# A Stability Control Method to Maintain Synchronization Stability of Wind Generation under Weak Grid

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## Abstract

**:**

## 1. Introduction

- (a)
- A clear exposition of the coupling mechanism among the PLL, weak grid, and current control is provided, laying the groundwork for forthcoming stability enhancement techniques.
- (b)
- An equivalent parallel resistance compensation method integrated into PLL is proposed to improve synchronization stability. It reshapes the qq-axis impedance and will not decrease the dynamic performance of PLL with a proper design of virtual resistance.
- (c)
- The compensation method demonstrates robustness against system parameter variations and grid impedance measurement errors.

## 2. Impedance Model and Stability Analysis of the Grid-Connected System

#### 2.1. Description and Model of System

_{f}is utilized to filter the GCI output voltage V

_{iabc}to PCC voltage V

_{abc}. By controlling the dq-component (${i}_{d}^{c},{i}_{q}^{c}$) of the current I

_{abc}and synchronizing with the voltage of PCC, the grid-following control based on PLL can be achieved. Table 1 lists the introduction and the value of system parameters. SCR is usually used to define the grid strength, and the equation of SCR can be expressed as:

_{g}is the grid voltage magnitude, Z

_{g}is the grid impedance, and S is the rated capacity of the grid-connected system. When SCR > 3, it is considered a strong grid; 2 < SCR < 3, it is considered a weak grid; and SCR < 2, it is considered a very weak grid.

**V**

_{abc}rather than

**V**

_{gabc}under the weak grid case. This leads to distinct system frames and control frames. The dq-component of variables in the system and control frame are equal in steady state, namely ${\mathit{x}}_{dq}^{s}={\mathit{x}}_{dq}^{c}$, where the control frame and the system frame are designated with “c” and ”s”.

**i**

_{dq}and modulated voltage ∆

**e**

_{dq}in different frames is as follows [8]:

_{dqref}and V

_{idq}denote the steady-state values [8]. H

_{pll}(s) describes the relationship from the $\Delta {v}_{q}^{s}$ to the ∆θ:

**G**

_{i}(s) represents the matrix of current control,

**G**

_{dec}(s) is the matrix of decoupling term, and

**I**is the unit matrix.

**Z**

_{f}corresponds to the inverter side filter inductance L

_{f}and is expressed as:

**G**

_{del}(s) accounts for the digital control delay, with T

_{del}= 1.5 T

_{s}. It is defined as:

**Z**

_{out}can be obtained as:

**Z**

_{g}, can be obtained as follows:

#### 2.2. Stability and the Coupling Mechanism Analysis

**Z**

_{out}can be presented in a more intuitive way:

**Z**

_{out}denotes the impedance of the GCI incorporating the PLL feedforward effect, while

**Z**

_{plant}refers to the GCI impedance when the PLL feedforward path is not taken into account. ${\mathit{Z}}_{pll}^{m}(s)$ represents the impedance formulated by the feedforward effect.

_{g}= 16 mH and the ${{\mathit{L}}_{o}(s)={\mathit{Z}}_{g}\cdot \mathit{Z}}_{out}^{-1}$ when L

_{g}= 9 mH, L

_{g}= 16 mH. The eigen-traces λ

_{1p}, λ

_{2p}correspond to

**L**

_{p}(s), and the eigen-traces λ

_{1o}, λ

_{2o}correspond to

**L**

_{o}(s).

_{g}= 16 mH, the λ

_{1p}, λ

_{2p}do not intersect (−1, j0), where the stable operation can be maintained. When taking into account the PLL effect, it is evident that the system remains stable when L

_{g}= 9 mH, whereas instability arises when L

_{g}= 16 mH. Therefore, the system’s stability is primarily influenced by both the PLL and the grid impedance. The coupling between the weak grid and the PLL will be explored in the following.

**Z**

_{out}

_{,qq}exhibits capacitive impedance characteristics in qq-axis due to the PLL effect, which may couple with a weak grid and further contribute to instability. Thus, the PLL affects the stability of the studied system via current control by the feedforward effect.

**Z**

_{g}can be formulated as:

**Z**

_{g}and PLL. PLL further feedforwards the current control, resulting in the formulation of the qq-axis capacitive impedance area.

**G**denotes the coupling term, which can be mathematically obtained as:

## 3. Analysis and Design of Equivalent Parallel Compensation Method

#### 3.1. Analysis of the Equivalent Parallel Compensation Method

**Z**

_{g}, it leads to a modification in the reference of the PLL, consequently changing the coupling method. It results in a reduction in the coupling degree between the qq-axis capacitive impedance of the GCI and the inductive weak grid. Meanwhile, the damping of the studied system can be improved by the virtual resistance.

