# Analysis of DFIG Interval Oscillation Based on Second-Order Sliding Film Damping Control

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. System Dynamic Modeling

_{w}represents the power that the wind farm is actively producing. The angular difference between the two zones is linked to the active power delivered between them. Thus, reactive power transmitted from zone 1 to zone 2 is proportional to voltage, as seen in (3). For voltage fluctuations generated by reactive power, the bus voltage can be regulated by improving the way through the reactive power [18].

## 3. Controller Design

#### 3.1. Selecting Variables

#### 3.2. Damping Controller Design

_{ω}. Thus, the system oscillation equation may be solved using the relationship shown in (3) in position of bus 1 voltage to obtain the results shown below.

_{ref}can be the actual control action used for the sliding damping control. If reactive power is used directly in the control loop, the relative degree can become significant. Therefore, the reactive power control loop’s dynamic variables use a first-order inertia loop. That allows the designed control to perform better in terms of control robustness to bounded systems.

_{ref}) that is utilized to modify damping.

_{w}is the wind farm’s rated reactive power in nominal operation. As a result, the wind farm’s reactive power is ΔQ

_{w}+ Q

_{w}. The degree of tsliding variable (σ = ω

_{12}) is 2, where

_{w}) and the relative rotor ${\omega}_{12}$ are zero in the steady state. Consequently, the fixed part F(0,δ,0,t) equals 0. Many second-order sliding-mode controllers can be utilized to 0 in unlimited time according to Equation (11). Such a controller is obviously resilient to any perturbation because Equation (11) ignores external disturbances. Therefore, the goal is to design a feedback control $u$ that converges all trajectories in Equation (11) to the phase plane in finite time, at the origin $\sigma =\dot{\sigma}=0$.

## 4. Stability Derivation for Second-Order Sliding-Mode Controllers

#### 4.1. Figures, Tables and Schemes

_{m}− c > 0 and the inequality

^{1/2}. Then, the controller u is u = α(ρ − β)/(|ρ| + β), and owing to the symmetry of the issue, it suffices to examine the situation in which σ > 0. $\dot{\rho}\in \left([-C,C]-\left[{K}_{m},{K}_{M}\right]\alpha \frac{\rho -\beta}{\left|\rho \right|+\beta}+\frac{1}{2}{\rho}^{2}\mathrm{sign}\sigma \right)|\sigma {|}^{-1/2}$, −∞ < ρ < ∞.

_{1}< ρ < ρ

_{2}.

_{1}< β < ρ

_{2}. The phase trajectory of the power system in the state space is shown in Figure 3. It is clear that the controlled system reached the sliding surface when both the sliding variable and its first order were set to zero.

#### 4.2. Frequency Domain Analysis of DFIG

_{dc}and u

_{dc}are the DC bus capacitance and voltage; ${U}_{\mathrm{dc}}^{\mathrm{ref}}$ is the given DC bus voltage; u

_{a}, u

_{b}, u

_{c}and i

_{a}, i

_{b}, i

_{c}are the voltage and current at (point of common coupling)PCC; i

_{ga}, i

_{gb}, i

_{gc}and u

_{ia}, u

_{ib}, u

_{ic}are the output current and voltage at GSC port; i

_{sa}, i

_{sb}, i

_{sc}are the generator stator side port currents; u

_{sa}, u

_{sb}, u

_{sc}are the output modulation signals at RSC and GSC; and θ

_{pLL}is the output modulation signal at RSC and GSC obtained by PLL sampling voltage locking at PCC. m

_{ra}, m

_{rb}, m

_{rc}and m

_{ia}, m

_{ib}, m

_{ic}are the output modulation signals of RSC and GSC, respectively; θ

_{pLL}is the phase angle obtained by voltage locking at the PCC sampled by (phase-locked loop)PLL; and θ

_{r}is the rotor electric angle measured by the position sensor. H

_{si}(s), H

_{ri}(s) are the GSC and RSC current loop PI control transfer functions, respectively, and H

_{u}(s) is the voltage loop PI control transfer function. H

_{P}(s) and H

_{Q}(s) are the power loop PI control transfer functions, and K

_{rd}and K

_{sd}are the current loop decoupling coefficients of RSC and GSC, respectively. The rotor rotational angular velocity is ω

_{r}differential angular velocity ω

_{s}= ω

_{1}− ω

_{r}, and the differential rate is S = ω

_{s}/ω

_{r}.

_{a}= U

_{1}and i

_{a}= I

_{1}is overlaid with positive sequence voltage harmonics of frequency f

_{p}. Due to the dynamic DC bus procedure and the asymmetry of the dq control, harmonic small signal components will pass each other and couple multiple frequency harmonics within the unit [17,18]. The positive-sequence current harmonic I

_{p1}with f

_{p}and the negative-sequence current harmonic Î

_{p2}with symmetric frequency f

_{p}− 2f

_{1}about f

_{1}are generated at the PCC and the voltage harmonic in the negative series Û

_{ip2}with f

_{p}− 2f

_{1}at the port, respectively. Define the voltage and current harmonics at the PCC as

_{ip2}, φ

_{up}and Û

_{p}= U

_{p}e

^{±j}

^{φ}

^{up}are the amplitude and phase, respectively, and the other variables are defined similarly.

