# Direct Power Control of a Bipolar Output Active Rectifier for More Electric Aircraft Based on an Optimized Sector Division

^{*}

## Abstract

**:**

## 1. Introduction

- A set of new voltage vectors are synthesized in the proposed DPC strategy to extend the eight basic voltage vectors. Based on the new synthesized voltage vectors, the zero-sequence current in TCI can be controlled in a stable manner while implementing the hysteresis power control of the TCIBAR.
- Based on the derived power model of TCIBAR in the synchronous rotating coordinate system, the effect of the new synthesized voltage vectors on the power variation of TCIBAR is quantitatively analyzed. On this basis, an optimized sector division method is proposed to establish a new switching table for TCIBAR, which can improve the quality of the phase currents in TCIBAR.
- A ZSV generation method is developed in the proposed DPC strategy. Based on the ZSV generation method, the voltage balance control of the bipolar DC ports in TCIBAR can be realized, even under unbalanced load conditions.

## 2. TCIBAR Model Based on Instantaneous Power Theory

_{s}and R are the winding resistance of the source inductor and the TCI; e

_{a}, e

_{b}, and e

_{c}are the three-phase AC source voltages; i

_{sa}, i

_{sb}, and i

_{sc}are the three-phase input currents; i

_{la}, i

_{lb}, and i

_{lc}are the three-phase currents in TCI; u

_{p}and u

_{n}are the port voltages; i

_{p}and i

_{n}are the load currents of positive and negative ports; and U

_{dc}is the total DC bus voltage.

_{d}, e

_{q}, e

_{0}, i

_{sd}, i

_{sq}, i

_{s}

_{0}, and u

_{d}, u

_{q}, u

_{0}are the d-axis, q-axis, and zero-sequence components of source voltages, input currents, and VSC voltages, respectively, and ω is the angular frequency of the AC source. The zero-sequence equation in Equation (1) can be ignored since i

_{s}

_{0}= 0.

_{q}) is equal to zero. Meanwhile, by ignoring some small components in Equation (4), the simplified power model of the TCIBAR can be obtained, as expressed in Equation (5).

_{aN}, u

_{bN}, and u

_{cN}are the three-phase voltages of the TCI.

_{ld}, u

_{lq}, and u

_{l}

_{0}and i

_{ld}, i

_{lq}, and i

_{l}

_{0}are the d-axis, q-axis, and zero-sequence components of the three-phase voltages and currents of the TCI, respectively.

## 3. Proposed DPC Based on Optimized Sector Division

#### 3.1. Extension of Voltage Vector

_{d}and u

_{q}) with the switching functions of voltage vectors, as shown in Equation (8),

_{d}and S

_{q}are the switching functions in the dq0 coordinate system.

_{l}

_{0}of the TCI can be calculated based on the Park transformation, as shown in Equation (10),

_{x}(x = a, b, c) are the switching states of the three-phase bridges in TCIBAR.

**V**

_{0}–

**V**

_{7}) into Equation (11), the switching functions of the basic voltage vector are obtained, as shown in Table 1.

_{0}, which leads to differences in the ZSV components in basic voltage vectors. Therefore, if the basic voltage vectors are directly adopted to establish the switching table for hysteresis power control, regardless of its effect on the ZSV, the zero-sequence current in TCI will be uncontrollable, which will lead to voltage imbalance between the bipolar DC ports of the TCIBAR. In the meantime, due to the limited number of basic voltage vectors in TCIBAR, there are not enough redundant voltage vectors that can be selected to realize hysteresis power control while maintaining the voltage balance between the bipolar DC ports. To overcome this problem, a simple and effective solution is to extend the basic voltage vectors and find new voltage vectors that have the same ZSV component to establish the switching table for the TCIBAR.

**U**

_{1}–

**U**

_{6}) can be synthesized by the adjacent non-zero basic voltage vectors, and the adjacent non-zero basic voltage vectors each act for half of one control cycle. In addition, the equivalent switching states of the synthesized voltage vectors can be represented by 0, 0.5, and 1. Similarly, the switching functions of the synthesized voltage vectors can be deduced as shown in Table 2.

