# Small-Signal Performance of Type 4 Wind Turbine Generator-Based Clusters in Power Systems

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## Abstract

**:**

## 1. Introduction

## 2. Theory of Small-Signal Stability Analysis with WTGs

#### 2.1. Small-Signal Stability Theory

**A**is the Jacobian matrix of the linearization equation of differential algebraic equations.

**A**; ${v}_{\mathrm{i}}$ is the nth row of the left eigenvector matrix, v of

**A**; they are respectively called the right and left eigenvectors. The relationship between the right and the left eigenvectors can be expressed as v = u

^{−1}.

**u**

_{i}and the left eigenvector

**v**

_{i}are combined to form the participation matrix p, which is used to measure the degree of correlations between modes and state variables. The kth row ith column of elements of the participating matrix p can be expressed as,

_{ki}is called the participation factor. The participation factor is dimensionless and used to measure the degree of mutual involvement between the ith mode and the kth state variable.

#### 2.2. Impact of Type 4 WTGs on Traditional Electromechanical Modes

#### 2.3. Identification of Electromechanical Modes Dominated by Type 4 WTGs

_{t}, ω

_{g}, δ

_{t}and δ

_{g}. However, a large number of oscillation modes can be solved by Equation (3), but only a few of them are dominated by the PMSGs of Type 4 WTGs. To identify the electromechanical modes dominated by Type 4 WTGs, the identification factor (IF) of the mechanical state variables of Type 4 WTGs in mode i, λ

_{i}, is defined using participation factors:

_{t}, ω

_{g}, δ

_{t}and δ

_{g}) of WTGs affecting mode i.

_{i}<< 1, it is a non-electromechanical mode. The performance of the new electromechanical modes participating in power oscillations can be verified by time-domain simulation and signal detection.

## 3. Second-Generation Generic Models of Type 4 WTGs

_{pcmd}and I

_{qcmd}are the active and reactive current commands; U

_{term}is the generator terminal voltage; U

_{ref}/U

_{reg}is the optional remote control bus voltage.

_{efFlag}is set in repc_a. The voltage control is implemented when R

_{efFlag}= 1, while the constant reactive power control is realized when R

_{efFlag}= 0.

_{ref}and the reactive power reference Q

_{ext}, both of which are obtained by the repc_a. The output of reec_a is the active current command (I

_{pcmd}) and the reactive current command (I

_{qcmd}), which transfer to the repc_a. The internal part of reec_a consists of three parts: active power control (generates the command I

_{pcmd}), reactive power control (generates the command I

_{qcmd}) and converter current limit logic, which limits the active and reactive currents to the current rating of the converter inside.

_{pcmd}and the reactive current command I

_{qcmd}are the inputs of regc_a. The outputs of the model are the active current I

_{p}and reactive current I

_{q}injected into the grid model. In Figure 1e, T

_{g}represents the active and reactive current injection time constants and the T

_{filt}represents the voltage filter time constant.

_{t}is the turbine inertia; H

_{g}is the generator inertia time constant; F

_{req1}is the frequency of the first torsional mode. D

_{shaft}is the coefficient of the shaft mechanical damping; K

_{shaft}is the shaft spring coefficient; P

_{mech}is the mechanical power from the wind turbine; P

_{gen}is the electromagnetic output power of the generator; ω

_{t}and ω

_{g}are the speeds of the wind turbine and the generator; Δω

_{tg}= ω

_{t}− ω

_{g}; δ

_{t}is the angle deviation of the turbine; δ

_{tg}is the angle deviation of the generator and δ

_{tg}= δ

_{t}− δ

_{g}.

## 4. Simulation Analysis and Verification

#### 4.1. Impact of the Increasing Penetration Levels of Type 4 WTGs on Traditional Interarea Modes

#### 4.2. Investigation of New Electromechanical Modes Dominated by Type 4 WTGs

_{i}, defined by Equation (5). Table 4 lists the new modes for each penetration level, participating WTGs in descending order of participation factors and identification factors of these new modes.

_{i}, are equal to their denominators in Table 4. This shows that the new electromechanical modes are dominated by the mechanical state variables of Type 4 WTGs. For each penetration level, there exists one electromechanical mode that has the lightest damping and strong interaction with most Type 4 WTGs. They are akin to the conventional interarea modes, but with frequencies higher than conventional interarea modes, here called strong-interaction modes in bold in Table 4. The other several modes are only related to a few WTGs with heavier damping, and they are akin to the conventional local modes, here called weak-interaction modes. The new electromechanical modes involving both strong- and weak-interaction modes have a weak correlation with the electrical control of full converters of Type 4 WTGs.

