# Virtual Vector-Based Direct Power Control of a Three-Phase Coupled Inductor-Based Bipolar-Output Active Rectifier for More Electric Aircraft

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of TCIBAR

_{A}~S

_{C}and ${S}_{A}^{\prime}$~${S}_{C}^{\prime}$ are the upper and lower switches of the VSC, respectively, and R

_{s}and R are the winding resistance of the filter inductor and the TCI. In addition, a three-phase AC source is used as the power supply instead of a generator.

_{a}, e

_{b}, and e

_{c}are the three-phase voltages of the AC source, u

_{ao}, u

_{bo}, and u

_{co}are the three-phase output voltages of the VSC, and i

_{sa}, i

_{sb}, and i

_{sc}are the three-phase input currents of the TCIBAR.

_{s0}= 0.

_{q}= 0. Then, by differentiating both sides of Equation (3), the active and reactive power variation rate of the TCIBAR can be written as

_{d}is a constant value, that is, de

_{d}/dt = 0. Substituting Equation (2) into Equation (4), the power model of the TCIBAR in the dq coordinate system can be deduced as

_{aG}, u

_{bG}, and u

_{cG}are the three-phase voltages across the TCI, and L

_{TCI}and R

_{TCI}are the inductance matrix and winding resistance matrix of the TCI, respectively.

## 3. Limitations of Classic DPC in TCIBAR Control

^{*}is obtained by the closed-loop control of DC bus voltage, and the reactive power reference q

^{*}is usually set as zero to achieve the unit power factor of the converter. According to the instantaneous power errors, the outputs of the power hysteresis comparators are determined. Then, in order to identify the angular position of the AC-source voltage vector, the vector space is evenly divided into twelve sectors, as shown in Figure 3, and the sector boundaries are defined as

_{n}is the sector number.

**V**

_{0}~

**V**

_{7}) on the active and reactive power of the rectifier. The switching table of the classic DPC strategy is shown in Table 1, where s

_{P}and s

_{Q}are the outputs of the active and reactive hysteresis comparators, respectively. In [30,31,32], several new switching tables have been proposed for PWM rectifiers based on different optimization objectives, and it can be seen that the switching table affects the performance of the DPC strategy directly and is the key to the DPC strategy.

## 4. Proposed VVB-DPC for TCIBAR

#### 4.1. Derivation of Virtual Vector

_{x}= 0 and S

_{x}= 1 correspond to the on-state and off-state of the upper switch in the phase bridge x, respectively. Besides, ε (0 < ε < 1) is defined as the voltage coefficient of the DC-side neutral point, and εU

_{dc}is the actual voltage of the negative DC port of TCIBAR.

**V**

_{0}, $(1/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ in

**V**

_{1},

**V**

_{3},

**V**

_{5}, $(2/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ in

**V**

_{2},

**V**

_{4},

**V**

_{6}and $\sqrt{3}(1-\epsilon ){U}_{dc}$ in

**V**

_{7}. Meanwhile, it can be found that, among the six non-zero voltage vectors, any two adjacent non-zero voltage vectors always generate the ZSVs of $(1/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ and $(2/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$. Based on this observation, a set of virtual vectors, which are synthesized by every two adjacent non-zero voltage vectors, are proposed to extend the original voltage vectors, and the virtual vector

**V**

_{mn}is denoted as

**V**

_{m}and

**V**

_{n}are the adjacent non-zero voltage vectors, T

_{s}is the control cycle, and t

_{m}and t

_{n}are the action time of

**V**

_{m}and

**V**

_{n}in one control cycle. Meanwhile, the sum of t

_{m}and t

_{n}is equal to T

_{s}.

**V**

_{1},

**V**

_{3}, and

**V**

_{5}as odd vectors and

**V**

_{2},

**V**

_{4}, and

**V**

_{6}as even vectors, it can be seen that the virtual vectors are synthesized by adjacent odd and even vectors in one control cycle. At the same time, the ZSV components in the odd and even vectors are $(1/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ and $(2/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$, respectively. Therefore, in order to make the ZSV component in each virtual vector equal, the odd and even vectors of each virtual vector should have the same action time ratio in one control cycle. In this paper, the action time of the even and odd vectors are set as $\lambda {T}_{s}$ and $(1-\lambda ){T}_{s}$, respectively, where λ ($0\le \lambda \le 1$) is the time coefficient. On this basis, the switching functions of the virtual vectors, as well as the corresponding ZSV components, can be deduced based on the volt-second equivalent principle, as shown in Table 3.

