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Identification of demagnetization faults in axial flux permanent magnet synchronous 1 machines using an inverse problem coupled with an analytical model Jan De Bisschop1,2 , Ahmed Abdallh1,3 , Peter Sergeant1,2 , and Luc Dupré1 1 Department of Electrical Energy, Systems and Automation, Ghent University, Belgium 2 Department of Industrial Technology and Construction, Ghent University, Belgium 3 Electrical Power and Machines Department, Cairo University, Egypt Abstract—This paper presents an analytical method for permanent magnet demagnetization detection in an axial flux permanent magnet synchronous machine. Firstly, a forward model is built to calculate the three phase terminal voltages with as inputs the three phase currents, the geometry of the machine and the electromagnetic properties of the machine. To model the demagnetization fault, the scalar potential equation is solved. The magnetization waveform is a square wave corrected with magnetization factors which represent the individual magnetization of every magnet. Secondly, the forward model is inverted so that the magnetization factors are computed out of the three phase currents and the three phase terminal voltages. The forward model is validated with experimental data and FEM simulations. Index Terms—Axial flux permanent magnet synchronous machine, demagnetization faults, inverse problem. y I. I NTRODUCTION L XIAL flux permanent magnet synchronous machines (AFPMSMs) are widely used nowadays in industry thanks to their high efficiency and high power density. However, these types of machines are commonly affected by several types of faults, such as a rotor eccentricity and permanent magnet demagnetization [1]. An early detection of these faults is an essential criterion of online continuous condition monitoring systems. Specifically, in this paper we focus only on demagnetization faults which may occur due to the temperature rise of the machine, in particular of the rotor, when it works extensively at high load conditions. If the magnet temperature exceeds a critical value, a partial demagnetization of the permanent magnets may happen leading to a dramatic full demagnetization fault. In order to avoid large safety margins or extra temperature measuring equipment, we develop a real-time algorithm to detect the magnets demagnetization. Detection can be done by flux probes [2] or other sensory techniques, but extra sensors result in extra production costs. Another technique is based on measuring the negative/zero sequence current, impedance or voltage [3]. Although this information is good for detecting any asymmetry in the machine, it is also sensitive for unbalance in the power supply. In this paper, we present an electromagnetic inverse problem to detect the demagnetization faults in AFPMSM. The identification procedure is performed based on an analytical model of the considered machine that solves Maxwell’s equations. This involves no extra sensors and takes the three phase current and voltages into account. N A II. A NALYTICAL MODEL A. Model construction The machine model for calculating the no-load magnetic field of the AFPMSM is shown in Fig. 1. A Cartesian coordinate system is chosen where x and y denote the circumferential and the axial direction, respectively. This defines a 2-D plane where the scalar magnetic potential is solved. The plane is a cylindrical cutting plane with a constant radius. The differential equation that needs to be solved is: ∂2ϕ ∂2ϕ + =0 ∂x2 ∂y 2 (1) The authors gratefully acknowledge the financial support of the Special Research Fund “Bijzonder Onderzoeksfonds” of Ghent University. Corresponding author: A. Abdallh (e-mail: [email protected]) L S hm Region I δ δ Region II 0 δ -δ S N L hm Region III -L x (a) (b) Fig. 1. (a) The studied AFPMSM. (b) Model for solving of no-load magnetic field of AFPMSM showing the different regions. with ϕ being the scalar magnetic potential. Region I and III represent the area with the permanent magnets. Region II is a combination of the two air gaps of the machine. The soft magnetic materials are assumed to have an infinite permeability. The soft magnetic material itself and the stator slots are not included in this model. This means that this model only considers the no-load situation. The modeling of stator currents is explained further in this paper. Moreover, a permeance function is used to account for the stator slot effects. The magnetization of the rotor is a square wave where the height is dependent on the remanent induction of the magnets. Demagnetization is easily included as the height of the square representing a certain magnet can be lowered. The magnetization pattern of each magnet is found by multiplying the healthy state magnetization with a magnetization factor, for each magnet ranging from 0 to 1 with 0 and 1 being the full demagnetization and healthy state, respectively. These magnetization waves can be described as a Fourier series. ( M1 (x) = M2 (x) = P+∞ n=−∞ P+∞ n=−∞ Mn1 e Mn2 e inπx τp inπx τp ∗ with M−n1 = Mn1 ∗ with M−n2 = Mn2 (2) M1 and M2 represent the magnetization of