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Enhanced rotor field‐oriented control of multiphase induction machines based on symmetrical components theory

2018; Institution of Engineering and Technology; Volume: 12; Issue: 4 Linguagem: Inglês

10.1049/iet-pel.2018.5710

1755-4543

Zicheng Liu, Zedong Zheng, Qingfeng Wang, Yongdong Li,

Multilevel Inverters and Converters

IET Power ElectronicsVolume 12, Issue 4 p. 656-666 Research ArticleFree Access Enhanced rotor field-oriented control of multiphase induction machines based on symmetrical components theory Zicheng Liu, Zicheng Liu State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, Hubei Province, People's Republic of ChinaSearch for more papers by this authorZedong Zheng, Corresponding Author Zedong Zheng zzd@mail.tsinghua.edu.cn Department of Electrical Engineering, Tsinghua University, Haidian District, Beijing, People's Republic of ChinaSearch for more papers by this authorQingfeng Wang, Qingfeng Wang Department of Electrical Engineering, Beijing Jiaotong University, Haidian District, Beijing, People's Republic of ChinaSearch for more papers by this authorYongdong Li, Yongdong Li Department of Electrical Engineering, Tsinghua University, Haidian District, Beijing, People's Republic of ChinaSearch for more papers by this author Zicheng Liu, Zicheng Liu State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, Hubei Province, People's Republic of ChinaSearch for more papers by this authorZedong Zheng, Corresponding Author Zedong Zheng zzd@mail.tsinghua.edu.cn Department of Electrical Engineering, Tsinghua University, Haidian District, Beijing, People's Republic of ChinaSearch for more papers by this authorQingfeng Wang, Qingfeng Wang Department of Electrical Engineering, Beijing Jiaotong University, Haidian District, Beijing, People's Republic of ChinaSearch for more papers by this authorYongdong Li, Yongdong Li Department of Electrical Engineering, Tsinghua University, Haidian District, Beijing, People's Republic of ChinaSearch for more papers by this author First published: 01 April 2019 https://doi.org/10.1049/iet-pel.2018.5710Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Fault-tolerant capability greatly promoted the application of multiphase machines on safety-critical occasions, and fault-tolerant control strategies are required to suppress the torque ripples. Based on the generalised symmetrical components (SCs) theory, a general expression of the independent SCs is derived during the fault-tolerant operation of symmetrical multiphase machines in this study, and coefficients of the four basic rotating components in the general expressions are calculated for specific open-circuit conditions. Then the bidirectional rotating proportional–integral controllers are designed to control all the rotating components in each SC. Considering control loops for all the independent SCs, an enhanced rotor field-oriented control fault-tolerant strategy is proposed for symmetrical multiphase induction machines (IMs) with any phase number m. Furthermore, additional rotating current controllers in the first SC control loop are added to reduce the low-order current harmonics during the fault-tolerant operation. Experimental evaluations in terms of the transient, dynamic and harmonic performances on both five-phase and nine-phase IM drive platforms are provided to verify the effectiveness of the proposed fault-tolerant strategy. 1 Introduction The last few decades have witnessed the unprecedented development of multiphase machines. Compared with traditional three-phase counterparts, multiphase machines have higher reliability [1], improved torque density and reduced torque ripple. Intensive research has been conducted to apply multiphase machines on some safety-critical occasions, such as electric ships [2], more-electric aircraft [3] and spacecraft [4]. Fault-tolerant ability is the key factor that improves the reliability of multiphase machines. The increase in phase number leads to additional degrees of freedom, which enables the multiphase drive system to operate with faults in one or even more phases [5]. The most widely studied faulty cases are open-circuit cases, while even the short-circuit faults can be turned into open-circuit ones by the blowing out of the fast-acting fuse [6]. However, once the open-circuit fault occurs, the remaining healthy phases of the multiphase machine would be asymmetric in the space distribution of stator windings, which could result in irregular air-gap magnetic motive force (MMF) and finally lead to serious torque pulsations [7]. Therefore, fault-tolerant control strategies are needed to suppress the torque ripples by compensating the