# Small-Signal Modelling and Stability Assessment of Phase-Locked Loops in Weak Grids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- A low-complexity small signal model is proposed for a grid connected converter controlled in the dq reference frame and synchronised with a dq-PLL. This model simplifies the design tasks so that issues such as the PLL bandwidth and the effects produced by a variation of the SCR on the stability of the system can be considered. (the SCR is used to describe the “weakness” of the grid).
- Based on the proposed model, a systematic PLL design process is proposed which can be used for balanced, three-phase, three-wire weak grids to ensure system stability.
- With the proposed PLL design scheme it is possible to find the maximum bandwidth of the PLL which can be used in the control system for a typical grid-connected power converter, without affecting the system stability.
- A comprehensive study of the effects of PLL bandwidth on the system stability is performed. These effects are also studied for different levels of grid weakness. The study presented in this paper has been verified through simulation and validated through extensive experimental work.

## 2. Enhanced PLL Design Process

#### 2.1. Proposed PLL Design for Weak Grids

#### 2.2. System under Study

## 3. Linearised State-Space Model

#### 3.1. Converter Model in the CRF

#### 3.1.1. Current Control Loop

#### 3.1.2. LC Filter Model

#### 3.1.3. PLL and Converter Models

#### 3.2. Converter Model in the ARF

#### 3.3. System Equations

#### 3.3.1. Capacitor Equations

#### 3.3.2. Grid Equations

#### 3.4. Steady State Operating Points

#### 3.5. Whole System Model

## 4. Stability Analysis and Simulation Results

#### 4.1. Comparison between Model and Simulation Results

^{®}(version 4.1.2, Plexim, Zurich, Swiss) and MATLAB

^{®}(version R2017b, MathWorks, Massachusetts, USA) software, is almost perfect. The figure confirms that the proposed small signal model (44) can effectively represent the effects of the PLL dynamics when the converter operates connected to a weak grid. Moreover, it is possible to conclude that with a high PLL bandwidth and a small SCR, the converter is not able to inject the nominal active current into the grid. In fact, in some conditions, the inverter can supply less than 50% of the rated active current, leading to under-utilisation of the installed power capacity. The proposed design process, shown in Figure 2, can effectively prevent incorrect PLL designs. For example, if ta renewable energy system is connected to a weak grid with a SCR equal to 1.6265, the proposed model shows that the fastest PLL that can be implemented without affecting the stability and power transfer capability has a bandwidth of about 30 Hz (see Figure 5).

#### 4.2. Eigenvalue Analysis

## 5. Experimental Results

- ✓
- Step 1: The reference value ${I}_{1d}^{*c}$ in the converter is changed from ${I}_{1d}^{*c}=14\text{}\mathrm{A}$ to ${I}_{1d}^{*c}=15\mathrm{A}$
- ✓
- Step 2: The reference value ${I}_{1d}^{*c}$ in the converter is changed from ${I}_{1d}^{*c}=15\mathrm{A}$ to ${I}_{1d}^{*c}=16\mathrm{A}$
- ✓
- Step 3: The reference value ${I}_{1d}^{*c}$ in the converter is changed from ${I}_{1d}^{*c}=16\mathrm{A}$ to ${I}_{1d}^{*c}=17\mathrm{A}$
- ✓
- Step 4: The reference value ${I}_{1d}^{*c}$ in the converter is changed from ${I}_{1d}^{*c}=17\mathrm{A}$ to ${I}_{1d}^{*c}=18\mathrm{A}$

_{model}− ζ

_{experimental}| < 0.1 in all the operating range. The error is much lower when the system is operating with damping coefficients above a given threshold (e.g., 0.25).

_{s}≈ 4/(ζω

_{n}) where $\zeta $ is the damping coefficient and ${\omega}_{n}$ is the natural frequency of the dominant poles.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Acronyms and Symbols

ARF | Actual Reference Frame |

BW | Bandwidth |

CRF | Converter Reference Frame |

KCL | Kirchhoff’s Current Law |

KVL | Kirchhoff´s Voltage Law |

LTI | Linear Time Invariant |

PCC | Point of Common Coupling |

PI | Proportional Integral |

PLL | Phase-locked loop |

PM | Phase Margin |

SCR | Short Circuit Ratio |

${\omega}_{nat}$ | Natural frequency |

$\zeta $ | Damping ratio |

${I}_{1dq}^{c}$ | Inverter output current in the converter reference frame |

${I}_{1dq}^{a}$ | Inverter output current in the actual reference frame |

${I}_{1dq}^{*c}$ | Reference values of currents in the converter reference frame |

