# A Torque Impulse Balance Control for Multi-Tooth Fault Tolerant Switched-Flux Machines under Open-Circuit Fault

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## Abstract

**:**

## 1. Introduction

- (1)
- Excellent torque capability and high power density.
- (2)
- High reliability and strong fault tolerant capability.
- (3)
- Superior steady-state and dynamic performance for the system under both healthy condition and open-circuit faults.

- (1)
- The PMs are mounted on the stator, being free from centrifugal force.
- (2)
- Simple and robust rotor.
- (3)
- The PM field and the armature field distribute in parallel, leading to very low risk of demagnetization.
- (4)
- The FSPM machine can suppress the short-circuit current by increasing self-inductance, whereas the rotor-PM machine needs to increase the leakage inductance to suppress the short-circuit current. Therefore, the SFPM machine can guarantee relatively high torque density while enhancing the ability to suppress the short-circuit [12].

- (1)
- The mutual inductance of the SRM is basically zero, whereas the FTFSPM needs the alternate pole-wound structure to reduce the mutual inductance.
- (2)
- The short-circuit current of the SRM is basically zero due to the absence of PMs, whereas the FTFSPM needs a special design to suppress the short-circuit current.
- (3)
- The SRM operates in half-cycle mode, whereas the FTFSPM operates in whole-cycle mode owing to the PMs, making a relatively high torque density and efficiency. In addition, although with doubly salient structure and concentrated winding, the SFPM machine can provide high sinusoidal PM back-electromotive force (back-EMF) due to the complementary winding structure; thus, this kind of machine is suitable for operation in BLAC mode [11,12].

## 2. System Structure and Operation Principle

#### 2.1. Machine Topology

- (1)
- The stagger degree between the two parts of the rotor is 180° (electrical degree).
- (2)
- The two-part PMs in a stator tooth have opposite magnetization direction.

- (1)
- The multi-tooth structure can effectively improve the ability to restrain short-circuit currents and maintain a relatively high torque density.
- (2)
- Based on the twisted-rotor, the phase back-EMF of high sinusoidal and symmetric can be obtained without amplitude decrease.

#### 2.2. Influence of PI Controller on Rotor Speed Dynamic Performance

_{2}, t

_{4}, t

_{6}, t

_{8}, t

_{10}and t

_{12}, ${n}_{r}={n}_{r}^{*}$ because ${T}_{e}\ne {T}_{l}$, n

_{r}will continue to change. At t

_{1}, t

_{3}, t

_{5}, t

_{7}, t

_{9}and t

_{13}, the change rate of the speed is zero because ${T}_{e}={T}_{l}$, and the change of T

_{e}will continue with ${T}_{e}^{*}$, because ${n}_{r}\ne {n}_{r}^{*}$.

- (1)
- For the control system, the electromagnetic torque and the rotor speed converge at the same time are sufficient conditions for restoring to the steady-state.
- (2)
- The PI parameters of the speed loop in the RDTC always affect the dynamic response of both electromagnetic torque and rotor speed. The PI controller cannot possibly guarantee the optimal dynamic performance.
- (3)
- A group of switching vector sequences can be found to simultaneously restore the electromagnetic torque and the rotor speed at t
_{1}. Thus, the speed can be restored to the reference after a decrease and increase only once without overshooting, that is, the possible optimal dynamic response of n_{r}can be obtained.

#### 2.3. Torque Impulse Balance Control

_{e}is the electromagnetic torque of the machines, T

_{l}is the load torque, and the integral form of (3) is

_{l}changes suddenly at t

_{0}; from t

_{0}to t

_{2}, the forward vectors are employed and T

_{e}increases continuously. From t

_{2}to t

_{3}, the zero vectors are employed and T

_{e}decreases continuously.

_{e}and n

_{r}can respectively converge to their corresponding reference values at t

_{3}, if a proper t

_{2}is obtained to satisfy A

_{1}= A

_{2}+ A

_{3}. Thus, the shortest restoration time of n

_{r}and the minimum ripple of the speed are reached.

_{eH}+ T

_{s}can be obtained as:

_{r}represents the rotor pole pairs, L

_{q}is the inductance of q-axis, ψ

_{s}is the amplitude of the stator flux linkage, ψ

_{pm}is the amplitude of the PM flux linkage, respectively, and θ is the torque angle.

_{e}is linearly related to sinθ when ψ

_{s}is kept constant, thus, Figure 6 is identical to Figure 13. In Figure 13, $\mathrm{sin}\gamma =2{T}_{l}{L}_{q}/6{P}_{r}{\psi}_{s}{\psi}_{pm}$.