asymmetry in the air-gap MMF of the machine. The most widely used fault-tolerant strategy is the optimal current control strategy, which aims to generate a smooth rotating air-gap MMF by properly regulating the currents in the remaining healthy phases. Meanwhile, redundant degrees of control freedom could be utilised to optimise the current references. The optimisation goals could be the minimisation of stator copper losses [8-10], the maximisation of the output torque [11, 12] and the uniformity of stator copper loss distribution [8]. For the optimal current control strategies, the remaining phase currents are usually asymmetrical in amplitudes and phase angles, and the control of currents is the key issue. To track the irregular phase current references, hysteresis controllers [9, 10, 12, 13], proportional–resonant (PR) controllers [14], model predictive controllers (MPC) [15, 16], sliding mode controllers (SMCs) [17] and fuzzy logic controllers (FLC) [18] are preferred. However, hysteresis controllers are susceptible to the noise, and they inevitably lead to random switching frequencies, which deteriorate the subharmonics, switching loss and electromagnetic interference emission [19]. The standard PR controllers have narrow band and infinite gain, which may cause stability problems when realised on digital controllers [20]. Additionally, the frequency of motor currents is time varying in the variable-speed drive applications, which increases the complexity of PR controller design. Though with a faster control response, the MPC controllers require too much computation cost, and their high dependence on model accuracy provides less robustness during the unavoidable fault detection delays [16]. SMCs are suggested in [17], but the Clarke transformation needs to be modified depending on the specific fault type, which limits the extension to different fault conditions. Besides, the chatter problems of SMC might cause stability problems when implemented digitally. Though the robustness and extensibility of FLC-based fault-tolerant control is quite good [18], its torque ripples are obvious especially in the fault-tolerant conditions, along with rather complicated computation. Therefore, some literatures [8, 11, 21-23] turned back to the proportional–integral (PI) controllers, which are widely adopted in the industry applications because of their simple structures and stable characteristics. Proper decoupling and rotating transformations are required when applying PI controllers in fault-tolerant operation. So far, there are mainly two kinds of solutions. The first solution is to find orthogonal reduced-order transformation matrices for the remaining healthy phases [21, 22]. However, this solution usually results in asymmetrical machine model with non-constant parameters [21], which requires very complicated computation. Even worse, different kinds of faulty conditions call for different orthogonal reduced-order matrices [22], corresponding to different machine models. The second solution is to remain the decoupling transformation on the fundamental plane that is adopted in the healthy condition, but add additional compensation transformations in non-torque contributing planes, which yields to the zero current constrains of the open-circuit faulty phases [8, 11, 23]. Dual PI controllers in x–y and 0+–0− planes are adopted in [11, 23] to achieve fault-tolerant operation, but only apply to the six-phase machines, including the symmetrical and asymmetrical ones. Multiple space vector representation is employed in [8], and the proposed fault-tolerant algorithm can be extended to any multiphase induction machine (IM) with an odd phase number. However, the low-order harmonics in the phase current are uncontrolled, which bring additional losses and are usually unwanted in multiphase drive [24]. Moreover, the current vectors are calculated in the stationary reference frame, which leads to the coupling of the reference values for clockwise and counter-clockwise PI controllers. The symmetrical somponents (SCs) theory was first proposed by Charles L. Fortescue in the year 1918, to analyse the steady-state behaviour of the three-phase rotating machines operating on unsymmetrical circuits [25]. This theory was then developed into an intuitive and powerful method to identify the imbalance of the three-phase power system, and even to rebalance the three phase voltages [26]. In recent years, the SC theory has been extended to multiphase drive applications to analyse and regulate the relations between different phases. A current balance control strategy for general m-phase drive systems was proposed in [27], which can effectively redress the imbalance of current