Fault Tolerant Capability of Symmetrical Multiphase Machines under One Open-Circuit Fault W.N.W.A. Munim *†, Hang Seng Che *, Wooi Ping Hew * * UMPEDAC, University of Malaya, Kuala Lumpur, Malaysia [email protected] [email protected] Faculty of Electrical Engineering, Universiti Teknologi MARA (UiTM), Malaysia [email protected] † Keywords: Multi-phase drives, fault-tolerance, current optimization, symmetrical components. references for different multiphase machines, based on different optimization targets. Open-circuit faults for dedicated phase such as five-phase machine has been reported by Ryu et al. [14] highlighting both minimum loss and maximum torque whilst Khalik et al. [15] has proposed the calculation of derating factor to avoid overheating of the machine after fault occurrence and maximum torque in order to maintain equal phase currents. On the other hand, different approach using Lagrange multiplier method has been investigated in [16] for six-phase focusing on minimum loss only and in [17] for seven-phase machine solving minimum loss and maximum torque in transient and steady-state operating conditions. Nevertheless, the overall comparison of post-fault performance for faulted n-phase indicating minimum loss, maximum torque as well as derating factor still not been stated yet in literature. To obtain a better overall view on the post-fault performance for machines with different phase number, some researchers have perform analysis on a range of multiphase machines. Fu et al. [18] presented post-fault current analysis for three- to seven-phase induction machines with the aim of equalising amplitude of all remaining currents after fault whereas in [19], Zheng et al. uses the symmetrical component transformation based on Fortescue to analyse the stator copper loss minimization and the inverter peak current minimization for three- to nine-phase machines in steadystate operating conditions. In addition, Baudart et al. [16] addressed the minimum loss of three- to eight-phase machines operating under open-phase conditions. However, the paper has not presented the derating factor which is the restriction of the maximum post-fault current does not exceed the rated phase current to maintain the reliability of the system. Based on the literature reviewed, three optimizations objectives are found to be common: equalising current amplitudes after fault, minimising copper losses and maximising torque/power under a given current limit. Among these, minimum loss and maximum torque (hereafter referred to as ML and MT respectively) are of particular importance. In this paper, the fault tolerant capability of symmetrical fivephase (S5), six-phase (S6), seven-phase (S7), and nine-phase (S9) in single isolated neutral under one open-circuit fault (OCF) is investigated. The optimized post-fault currents under ML and MT modes are first obtained, and the performances of the machines in terms of derating factor (a) and stator copper loss under both modes are evaluated. The Abstract This paper presents a fault-tolerant investigation of symmetrical multi-phase induction machines (five-phase, sixphase, seven-phase and nine-phase) in terms of minimum loss and maximum torque with single isolated neutral point under open-circuit fault at one of the phases. Minimum reconfiguration of the controller attainable for the post-fault control by using normal decoupling transformation. The postfault performance is carried out based on derating factor, minimum stator copper loss and minimum peak current. The simulation results confirm the validity of the theoretical postfault current limits for symmetrical multi-phase induction machines under one open-phase fault. 