_{0}; thus, the ZSV components in the synthesized voltage vectors are all the same, according to Equation (10). Meanwhile, when the DC voltages of the bipolar DC ports are balanced (that is, η = 0.5), the ZSV applied to the TCI will be equal to zero under the action of the synthesized voltage vectors. Therefore, if the synthesized voltage vectors are used to establish the switching table, the TCIBAR can realize hysteresis power control without causing the runaway of the zero-sequence current in the TCI.

#### 3.2. Effect of Voltage Vector on Power Variation

#### 3.2.1. Vector Space Division for Reactive Power

_{s}, U

_{dc}, and e

_{d}are always positive, the sign of the reactive power variation rate (dq/dt) is directly dependent on the switching function S

_{q}.

**U**

_{1}as an example and substitute its switching function S

_{q}into the expression of dq/dt to get the following equation:

**U**

_{1}has the opposite effect on the reactive power of the TCIBAR.

_{q}of the other voltage vectors are successively substituted into the expression of dq/dt in Equation (8), and the areas divided by different voltage vectors’ effect on reactive power can be obtained, as shown in Figure 3.

#### 3.2.2. Vector Space Division for Active Power

_{d}, but is also related to the amplitude of the AC source voltage e

_{d}and the DC bus voltage U

_{dc}. Therefore, by analyzing the value of $-{S}_{d}+{e}_{d}/{U}_{dc}$ corresponding to each synthesized voltage vector, the vector space division for active power can be obtained.

**U**

_{1}as an example and substituting its switching function S

_{d}into $-{S}_{d}+{e}_{d}/{U}_{dc}$, the following expression is obtained:

**U**

_{1}. As shown in Figure 5, the vector space division for active power under the action of other synthesized voltage vectors can be deduced using the same method, and the boundaries can be expressed as:

**U**

_{1},

**U**

_{2},

**U**

_{3},

**U**

_{4},

**U**

_{5}, and

**U**

_{6}, respectively.

#### 3.3. Optimized Sector Division and Switching Table

_{P}and s

_{Q}are the outputs of the active and reactive hysteresis comparators.

#### 3.4. Voltage Balance Control under DPC Architecture

_{0}of the synthesized voltage vectors into Equation (10), the ZSV components in the synthesized voltage vectors are all equal to $\sqrt{3}(1-2\eta ){U}_{dc}/2$. The required ZSV cannot be accurately generated only by the six synthesized voltage vectors.

**V**

_{0}and

**V**

_{7}are $-\sqrt{3}\eta {U}_{dc}$ and $\sqrt{3}(1-\eta ){U}_{dc}$, respectively. Due to the constraint $0<\eta <1$, the signs of $-\sqrt{3}\eta {U}_{dc}$ and $\sqrt{3}(1-\eta ){U}_{dc}$ are opposite. Meanwhile, the ZSV components in

**V**

_{0},

**V**

_{7}, and the synthesized voltage vector always satisfy the following inequality:

**V**

_{0}and

**V**

_{7}can be inserted into the synthesized voltage vector to generate the desired ZSV.

**V**

_{0}is inserted. In this case, the action time of voltage vectors in one control cycle can be calculated as:

_{m}and t

_{0}are the action time of the synthesized voltage vector and

**V**

_{0}, respectively.

**V**

_{7}is inserted. In this case, the action time of voltage vectors can be calculated as:

_{7}is the action time of

**V**

_{7}.

## 4. Experimental Results

#### 4.1. Experimental Parameters

_{s}was equal to 50 μs.

#### 4.2. Steady State Experimental Study

_{ln}, and the voltages of the bipolar DC ports (u

_{p}and u

_{n}) were not balanced. The proposed DPC strategy effectively controlled the zero-sequence current and stabilized i

_{ln}at 0, as shown in Figure 10b. Meanwhile, the voltage balance between the bipolar DC ports was realized under the no load condition.

_{ln}was almost out of control, and the voltages of the positive and negative ports fluctuated greatly and could not maintain balance. As Figure 12b shows, based on the proposed DPC strategy, the total zero-sequence current i

_{ln}was controlled to provide the unbalanced current for the negative port. Meanwhile, the voltage fluctuations of the bipolar DC ports were suppressed, and the voltage balance between the bipolar DC ports was realized, even under an unbalanced load condition.