amplitudes in different stator phases by eliminating the fundamental currents in redundant SCs. Moreover, by analysing the features of SCs under faulty conditions, an accurate and robust fault detection method was proposed for a five-phase machine in [28]. This paper concentrates on the fault-tolerant field-oriented control design for multiphase machines, and utilises the generalised SC theory to regulate the relations among the stator phase currents. The discussion here differs from other fault-tolerant methods in previous literatures in the following aspects: (i) Based on the SC theory, the positive- and negative-sequence reference currents in non-torque/flux producing SCs are separately related to the decoupled d and q currents in the fundamental components, which ensures a smooth dynamic transition from pre-fault to post-fault conditions. (ii) The orthogonality of selected transformations allows the decoupled control of low-order current harmonics in the time domain and symmetrical current components in the space domain. Therefore, low-order harmonics can be suppressed during the fault-tolerant operation. The generalised SC theory for multiphase systems was introduced in Section 2. To achieve fault-tolerant control, constrains on different SCs under different open-circuit faulty conditions were derived in Section 3. Then enhanced rotor field-oriented control (eRFOC) with fault-tolerant ability was proposed in Section 4, which contains PI control loops for both the independent SCs and the low-order current harmonics. Additionally, the computation cost comparison between conventional rotor field-oriented control (RFOC) and eRFOC is conducted in Section 5. Finally, experimental results on both 5PIM and 9PIM in Section 6 demonstrated the validity of the proposed fault-tolerant strategy in terms of transient, dynamic and harmonic performances. 2 Symmetrical components theory for multiphase systems In analogy to the three-phase system, the generalised SC theory for a symmetrical m-phase system can be expressed in (1). is the current phasor of the kth phase in the m-phase stationary reference frame, and is the current phasor of the kth SC. The complex transformation matrix is expressed as S in (2), and defined in (3) is the kth row vector. The quantity a in the transformation matrix stands for a rotation of rad, shown in (4) (1) (2) (3) (4)It is interesting to find that is orthogonal to all the other row vectors in S except , which is shown in the equation of the dot product (5). In analogy to the three-phase system, and can be called positive- and negative-kth sequence SCs, because they rotate in the opposite directions (5) Symmetrical multiphase machines are usually with odd phase numbers, such as five-phase [9, 10, 13-15], seven-phase [8] and nine-phase [12, 18] machines. For less torque pulsations, machines with even phase numbers are usually in asymmetrical structures, which can be regarded as the combination of two symmetrical m/2-phase systems [11, 21]. Therefore, the m-phase system with odd phase number is the typical condition in multiphase drive applications, and the relationship between the SCs of an m-phase system can be shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Relationship between SCs in the symmetrical m-phase system (m is odd) In analogy to the three-phase system, the generalised Clarke transformation is given in (6). So , … are chosen as the independent SCs, which is expressed in (7), (6) (7) 3 Constraints on SCs under open-circuit conditions When the open-circuit fault occurs, current in the faulty phase disappears, so the air-gap MMF is distorted. Fortunately, the redundant control degrees of multiphase systems potentiate the compensation of MMF by optimising the currents in remaining phases. However, for multiphase machines with a single neutral point, at least three healthy phases are required to generate a smooth rotating MMF. Taking the five- and nine-phase systems as examples, the faulty conditions with one or two open-circuit phases can be summarised in Figs. 2 and 3. Fig. 2Open in figure viewerPowerPoint 5PIM under (a) Healthy condition, (b) One-phase faulty condition 1F, (c) Two-phase faulty conditions of 2F-I, (d) Two-phase faulty conditions of 2F-II Fig. 3Open in figure viewerPowerPoint 9PIM under (a) Healthy condition, (b) One-phase faulty condition 1F, (c) Two-phase faulty conditions of 2F-I, (d) Two-phase faulty conditions of 2F-II, (e) Two-phase faulty conditions of 2F-III, (f) Two-phase faulty conditions of 2F-IV For multiphase machines with distributed windings, a regular smooth MMF depends on a circular shape of current trajectory in the plane (the flux and torque producing plane), as shown in (8). The phase currents under open-circuit faults are constrained to zero, shown in (9). Additionally, the connected neutral point calls for the Kirchhoff equation in (10). If the minimum loss strategy is adopted, then the goal function is expressed in (11) (8-10) (11)Solving the above optimisation problems, we can obtain the fault-tolerant phase currents under respective faulty conditions for the 5PIM and 9PIM, shown in Figs. 4 and 5. Fig. 4Open in figure viewerPowerPoint Fault-tolerant current references for the 5PIM under (a) Healthy condition, (b) One-phase faulty condition 1F, (c) Two-phase faulty conditions of 2F-I, (d) Two-phase faulty conditions of 2F-II Fig. 5Open in figure viewerPowerPoint Fault-tolerant current references for the 9PIM under (a) Healthy condition, (b) One-phase faulty condition 1F, (c) Two-phase faulty conditions of 2F-I, (d) Two-phase faulty conditions of 2F-II, (e) Two-phase faulty conditions of 2F-III, (f) Two-phase faulty conditions of 2F-IV Applying the transformation in (1) to the above fault-tolerant phase currents, the SCs under open-circuit constrains can be obtained. As the kth and (m–k)th components are mirror symmetrical, only the (m − 1)/2 orthogonal SCs are considered. To better demonstrate the relationship between different SCs, the general expression of the kth SC is shown in (12). Tables 1 and 2 list the constraints on SCs for the 5PIM and 9PIM, respectively (12) Table 1. Fault-tolerant constrains on SCs for the 5PIM Fault type 1F 1.0 0 0 0 −0.5 0 −0.5 0 2F-I 1.0 0 0 0 0.3090 −0.9511 −1.3090 −0.9511 2F-II 1.0 0 0 0 −0.8090 −0.5878 −0.1910 −0.5878 Table 2. Fault-tolerant constrains on SCs for the 9PIM Fault type 1F 1.0 0 0 0 −0.1667 0 −0.1667 0 −0.1667 0 −0.1667 0 −0.1667 0 −0.1667 0 2F-I 1.0 0 0 0 −0.2608 0.2188 −0.0350 0.1985 0.0026 −0.0149 −0.0865 −0.1499 −0.1818 −0.3149 −0.4385 −0.1596 2F-II 1.0 0 0 0 0.0023 −0.0129 −0.2971 −0.1081 −0.3470 −0.1263 −0.1248 0.2161 −0.0576 0.0997 −0.1758 −0.1475 2F-III 1.0 0 0 0 −0.0833 −0.1443 −0.0833 0.1443 −0.0833 0.1443 −0.3333 0 −0.3333 0 −0.0833 −0.1443 2F-IV 1.0 0 0 0 −0.3367 −0.1226 −0.2358 −0.1979 −0.2190 −0.1838 −0.1043 −0.1806 −0.0895 −0.1550 −0.0146 −0.0828 4 Enhanced RFOC design based on SC theory 4.1 Control of SCs As illustrated in Section 2, the independent SCs are orthogonal to each other, so they can be controlled separately. Each SC contains several basic rotating components, as shown in (12). By proper rotating transformation, these rotating components can be linearised, and therefore can be controlled by PI controllers. The rotating components and respective rotating transformations are shown in Table 3, where is given in (13) and is the initial electric angle. For each of the independent SCs, the control diagram can be designed in Fig. 6, where dual rotating transformations are employed. The corresponds to plane, which is the plane after rotating transformation (13) Table 3. Rotating components and respective rotating transformations Rotating component Currents in the plane Rotating transformation Currents in the plane Fig. 6Open in figure viewerPowerPoint General control diagram for When currents in different planes are controlled simultaneously, it would be much convenient if their numerical relationships were established. Equation (8) ensures that contains only the component, so and currents in the plane are much simple, and all the other , can be expressed in terms of , . Then according to (8), (12) and Table 3, currents in planes can be expressed as (14) 4.2 Design of eRFOC Due to the superior dynamic performance, RFOC is preferred in the inverter-fed motor drive applications, which could achieve the decoupling control of the torque and the flux linkage. For multiphase machines, the mutual inductance in the plane is much smaller than in the plane, which means and do not contribute much to the electromagnetic torque and flux, and most of the electromechanical energy conversion occurs on the plane. For multiphase machines with ideal sinusoidal winding distribution, there is even no mutual inductance in the plane. Therefore, the reference values of and in RFOC are given by the rotor flux command and the torque command , shown in (15) and (16). The symbol is the number of polar pairs, is the rotor leakage inductance (15) (16)Actually, in both healthy and fault-tolerant conditions, only the basic rotating component exists in , whose linearisation only needs the transformation. However, along with the fundamental current, other low-order harmonics exist in the phase current of multiphase machines. Considering that in most of the fault-tolerant conditions, the amplitude of is much higher than that of other SCs, low-order harmonics elimination