# Research on Mechanism and Damping Control Strategy of DFIG-Based Wind Farm Grid-Connected System SSR Based on the Complex Torque Method

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Assumptions

#### 2.2. System Model and Equations

_{s}and P

_{s}represent the stator terminal voltage and active power, respectively; R

_{L}, X

_{L}, and X

_{C}represent the resistance, reactance, and series-compensated capacitor of the transmission line. The parameters of the equivalent DFIG wind generator in per unit values are shown in Table A1 in Appendix A.

_{sq}, u

_{sd}, i

_{sq}, i

_{sd}, u

_{rq}, u

_{rd}, i

_{rq}, and i

_{rd}are the voltages and currents of the stator and rotor q,d-axis component, respectively; ψ

_{sq}, ψ

_{sd}, ψ

_{rq}, and ψ

_{rd}are the flux linkages of the stator and rotor q,d-axis component, respectively; R

_{s}and R

_{r}are the resistance of the stator and rotor, respectively; ω

_{1}and ω

_{r}are the synchronous rotating angular velocity and rotor rotating angular velocity, respectively; L

_{s}, L

_{r}, and L

_{m}are the self-inductance and mutual inductance of the stator and rotor; p is the differential operator.

#### 2.3. Transfer Function of RSC Control

_{s}and Q

_{s}represent the stator active and reactive power; P

_{s-ref}and ψ

_{s}represent the stator active power reference value and flux linkage, respectively; k

_{p2}, k

_{i2}, k

_{p3}, and k

_{i3}represent the proportional and integral coefficients of the RSC outer and inner controller, respectively. In the following equations and pictures, the parameters with -ref subscript indicate reference values, those with 0 are steady-state values, and those with Δ are micro variables.

#### 2.4. Transfer Function of Rotor Speed Control

_{rmin}, ω

_{rmax}, P

_{s1}, and P

_{s2}represent the minimum and maximum value of the rotor speed and stator active power under the MPPT operating area, respectively; a, b, and c are the quadratic term fitting coefficients.

_{p1}and k

_{i1}represent the proportional and integral coefficients of the rotor speed controller. The active power reference command value can be described by Figure 4:

_{s}< P

_{s1}or P

_{s}> P

_{s2}, the rotor speed reference value is constant and has nothing to do with the stator active power, which leads to the disappearance of the feedback branch in Figure 5. At this moment, Equation (11) is not applicable. At the subsynchronous frequency, the slip of the generator is negative, which causes the rotor equivalent resistance to be negative. When P

_{s}< P

_{s1}, the rotor speed is small in a low wind speed, which makes the slip become small and the rotor equivalent resistance become large. When the rotor equivalent negative resistance is larger than the sum of the stator resistance and the resistance in transmission lines at the resonance frequency, SSR will occur [5]. When P

_{s}> P

_{s1}, the rotor speed is the maximum value because of the high wind speed. The slip is a large negative value, so the rotor equivalent resistance becomes a small negative value. The total resistance of the DFIG system is positive, and the system can maintain stability when the perturbation occurs.

## 3. Results

#### 3.1. Electrical Damping Expression

_{e}(λ)Δδ is the synchronous torque component related to the angular displacement and the transfer function K

_{e}(λ) is the electrical synchronous torque coefficient; D

_{e}(λ)Δω is the damping torque component related to the angular velocity; and the transfer function D

_{e}(λ) is the electrical damping torque coefficient.

_{e}) and the mechanical damping torque coefficient (D

_{m}) is less than zero, the system SSR will occur [38]. In general, the mechanical damping coefficient value of DFIG is much smaller than the electrical damping coefficient, so the mechanical damping effect of the wind turbine is ignored, and the SSR stability of the DFIG wind farm grid-connected system is evaluated based on the electrical damping coefficient [15].

_{p}is the number of pole pairs of the generator.