_{l}

_{0}= 0. Meanwhile, considering that the bipolar DC voltages of the TCIBAR should be symmetrical in the HVDC power system of MEA, the voltage coefficient ɛ should be equal to 0.5 in the steady state. Substituting u

_{l}

_{0}= 0 and ɛ = 0.5 into Equation (13), the time coefficient λ can be calculated as

#### 4.2. Switching Table Based on Virtual Vectors

_{s}, the AC-source branch of TCIBAR satisfies the vector equation in Equation (15),

**I**

_{s}(0) and

**I**

_{s}(t) are the AC-source current vector at time 0 and time t, and

**E**is the AC-source voltage vector.

**E**−

**V**

_{mn}remains almost unchanged in one control cycle T

_{s}, the variation of the AC-source current vector (Δ

**I**

_{s}) in a control cycle can be deduced according to Equation (15), as shown in Equation (16).

_{d}and ΔI

_{q}are the projection components of Δ

**I**

_{s}on the d-axis and q-axis, and γ is the angle between Δ

**I**

_{s}and the d-axis.

**E**−

**V**

_{mn}and the power variation can be obtained as follows:

**E**−

**V**

_{mn}on the d-axis, while the reactive power variation is determined by the projection of

**E**−

**V**

_{mn}on the q-axis. Therefore, the effect of each virtual vector on the power variation of TCIBAR can be obtained according to the direction and amplitude of the projection of

**E**−

**V**

_{mn}on the d-axis and q-axis. Taking the AC-source voltage vector

**E**in the sector θ

_{3}as an example, the effect of different voltage vectors on the power variation of TCIBAR can be analyzed, and Figure 5a shows the effect of the virtual vector

**V**

_{56}. In the meantime, the vector space can be divided into four areas according to the signs of the active and reactive power variation rates, as shown in Figure 5b.

**E**is in the sector θ

_{3}, the virtual vectors

**V**

_{34},

**V**

_{45},

**V**

_{56}, and

**V**

_{61}will increase the active power of TCIBAR, while

**V**

_{12}and

**V**

_{23}will decrease the active power. At the same time, the reactive power of the TCIBAR will be increased by the virtual vectors

**V**

_{23},

**V**

_{34}, and

**V**

_{45}and decreased by

**V**

_{56},

**V**

_{61}, and

**V**

_{12}. Thus, the appropriate virtual vector can be selected according to the outputs of the power hysteresis comparators. Similarly, when the AC-source voltage vector

**E**is in other eleven sectors, the same analysis method is used to select the corresponding virtual vectors and the complete virtual-vector switching table can be established, as shown in Table 4.

#### 4.3. Neutral-Point Potential Control Method Based on DPC Architecture

_{l}

_{0}), respectively. Meanwhile, the PI controllers of Δu and i

_{l}

_{0}can be designed by referring to the classical PI parameter design method in [33,34]. Based on this control algorithm, the reference value of ZSV can be obtained. However, since there is no voltage modulation module in the DPC architecture, the accurate generation of the actual ZSV is the critical problem in the neutral-point potential control of TCIBAR.

**V**

_{7}; and (3) $-\sqrt{3}\epsilon {U}_{dc}$ in

**V**

_{0}. Because of the constraint that $0<\epsilon <1$, the ZSV components in

**V**

_{0}and

**V**

_{7}always have the opposite signs, and the three cases of the ZSV components satisfy the following inequality:

**V**

_{0}or

**V**

_{7}) with a certain duty ratio into the selected virtual vector within one control cycle. Meanwhile, the inserted zero vector and its action time are dependent on the reference value of the ZSV (${u}_{l0}^{*}$) and can be categorized as the following two cases.

**V**

_{7}is inserted, and the action time can be calculated as follows:

_{mn}and t

_{7}are the action time of the selected virtual vector and the zero vector

**V**

_{7}.