1 Introduction Since beginning of the 20th century, the use of multi-phase (more than three-phase) machines have been explored [1] for their various advantages over conventional three-phase machines, such as reducing the current per phase without increasing the voltage per phase, lowering dc harmonics and increasing reliability [2]. Even though multiphase grid does not exist by default, advanced applications of power electronic these days enables the use of multiphase electric drives with literally no restriction on the number of phases. This has opened the possibility for the utilisation of multiphase machines on applications such as electric vehicles [3], “more electric’ ships and aircraft, offshore wind farms generators, high speed elevators and etc. In the light of this, various research works have been devoted to the development and study of multiphase machines and drives, particularly in terms of fault tolerant design [4]–[8], modelling [9], [10] and control [11]–[13]. One of the most acclaimed features of multiphase machine is its ability to tolerate open-circuit faults in its phase windings. To do this, a suitable set of new (post-fault) current references is necessary to avoid negative sequence component in the fundamental magneto-motive force (MMF) of the machine, which will otherwise manifest as undesirable torque and speed oscillations. Over the years, many works have been devoted to the optimization of such post-fault current 1 results provide a guideline on the choice of multiphase machine and mode of operation for a given fault tolerance requirement. This paper is organized in the following manner: Section 2 outlines the concepts of current optimization in post-fault mode, while Section 3 explains the indicators used to evaluate fault tolerant capability of a given machine. The theoretical current waveforms and post-fault performances of the machines are simulated and shown in Section 4, before conclusions are given in Section 5. 2 Current Optimization in Post-Fault Mode This section describes the healthy and faulted operation of S5, S6, S7 and S9 drives and the optimization technique to achieve disturbance-free fault-tolerant operation. 2.1 Overview of multi-phase drives Symmetrical multi-phase induction machine is controlled using vector space decomposition (VSD) model approach and the generalized Clarke transformation matrix [2] as shown in Equation (1). Tn 2 ... n x( n 4)/2 y( n 4)/2 0 0 x1 y1 x2 y2 ... 1 cos cos 2 cos 3 ... 0 sin sin 2 sin 3 ... 1 cos 2 cos 4 cos 6 ... 0 sin 2 sin 4 sin 6 ... 1 cos 3 cos 6 cos 9 ... 0 sin 3 sin 6 sin 9 ... ... ... n2 n2 n2 1 cos cos 2 cos 3 ... 2 2 2 n2 n2 n2 0 sin sin 2 sin 3 ... 2 2 2 1 1 1 1 ... 2 2 2 2 1 1 1 1 ... 2 2 2 2 where α = 2/n. cos n 1 cos 2 n 1 sin 2 n 1 cos 3 n 1 sin 3 n 1 n2 cos n 1 2 n2 sin n 1 2 1 2 1 2 sin n 1 (1) Decoupling transformation [Tn] will be used to transform the n-phase variables to the n decoupled variables namely α-β components which related to flux and torque, x-y currents ((n4)/2 for n=even or (n-3)/2 for n=odd) and zero sequence components 0+0- which are not involved in the energy conversion process and only contribute to loss. For odd phase numbers, 0- is omitted as it is only exist in even number of phases. Fault-tolerant control targeting to preserve the prefault torque with no additional torque ripple as in [11] is assumed here. In order to do that, new optimized post-fault current references need to be defined, based on the Minimum loss (ML) and maximum torque (MT) optimization objectives commonly used in literature. 