_{ia}= U

_{i1}, i

_{ga}= I

_{g1}, and the harmonics are

_{dc}= ${U}_{\mathrm{dc}}^{\mathrm{ref}}$ and harmonics ${\widehat{\mathrm{u}}}_{\mathrm{dc}}$:

_{ra}= U

_{r1}, i

_{ra}= I

_{r1}and harmonics as

_{sa}= I

_{s1}and the harmonics are

#### 4.3. Asynchronous Induction Generator Impedance Modeling

_{s}, R

_{r}are the resistance values of each phase of the stator rotor winding; Ψ

_{sabc}, Ψ

_{rabc}are the three-phase magnetic chains of the stator-rotor; L

_{ss}, L

_{sr}, L

_{rs}, the stator-rotor winding’s mutual-inductance is denoted by L

_{rr}; K

_{e}is the stator-rotor turns ratio. L

_{ss}, L

_{sr}, L

_{rs}, K

_{e}are the stator-rotor turns ratio. From the voltage equation and magnetic chain equation, we can obtain the generator. The impedance model is

_{p}and Y

_{c}are

_{n}and Y

_{r}can be obtained from Y

_{p}and Y

_{c}after discounting, obtained as:

## 5. Simulation Research

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

DFIG | doubly fed induction generator |

SSCI | sub-synchronous control interaction |

STATCOM | static synchronous compensator |

SVC | static var compensator |

δ | generator rotor angle |

δ_{12} | relative rotor angle |

ω_{12} | relative rotor speed |

H_{1} | inertia of area 1 |

H_{2} | inertia of area 2 |

P_{w} | wind farm power |

Q_{1} | moved power |

Q_{s0} | SG reactive power |

Q_{w} | DFIG reactive power |

s | sole output of n |

a, b, $\sigma $ | smooth functions |

GSC | grid side converter |

RSC | rotor side converter |

PCC | point of common coupling |

C_{dc} | DC bus capacitance |

U_{dc} | DC bus voltage |

${\mathrm{U}}_{\mathrm{dc}}^{\mathrm{ref}}$ | given DC bus voltage |

u_{a}, u_{b}, u_{c} | voltage at PCC |

i_{a}, i_{b}, i_{c} | current at PCC |

i_{ga}, i_{gb}, i_{gc} | output voltage at GSC port |

u_{ia}, u_{ib}, u_{ic} | output current at GSC port |

i_{sa}, i_{sb}, i_{sc} | generator stator side currents |

u_{sa}, u_{sb}, u_{sc} | generator stator side voltage |

θ_{pLL} | phase angle obtained by PLL |

m_{ra}, m_{rb}, m_{rc} | output modulation signals of RSC |

m_{ia}, m_{ib}, m_{ic} | output modulation signals of GSC |

Rs, Rr | resistance values of winding |

Yp | positive sequence conductance |

Yc | negative sequence conductance |

H_{P}(s), H_{Q}(s) | loop PI control transfer functions |

## Appendix A

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Rated capacity S/MW | 1.5 |

DC voltage U_{dc}/kV | 1.5 |

AC voltage V_{1}/V | 563 |

Polar logarithm p | 2 |

Mutual inductance Lm(pu) | 4.1 |

Stator resistance r_{s}(pu) | 0.007 |

Rotor resistance R_{r}(pu) | 0.005 |

Rotor inductance L_{lr}(pu) | 0.11 |

line inductor inductance X_{L}/km | 0.25 |

line inductor resistance R/km | 0.023 |

line capacitance resistance nF /km | 12 |

Coupling inductor inductance L(pu) | 0.00178 |

Coupling inductor resistance R(pu) | 0.000929 |

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**MDPI and ACS Style**

Liu, Q.; Wu, J.; Wang, H.; Zhang, H.; Yang, J. Analysis of DFIG Interval Oscillation Based on Second-Order Sliding Film Damping Control. *Energies* **2023**, *16*, 3091.
https://doi.org/10.3390/en16073091

**AMA Style**

Liu Q, Wu J, Wang H, Zhang H, Yang J. Analysis of DFIG Interval Oscillation Based on Second-Order Sliding Film Damping Control. *Energies*. 2023; 16(7):3091.
https://doi.org/10.3390/en16073091

**Chicago/Turabian Style**

Liu, Qi, Jiahui Wu, Haiyun Wang, Hua Zhang, and Jian Yang. 2023. "Analysis of DFIG Interval Oscillation Based on Second-Order Sliding Film Damping Control" *Energies* 16, no. 7: 3091.
https://doi.org/10.3390/en16073091