_{t}and ω

_{g}of Type 4 WTGs. Therefore, the strong-interaction modes dominated by Type 4 WTGs result from a strong interaction between the power angle and speed of the Type 4 WTGs.

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

WTG | Wind turbine generator |

TMMC | Ten million megawatt cluster |

GM | Generic model |

WECC | Western Electricity Coordinating Council |

IF | Identification factor |

regc_a | Generator/converter model |

reec_a | Electrical control model |

wtgt_a | Shaft model of wind turbine generator |

repc_a | Plant-level controller model |

SG | Synchronous generator |

AC | Alternative current |

PI | Participation index |

PMSG | Permanent magnet synchronous generators |

NETS | New England test system |

NYPS | New York power system |

PSS | Power system stabilizer |

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**Figure 1.**Second-generation generic model (GM) of Type 4 wind turbine generators (WTGs): (

**a**) Type 4A; (

**b**) Type 4B; (

**c**) repc_a; (

**d**) reec_a; (

**e**) regc_a; (

**f**) wtgt_a.

**Figure 2.**IEEE 16-machine 68-bus system diagram: with the increasing of wind penetration, the synchronous generators (SGs) of the New England test system (NETS) are gradually replaced by Type 4 WTGs; its behavior is similar to the 10 million megawatt cluster (TMMC).

**Figure 4.**Frequencies of interarea modes vs. wind power penetration. (

**a**) Mode 1; (

**b**) Mode 2; (

**c**) Mode 3; (

**d**) Mode 4.

**Figure 5.**Damping ratio and frequency of the strong-interaction mode vs. wind power penetration. (

**a**) the damping ratios of the strong-interaction modes; (

**b**) the frequencies of the strong-interaction modes.

**Figure 6.**Power oscillation of WTGs G9 and G7 under the penetration level of 50.17%. (

**a**) the active power curves of the WTGs G9; (

**b**) the reactive power curves of the WTGs G9; (

**c**) the active power curves of the WTGs G7; (

**d**) the reactive power curves of the WTGs G7.

**Figure 7.**Power curve of tie lines for 27–53 and 60–61 under 50.17% wind power penetration. (

**a**) the active power curves of the interarea tie lines 27–53; (

**b**) the active power curves of the interarea tie lines 60–61.

Mode | Eigenvalues | Frequency (Hz) | Damping Ratio | Participation Generator |
---|---|---|---|---|

1 | −0.2625 ± 4.9801 | 0.7926 | 0.0526 | 15, 14, 16 |

2 | −0.1878 ± 3.6787 | 0.5855 | 0.0510 | 13, 16, 14, 6, 12, |

3 | −0.1645 ± 3.2708 | 0.5206 | 0.0502 | 16, 14, 13 |

4 | −0.0642 ± 2.2737 | 0.3619 | 0.0282 | 15, 14, 16, 13 |

5 | −0.4311 ± 6.3543 | 1.0113 | 0.0677 | 3, 2, 6, 5, 7, 4 |

6 | −0.1499 ± 6.6674 | 1.0612 | 0.0225 | 9, 3, 8 |

7 | −0.4443 ± 6.9603 | 1.1078 | 0.0637 | 12, 13 |

8 | −0.5850 ± 7.0540 | 1.1227 | 0.0826 | 5, 6, 7, 4 |

9 | −0.2895 ± 7.7299 | 1.2303 | 0.0374 | 10, 9, 1, 8, 12 |

10 | −0.6337 ± 7.8369 | 1.2473 | 0.0806 | 2, 3 |

11 | −0.3960 ± 8.1306 | 1.2940 | 0.0487 | 10, 1, 8, 9 |

12 | −0.9023 ± 9.0836 | 1.4457 | 0.0989 | 7, 6 |

13 | −0.7164 ± 9.2516 | 1.4724 | 0.0772 | 8, 1 |

14 | −0.9341 ± 9.5130 | 1.5140 | 0.0977 | 4, 5 |

15 | −0.7562 ± 11.2750 | 1.7944 | 0.0669 | 11, 10 |

**Table 2.**Replaced units and corresponding penetration levels in the New England test system (NETS) area.

Penetration Levels | 15.44% | 25.86% | 36.67% | 50.17% | 59.92% | 72.12% | 84.66% | 95.18% |

Replaced Generators | G9 | G9, G8 | G9, G8, G7 | G9, G8, G7, G6 | G9, G8, G7, G6, G5 | G9, G8, G7, G6, G5, G4 | G9, G8, G7, G6, G5, G4, G3 | G9, G8, G7, G6, G5, G4, G3, G2 |

**Table 3.**WTG participation index (PI

_{i}) of interarea modes for the case with the penetration level 50.17% in the NETS area.