the rotor of region I and III respectively. τp is the half of the total period, which is the circumference of the machine. The constitutive relationship (3) connects the magnetization with the magnetic flux density. The magnetization vector is directed along the axial direction. µ0 is the permeability of vacuum, µ is the absolute permeability of the magnets. ~ = µ0 M ~ + µH ~ B (3) 2 The magnetic field strength can be derived from this solution by: Hx = −∂ϕ/∂x, Hy = −∂ϕ/∂y. (5) The solution must satisfy the following boundary conditions: ( HxI (x, y)|[y=L] = 0 (6) HxIII (x, y)|[y=−L] = 0 ( ByI (x, y)|[y=δ] = ByII (x, y)|[y=δ] (7) HxI (x, y)|[y=δ] = HxII (x, y)|[y=δ] ( ByII (x, y)|[y=−δ] = ByIII (x, y)|[y=−δ] (8) HxII (x, y)|[y=−δ] = HxIII (x, y)|[y=−δ] This results in a system of six equations. By solving this system the constants C1 to C6 can be determined. The flux density in the air gap (region II) becomes: +∞ inπx X nπy nπ − nπy C 3 e τp − C 4 e τp e τp τ n=−∞ p (9) +∞ X nπy nπy inπx inπ − BxII (x, y) = −µ0 C 3 e τp + C 4 e τp e τp τ n=−∞ p (10) Although (9) and (10) contain complex functions and coefficients, the functions ByII (x, y) and BxII (x, y) are real functions. The armature reaction is based on [4]. BAR (x) uses the current sheet A(x) to compute the air gap field generated by the stator coils. The current sheet is a square wave based on the injected three phase currents. The squares represent the current going through the stator windings. The three phase currents are in-phase with the three phase terminal voltages at full-load. +∞ n X inπx cosh nr z1 (z − z) iAn e τp BAR (x) = µ0 cosh 2 n r sinh r z2 n=−∞ (11) +∞ X inπx (12) An e τp A(x) = 200 Analytical FEM Measured Phase Voltage [V] 150 100 50 0 −50 −100 −150 −200 0 50 100 150 200 Rotor Position [◦ ] 250 300 350 Fig. 2. No-load terminal voltage of one phase from the results of the analytical model verified with the FEM and experimental data for a healthy machine. Terminal Voltages 400 Va -Num. Vb -Num. Vc -Num. Va -Anal. Vb -Anal. Vc -Anal. 300 200 Voltage [V] The solution of the differential equation for each region is: inπx nπy P+∞ − nπy ϕ1 = n=−∞ C1 e τp + C2 e τp e τp inπx nπy P+∞ − nπy ϕ2 = n=−∞ C3 e τp + C4 e τp e τp (4) nπy nπy inπx P − +∞ τp ϕ3 = + C 6 e τp e τp . n=−∞ C5 e 100 0 −100 −200 −300 −400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Rotor Position [rad] Fig. 3. Full-load three phase terminal voltages of analytical and FEM simulations with - for the clarity of the figure - quite extreme demagnetization faults. Rotor 1: magnet 4 = 0% and Rotor 2: Magnet 7 = 0%. ByII (x, y) = −µ0 n=−∞ The magnetic field of the permanent magnets and of the armature reaction are combined with a permeance function to result in the total air gap field. The permeance function is based on the function in [5]. From this total air gap field, the back-EMF for every tooth can be calculated. The model is built in the frequency domain. Five stator coils are combined to become the phase back-EMF. After accounting for the slot leakage inductances, the end winding inductances and the armature resistances, the three phase terminal voltages are obtained. The inductances are computed based on [6]. Va,n = En − Rs Ia,n − inωm Lend Ia,n − inωm Lslot Ia,n (13) with Lend and Lslot the end winding inductance and the slot and tooth tip leakage inductance resp. The air gap stray field is accounted for in the computed total air gap field. The three phase currents and the magnetization coefficients are the inputs of the forward model. The magnetization coefficients determine the back-EMF. The three phase terminal voltages are the output of the model. In the inverse model, the three phase currents and terminal voltages are the inputs, while the magnetization coefficients are the output. B. Model validation Fig. 2 shows that the results of the forward model are confirmed by the numerical and experimental data of [7]. This is for a healthy machine. A FEM model is made to confirm the results with demagnetization faults. It solves the vector potential differential equation in a 2-D cylindrical cutting plane with constant radius. Because demagnetization faults will not be symmetrical, the model solves the total motor. Several cutting planes are simulated and combined to compute the three phase terminal voltages via the 2D multislice technique [4]. Fig. 3 shows a good correspondence between the numerical and analytical data. The three phases do not have the same peak value. Thanks to the demagnetization faults the peak values of a phase is not constant, it oscillates over a period of one rotation. III. D EMAGNETIZATION IDENTIFICATION The magnetization coefficients of permanent magnets are identified by solving an electromagnetic inverse problem [8], where an accurate interpretation of the time-domain three phase voltages into the analytical model is performed. 