**Z**

_{g}in complex space is expressed as follows [26]:

_{v}is parallel with

**Z**

_{g}, the expression for the grid impedance

**Z**

_{g}

_{,com}after parallel connection can be expressed as:

_{mg}is the grid impedance value measured by the impedance measuring device,

**G**can be expressed as another expression:

_{pll}to H

_{pll}

_{,com}, namely modify the input of the PLL as shown in Formula (18). Where G

_{vr}

_{,dq}in dq domain can be converted to G

_{vr}

_{,αβ}in αβ domain:

_{vr}

_{,αβ}with the PLL input in the αβ domain, the transfer function of H

_{pll}becomes H

_{pll}

_{,com}, which equivalently parallels the virtual resistance R

_{v}with the grid impedance. The equivalent parallel resistance compensation method alters the coupling method in the studied system. Adopting the equivalent parallel resistance compensation method, the coupling degree will decrease, which theoretically improves the damping of the studied system.

**G**

_{vr}

_{,αβ}= G

_{vr}

_{,αβ}·

**I**, and

**G**

_{bpf}= G

_{bpf}·

**I**. G

_{bpf}is the band-pass filter aimed at preserving high-frequency characteristics and suppressing additional harmonics, expressed as:

_{1}denotes the grid angular frequency. The introduction of

**G**

_{vr}

_{,αβ}changes the phase reference of PLL input, consequently altering the stable operational point. This leads to an undesirable reactive power. Therefore, the dq transform angle should be modified, formulated as:

_{pll}

_{,com}represents the corrected dq transform angle. Since the phase shift of the G

_{bpf}is 0 at the fundamental frequency, θ

_{pll}

_{,com}can be formulated as:

#### 3.2. Dynamic Performance of Compensation Method

_{v}(R

_{v}= 4 and R

_{v}= 1) is illustrated in Figure 7. The improved PLL exhibits reduced undershoot compared to the conventional PLL. Furthermore, as R

_{v}decreases, the undershoot is further minimized. Therefore, the improved PLL effectively mitigates the overshoot and undershoot observed in the conventional PLL.

_{v}= 4. But it will further decrease with improved PLL when R

_{v}= 1. Thus, the R

_{v}will not be designed too small for a better dynamic response time of the PLL.

_{v}, the proposed compensation method effectively maintains the stability while preserving minimal impact on the dynamic performance of the frequency tracking. For the selection of R

_{v}, if R

_{v}is too high, the coupling degree cannot be decreased. Conversely, if R

_{v}is set too low, it may adversely affect the dynamic performance of the PLL.

#### 3.3. Design and Stability Analysis

_{v}is (0, 6) by employing the GNC to ${{\mathit{L}}_{c}\left(s\right)={\mathit{Z}}_{g}\cdot \mathit{Z}}_{out,com}^{-1}$. In this study, R

_{v}is chosen as 2, as it offers appropriate damping while preserving the dynamic performance of the PLL. Figure 8 illustrates the GNC plot for L

_{g}= 16 mH and R

_{v}= 2. The GNC plot clearly indicates that (−1, j0) lies outside the λ

_{1oc}, λ

_{2oc}, indicating the stable operation of the interconnection system.

#### 3.4. Robustness Analysis

_{mg}of G

_{vr}

_{,dq}also influence the effect of the compensation method. Therefore, it is imperative to conduct further analysis on the robustness of the proposed compensation method.

_{mg}and the L

_{g}. Therefore, it is necessary to analyze whether the equivalent parallel resistance compensation can achieve stable control when there are errors in grid impedance measurement.

_{mg}, as revealed by Equation (19), on the compensation method can be inferred. Case IV and case V demonstrate that the compensation method can maintain stability even when the error in grid impedance measurement reaches −70% L

_{g}and +160 L

_{g}.

## 4. Experiment and Simulation Verification

_{g}. It can be observed from Figure 12b that when the SCR is switched from 4.6 to 2.59, the oscillations will occur. Notably, a positive-sequence frequency of 111.25 Hz and a negative-sequence frequency of 11.25 Hz are observed due to the mirror frequency effect, as illustrated in Figure 12c.

_{v}= 1, R

_{v}= 2, R

_{v}= 4. As depicted in Figure 13, the utilization of the compensation method reduces voltage and current distortion. Consequently, the studied system demonstrates stable operation with the utilization of the compensation method. Additionally, Figure 14 presents the fast Fourier transform (FFT) analysis of the current, indicating the damping of oscillation frequencies in all three cases.

_{v}= 2 under case I. The parameter variations of the studied system alter the oscillation frequency. From Figure 15a, when the compensation is not utilized, the investigated system experiences resonance. When the compensation method is adopted, the resonance can be mitigated. But the reference of PLL will be changed. Thus, there is a transient current overshoot and reactive power before the phase lock process, as depicted in Figure 15c. Figure 16 shows the damping process of i

_{a}, which is based on discrete Fourier transformation (DFT). The oscillation frequency of 16 Hz and 116 Hz will be attenuated after the adoption of the compensation method.