is added in the control loop. As shown in Fig. 7, is employed to suppress the kth order harmonic. Fig. 7Open in figure viewerPowerPoint Control diagram for with low-order harmonics elimination In healthy conditions, the multiphase IM only calls for the control of . However in faulty conditions, the fault-tolerant operation requires the constrains on other SCs discussed in Section 3, and therefore calls for control loops of all the independent SCs. Evolved from traditional RFOC, the eRFOC can be designed in Fig. 8. It should be noted that the orthogonality of independent SCs ensures that the regulation of would not affect other independent SCs. Therefore, the final phase voltage reference would be the sum of respective reference values from (m − 1)/2 groups of current controllers, shown in the following equation: (17) Fig. 8Open in figure viewerPowerPoint eRFOC diagram for the multiphase IM 5 Computation cost analysis The difference of conventional RFOC and eRFOC mainly lies in the current control block, so the computation cost analysis concentrates on the control of symmetrical current components. In the conventional RFOC, which is directly extended from three-phase to multiphase occasions, only and of are controlled. However, in the eRFOC, all the independent SCs are taken into consideration. Furthermore, dual rotating transformations are required for , as shown in Fig. 6, while both low-order harmonics and the fundamental component need to be controlled for , as shown in Fig. 7. Table 4 compares the number of function blocks in the two control strategies, and Table 5 lists the number of addition and multiplication operations of each function block. Therefore, the current control of RFOC needs (4m + 14) multiplications and (3m + 8) additions, while that of eRFOC needs multiplications and additions. Table 4. Number of function blocks in control strategies for m-phase machine Control strategies SCs Clarke transformation Inverse Clarke transformation Rotating transformation PI controllers (m by m) (m by 2) (m by m) (2 by m) conventional RFOC 0 1 0 1 2 2 eRFOC 1 0 1 0 (k > 1) 0 1 s 0 1 4 4 Table 5. Number of multiplications and additions in respective function blocks Clarke transformation Inverse Clarke transformation Rotating transformation PI controllers (m by m) (m by 2) (m by m) (2 by m) multiplications 4 3 additions m 2 3 As is tested on the DSP (TMS320F28335) control board, each floating-point addition or multiplication operation would take seven cycles of the 150 MHz system clock. Therefore, for an m-phase drive system, the conventional RFOC costs about cycles, while eRFOC costs about cycles. The multiphase machines studied nowadays are usually with no more than 15 phases. In the symmetrical 15-phase case, which is the most complicated case, the computation in conventional RFOC and eRFOC would cost about 889 and 13,609 cycles, respectively. Considering that the 5 kHz sampling frequency is adopted in this control system, which corresponds to 30,000 cycles, the computation cost is not a very serious problem for the floating-point DSP. 6 Experiment results To check the fault-tolerant performance of the eRFOC, experiments were conducted on both 5PIM and 9PIM, whose photographs are shown in Figs. 9 and 10. Both the multiphase IMs are manufactured by rewinding the stator coils of traditional three-phase IMs, and their parameters are listed in Table 6. Table 6. Parameters of the multiphase IMs 5PIM 3.3 Ω 682 mH 2.2 Ω 20 mH 1 25.6 mH 9PIM 1.1 Ω 167 mH 0.7 Ω 10 mH 1 9 mH Fig. 9Open in figure viewerPowerPoint 5PIM experimental platform Fig. 10Open in figure viewerPowerPoint 9PIM experimental platform An extensible multiphase converter is built in the lab to feed the IMs. The block diagram of the multiphase drive experimental platform is shown in Fig. 11a. The combination of DSP and FPGA acts as the Master-controller, which is in charge of almost all the computation and generates the sinusoidal modulation waves for each inverter phase. Each phase module of the inverter is composed of a complex programmable logic device sub-controller and one inverter phase leg. The sub-controller generates the carrier wave and then compares the carrier to the modulation wave to give out pulse-width modulation signals, while it also samples the phase current by Hall sensors and sends the value back to the Master-controller. Fig. 11b is the photograph of the multiphase inverter. Fig. 11Open in figure viewerPowerPoint The experiment platform (a) Block diagram of the multiphase converter, (b) Its photograph 6.1 Transient performance In this part, the drive performance before and after fault-tolerant control is investigated under various open-circuit faulty scenarios, when