_{mn}and t

_{7}can be obtained as

**V**

_{0}is inserted to synthesize the reference ZSV, and the action time can be calculated as

_{0}is the action time of

**V**

_{0}.

_{mn}and t

_{0}can be obtained as follows:

#### 4.4. Overall Control Block Diagram of the Proposed VVB-DPC

## 5. Simulation Results

_{ln}) in the TCI and leads to a voltage imbalance between the bipolar DC ports. However, the proposed switching table does not affect the zero-sequence current in TCI and can realize the effective control of both the DC bus voltage and neutral-point potential, which verifies the feasibility of the proposed VVB-DPC strategy.

## 6. Experimental Results

#### 6.1. Experimental Prototype and Parameters

#### 6.2. Steady-state Experimental Research

_{l}

_{0}in TCI is uncontrolled. Thus, the total zero-sequence current (i

_{ln}) injected into the DC-side neutral point fluctuates greatly, which further leads to the fluctuation of the DC-side neutral-point potential and the voltage imbalance between the positive and negative DC ports.

_{ln}can be maintained at 0 A stably, and the voltage balance between the bipolar DC ports is achieved. Therefore, the experimental results in Figure 11 validate the feasibility and effectiveness of the proposed virtual-vector switching table.

_{ln}. Meanwhile, the DC voltages of the positive and negative ports are not balanced, and the voltage difference is about 20 V.

#### 6.3. Dynamic Experimental Research

_{ln}can be controlled at 0 A stably.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Control diagram of the classic DPC strategy. The superscript * represents the reference value of the variable.

**Figure 5.**Effect of voltage vectors on power variation of TCIBAR when

**E**in sector θ

_{3}. (

**a**) Effect of

**V**

_{56}. (

**b**) Area division based on power variation rate.

**Figure 6.**Control block diagram of the proposed VVB-DPC strategy. The superscript * represents the reference value of the variable.

**Figure 7.**Simulation results of different switching tables under no-load condition. (

**a**) Classic switching table. (

**b**) Proposed virtual-vector switching table.

**Figure 8.**Simulation results under unbalanced load conditions. (

**a**) Without proposed neutral-point potential control method. (

**b**) With proposed neutral-point potential control method.

**Figure 9.**Dynamic simulation results of the proposed VVB-DPC strategy. (

**a**) Under balanced load condition. (

**b**) Under unbalanced load condition.

**Figure 11.**Steady-state experimental results of different switching tables under no-load condition. (

**a**) Classic switching table. (

**b**) Proposed virtual-vector switching table.

**Figure 12.**Steady-state experimental results of TCIBAR without the proposed neutral-point potential control method under unbalanced load condition.

**Figure 13.**Steady-state experimental results of TCIBAR with the proposed neutral-point potential control method under unbalanced load condition.