_{Te}(jω

_{er})]. According to the principle of the complex torque coefficient method, when the real part of the transfer function from electromagnetic torque is larger than zero, that is, Re[G

_{Te}(jω

_{er})] > 0 or |∠G

_{Te}(jω

_{er})| < 90°, the system electrical damping is positive, and the system can become stable after suffering a perturbation. On the contrary, when Re[G

_{Te}(jω

_{er})] < 0 or |∠G

_{Te}(jω

_{er})| > 90°, the system electrical damping become negative and the system will have risks of SSR. When the real part of the gain transfer function is equal to zero, that is Re[G

_{Te}(jω

_{er})] = 0 or |∠G

_{Te}(jω

_{er})| = 90°, the system electrical damping is equal to zero, and the electromagnetic torque of DFIG will produce the constant amplitude oscillation. At this moment, the angular frequency from the rotational speed is called the critical stable angular frequency. The DFIG system connected to a series complementary line can be equated to a second-order oscillation circuit, and the oscillation frequency of the system can be calculated. When this oscillation frequency is greater than the critical stability frequency, the electrical damping corresponding to the oscillation frequency is negative, which leads to the instability of the system.

#### 3.2. Design of Proposed Damping Controller

_{SDC}represents the transfer function of SDC and G

_{Te-SDC}represents the transfer function between the electromagnetic torque and the rotor speed with SDC.

_{c}and B represent the cut-off frequency and the bandwidth; T

_{1}and T

_{2}represent the leading and lagging phase compensation parameters; and K represents the parameter of the gain link.

_{p1}, k

_{p2}, and k

_{p3}are 1.3, 1.6, and 0.47 p.u., respectively; k

_{i1}, k

_{i2}, and k

_{i3}are 20, 40, and 10 p.u., respectively. The cut-off frequency of the band pass filter is set to 35 Hz. The number of the particle population n is 100. The maximum number of the iterations t

_{max}is 200. The initial inertial weight ω is 0.6. The learning factors c

_{1}and c

_{2}are both 0.5. The upper and lower speed limits are 20 and −20, respectively. After iterative calculation based on changing the system operating conditions many times, the optimal parameters of the subsynchronous damping controller are finally obtained as follows: ξ = 0.07142, T

_{1}= 0.01, T

_{2}= 0.001, K = 17.

## 4. Discussion

#### 4.1. Description of Analysis and Simulation Parameters Setting

#### 4.2. Impact Parameters and Sensitivity Analysis

#### 4.2.1. Impact of Wind Speed on SSR Electrical Damping

_{er}), the system stable phase margin and electrical damping decrease, and the SSR stability of the system gradually deteriorates. On the other hand, under the same angular frequency, the higher the wind speed, the larger the phase and the real part of the transfer function, indicating that the greater the stable phase margin and the electrical damping, the better the system SSR stability. In addition, the critical stable angular frequency (ω

_{er0}) of the system electrical damping increases with the increase of the wind speed. This result shows that at high wind speed, only when a higher series compensation capacitor is connected to the transmission line can SSR be induced, which verifies the result that higher the wind speed, the better the system SSR stability.

#### 4.2.2. Impact of Controller Parameters on SSR Electrical Damping

_{p3}) and integral coefficient (k

_{i3}) from the RSC inner loop controller on SSR electrical damping. In Figure 8a, when the angular frequency is lower than 54.5 rad/s, the phase of the electromagnetic torque does not change much with k

_{p3}increasing; when the angular frequency is higher than 54.5 rad/s, the larger the k

_{p3}, the smaller the phase. These results indicate that the system stable phase margin decreases with the increase of k

_{p3}. In Figure 8b, when the angular frequency is at the range of the 24 to 59 rad/s, the real part of the electromagnetic torque increases with the increase of k

_{p}

_{3}, but the system electrical damping is all positive, and the system is stable. When the angular frequency is higher than 59 rad/s, as k

_{p}

_{3}increases, the real part of the electromagnetic torque and the critical stable angular frequency are smaller. These results show that the larger the k

_{p}

_{3}, the smaller the system electrical damping and the worse the system SSR stability.