_{v}= 2 under case II and case III. It can be deduced that the compensation method effectively dampens the resonance even when the oscillation frequency has shifted, indicating the robustness of the method against the introduction of reactive power, fluctuations in PLL bandwidth, and grid impedance. These experimental results provide further evidence of the effectiveness and robustness of the compensation method.

_{v}= 2 under case IV and case V. Given Formula (23), the error in grid impedance measurement does not affect the effectiveness of the phase correction. In addition, the error of grid impedance measurement can be up to −70% and +160%, demonstrating the robustness of the compensation method to errors in grid impedance measurement, as verified experimentally.

_{v}= 2 are shown in Figure 20, which is conducted in MATLAB/Simulink. The system can maintain stability in both cases, which indicates the proposed compensation method can deal with the change of grid frequency.

## 5. Conclusions and Future Work

_{g}and +160% L

_{g}. Experimental waveforms further validate the effectiveness of this method.

- (1)
- The instability mechanism of the grid-connected system can be deeply analyzed, which enables us to propose a more effective stability control method.
- (2)
- The effectiveness and implementation of the equivalent parallel resistance compensation method in scenarios such as multiple inverters should be explored.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The GNC plot under weak grid (

**a**) L

_{g}= 16 mH, without PLL effect (

**b**) L

_{g}= 9 mH, with PLL effect (

**c**) L

_{g}= 16 mH, with PLL effect.

**Figure 5.**(

**a**) PLL considering grid impedance (

**b**) current control of GCI with PLL effects considering grid impedance.

**Figure 10.**The GNC plot under R

_{v}= 2 (

**a**) case I; (

**b**) case II; (

**c**) case III; (

**d**) case IV; (

**e**) case V.

**Figure 12.**Waveforms under (

**a**) L

_{g}= 9 mH and (

**b**) L

_{g}= 16 mH (

**c**) FFT analysis of unstable current.

**Figure 13.**Waveforms with proposed compensation under (

**a**) R

_{v}= 1, (

**b**) R

_{v}= 2, and (

**c**) R

_{v}= 4.

**Figure 15.**Case I (

**a**) voltage and current waveforms (

**b**) the stable waveforms (

**c**) the unstable waveforms.

**Figure 20.**R

_{v}= 2. (

**a**) The waveforms of frequency changes 0.5 Hz; (

**b**) the stable waveforms of frequency changes −0.5 Hz.

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

DC-link voltage V_{dc} | 400 V |

Grid voltage RMS value V_{g} | 130 V |

Rated power P_{n} | 2 kW |

Filter inductors L_{f} | 3 mH |

Filter capacitors C_{f} | 20 μF |

Damping resistors R_{f} | 10 Ω |

Grid inductors L_{g} | 9/16 mH |

Switching frequency f_{sw} | 10 kHz |

Sampling frequency f_{s} | 10 kHz |

Proportional gain of G_{i} k_{pi} | 5.24 |

Integral gain of G_{i} k_{ii} | 1370 |

Proportional gain of G_{pll} k_{ppll} | 4.2 |

Integral gain of G_{pll} k_{ipll} | 384 |

Cases | L_{mg} (mH) | L_{g} (mH) | PLL Parameters | dq-Axis Current (A) |
---|---|---|---|---|

I | 16 | 16 | k_{ppll} = 4.2, k_{ipll} = 384 | I_{dref} = 10, I_{qref} = 2 |

II | 16 | 16 | k_{ppll} = 4.2, k_{ipll} = 484 | I_{dref} = 10, I_{qref} = 0 |

III | 16 | 17.5 | k_{ppll} = 4.2, k_{ipll} = 384 | I_{dref} = 10, I_{qref} = 0 |

IV | 4.8 (−70% L_{g}) | 16 | k_{ppll} = 4.2, k_{ipll} = 384 | I_{dref} = 10, I_{qref} = 0 |

V | 41.6 (+160% L_{g}) | 16 | k_{ppll} = 4.2, k_{ipll} = 384 | I_{dref} = 10, I_{qref} = 0 |

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## Share and Cite

**MDPI and ACS Style**

Wu, M.; Zeng, J.; Ying, G.; Xu, J.; Yang, S.; Zhou, Y.; Liu, J.
A Stability Control Method to Maintain Synchronization Stability of Wind Generation under Weak Grid. *Energies* **2024**, *17*, 4450.
https://doi.org/10.3390/en17174450

**AMA Style**

Wu M, Zeng J, Ying G, Xu J, Yang S, Zhou Y, Liu J.
A Stability Control Method to Maintain Synchronization Stability of Wind Generation under Weak Grid. *Energies*. 2024; 17(17):4450.
https://doi.org/10.3390/en17174450

**Chicago/Turabian Style**

Wu, Minhai, Jun Zeng, Gengning Ying, Jidong Xu, Shuangfei Yang, Yuebin Zhou, and Junfeng Liu.
2024. "A Stability Control Method to Maintain Synchronization Stability of Wind Generation under Weak Grid" *Energies* 17, no. 17: 4450.
https://doi.org/10.3390/en17174450