3 TABLE I T HE INVERSE PROBLEM RECOVERY SOLUTION STARTING FROM NOISY ‘ NUMERICAL’ MEASUREMENTS , WITH 10% NOISE LEVEL . Inverse Algorithm Measured 3-phase current Geometrical parameters Rotor speed Analytical model ? Termination Criteria? Magnetization coefficients + Case Original defect Recovered defect 1 [6, L, 0.5] [6, L, 0.4964] 2 {[6, L, 0.5], [12, R, 0.5]} {[6, L, 0.5009], [12, R, 0.5011]} Measured 3-phase terminal voltage Fig. 4. The schematic diagram of the inverse problem algorithm. A. Inverse problem formulation The inputs for the proposed inverse problem are the geometrical and circuit parameters of the machine, the three phase current waveforms, and the rotor speed. In addition, the ‘measured’ three phase terminal voltages are inputs, see Fig. 4. The magnetization coefficients of permanent magnets K are recovered by minimizing iteratively an objective function: e = arg min OF (K) K (14) K e is the vector of the recovered magnetization coefwhere K ficients using the inverse approach. The objective function OF is formulated as the Euclidean distance between the ‘measured’ and the corresponding ‘simulated’ three phase terminal voltages, Vabc,m and Vabc,s , respectively: OF = kVabc,m − Vabc,s (K)k 2 (15) Several case studies are carried out to validate the proposed inverse approach, see section IV. Here, p vector contains all information related to the threephase currents Iabc , rotor speed, magnetic material parameters, and all geometrical parameters. The vector u explicitly concerns the considered two uncertain circuit model parameters, u = [Rs , Ls ]. It is obvious that the inverse problem solution is influenced by the uncertainty in u. In other words, solving the inverse problem based on the traditional objective function formulation, i.e. (15), may result in inaccurate results. Therefore, the objective function formulation needs to be modified in order to compensate this error. In this paper, the Minimum Path of the Uncertainty (MPU) technique is implemented, which is based on adapting iteratively the objective function to be minimized by the sensitivity of the forward model response to the uncertain circuit parameters u: ∂Vabc,s (p, u, K) 2 OFM P U = Vabc,m − Vabc,s (p, u• , K) − α (17) ∂u where u• is the mean value of the uncertain parameter. In (17), α is the fitting constant that can be obtained by setting the objective function OFM P U equal to zero [10]. B. Effect of the model uncertainties In the previous section, the forward analytical model is assumed to be ideal, i.e. all physical phenomena are taken into consideration and all model parameters are assumed perfect. However, in practice, it is well-known that the knowledge of some model parameters is uncertain. For example, some geometrical parameters are exposed to a small deviation when the machine is manufactured [9], one of the most uncertain geometrical parameters is the air gap. The effect of the uncertainty in the geometrical parameters on the accuracy of the inverse problem results has been extensively studied [10]. In addition, the values of some circuit parameters of the electromagnetic device, such as resistance and inductance, are uncertain. The effect of such uncertainties of circuit parameters in the inverse problem resolution has not been studied in the literature. In this section, the effects of the uncertainty in the values of both the armature resistance (Rs ) and leakage inductance (Ls ) per phase are presented in details. In order to take the effect of the circuit parameters uncertainty into account, let us formulate the forward problem as follows. The output three phase terminal voltages (Vabc ) can be written as a function of the ‘well-known’ geometrical and circuit model parameters p, ‘uncertain’ circuit model parameters u, and magnetization coefficients of the permanent magnets K: Vabc = Γ(p, u, K) (16) IV. R ESULTS AND DISCUSSION A. Idealized model In this section, we assume that the values of all circuit parameters are precisely known. Several case studies are performed. For the sake of simplicity, the measurement data are obtained numerically by adding some noise level (nl) to the output of the forward model. One and two defected magnets are studied with different noise levels. The error in the inverse problem solution E is calculated as the percentage of the difference between the ‘actual’ and ‘recovered’ magnetization e respectively: coefficients of permanent magnets, K∗ and K, ∗ e (18) E = (K∗ − K)/K × 100% Table I shows the results for several possible combinations of the defects. The results shown in this table are obtained by solving (14) and (15) starting from noisy ‘numerical’ measurements, with nl = 10%, i.e. Vabc,(nl) = Vabc,(nl=0) × (1 ± nl · N [0, 1]), where N [0, 1] is a normally distributed random number with zero mean and a standard deviation of unity. The defect, in the second column in the table, is defined as a set of three parameters D = [a1 , a2 , a3 ], where a1 is the order of the defected magnet in the model, i.e. a1 ∈ [1, · · · , 16]. a2 is the defect side, i.e. a2 ∈ [L, R], where L and R refer to as the left and right hand side of the rotor, respectively. a3 is the magnetization factor, i.e. 0 ≤ a3 ≤ 1. 