# Wide Frequency PWM Rectifier Control System Based on Improved Deadbeat Direct Power Control

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PWM Rectifier Model and Traditional Deadbeat Direct Power Control Strategy

#### 2.1. Mathematical Model of Voltage Type PWM Rectifier

_{a}, V

_{b}and V

_{c}are three-phase symmetric AC voltages, i

_{a}, i

_{b}and i

_{c}are corresponding three-phase AC currents, S

_{1}–S

_{6}are six way switches, V

_{dc}is the DC side voltage of the rectifier, L

_{f}is the filter inductor, R

_{f}is the sum of the resistance in the filter inductor and the equivalent resistance of the switch loss, and R

_{dc}is the load resistance.

_{Ra}, V

_{Rb}and V

_{Rc}represent the three-phase voltage value of the voltage type PWM rectifier side. Using the method of coordinate transformation in motor control, the parameters in the above equation are transformed into the αβ two-phase stationary coordinate system by orthogonal transformation. The mathematical model of the voltage type PWM rectifier in the stationary coordinate system is identified and obtained as [21]

#### 2.2. Deadbeat Direct Power Control Strategy

_{α}[k + 1] and i

_{β}[k + 1] are the components of the line current at time (k + 1)T

_{s}in the stationary coordinate system. The instantaneous active power p and the instantaneous reactive power q at time (k + 1)T

_{s}are expressed as

_{α}[k + 1] and i

_{β}[k + 1] can be expressed by instantaneous active power and instantaneous reactive power, as below

_{α}+ jx

_{β}, and the expected value of instantaneous active power p[k + 1] = p

_{ref}[k + 1] is set, which is output by PI controller. In order to make the power factor 1, the expected value of instantaneous reactive power q[k + 1] = q

_{ref}[k + 1] = 0 at the time of (k + 1)T

_{s}can be obtained after sorting

**V**[k + 1] represents the composite vector of the power supply voltage vector at time k + 1 and can be represented by the rotation ωT

_{s}Angle of the composite vector

**V**[k] at time k

#### 2.3. Parameter Sensitivity Analysis under Wide Frequency Conditions

#### 2.3.1. Analysis of Resistive Sensitivity

_{s}time is:

**i**

^{*}[k + 1] is the estimated value of the actual current, ${R}_{\mathrm{f}}^{*}$ is the actual value after the change of the line resistance, ${L}_{\mathrm{f}}^{*}$ is the actual value after the change of the line resistance, ${R}_{\mathrm{f}}^{*}$ = R

_{f}+ ΔR

_{f}, ${L}_{\mathrm{f}}^{*}$ = L

_{f}+ ΔL

_{f}, where ΔR

_{f}and ΔL

_{f}are the changes of the line resistance and filter inductance from the actual and given values, respectively.

**e**of current estimation basically does not change, so the influence of resistance on the model is ignored. When the inductance changes from 0.5 mH to 1.5 mH, the error of current estimation will be significantly affected. The reason for the change of inductance value is specifically explored through the expression of inductance, namely:

#### 2.3.2. Frequency Sensitivity Analysis

_{s}is f

_{1}, since the frequency changes to f

_{2}at time (k + 1)T

_{s}, the frequency used to calculate the supply voltage vector at (k + 1)T

_{s}at time k is f

_{1}. There is an error with the actual frequency at time (k + 1)T

_{s}and the error is Δω = 2π(f

_{2}−f

_{1}). Then, it will affect the calculation of

**V**[k + 1] and the estimation of

**i**[k + 1] and then there will be errors in the calculation of Equation (4). Due to the existence of the control lag, in fact, the vector V

_{R}[k] calculated at kT

_{s}will be applied at (k + 1)T

_{s}, and its reference value can only be reached at k + 2. The two-step prediction will lead to the frequency variation continuing to expand, increasing the effect on the calculation result.

## 3. Improved Deadbeat Direct Power Control Strategy under Wide Frequency

#### 3.1. Repetition Control Strategy

_{q}z

^{−N}, where k

_{q}is the quasi-integral coefficient. In order to ensure the stability of the system, 0 < k

_{q}< 1 is generally taken. In this paper, k

_{q}= 0.95 is taken, and N is the sampling times in a fundamental wave period. In k

_{r}z

^{−N + 1}, z

^{−N}is the k

_{r}times of the error integral at the same sampling time of the previous fundamental wave period superimposed to the current fundamental wave period, which is used to compensate the predicted value of the fundamental wave period.

_{r}is the proportionality coefficient, e

_{i}and e

_{o}are error signals. No matter what the input signal waveform is, if it appears repeatedly in the form of fundamental wave period, the output will be the periodic accumulation of the input signal. The discretization form of the transfer function difference is as follows:

_{s}moment of power projection q

_{ref}[k + 1] and k times before sampling period prediction error values, the sum of the forecast revisions to instantaneous active power and the instantaneous reactive power for each cycle can inhibit periodic disturbance caused by the frequency change, making its reactive steady-state error close to zero.