both the 5PIM and 9PIM are rotating at 300 rpm. The five-phase one has no load, but the nine-phase one has a load torque of about 4.3 Nm generated by the magnetic powder brake. Respective results are shown in Figs. 12 and 13. Fig. 12Open in figure viewerPowerPoint 5PIM's transient waveforms of (a) Phase currents, (b) Zoom-in view of the phase currents, (c) iq1 current, (d) Output torque under three different faulty conditions Fig. 13Open in figure viewerPowerPoint 9PIM's transient waveforms of (a) Phase currents, (b) Zoom-in view of the phase currents, (c) iq1 current, (d) Output torque under five different faulty conditions The current waveforms of phase currents are shown in Figs. 12a and 13a, and zoom-in views of 1 s are shown in Figs. 12b and 13b. The waveforms before and after the switch point demonstrate that the transition of the phase currents is smooth without surge currents. As currents in the plane contribute most of the flux and torque, the quality of torque is mainly determined by . Waveforms of shown in Figs. 12c and 13c declare that under the various kinds of the open-circuit fault conditions, the oscillations of are gradually suppressed in <0.5 s after the fault-tolerant command is activated. The waveforms of output torque measured from the torque meter are shown in Figs. 12d and 13d, which correspond well with the currents in Figs. 12c and 13c, respectively. Furthermore, the amplitudes of torque ripples before and after fault-tolerant activation are listed in Table 7, which confirms the effectiveness of the proposed strategy under all the experimented faulty conditions. Table 7. Comparison of torque ripples before and after the activation of eRFOC Fault type Amplitudes of torque ripples, Nm Fault type Amplitudes of torque ripples, Nm Without FT control With FT control Without FT control With FT control 5PIM 9PIM 1F 1.42 0.17 1F 0.59 0.35 2F-I 3.14 0.26 2F-I 2.12 0.46 2F-II 2.31 0.28 2F-II 0.69 0.38 2F-III 0.65 0.31 2F-IV 1.76 0.51 Additionally, current trajectories of the SCs are shown in Figs. 14 and 15. Due to the limited size of the manuscript, only the trajectories under two faulty types are plotted for each of the multiphase IMs. Under faulty conditions without fault-tolerant control in (1F-a), (2FII-a) of Fig. 14 and (1F-a), (2FI-a) of Fig. 15, the trajectories of are no longer a regular circle, which would result in irregular MMF and obvious torque oscillations. It should be noted that the trajectories with two faulty phases are more distorted than those with one faulty phase in both Figs. 14 and 15, which correspond with the more serious torque ripples under 2F conditions in Figs. 12 and 13. The activation of the proposed fault-tolerant control reshaped the current trajectory of to a near circle by adjusting the trajectories of other SCs shown in (1F-b), (2FII-b) of Fig. 14 and (1F-b), (2FI-b) of Fig. 15. Therefore, the torque ripples, mainly affected by , could be effectively suppressed after the fault-tolerant switch point. Fig. 14Open in figure viewerPowerPoint Current trajectories of SCs for the 5PIM under respective faulty conditions (a) With fault-tolerant control, (b) Without fault-tolerant control Fig. 15Open in figure viewerPowerPoint Current trajectories of SCs for the 9PIM under respective faulty conditions (a) With fault-tolerant control, (b) Without fault-tolerant control 6.2 Dynamic performance In this part, dynamic responses of the fault-tolerant strategy are investigated. For the 5PIM, the reference speed steps from 300 to −300 rpm with no load in the faulty condition of 1F type. The speed performance is shown in Fig. 16a, where the transition cost about 10 s. Phase currents in Fig. 16b show the dynamic process of currents regulation. The amplitudes of the steady-state currents after the reversal are almost the same as before, but the phase position relationship of the four phase currents is inversed, when compared the current waveforms at 300 rpm to those at −300 rpm. Fig. 16Open in figure viewerPowerPoint Dynamic responses of the 5PIM (a) Speed of the 5PIM in 1F scenario, (b) Phase current dynamic responses of the 5PIM in 1F scenario For the 9PIM with a load torque of 4 Nm, a step in the reference speed from 150 to 300 rpm is implemented under the faulty condition with two adjacent open-circuit phases (type 2F-I). It is clear that the controlled system presents a speed response of about 10 s in Fig. 17a. Fault-tolerant control with absent of two phases calls for unequal amplitudes of the phase currents in Fig. 17b. This complies with the optimal current references in Fig. 5c, where the amplitudes of currents in the third and ninth phases are about 1.8 times as large as those in the fourth and