_{ln}was stably maintained at 0 A. In addition, when compared with the experimental results in Figure 13b, the total harmonic distortion (THD) of the phase current in Figure 14b reduced from 9.61% to 6.95% when the switching table based on the optimized 18-sector division was used. Therefore, the optimized sector division method improved the quality of the phase currents in the TCIBAR and had better steady-state performance, which created the advantage of a lower filter weight and was beneficial to the application of the TCIBAR in MEA.

#### 4.3. Dynamic Experimental Study

_{ln}was stabilized at 0 A, and the voltage balance between the bipolar DC ports was maintained during the dynamic process. Different from the balanced step load, the unbalanced step load led to a voltage imbalance between the bipolar DC ports, as shown in Figure 16. Based on the proposed strategy, the TCIBAR actively regulated the zero-sequence current, and the voltage balance between the bipolar DC ports was restored in 30 ms.

## 5. Conclusions

- Based on the instantaneous power theory, the power model of the TCIBAR in the synchronous rotating coordinate system was established and verified.
- Based on the new synthesized voltage vectors, the proposed DPC strategy realized the hysteresis power control of TCIBAR without causing the runaway of the zero-sequence current in TCI.
- The optimized sector division method in the proposed DPC strategy effectively reduced the THD of the phase currents and improved the steady-state performance of the TCIBAR.
- Based on the proposed ZSV generation method, the proposed DPC strategy realized the voltage balance control of the bipolar DC ports in TCIBAR and maintained the voltage balance between the bipolar DC ports, even under unbalanced load conditions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DPC | Direct power control |

MEA | More electric aircraft |

TCIBAR | Three-phase coupled inductor-based bipolar output active rectifier |

TCI | Three-phase coupled inductor |

ZSV | Zero-sequence voltage |

EPS | Electrical power system |

HVDC | High-voltage direct current |

ATRU | Auto transformer rectifier unit |

VSC | Voltage source converter |

VOC | Voltage-oriented control |

PI | Proportional–integral |

FFT | Fast Fourier transform |

THD | Total harmonic distortion |

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**Figure 3.**Illustration of areas divided by the synthesized voltage vectors’ effect on reactive power. (

**a**)

**U**

_{1}, (

**b**)

**U**

_{2}, (

**c**)

**U**

_{3}, (

**d**)

**U**

_{4}, (

**e**)

**U**

_{5}, and (

**f**)

**U**

_{6}.

**Figure 5.**Illustration of areas divided by the synthesized voltage vectors’ effect on active power. (

**a**)

**U**

_{1}, (

**b**)

**U**

_{2}, (

**c**)

**U**

_{3}, (

**d**)

**U**

_{4}, (

**e**)

**U**

_{5}, and (

**f**)

**U**

_{6}.

**Figure 10.**Experimental results of different DPC strategies under a no load condition. (

**a**) Classic DPC strategy and (

**b**) proposed DPC strategy.

**Figure 11.**Experimental results of different DPC strategies under a balanced load condition. (

**a**) Classic DPC strategy and (

**b**) proposed DPC strategy.

**Figure 12.**Experimental results of different DPC strategies under an unbalanced load condition. (

**a**) Classic DPC strategy and (

**b**) proposed DPC strategy.

**Figure 13.**Experimental results based on the traditional 12-sector division. (

**a**) Voltage and current waveforms of TCIBAR and (

**b**) fast Fourier transform (FFT) results of the phase current.

**Figure 14.**Experimental results based on the optimized 18-sector division. (

**a**) Voltage and current waveforms of TCIBAR and (

**b**) fast Fourier transform (FFT) results of the phase current.

Vectors | S_{a} | S_{b} | S_{c} | S_{d} | S_{q} | S_{0} |
---|---|---|---|---|---|---|