2.2 Minimum Loss (ML) When the phase A current is restricted to zero, the objective of ML is to minimize the stator copper losses defined by the cost function 𝐽𝑀𝐿 : 2 2 𝐽𝑀𝐿 = min{√(𝑖𝛼2 + 𝑖𝛽2 + 𝑖𝑥2 + 𝑖𝑦2 + 𝑖0+ + 𝑖0− )} (2) However, ML will results in unequal phase currents eg: S5 produces two sets of equal phase currents with different amplitude, S6 results in two sets of equal phase currents and one phase current with different amplitude. This mode also leads to the reduced of maximum achievable torque. 2.3 Maximum Torque (MT) Since inverter is usually designed to operate with current limited to the nominal value, this current limit should be obeyed even during fault to avoid damaging the drive. Hence, in MT post-fault operation, the aim is to minimize the maximum phase current of the remaining healthy phases. This is equivalent to the inverter peak current minimization used in [19]. In this case, the cost function 𝐽𝑀𝑇 targets to maximize the torque, which in turn indicates maximizing the amplitude of the α-β phasor: 𝐽𝑀𝑇 = max(|𝐼𝛼𝛽 |) (3) Subjectto: 𝐼𝑓𝑎𝑢𝑙𝑡 = 0 ∈ {Faultedphases} 𝑖0+ = 0 minthemaximumphasecurrent ∈ {Healthyphases} (4) 2.4 Optimization There are various ways to optimization post-fault currents, and the approach in [11], i.e. based on decoupled variables, have been adopted here in this paper. Coefficients “K” are used to relate the non-energy-converting currents with the α-β references. The number of optimized currents depends on the phase number n, where there are (n-4)/2 or (n-3)/2 x-y currents in addition to 2 or 1 zero sequence currents for even or odd phase machines respectively. Equations (5) shows the coefficients used in S9: ∗ 𝑖𝑥1 = 𝐾1 · 𝑖𝛼∗ + 𝐾2 · 𝑖𝛽∗ ∗ 𝑖𝑦1 = 𝐾3 · 𝑖𝛼∗ + 𝐾4 · 𝑖𝛽∗ ∗ 𝑖𝑥2 = 𝐾5 · 𝑖𝛼∗ + 𝐾6 · 𝑖𝛽∗ ∗ 𝑖𝑦2 = 𝐾7 · 𝑖𝛼∗ + 𝐾8 · 𝑖𝛽∗ ∗ 𝑖𝑥3 = 𝐾9 · 𝑖𝛼∗ + 𝐾10 · 𝑖𝛽∗ ∗ 𝑖𝑦3 = 𝐾11 · 𝑖𝛼∗ + 𝐾12 · 𝑖𝛽∗ (5) In the case of S9, the x1y1, x2y2, and x3y3, currents references for the corresponding post-fault modes can be obtained by optimizing the coefficients K1, K2,…, K12, based on different optimization objective (ML or MT). Zero sequence 0+ is set to be zero. Meanwhile, for S6, only one set of x-y currents and two zero sequence currents 0+0- need to be optimized, as shown in Equation (6). 𝑖𝑥∗ = 𝐾1 · 𝑖𝛼∗ + 𝐾2 · 𝑖𝛽∗ 𝑖𝑦∗ = 𝐾3 · 𝑖𝛼∗ + 𝐾4 · 𝑖𝛽∗ ∗ 𝑖0+ = 𝐾5 · 𝑖𝛼∗ + 𝐾6 · 𝑖𝛽∗ ∗ 𝑖0− = 𝐾7 · 𝑖𝛼∗ + 𝐾8 · 𝑖𝛽∗ (6) This paper using a non-linear optimization method, i.e. the generalized reduced gradient (GRG) approach, provided by “Solver”, an add-on in MS Office Excel. The optimization objectives for ML and MT modes based on (2) and (3) respectively and are subjected to (4). The coefficients will be varied at each iteration and the subsequent phase currents amplitudes are obtained by applying [Tn]-1 onto the VSD currents. Derating factor, a (%) 100 𝑎= |𝐼𝛼𝛽| (7) 𝑅𝑎𝑡𝑒𝑑 The mode of operation factor (kM): enhancement of the derating factor a with maximum torque (MT) compared to minimum loss (ML) criterion: 𝑘𝑀 = 𝑎𝑀𝑇 −𝑎𝑀𝐿 𝑎𝑀𝐿 x100 (8) While ML mode improves efficiency, it gives less post-fault torque/power than MT mode. The value of kM hence indicates the gain in torque/power if MT mode is chosen over ML mode. Phase S5Ø a 0.6813 S6Ø 0.6882 S7Ø 0.7043 S9Ø 0.7403 Minimum Loss, ML (p.u) Ploss Coefficients, 𝑲 1.4999 𝑲𝟏 = -1 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 1.3333 𝑲𝟏 = -0.6667 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 𝑲𝟓 = 0 𝑲𝟔 = 𝑲𝟕 = -0.4714 𝑲𝟖 = 1.2499 𝑲𝟏 = -0.5 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 𝑲𝟓 = -0.5 𝑲𝟔 = 𝑲𝟕 = 0 𝑲𝟖 = 1.1666 𝑲𝟏 = -0.3333 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 𝑲𝟓 = -0.3333 𝑲𝟔 = 𝑲𝟕 = 0 𝑲𝟖 = 𝑲𝟗 = -0.3333 𝑲𝟏𝟎 = 𝑲𝟏𝟏 = 0 𝑲𝟏𝟐 = 50 40 30 20 10 4 5 6 7 Number of phases Maximum Torque 8 9 10 Current Limit Figure 1. Comparison of derating factor for S5, S6, S7 and S9. 16.56 Derating factor can be used to evaluate the post-fault torque available for given faulted machine without violating nominal current limit. A higher derating factor indicates higher maximum torque can be obtained for a given current limit. 