Mode # | WTG PI_{i} |
---|---|

1 | 0/5.428 |

2 | 0/8.239 |

3 | 0/3.2 |

4 | 0/5.55 |

Penetration Levels | Eigenvalues | Damping Ratio | Frequency | Participating WTGs | IF_{i} |
---|---|---|---|---|---|

15.44% | −0.4440 ± 13.701 | 0.0324 | 2.1805 | G9 | 1.97/1.97 |

25.86% | −0.4428 ± 13.702 | 0.0323 | 2.1807 | G9, G8 | 2.1/2.1 |

−0.4638 ± 13.693 | 0.0339 | 2.1793 | G8 | 1.97/1.97 | |

36.67% | −0.4425 ± 13.702 | 0.0323 | 2.1807 | G9, G8, G7 | 2.1/2.1 |

−0.4591 ± 13.694 | 0.0335 | 2.1795 | G7 | 1.97/1.97 | |

−0.4638 ± 13.693 | 0.0338 | 2.1793 | G8 | 1.97/1.97 | |

50.17% | −0.4378 ± 13.706 | 0.0319 | 2.1813 | G9, G7, G6 | 3.43/3.43 |

−0.4479 ± 13.701 | 0.0327 | 2.1806 | G9, G7, G6 | 4.63/4.63 | |

−0.4627 ± 13.693 | 0.0338 | 2.1793 | G9, G7 | 3.55/3.55 | |

−0.4637 ± 13.693 | 0.0339 | 2.1793 | G8 | 1.97/1.97 | |

59.92% | −0.4374 ± 13.706 | 0.0319 | 2.1814 | G9, G7, G6 | 3.24/3.24 |

−0.4478 ± 13.701 | 0.0327 | 2.1806 | G9, G7, G6 | 5.1/5.1 | |

−0.4627 ± 13.693 | 0.0338 | 2.1793 | G6, G7 | 3.59/3.59 | |

−0.4636 ± 13.693 | 0.0338 | 2.1793 | G8 | 1.97/1.97 | |

−0.4657 ± 13.691 | 0.0340 | 2.1791 | G5 | 1.97/1.97 | |

72.12% | −0.3981 ± 13.729 | 0.0290 | 2.1850 | G6, G7, G9, G4, G5 | 6.91/6.91 |

−0.4452 ± 13.701 | 0.0325 | 2.1806 | G9, G7 | 2.19/2.19 | |

−0.4575 ± 13.696 | 0.0334 | 2.1798 | G4, G5, G7, G6 | 4.4/4.4 | |

−0.4636 ± 13.693 | 0.0338 | 2.1793 | G8 | 1.97/1.97 | |

−0.4625 ± 13.693 | 0.0338 | 2.1793 | G6, G7 | 3.61/3.61 | |

−0.4641 ± 13.692 | 0.0339 | 2.1792 | G5, G4 | 3.15/3.15 | |

84.66% | −0.3601 ± 13.743 | 0.0262 | 2.1872 | G7, G6, G9, G4, G5, G3 | 7.28/7.28 |

−0.4450 ± 13.701 | 0.0325 | 2.1806 | G9, G7 | 2.07/2.07 | |

−0.4571 ± 13.696 | 0.0334 | 2.1798 | G4, G5, G7, G3, G6 | 6.06/6.06 | |

−0.4582 ± 13.696 | 0.0334 | 2.1797 | G3, G4 | 2.26/2.26 | |

−0.4635 ± 13.693 | 0.0338 | 2.1793 | G8 | 1.97/1.97 | |

−0.4624 ± 13.693 | 0.0337 | 2.1793 | G6, G7 | 3.55/3.55 | |

−0.4640 ± 13.692 | 0.0339 | 2.1792 | G5, G4 | 3.17/3.17 | |

95.18%. | −0.0854 ± 13.747 | 0.0062 | 2.1878 | G6, G7, G9, G4, G3, G5, G2 | 8.53/8.53 |

−0.4448 ± 13.701 | 0.0324 | 2.1806 | G9 | 1.97/1.97 | |

−0.4510 ± 13.699 | 0.0329 | 2.1803 | G3, G2, G7, G6 | 4.13/4.13 | |

−0.4573 ± 13.696 | 0.0334 | 2.1798 | G4, G5, G7, G6 | 3.17/3.17 | |

−0.4616 ± 13.693 | 0.0337 | 2.1794 | G7, G3 | 3.32/3.32 | |

−0.4635 ± 13.693 | 0.0338 | 2.1793 | G8 | 1.97/1.97 | |

−0.4621 ± 13.692 | 0.0337 | 2.1792 | G6, G7 | 3.58/3.58 | |

−0.4638 ± 13.692 | 0.0339 | 2.1791 | G5, G4 | 3.17/3.17 |