V_{0} | 0 | 0 | 0 | $0$ | $0$ | $0$ |

V_{1} | 1 | 0 | 0 | $\sqrt{\frac{2}{3}}\mathrm{cos}\omega t$ | $-\sqrt{\frac{2}{3}}\mathrm{sin}\omega t$ | $\frac{1}{\sqrt{3}}$ |

V_{2} | 1 | 1 | 0 | $\sqrt{\frac{2}{3}}\mathrm{cos}(\omega t-\frac{\pi}{3})$ | $-\sqrt{\frac{2}{3}}\mathrm{sin}(\omega t-\frac{\pi}{3})$ | $\frac{2}{\sqrt{3}}$ |

V_{3} | 0 | 1 | 0 | $\sqrt{\frac{2}{3}}\mathrm{cos}(\omega t-\frac{2\pi}{3})$ | $-\sqrt{\frac{2}{3}}\mathrm{sin}(\omega t-\frac{2\pi}{3})$ | $\frac{1}{\sqrt{3}}$ |

V_{4} | 0 | 1 | 1 | $-\sqrt{\frac{2}{3}}\mathrm{cos}\omega t$ | $\sqrt{\frac{2}{3}}\mathrm{sin}\omega t$ | $\frac{2}{\sqrt{3}}$ |

V_{5} | 0 | 0 | 1 | $\sqrt{\frac{2}{3}}\mathrm{cos}(\omega t+\frac{2\pi}{3})$ | $-\sqrt{\frac{2}{3}}\mathrm{sin}(\omega t+\frac{2\pi}{3})$ | $\frac{1}{\sqrt{3}}$ |

V_{6} | 1 | 0 | 1 | $\sqrt{\frac{2}{3}}\mathrm{cos}(\omega t+\frac{\pi}{3})$ | $-\sqrt{\frac{2}{3}}\mathrm{sin}(\omega t+\frac{\pi}{3})$ | $\frac{2}{\sqrt{3}}$ |

V_{7} | 1 | 1 | 1 | $0$ | $0$ | $\sqrt{3}$ |

Vectors | S_{a} | S_{b} | S_{c} | S_{d} | S_{q} | S_{0} |
---|---|---|---|---|---|---|

U_{1} | 1 | 0.5 | 0 | $\frac{1}{\sqrt{2}}\mathrm{cos}(\omega t-\frac{\pi}{6})$ | $-\frac{1}{\sqrt{2}}\mathrm{sin}(\omega t-\frac{\pi}{6})$ | $\frac{\sqrt{3}}{2}$ |

U_{2} | 0.5 | 1 | 0 | $\frac{1}{\sqrt{2}}\mathrm{cos}(\omega t-\frac{\pi}{2})$ | $-\frac{1}{\sqrt{2}}\mathrm{sin}(\omega t-\frac{\pi}{2})$ | $\frac{\sqrt{3}}{2}$ |

U_{3} | 0 | 1 | 0.5 | $\frac{1}{\sqrt{2}}\mathrm{cos}(\omega t-\frac{5}{6}\pi )$ | $-\frac{1}{\sqrt{2}}\mathrm{sin}(\omega t-\frac{5}{6}\pi )$ | $\frac{\sqrt{3}}{2}$ |

U_{4} | 0 | 0.5 | 1 | $\frac{1}{\sqrt{2}}\mathrm{cos}(\omega t+\frac{5}{6}\pi )$ | $-\frac{1}{\sqrt{2}}\mathrm{sin}(\omega t+\frac{5}{6}\pi )$ | $\frac{\sqrt{3}}{2}$ |

U_{5} | 0.5 | 0 | 1 | $\frac{1}{\sqrt{2}}\mathrm{cos}(\omega t+\frac{\pi}{2})$ | $-\frac{1}{\sqrt{2}}\mathrm{sin}(\omega t+\frac{\pi}{2})$ | $\frac{\sqrt{3}}{2}$ |