_{0}to t

_{2}can be expressed as:

- In sector II and V of the stator flux linkage, see (8).$$\begin{array}{cc}\hfill {k}_{1}& =\frac{\mathrm{sin}({\theta}_{0}+\omega nT-{\omega}_{r}nT)-\mathrm{sin}[{\theta}_{0}+\omega (n-1)T-{\omega}_{r}(n-1)T]}{T}\hfill \\ & =\mathrm{cos}\frac{2{\theta}_{0}+\omega (2n-1)T-{\omega}_{r}(2n-1)T]}{T}(\frac{1}{T}arctg\frac{2\sqrt{3}{U}_{dc}\times T}{\pi {\psi}_{s}}-{\omega}_{r})\hfill \end{array}\text{}$$
- In sector I, III, IV and VI of the stator flux linkage, see (9).$$\begin{array}{cc}\hfill {k}_{1}& =\frac{\mathrm{sin}({\theta}_{0}+\omega nT-{\omega}_{r}nT)-\mathrm{sin}[{\theta}_{0}+\omega (n-1)T-{\omega}_{r}(n-1)T]}{T}\hfill \\ & =\mathrm{cos}\frac{2{\theta}_{0}+\omega (2n-1)T-{\omega}_{r}(2n-1)T]}{T}(\frac{1}{T}arctg\frac{3\sqrt{3}{U}_{dc}\times T}{2\pi {\psi}_{s}}-{\omega}_{r})\hfill \end{array}$$

_{dc}is the converter direct current (DC) link voltage, ω is the angular frequency of the stator flux linkage vector when the forward vectors are applied to the machine, T and n are the interrupt period and the number of interrupt periods, respectively, ω

_{r}is the rotor angular frequency, and θ

_{0}is the torque angle at t

_{0}.

_{2}to t

_{3}can be expressed as:

- In sector II and V of the stator flux linkage, see (10).$$\begin{array}{cc}\hfill {k}_{2}& =\frac{\mathrm{sin}({\theta}_{0}-\omega nT-{\omega}_{r}nT)-\mathrm{sin}[{\theta}_{0}-\omega (n-1)T-{\omega}_{r}(n-1)T]}{T}\hfill \\ & =\mathrm{cos}\frac{2{\theta}_{0}-\omega (2n-1)T-{\omega}_{r}(2n-1)T]}{T}(-\frac{1}{T}arctg\frac{2\sqrt{3}{U}_{dc}\times T}{\pi {\psi}_{s}}-{\omega}_{r})\hfill \end{array}$$
- In sector I, III, IV and VI of the stator flux linkage, see (11).$$\begin{array}{cc}\hfill {k}_{2}& =\frac{\mathrm{sin}({\theta}_{0}-\omega nT-{\omega}_{r}nT)-\mathrm{sin}[{\theta}_{0}-\omega (n-1)T-{\omega}_{r}(n-1)T]}{T}\hfill \\ & =\mathrm{cos}\frac{2{\theta}_{0}-\omega (2n-1)T-{\omega}_{r}(2n-1)T]}{T}(-\frac{1}{T}arctg\frac{3\sqrt{3}{U}_{dc}\times T}{2\pi {\psi}_{s}}-{\omega}_{r})\hfill \end{array}$$
_{2}is the torque angle at t_{2}.

_{2}can be achieved via the following steps:

- (1)
- First, achieve the double integral value of k
_{1}from t_{0}to t_{1}, where t_{1}is the moment when $d{\omega}_{r}/dt=0$. - (2)
- Then, calculate the double integral value of $\left({k}_{1}-{k}_{1}^{2}/{k}_{2}\right)$ using t
_{1}, the moment when this integral value is equal to $\int {\displaystyle {\int}_{{t}_{0}}^{{t}_{1}}{k}_{1}dt}}dt$ is t_{2}.

## 3. Experimental Results

#### 3.1. Experimental Results under Open-Circuit Fault

#### 3.1.1. Steady-State Performance

_{e}and n

_{r}by using DTC and RDTC, respectively. The torque ripple peak value in Figure 17 is 2.4 N∙m while that in Figure 18 is only 0.9 N·m, indicating that the RDTC can effectively suppress the torque ripple.

#### 3.1.2. Dynamic Performance

_{r}has a long period of oscillation before it converges, and the peak value of the oscillation is high, whereas when the poles are far away from the imaginary axis, the adjustment of n

_{r}lasts for a long time.

_{r}is improved, as shown in Figure 19.

_{r}is excellent, as seen in Figure 20.

#### 3.2. Experimental Results under Healthy Condition

_{r}under DTC and TIBC-DTC are shown in Figure 25, and the performance metrics of DTC and TIBC-DTC are given in Table 5.