_{i}

_{3}, the larger the system stable phase margin and electrical damping. When the angular frequency is higher than 70 rad/s, the phase and the real part of the electromagnetic torque decreases with the increase of k

_{i3}. However, the system electrical damping is negative at this moment, and increasing k

_{i}

_{3}will only make the system SSR stability worse.

#### 4.2.3. The Sensitivity of the Controller Parameters

_{Te}(s,X)] represents the system electrical damping.

_{p}

_{1}. The sensitivities of k

_{p}

_{2}and k

_{p}

_{3}are positive in the range of 24 to 59 rad/s. In this range, the system electrical damping itself is positive, indicating that increasing the proportional coefficient at this range can increase the system stability. When the angular frequency is higher than 59 rad/s, the sensitivities of k

_{p}

_{2}and k

_{p}

_{3}are negative, which indicate that increasing the proportional coefficients at this range will decrease the SSR electrical damping and make the system unstable.

#### 4.2.4. Comparison with Other Literature Results

#### 4.3. Time-Domain Simulation Verification

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Parameters | Value |
---|---|

Rated power | 200 MW |

Rated voltage | 690 V |

Ls(p.u.) | 4.60 |

Lr(p.u.) | 4.61 |

Lm(p.u.) | 4.50 |

Rs(p.u.) | 0.0054 |

Rr(p.u.) | 0.00607 |

Dm(p.u.) | 0.01 |

Parameters | Value |
---|---|

k_{p1}(p.u.) | 1.5 |

k_{i1}(p.u.) | 10 |

k_{p2}(p.u.) | 1.8 |

k_{i2}(p.u.) | 20 |

k_{p3}(p.u.) | 0.5 |

k_{i3}(p.u.) | 2.0 |

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**Figure 7.**Frequency response of the gain of ΔT

_{e}under different wind speeds: (

**a**) phase vary with ω

_{er}; (

**b**) real part vary with ω

_{er}.

**Figure 8.**Frequency response of the gain of ΔT

_{e}under different control parameters of RSC current control loop: (

**a**) phase vary with ω

_{er}under k

_{p3}; (

**b**) real part vary with ω

_{er}under k

_{p3}; (

**c**) phase vary with ω

_{er}under k

_{i3}; (

**d**) real part vary with ω

_{er}under k

_{i3}.

**Figure 9.**Electrical damping sensitivity of system under different proportional and integral parameters: (

**a**) proportional coefficients; (

**b**) integral coefficients.

**Figure 10.**Electromagnetic torque at different wind speeds and compensation degrees: (

**a**) different wind speed; (

**b**) different compensation degree.

**Figure 11.**Electromagnetic torque in different RSC inner-loop proportional and integral coefficients: (

**a**) different k

_{p3}; (

**b**) different k

_{i3}.

**Figure 12.**Effect comparison of the system with and without SDC under different operating conditions: (

**a**) compensation degree of 20%; (

**b**) k

_{p3}= 0.52; (

**c**) k

_{i3}= 1; (

**d**) k

_{p1}= 1.3.

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**MDPI and ACS Style**

Peng, X.; Chen, R.; Zhou, J.; Qin, S.; Bi, R.; Sun, H.
Research on Mechanism and Damping Control Strategy of DFIG-Based Wind Farm Grid-Connected System SSR Based on the Complex Torque Method. *Electronics* **2021**, *10*, 1640.
https://doi.org/10.3390/electronics10141640

**AMA Style**

Peng X, Chen R, Zhou J, Qin S, Bi R, Sun H.
Research on Mechanism and Damping Control Strategy of DFIG-Based Wind Farm Grid-Connected System SSR Based on the Complex Torque Method. *Electronics*. 2021; 10(14):1640.
https://doi.org/10.3390/electronics10141640

**Chicago/Turabian Style**

Peng, Xiaotao, Renjie Chen, Jicheng Zhou, Shiyao Qin, Ran Bi, and Haishun Sun.
2021. "Research on Mechanism and Damping Control Strategy of DFIG-Based Wind Farm Grid-Connected System SSR Based on the Complex Torque Method" *Electronics* 10, no. 14: 1640.
https://doi.org/10.3390/electronics10141640