60 Minimum Loss km % 15 The derating factor (a): per unit value of the postfault α-β current phasor modulus, with limitation that the maximum post-fault phase current does not exceed the rated phase current [11]. 𝑃𝑜𝑠𝑡−𝑓𝑎𝑢𝑙𝑡 70 20 Several performance indicators have been selected to examine the performance of the machines after fault. |𝐼𝛼𝛽| 80 0 3 3 Post-Fault Performance Indicator 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15.28 12.03 10 6.21 5 0 3 4 5 6 7 Number of phases 8 9 10 Figure 2. Mode of operation factor (kM) versus number of phases for 1 OCF. Normalised Post-fault Stator Copper Loss (Ploss): since different mode gives different phase currents, the stator copper loss is expected to differ. The stator copper loss, normalised to that of a healthy machine, can be calculated based on: 𝑃𝑙𝑜𝑠𝑠 = 2 +𝑖 2 +𝑖 2 +𝑖 2 +𝑖 2 +𝑖 2 } {𝑖𝛼 𝛽 𝑥 𝑦 0+ 0− (9) 2 2 𝑖𝛼_𝑅𝑎𝑡𝑒𝑑 +𝑖𝛽_𝑅𝑎𝑡𝑒𝑑 Figure 2 shows the mode of operation factor (kM) versus number of phases for 1 OCF. It shows that MT gives approximately 6-17% of improvement in a compared to ML mode, with the improvement being increasingly significant for higher number of phases. a 0.7236 0.7710 0.8119 0.8629 Maximum Torque, MT (p.u) Ploss Coefficients, 𝑲 1.5279 𝑲𝟏 = -1 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 1.4013 𝑲𝟏 = -0.6484 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 𝑲𝟓 = 0 𝑲𝟔 = 𝑲𝟕 = -0.4972 𝑲𝟖 = 1.3003 𝑲𝟏 = -0.5864 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 𝑲𝟓 = -0.4136 𝑲𝟔 = 𝑲𝟕 = 0 𝑲𝟖 = 1.1937 𝑲𝟏 = -0.4250 𝑲𝟐 = 𝑲𝟑 = 0 𝑲𝟒 = 𝑲𝟓 = -0.3747 𝑲𝟔 = 𝑲𝟕 = 0 𝑲𝟖 = 𝑲𝟗 = -0.2003 𝑲𝟏𝟎 = 𝑲𝟏𝟏 = 0 𝑲𝟏𝟐 = 0 -0.2361 0 -0.3681 0 0 0 -0.1935 0 -0.2198 0 -0.1355 0 -0.0820 0 0.0347 Table 1: Reference x-y, 0+-0- Currents under Fault Tolerant ML and MT Modes of Operation for Symmetrical Five-, Six-, Seven- and Nine-Phase Machine. The overall derating factor performance with 1OCF comparing S5, S6, S7 and S9 is illustrated in Fig. 1. Since the post-fault current and torque production is proportional to a and a2 respectively in induction machine [11], the derating factor becomes significant as the number of phases reduced in ML mode, with more improvement for higher number of phases. However, with each additional phase added, the improvement gained becomes increasing marginal where the gain is seen to saturate from S7 to S9. 2 ia 1.5 ib ic id The outcomes of the optimization results are given in Table 1, including the optimized coefficients as well as the performance indicators. 4 Simulation Results To visualise and verify the optimization results in Table I, simulation studies have been conducted using Matlab Simulink. Phase currents for the studied machines pre- and 2 ie Currents (p.u.) Currents (p.u.) 0 -0.5 -1 0.005 0.01 0.015 0.02 0.025 Time (s) id ie if 0.5 0 -0.5 -1 0.03 0.035 -2 0 0.04 0.005 0.01 0.015 (a) 2 0.02 Time (s) 0.025 0.03 0.035 0.04 (b) ia 1.5 ib ic id ie if 2 ig ia 1.5 ib ic id ie if ig ih ii 1 Currents (p.u.) 1 Currents (p.u.) ic -1.5 -1.5 0.5 0 -0.5 -1 0.5 0 -0.5 -1 -1.5 -1.5 -2 0 ib 1 1 0.5 -2 0 ia 1.5 0.005 0.01 0.015 0.02 0.025 Time (s) 0.03 0.035 -2 0 0.04 0.005 0.01 0.015 0.02 0.025 Time (s) 0.03 0.035 0.04 (c) (d) Figure 3. Stator phase current waveforms for healthy and ML modes (with 1OCF) for symmetrical multiphase machines a) S5, b) S6, c) S7 and d) S9. 2 ia 1.5 ib ic id 2 ie 0.5 0 -0.5 -1 -1.5 id ie if 0 -0.5 -1 -1.5 0.005 0.01 0.015 0.02 0.025 Time (s) 2 0.03 0.035 -2 0 0.04 0.005 0.01 0.015 0.02 Time (s) 0.025 0.03 ic ie 0.035 0.04 (b) ia 1.5 ib ic id ie if 2 ig ia 1.5 1 ib id if ig ih ii 1 Currents (p.u.) Currents (p.u.) ic 0.5 (a) 0.5 0 -0.5 -1 -1.5 -2 0 ib 1 Currents (p.u.) Currents (p.u.) 1 -2 0 ia 1.5 0.5 0 -0.5 -1 -1.5 0.005 0.01 0.015 0.02 Time (s) 0.025 0.03 0.035 0.04 -2 0 0.005 0.01 0.015 0.02 Time (s) 0.025 0.03 (c) (d) Figure 4. Stator phase current waveforms for healthy and MT modes (with 1OCF) for symmetrical multiphase machines a) S5, b) S6, c) S7 and d) S9. 