## 4. Conclusions

- Compared with the vector control with current vector compensation technology, the direct torque control with voltage vector reconstruction technique (RDTC) can achieve the electromagnetic torque with good dynamic performance, but for the rotor speed, its dynamic performance is always influenced by the PI parameters of the speed loop.
- In the dynamic process, the torque impulse balance control (TIBC) can obtain an optimal voltage vector sequence which has only one switch between forward vectors and zero vectors.
- The optimal voltage vector can achieve excellent dynamic performance of the rotor speed: (1) No overshoot in the speed; (2) Only one adjustment; (3) The shortest settling time.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Fault tolerant PM machines. (

**a**) Fault tolerant rotor-PM(FTRPM) machine; (

**b**) Fault-tolerant FSPM (FTFSPM) machine.

**Figure 4.**MTFTSFM: (

**a**) The machine structure; (

**b**) The stator structure; (

**c**) Prototype stator; (

**d**) The rotor structure; (

**e**) Prototype rotor.

**Figure 11.**Optimal inductor current path for load current change (top: inductor current, bottom: capacitor voltage).

Torque and Flux-Linkage | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

${T}_{e}\uparrow $${\psi}_{s}\uparrow $ | ${V}_{2}$ | ${V}_{3}$ | ${V}_{4}$ | ${V}_{5}$ | ${V}_{6}$ | ${V}_{1}$ |

${T}_{e}\uparrow $${\psi}_{s}\downarrow $ | ${V}_{3}$ | ${V}_{4}$ | ${V}_{5}$ | ${V}_{6}$ | ${V}_{1}$ | ${V}_{2}$ |

${T}_{e}\downarrow $${\psi}_{s}\uparrow $ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ |

${T}_{e}\downarrow $${\psi}_{s}\downarrow $ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ | ${V}_{0}$ |

Torque and Flux-Linkage | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

${T}_{e}\uparrow $${\psi}_{s}\uparrow $ | ${V}_{o2}$ | ${V}_{o3}$ | ${V}_{o4}$ | ${V}_{o5}$ | ${V}_{o6}$ | ${V}_{o1}$ |

${T}_{e}\uparrow $${\psi}_{s}\downarrow $ | ${V}_{o3}$ | ${V}_{o4}$ | ${V}_{o5}$ | ${V}_{o6}$ | ${V}_{o1}$ | ${V}_{o2}$ |

${T}_{e}\downarrow $${\psi}_{s}\uparrow $ | ${V}_{o6}$ | ${V}_{o1}$ | ${V}_{o2}$ | ${V}_{o3}$ | ${V}_{o4}$ | ${V}_{o5}$ |

${T}_{e}\downarrow $${\psi}_{s}\downarrow $ | ${V}_{o5}$ | ${V}_{o6}$ | ${V}_{o1}$ | ${V}_{o2}$ | ${V}_{o3}$ | ${V}_{o4}$ |

Terms for Comparison | Mechanical System | Electrical System |
---|---|---|

Coordinate | Angular displacement θ | Flux-linkage ψ |

Velocity | Angular velocity ω | Voltage u |

Force | Torque T | Current i |

Inertial element | Moment of inertia J | Capacitor C |

Elastic element | Torsional stiffness coefficient K_{θ} | Inductance reciprocal 1/L |

Resistance element | Rotation resistance coefficient R_{ω} | Conductance G |

**Table 4.**Performance metrics of the direct torque control with voltage vector reconstruction (RDTC) without torque impulse balance control (TIBC) and with TIBC.

Methods | Settling Time $({\mathit{t}}_{\mathbf{s}})$ | Peak Value Time $({\mathit{t}}_{\mathit{p}})$ | Speed Dip Δn-down | Over Shoot Amount $(\mathit{\sigma}\%)$ | Adjustment Times $\left(\mathit{Z}\right)$ |
---|---|---|---|---|---|

RDTC without TIBC (P = 7.2, I = 0.96) | 205 ms | 75 ms | 120 r/min | 16.67% | 3 |

RDTC without TIBC (P = 9.5, I = 1.1) | 95 ms | 60 ms | 70 r/min | 13.33% | 4 |

RDTC with TIBC (TIBC-RDTC) | 50 ms | 50 ms | 60 r/min | 0.00% | 1 |

Methods | ${\mathit{t}}_{\mathbf{s}}$ | ${\mathit{t}}_{\mathit{p}}$ | $\Delta \mathit{n}$-down | $\mathit{\sigma}\%$ | $\mathit{Z}$ |
---|---|---|---|---|---|

DTC (P = 20, I = 0.8) | 375 ms | 175 ms | 65 r/min | 21.67% | 6 |

DTC (P = 10, I = 1.2) | 200 ms | 160 ms | 48 r/min | 16.00% | 2 |

TIBC-DTC | 50 ms | 50 ms | 40 r/min | 13.33% | 1 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Hao, W. A Torque Impulse Balance Control for Multi-Tooth Fault Tolerant Switched-Flux Machines under Open-Circuit Fault. *Energies* **2018**, *11*, 1919.
https://doi.org/10.3390/en11071919

**AMA Style**

Wang Y, Hao W. A Torque Impulse Balance Control for Multi-Tooth Fault Tolerant Switched-Flux Machines under Open-Circuit Fault. *Energies*. 2018; 11(7):1919.
https://doi.org/10.3390/en11071919

**Chicago/Turabian Style**

Wang, Yu, and Wenjuan Hao. 2018. "A Torque Impulse Balance Control for Multi-Tooth Fault Tolerant Switched-Flux Machines under Open-Circuit Fault" *Energies* 11, no. 7: 1919.
https://doi.org/10.3390/en11071919