${E}_{1dq}^{c}$ | dq components of the capacitor voltage in the converter reference frame |

${E}_{1dq}^{a}$ | dq components of the capacitor voltage in the actual reference frame |

${I}_{gdq}^{c}$ | Grid current in the converter reference frame |

${I}_{gdq}^{a}$ | Grid current in the actual reference frame |

${V}_{gdq}^{c}$ | dq components of the grid voltage in the converter reference frame |

${V}_{gdq}^{a}$ | dq components of the grid voltage in the actual reference frame |

${I}_{C1dq}^{c}$ | Current in the capacitor (converter reference frame) |

${I}_{C1dq}^{a}$ | Current in the capacitor (actual reference frame) |

${V}_{1dq}^{c}$ | dq components of the voltage at the converter output (converter reference frame) |

${V}_{1dq}^{a}$ | dq components of the voltage at the converter output (actual reference frame) |

${L}_{1}$ | Inverter side filter inductor |

${C}_{1}$ | Filter capacitor |

${L}_{g}$ | Inductive component of grid impedance |

${R}_{g}$ | Resistive component of grid impedance |

${\omega}_{PLL}$ | Frequency obtained by the PLL |

${\omega}_{n}$ | Nominal frequency of the electrical system |

$\u2206\theta $ | Perturbation in the angle between the CRF and the ARF |

${\theta}_{e}$ | Phase angle of the electrical system |

${\theta}_{PLL}$ | Estimation of ${\theta}_{e}$ obtained by the PLL |

δ | Angle between ${E}_{1dq}^{a}$ and ${V}_{gdq}^{a}$ |

${Z}_{g}$ | Grid impedance |

## Appendix A

_{1}is a remainder that absorbs all the differences that the first order approximation cannot represent [47]. A similar process can be applied to g, obtaining a linear approximation of it.

## Appendix B

_{e}), corresponds to that shown in (A8)

## Appendix C

**Figure A1.**(

**a**) pole diagram of Equation (A9), and (

**b**) experimental measure of (A10) from the experimental system.

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**Figure 5.**Maximum active current that the converter can be injected into the grid as a function of the PLL bandwidth and the SCR of the grid—Comparison between PLECs

^{®}simulation and the proposed small signal model (nominal active current = 18 A).

**Figure 7.**Damping of modes of the proposed small signal model as a function of the PLL bandwidth and the following short circuit ratios (SCR) of the weak grid: (

**a**) SCR = 2.5942; (

**b**) SCR = 2.1652; (

**c**) SCR = 1.8577; (

**d**) SCR = 1.6265.

**Figure 8.**Damping of modes 7 and 8 of the proposed small signal model as a function of the PLL bandwidth and the active current injected by the converter into the weak grid, for the following short circuit ratios (SCR): (

**a**) SCR = 2.5942; (

**b**) SCR = 2.1652; (

**c**) SCR = 1.8577; (

**d**) SCR = 1.6265; (

**e**) SCR = 1.4463.

**Figure 10.**Experimental results: (

**a**) Maximum active current injected by the converter into the grid for SCR = 2.5942 and PLL bandwidth of 51.514 Hz (10 A/div); (

**b**) Same as a) but with SCR = 2.1652 (10 A/div).

**Figure 11.**Experimental results: (

**a**) Maximum active current injected by the converter into the grid with a PLL bandwidth of 40.723 Hz (10 A/div); (

**b**) Maximum active current injected with a PLL bandwidth of 51.514 Hz (10 A/div).

**Figure 12.**Experimental results: (

**a**) Maximum active current injected by the converter into the grid with PLL bandwidth of 30.898 Hz (10 A/div); (

**b**) Maximum active current with PLL bandwidth of 40.723 Hz (10 A/div), (

**c**) Maximum active current with PLL bandwidth of 51.514 Hz (10 A/div).

**Figure 13.**Experimental results: (

**a**) Maximum active current injected into the grid with PLL bandwidth of 20.334 Hz (10 A/div); (

**b**) Maximum active current with PLL bandwidth of 30.898 Hz (10 A/div); (

**c**) Maximum active current with PLL bandwidth of 40.723 Hz (10 A/div); (

**d**) Maximum active current with PLL bandwidth of 51.514 Hz (10 A/div).

**Figure 15.**Transient response of frequency at the PLL output for a SCR of 1.4463 and for a PLL BW of 20.334 Hz, in the four steps considered in for the validation—MatLab data logging of the experimental waveforms.