U_{6} | 1 | 0 | 0.5 | $\frac{1}{\sqrt{2}}\mathrm{cos}(\omega t+\frac{\pi}{6})$ | $-\frac{1}{\sqrt{2}}\mathrm{sin}(\omega t+\frac{\pi}{6})$ | $\frac{\sqrt{3}}{2}$ |

s_{P} | s_{Q} | θ_{1} | θ_{2} | θ_{3} | θ_{4} | θ_{5} | θ_{6} | θ_{7} | θ_{8} | θ_{9} | θ_{10} | θ_{11} | θ_{12} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | U_{6} | U_{6} | U_{1} | U_{1} | U_{2} | U_{2} | U_{3} | U_{3} | U_{4} | U_{4} | U_{5} | U_{5} |

0 | 1 | U_{1} | U_{1} | U_{2} | U_{2} | U_{3} | U_{3} | U_{4} | U_{4} | U_{5} | U_{5} | U_{6} | U_{6} |

1 | 0 | U_{4} | U_{5} | U_{5} | U_{6} | U_{6} | U_{1} | U_{1} | U_{2} | U_{2} | U_{3} | U_{3} | U_{4} |

1 | 1 | U_{2} | U_{3} | U_{3} | U_{4} | U_{4} | U_{5} | U_{5} | U_{6} | U_{6} | U_{1} | U_{1} | U_{2} |

s_{P} | s_{Q} | θ_{1} | θ_{2} | θ_{3} | θ_{4} | θ_{5} | θ_{6} | θ_{7} | θ_{8} | θ_{9} | θ_{10} | θ_{11} | θ_{12} | θ_{13} | θ_{14} | θ_{15} | θ_{16} | θ_{17} | θ_{18} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | U_{6} | U_{6} | U_{6} | U_{1} | U_{1} | U_{1} | U_{2} | U_{2} | U_{2} | U_{3} | U_{3} | U_{3} | U_{4} | U_{4} | U_{4} | U_{5} | U_{5} | U_{5} |

0 | 1 | U_{1} | U_{1} | U_{1} | U_{2} | U_{2} | U_{2} | U_{3} | U_{3} | U_{3} | U_{4} | U_{4} | U_{4} | U_{5} | U_{5} | U_{5} | U_{6} | U_{6} | U_{6} |

1 | 0 | U_{5} | U_{5} | U_{6} | U_{6} | U_{6} | U_{1} | U_{1} | U_{1} | U_{2} | U_{2} | U_{2} | U_{3} | U_{3} | U_{3} | U_{4} | U_{4} | U_{4} | U_{5} |

1 | 1 | U_{1} | U_{2} | U_{2} | U_{2} | U_{3} | U_{3} | U_{3} | U_{4} | U_{4} | U_{4} | U_{5} | U_{5} | U_{5} | U_{6} | U_{6} | U_{6} | U_{1} | U_{1} |

Parameter | Symbol | Value |
---|---|---|

Rated power | P | 5 kW |

RMS value of AC source phase voltage | E_{ac} | 115 V |

Frequency of AC source | f_{ac} | 400 Hz |

Rated DC bus voltage | U_{dc} | 360 V |

Rated positive voltage | u_{p} | 180 V |

Rated negative voltage | u_{n} | 180 V |

Positive port capacitance | C_{p} | 6600 μF |

Negative port capacitance | C_{n} | 6600 μF |

Filter inductance | L_{s} | 1.5 mH |

Self-inductance of TCI | L | 0.526 H |

Mutual inductance of TCI | M | 0.259 H |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Y.; Huang, W.; Bu, F.
Direct Power Control of a Bipolar Output Active Rectifier for More Electric Aircraft Based on an Optimized Sector Division. *World Electr. Veh. J.* **2023**, *14*, 89.
https://doi.org/10.3390/wevj14040089

**AMA Style**

Zhao Y, Huang W, Bu F.
Direct Power Control of a Bipolar Output Active Rectifier for More Electric Aircraft Based on an Optimized Sector Division. *World Electric Vehicle Journal*. 2023; 14(4):89.
https://doi.org/10.3390/wevj14040089

**Chicago/Turabian Style**

Zhao, Yajun, Wenxin Huang, and Feifei Bu.
2023. "Direct Power Control of a Bipolar Output Active Rectifier for More Electric Aircraft Based on an Optimized Sector Division" *World Electric Vehicle Journal* 14, no. 4: 89.
https://doi.org/10.3390/wevj14040089