**Table 5.**Participating WTGs and state variables of the strong-interaction mode for the penetration level 50.17% in the NETS area.

Participation Factor | Generator | Module | Related State Variables |
---|---|---|---|

1.00000 | G9 | wtgt_a | The twist angle of the shaft (δ_{t}) |

0.97162 | G9 | wtgt_a | Generator speed deviation (Δω_{g}) |

0.40640 | G7 | wtgt_a | The twist angle of the shaft (δ_{t}) |

0.39469 | G7 | wtgt_a | Generator speed deviation (Δω_{g}) |

0.33489 | G6 | wtgt_a | The twist angle of the shaft (δ_{t}) |

0.32535 | G6 | wtgt_a | Generator speed deviation (Δω_{g}) |

Measured Power | Main Modes | Frequency | Damping Ratio |
---|---|---|---|

Active power of G9 | −0.0250 ± 2.1603 | 0.344 | 0.0116 |

−0.0520 ± 13.8788 | 2.210 | 0.0037 | |

Reactive power of G9 | 0.0230 ± 2.1164 | 0.337 | −0.0109 |

−2.6000 ± 13.2508 | 2.110 | 0.1925 | |

Active power of G8 | 0.0240 ± 2.1478 | 0.342 | −0.0112 |

−0.0560 ± 13.6276 | 2.170 | 0.0041 | |

Reactive power of G8 | −0.0320 ± 2.1352 | 0.340 | 0.0150 |

−6.1000 ± 13.0624 | 2.080 | 0.4231 | |

Active power of G7 | 0.0620 ± 2.1540 | 0.343 | −0.0288 |

−0.6000 ± 13.5648 | 2.160 | 0.0442 | |

Reactive power of G7 | 0.0290 ± 2.1415 | 0.341 | −0.0135 |

−0.0350 ± 14.9464 | 2.380 | 0.0023 | |

Active power of G6 | 0.0320 ± 2.1415 | 0.341 | −0.0149 |

−1.2000 ± 14.8836 | 2.370 | 0.0804 | |

Reactive power of G6 | 0.0490 ± 2.1917 | 0.349 | −0.0224 |

−3.0000 ± 13.502 | 2.150 | 0.2169 | |

Active power of tie line 27–53 | 0.0100 ± 2.1729 | 0.346 | −0.0046 |

−1.8000 ± 13.8788 | 2.210 | 0.1286 | |

Active power of tie line 54–53 | 0.0150 ± 2.1854 | 0.348 | −0.0069 |

−2.0000 ± 15.4488 | 2.460 | 0.1284 | |

Active power of tie line 60–61 | 0.0230 ± 2.1603 | 0.344 | −0.0106 |

−2.1000 ± 13.7532 | 2.190 | 0.1509 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, W.; Teng, Z.; Zhao, J.; Qiu, J. Small-Signal Performance of Type 4 Wind Turbine Generator-Based Clusters in Power Systems. *Energies* **2018**, *11*, 1486.
https://doi.org/10.3390/en11061486

**AMA Style**

Chen W, Teng Z, Zhao J, Qiu J. Small-Signal Performance of Type 4 Wind Turbine Generator-Based Clusters in Power Systems. *Energies*. 2018; 11(6):1486.
https://doi.org/10.3390/en11061486

**Chicago/Turabian Style**

Chen, Wuhui, Zaixing Teng, Junhua Zhao, and Jing Qiu. 2018. "Small-Signal Performance of Type 4 Wind Turbine Generator-Based Clusters in Power Systems" *Energies* 11, no. 6: 1486.
https://doi.org/10.3390/en11061486