eighth phases. Fig. 17Open in figure viewerPowerPoint Dynamic responses of the 9PIM (a) Speed of the 9PIM in 2F-I scenario, (b) Phase current dynamic responses of the 9PIM in 2F-I scenario 6.3 Harmonic performance Additional controllers in harmonic planes for and are included in the control block of the proposed eRFOC strategy, but are absent in the previous strategies [8, 11]. For the 5PIM rotating at 300 rpm, the steady-state currents under faulty condition of 1F are shown in Fig. 18a when eRFOC in Fig. 8 is applied. If controllers on third harmonic plane are disabled, the steady-state currents are shown in Fig. 18b, where the phase currents are visibly distorted. Detailed fast Fourier transform (FFT) analysis of the current in phase C is shown in Fig. 19. Without harmonic control, the third harmonic content can be as high as 9.89%; while with harmonic control, the third harmonic can be suppressed to about 2.12%. Fig. 18Open in figure viewerPowerPoint Phase currents of the 5PIM under the 1F faulty condition (a) With low-order harmonics suppression, (b) Without low-order harmonics suppression Fig. 19Open in figure viewerPowerPoint FFT analysis of (a) With low-order harmonics suppression, (b) Without low-order harmonics suppression In the nine-phase case, comparative experiments between with and without harmonic controllers in the control block were conducted under the faulty type of 2F-I, when the machine is rotating at 300 rpm with a load torque of about 4 Nm. To better show the difference, only three out of the nine phase currents are shown in Fig. 20. Clearly, phase currents in Fig. 20b present more distorted waveforms than those in Fig. 20a, which is due to the absence of third, fifth and seventh harmonic current controllers. The FFT results of in Fig. 21 further verify that the third, fifth and seventh current harmonics can be effectively suppressed to <4% with respective harmonic control loops, and the total harmonic distortion (THD) of would be improved from 14.84 to 7.79%. Fig. 20Open in figure viewerPowerPoint Phase currents of the 9PIM under the 2F-I faulty condition (a) With low-order harmonics suppression, (b) Without low-order harmonics suppression Fig. 21Open in figure viewerPowerPoint FFT analysis of (a) With low-order harmonics suppression, (b) Without low-order harmonics suppression 6.4 Comparison with previous methods The comparison between previous fault-tolerant methods and the proposed eRFOC in terms of phase current THD, torque ripples, anti-noise interference ability and so on is summarised in Table 8. As illustrated in this table, the proposed eRFOC has achieved satisfactory performance in all the listed indexes, while previous methods have one or more significant drawbacks. Table 8. Comparison between the proposed eRFOC and previous fault-tolerant methods Current controllers Hysteresis [9, 10, 12, 13] PR [14] MPC [15, 16] SMC [17] FLC [18] PI types — — — — — multi-space vector [8] dual-PI [11, 23] reduced-order model [21, 22] proposed eRFOC current THD H L M M M M L L L torque ripples L L L M H L L L L anti-noise interference ability L H M H H H H H H stability of digital implementation H L H M H H H H H robustness against parameter variation H H L H H H H H H computation cost L L H M H M L H M extensibility H H L L H H L L H H, high; M, medium; L, low. 7 Conclusion The SC theory is a powerful method to identify and regulate the relationship among different phases of the multiphase system. Evolved from the traditional RFOC for multiphase IMs, a generalised fault-tolerant eRFOC strategy was proposed in this paper to suppress the torque pulsations by regulating the independent symmetrical current components under various kinds of open-circuit faulty conditions. The separation of the positive- and negative-sequence reference currents for respective SCs enables a smooth dynamic transition without surge currents from pre- to post-fault conditions. Moreover, the decoupling of different time-domain rotating transformations ensures the suppression of low-order harmonics during the fault-tolerant operation. Experiment results of transient, dynamic and harmonic performances on 5PIM and 9PIM drive platforms confirmed the validity of the proposed fault-tolerant eRFOC strategy. Though compared with the traditional RFOC, the proposed eRFOC is of high complexity, whose computation cost increases exponentially as the phase number increases, it is practical on floating-point DSP for multiphase machines with commonly used phase numbers. 8 References 1Jahns T.M.: 'Improved reliability in solid-state AC drives by means of multiple independent phase drive units', IEEE Trans. Ind. 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