0.035 0.04 2 2 Ploss S5 1.5 Ploss (p.u.) Ploss (p.u.) 1.5 1 Healthy MT ML 1 Healthy 0.01 0.02 0.03 Time (s) 0.04 0.05 0 0 0.06 0.01 0.02 (a) 0.04 0.05 2 Ploss S7 0.06 Ploss S9 1.5 Ploss (p.u.) 1.5 Ploss (p.u.) 0.03 Time (s) (b) 2 1 Healthy MT ML 1 Healthy MT ML 0.5 0.5 0 0 MT ML 0.5 0.5 0 0 Ploss S6 0.01 0.02 0.03 Time (s) 0.04 0.05 0.06 0 0 0.01 0.02 0.03 Time (s) 0.04 0.05 0.06 (c) (d) Figure 5. Ploss under healthy, ML and MT modes (with 1OCF) for symmetrical multiphase machines a) S5, b) S6, c) S7 and d) S9. post-fault have been plotted based on the information in Table I. The simulation has been set to maintain the same α-β magnitude (hence torque/power) in pre- (𝑡 < 0.02𝑠) and post(𝑡 > 0.02𝑠) fault situations. Results for S5, S6, S7 and S9 under ML and MT mode in healthy and after faulty condition are shown in Fig. 3 and Fig. 4 respectively. Using ML criterion, the phase currents increases after fault with the pre-fault current being 0.6813, 0.6882, 0.7043 and 0.7403 times the maximum post-fault currents for S5, S6 S7 and S9 respectively, in accordance with their respective derating factor in Table 1. If MT criterion used, the phase currents increases after fault with the pre-fault current being 0.7236, 0.7710, 0.8119 and 0.8629 times the maximum post-fault currents for S5, S6 S7 and S9 respectively. It is worth noting that ML will leads to unequal phase currents. This can be seen from Fig. 3(a) whereas S5 produces two sets of currents (ib and ie) and (ic and id) with 1.4678 and 1.2631 times of its rated value respectively. S6 however results in two sets of equal phase currents (ib and if), (ic and ie) and one phase current (id) with different amplitude. On the other hand, MT mode gives rise to equal current amplitudes in the remaining phases after fault, even though equalisation of current amplitude was not specified as an optimization objective. Finally, Fig. 5 shows the comparison of normalised stator copper losses Ploss for S5-S9 under healthy, ML and MT modes respectively. The healthy state has been set to (𝑡 < 0.02𝑠), ML mode at (0.02𝑠 < 𝑡 < 0.04𝑠) and MT mode at (0.04𝑠 < 𝑡 < 0.06𝑠), such that instantaneous power loss for each mode over one fundamental cycle can be observed. It is worth noting that for all studied machines, fault tolerant operation under single OCF resulted in higher instantaneous power loss (than healthy case) which is oscillating with twice the fundamental frequency. As the number of phase increases, the increase in average Ploss becomes less significant. Furthermore, it can be seen that MT mode give more torque at a marginal increase in stator copper losses. For S5, MT mode gives 6.2% more torque increase at a cost of 1.9% additional loss; while for S9, a 16.6% gain in torque with 2.3% increase in loss is obtained when MT mode is chosen over ML mode. This implies that generally, MT mode gives better performance with marginal penalty in power loss. This is more so as the number of phases increase. 5 Conclusion In this paper, an analysis on the fault tolerant capability of several popular symmetrical multiphase machines under one open-circuit fault has been presented. The paper considers comparison of minimum loss, maximum torque and derating factor under fault scenario for S5, S6, S7 and S9. Based on the theoretical and simulation study, it can be concluded that MT mode is more favourable than ML mode, since the gain in torque/power is generally more significant than the increase in power losses. 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