4 (a) fD = [6; L; 0:5]; [7; L; 0:5]g 6 10 20 E% E% 30 10 5 0 0 (b) (a) Rotor right hand side 15 6 5 5 4 4 E% Rotor left hand side E% 40 3 Traditional OF MPU OF 3 2 -10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Magnet order 1.5 Rotor left hand side -5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 (b) fD = [6; L; 0:5]; [12; R; 0:5]g Magnet order Rotor right hand side 0.3 2 1 1 0 0 0.5 0 -0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Magnet order 0.2 E % E % 1 0.1 0 -0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Magnet order Fig. 5. The recovery error E for (a) two adjacent defected magnets, and (b) two far defected magnets. For the first case study, one defected magnet is studied. The error in the identified magnetization coefficient of the correct magnet is 0.72%. The magnetization coefficients of the remaining magnets, which were assumed in healthy conditions, are correctly identified. In the second case study, two magnets are assumed defected with specific magnetization factors. For example, magnets number 6 and 12, at different rotor sides, are defected with a magnetization factor of 0.5. The identification results of the correct magnets were 0.5009 and 0.5011, respectively. Again, the magnetization coefficients of the unaffected magnets are ‘almost’ recovered accurately. In order to check the validity of the inverse approach in a wide range of faults, Monte Carlo simulations are performed with 100 samples, the defect D of each sample is generated randomly. The solutions of the inverse problems follow ‘qualitatively’ the same trend as shown in Table I. However, we noticed that when the defected magnets are located close to each other, the accuracy of the inverse problem solution becomes worse than the case when the defected magnets are located far from each other. Fig. 5 depicts the error E in the both sides of the rotor for the two considered extreme cases. Although the recovery error is high in Fig. 5(a), the correct defected magnet is detected. Moreover, in order to study the effect of the measurement noise on the solution accuracy, the inverse problem is solved, for a case study with only one defected magnet, starting from 10 different ‘numerical’ noisy measurement data sets with a predefined noise level. Fig. 6(a) shows the effect of the measurement noise on the accuracy of the inverse problem, in an average way, for an assumed defect D = [6, L, 0.5]. It is clear that the accuracy of the inverse problem depends on the accuracy of the measurement, the larger measurement noise, the larger recovery error. B. Non-idealized model As described in section III-B, the accuracy of the inverse problem does not only depend on the accuracy of the measurements being used, but also on the accuracy of the forward model parameters. In the following, the effect of each uncertain model parameter is studied individually. For each uncertain parameter, two inverse problems are solved; i.e. based on the traditional and modified objective function formulations, (15) and (17), respectively. The sensitivity of the model response to the uncertain model parameter is calculated 0.1 0.2 nl 0.3 0.4 0 Rs Ls Fig. 6. (a) The effect of the measurement noise on the solution of the inverse problem, in an average way, for D = [6, L, 0.5]. (b) The comparison between the traditional and the MPU objective function formulation for the separate effect of the uncertainty in the values of Rs and Ls . using the finite difference approach, with a step of 10% of its nominal value. Fig. 6(b) depicts the comparison between the traditional and the MPU objective function formulation for the separate effect of the uncertainty in the values of Rs and Ls , for one defect D = [6, L, 0.5]. The results shown in Fig. 6(b) clarify the effectiveness of the MPU technique over the traditional one. V. C ONCLUSION In conclusion, the developed direct model without faults calculates the terminal voltages fast and accurately, with a good corresponding with finite element computations and with the experimental results. A coupled experimental-analytical electromagnetic inverse problem is proposed to identify the magnetization coefficient of the magnets. Several case studies are performed, and the obtained results reveal the effectiveness of the proposed methodology in condition monitoring of demagnetization defects. The proposed approach can be applied to real drive systems, where the inverse problem is solved at regular time instants, e.g. every minute, on the computer that does the master control of the drive. The operating setpoint of the motor is modified to avoid damage. R EFERENCES [1] P. Tavner, “Review of condition monitoring of rotating electrical machines,” IET Electr. Power App., vol. 2, pp. 215-247, 2008. [2] M. Sasic, et al., “Detecting turn shorts in rotor windings, a new test using magnetic flux monitoring,” IEEE Ind. Appl. Mag., vol. 29, pp. 63-69, 2013. [3] Y. Da, et al., “A new approach to fault diagnostics for permanent magnet synchronous machines using electromagnetic signature analysis,” IEEE Trans. Power Electr., vol. 28, pp. 4104-4112, 2013. [4] H. Vansompel, et al., “A multilayer 2-D-2-D coupled model for eddy current calculation in the rotor of an axial-flux PM machine,” IEEE Trans. Energy Convers., vol. 27, pp. 784-791, 2012. [5] D. 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