**Figure 14.**Dynamic experimental results of the proposed VVB-DPC strategy under balanced step load condition.

**Figure 15.**Dynamic experimental results of the proposed VVB-DPC strategy under unbalanced step load condition.

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

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

1 | 0 | V_{6} | V_{7} | V_{1} | V_{0} | V_{2} | V_{7} | V_{3} | V_{0} | V_{4} | V_{7} | V_{5} | V_{0} |

1 | 1 | V_{7} | V_{7} | V_{0} | V_{0} | V_{7} | V_{7} | V_{0} | V_{0} | V_{7} | V_{7} | V_{0} | V_{0} |

Vectors | S_{a} | S_{b} | S_{c} | u_{l}_{0} |
---|---|---|---|---|

V_{0} | 0 | 0 | 0 | $-\sqrt{3}\epsilon {U}_{dc}$ |

V_{1} | 1 | 0 | 0 | $(1/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ |

V_{2} | 1 | 1 | 0 | $(2/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ |

V_{3} | 0 | 1 | 0 | $(1/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ |

V_{4} | 0 | 1 | 1 | $(2/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ |

V_{5} | 0 | 0 | 1 | $(1/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ |

V_{6} | 1 | 0 | 1 | $(2/\sqrt{3}-\sqrt{3}\epsilon ){U}_{dc}$ |

V_{7} | 1 | 1 | 1 | $\sqrt{3}(1-\epsilon ){U}_{dc}$ |

Vectors | S_{a} | S_{b} | S_{c} | u_{l}_{0} |
---|---|---|---|---|

V_{12} | $1$ | $1-\lambda $ | $0$ | $(2-\lambda -3\epsilon ){U}_{dc}/\sqrt{3}$ |

V_{23} | $1-\lambda $ | $1$ | $0$ | $(2-\lambda -3\epsilon ){U}_{dc}/\sqrt{3}$ |

V_{34} | $0$ | $1$ | $1-\lambda $ | $(2-\lambda -3\epsilon ){U}_{dc}/\sqrt{3}$ |

V_{45} | $0$ | $1-\lambda $ | $1$ | $(2-\lambda -3\epsilon ){U}_{dc}/\sqrt{3}$ |

V_{56} | $1-\lambda $ | $0$ | $1$ | $(2-\lambda -3\epsilon ){U}_{dc}/\sqrt{3}$ |

V_{61} | $1$ | $0$ | $1-\lambda $ | $(2-\lambda -3\epsilon ){U}_{dc}/\sqrt{3}$ |

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 | V_{61} | V_{61} | V_{12} | V_{12} | V_{23} | V_{23} | V_{34} | V_{34} | V_{45} | V_{45} | V_{56} | V_{56} |

0 | 1 | V_{12} | V_{12} | V_{23} | V_{23} | V_{34} | V_{34} | V_{45} | V_{45} | V_{56} | V_{56} | V_{61} | V_{61} |

1 | 0 | V_{45} | V_{56} | V_{56} | V_{61} | V_{61} | V_{12} | V_{12} | V_{23} | V_{23} | V_{34} | V_{34} | V_{45} |

1 | 1 | V_{23} | V_{34} | V_{34} | V_{45} | V_{45} | V_{56} | V_{56} | V_{61} | V_{61} | V_{12} | V_{12} | V_{23} |

Vectors | S_{a} | S_{b} | S_{c} | u_{l}_{0} |
---|---|---|---|---|

V_{12} | 1 | 0.5 | 0 | $\sqrt{3}(1-2\epsilon ){U}_{dc}/2$ |

V_{23} | 0.5 | 1 | 0 | $\sqrt{3}(1-2\epsilon ){U}_{dc}/2$ |

V_{34} | 0 | 1 | 0.5 | $\sqrt{3}(1-2\epsilon ){U}_{dc}/2$ |

V_{45} | 0 | 0.5 | 1 | $\sqrt{3}(1-2\epsilon ){U}_{dc}/2$ |

V_{56} | 0.5 | 0 | 1 | $\sqrt{3}(1-2\epsilon ){U}_{dc}/2$ |

V_{61} | 1 | 0 | 0.5 | $\sqrt{3}(1-2\epsilon ){U}_{dc}/2$ |

V_{0} | 0 | 0 | 0 | $-\sqrt{3}\epsilon {U}_{dc}$ |

V_{7} | 1 | 1 | 1 | $\sqrt{3}(1-\epsilon ){U}_{dc}$ |

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

Rated power | P | 5 kW |

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

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

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

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

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

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. Virtual Vector-Based Direct Power Control of a Three-Phase Coupled Inductor-Based Bipolar-Output Active Rectifier for More Electric Aircraft. *Energies* **2023**, *16*, 3038.
https://doi.org/10.3390/en16073038

**AMA Style**

Zhao Y, Huang W, Bu F. Virtual Vector-Based Direct Power Control of a Three-Phase Coupled Inductor-Based Bipolar-Output Active Rectifier for More Electric Aircraft. *Energies*. 2023; 16(7):3038.
https://doi.org/10.3390/en16073038

**Chicago/Turabian Style**

Zhao, Yajun, Wenxin Huang, and Feifei Bu. 2023. "Virtual Vector-Based Direct Power Control of a Three-Phase Coupled Inductor-Based Bipolar-Output Active Rectifier for More Electric Aircraft" *Energies* 16, no. 7: 3038.
https://doi.org/10.3390/en16073038