#### 3.2. Power Compensation Strategy

_{f}and ΔL

_{f}are the changes of the actual line resistance and filter inductance relative to the model parameters. Let Δ

**V**

_{Rf}and Δ

**V**

_{Lf}be the voltage vectors generated by this variation, then the model after the resistance change is:

_{s}and kT

_{s}, then at time (k − 1)T

_{s}, the current at time kT

_{s}is estimated by deadbeat direct power control without considering the change of parameters, and the current value is expressed by vector

**i***, then the discrete model equation is:

_{s}time is:

**V**

_{f}[k] = Δ

**V**

_{Rf}+ Δ

**V**

_{Lf}, due to the inductance change, the voltage vector at the rectifier side becomes the sum of the voltage vector at the rectifier side when the inductance does not change, and the voltage vector generated by the resistive change. By subtracting Equations (17) and (18), we can obtain:

_{f}and Δp

_{f}represent the instantaneous active and reactive power generated by the resistive change, respectively. The instantaneous active and reactive power generated by the resistive and inductance change can be obtained by bringing the instantaneous active and reactive power into the power reference value after repeated change:

_{new}and q

_{new}are the expected value of instantaneous active power and the expected value of reactive power after improvement respectively.

## 4. Experimental Results and Analysis

_{p}= 5 and K

_{i}= 8.

#### 4.1. Influence of Frequency Variation on Inductance

#### 4.2. Feasibility Verification Analysis

#### 4.3. Dynamic Performance Comparison

## 5. Conclusions

- The proposed method solves the reactive steady-state error problem caused by the frequency change, and improves the power quality;
- The proposed method effectively solves the problem of inaccurate model calculation caused by the variation of resistive parameters and improves the stability in the application field.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**The change of permeability of L material: (

**a**) L Relationship between material permeability and temperature; (

**b**) L Relationship between material permeability and frequency.

**Figure 7.**PWM Rectifier Single-Phase System Model: (

**a**) Ideal single-phase system model. (

**b**) Actual single-phase system model.

**Figure 10.**Feasibility Analysis: (

**a**) three-phase AC voltage and three-phase AC current; (

**b**) DC-link voltage.

**Figure 11.**Comparison of reactive power variation under different frequency when inductance changes suddenly: (

**a**) Inductance change of traditional control scheme at 100 Hz; (

**b**) Inductance change of proposed control scheme at 100 Hz; (

**c**) Inductance change of traditional control scheme at 400 Hz; (

**d**) Inductance change of proposed control scheme at 400 Hz.

**Figure 12.**Comparison of three-phase current waveforms and THD: (

**a**) the three-phase current waveform of the conventional deadbeat direct power control; (

**b**) the three-phase current waveform of the proposed scheme; (

**c**) THD of conventional deadbeat direct power control current; (

**d**) THD of proposed scheme current.

**Figure 13.**Dynamic performance under supply frequency changes: (

**a**) Frequency variation and reactive power of conventional control schemes; (

**b**) Improved control scheme frequency variation and reactive power.

Parameter | Symbols | Values |
---|---|---|

DC-link voltage | V_{dc} | 270 V |

Effective value of AC voltage | V_{pcc} | 115 V |

Filter resistance | R_{f} | 0.25 Ω |

Filter inductance | L_{f} | 1.1 mH |

DC-link capacitance | C | 200 μF |

Frequency/Hz | Inductance Lf/mH |
---|---|

100 | 1.0479 |

300 | 1.0462 |

500 | 1.0457 |

700 | 1.0450 |

800 | 1.0448 |

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**MDPI and ACS Style**

Chen, W.; Li, S.; Sun, W.; Bi, K.; Lin, Z.; Zhang, G.
Wide Frequency PWM Rectifier Control System Based on Improved Deadbeat Direct Power Control. *World Electr. Veh. J.* **2022**, *13*, 230.
https://doi.org/10.3390/wevj13120230

**AMA Style**

Chen W, Li S, Sun W, Bi K, Lin Z, Zhang G.
Wide Frequency PWM Rectifier Control System Based on Improved Deadbeat Direct Power Control. *World Electric Vehicle Journal*. 2022; 13(12):230.
https://doi.org/10.3390/wevj13120230

**Chicago/Turabian Style**

Chen, Wei, Shaozhen Li, Wenbo Sun, Kai Bi, Zhichen Lin, and Guozheng Zhang.
2022. "Wide Frequency PWM Rectifier Control System Based on Improved Deadbeat Direct Power Control" *World Electric Vehicle Journal* 13, no. 12: 230.
https://doi.org/10.3390/wevj13120230