuit ratio μ is estimated with a short time transient. The simulation and experimental results have validated both the correction of the established models and the effectiveness of the proposed parameter estimation method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by the National Natural Science Foundation of China (Grant no. 51407143).
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