**Figure 16.**Transient response of frequency at the PLL output for a SCR of 1.6265 and for a PLL BW of 30.898 Hz, in the 4 steps considered in for the validation—Matlab data logging of the experimental waveforms.

**Figure 17.**Transient response of frequency at the PLL output for a SCR of 1.8577 and for a PLL BW of 40.723 Hz, in the four steps considered in for the validation—Matlab data logging of the experimental waveforms.

**Figure 18.**Transient response of frequency at the PLL output for a SCR of 2.1652 and for a PLL BW of 51.514 Hz, in the 4 steps considered in for the validation—Matlab data logging of the experimental waveforms.

L_{1} | R_{1} | C_{1} | R_{g} | V_{g} (Line to Line) |
---|---|---|---|---|

2.3 [mH] | 0.2 [Ω] | 10 [uF] | 0.8 [Ω] | 230$\sqrt{2}$ [V] |

${\mathit{k}}_{\mathit{p}\mathit{P}\mathit{L}\mathit{L}}$ | ${\mathit{k}}_{\mathit{i}\mathit{P}\mathit{L}\mathit{L}}$ | P.M. (Phase Margin) | PLL Bandwidth [Hz] |
---|---|---|---|

0.1388025 | 3.0845 | 65.5 | 10.277 |

0.2710840 | 12.322 | 64.7 | 20.334 |

0.4176300 | 27.842 | 65.6 | 30.898 |

0.5432020 | 49.382 | 64.7 | 40.723 |

0.6963750 | 77.375 | 65.6 | 51.514 |

0.8334000 | 111.12 | 65.5 | 61.697 |

0.9735680 | 152.12 | 65.5 | 72.136 |

1.1116560 | 198.51 | 65.5 | 82.388 |

1.24620000 | 249.24 | 65.5 | 92.336 |

1.38564000 | 307.92 | 65.5 | 102.648 |

${\mathit{L}}_{\mathit{g}}\text{}\left(\mathbf{mH}\right)$ | SCR |
---|---|

25.2 | 2.5942 |

30.4 | 2.1652 |

35.4 | 1.8577 |

40.4 | 1.6265 |

45.6 | 1.4463 |

PLL Bandwidth [Hz] | Maximum Active Current—Proposed Model [A] | Maximum Active Current—Experimental Rig [A] |
---|---|---|

10.277–20.334–30.898–40.723 | 18 | 18 |

51.514 | 15.7 | 14.5 |

PLL Bandwidth [Hz] | Maximum Active Current—Proposed Model [A] | Maximum Active Current—Experimental Rig [A] |
---|---|---|

10.277–20.334–30.898 | 18 | 18 |

40.723 | 17.5 | 16.5 |

51.514 | 11.8 | 10 |

PLL Bandwidth [Hz] | Maximum Active Current—Proposed Model [A] | Maximum Active Current—Experimental Rig [A] |
---|---|---|

10.277–20.334 | 18 | 18 |

30.898 | 18 | 16.5 |

40.723 | 13.2 | 13.5 |

51.514 | 8.7 | 8.5 |

**Table 7.**Comparison between the damping of modes 7 and 8 given by the proposed model and the those found in the experimental system, for a SCR = 1.4463 and PLL bandwidth equal to 20.344 Hz.

Reference Value in the Converter [A] $({\mathit{I}}_{1\mathit{d}}^{*\mathit{c}})$ | Damping of Modes 7 and 8—Proposed Model $({\mathit{\zeta}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}})$ | Damping of Modes 7 and 8—Experimental Rig $({\mathit{\zeta}}_{\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}})$ |
---|---|---|

14 | 0.153 | 0.176 |

15 | 0.146 | 0.119 |

16 | 0.140 | 0.088 |

17 | 0.137 | 0.050 |

**Table 8.**Comparison between the damping of modes 7 and 8 given by the proposed model and the those found in the experimental system, for SCR = 1.6265 and PLL bandwidth equal to 30.898 Hz.

Reference Value in the Converter [A] $({\mathit{I}}_{1\mathit{d}}^{*\mathit{c}})$ | Damping of Modes 7 and 8—Proposed Model $({\mathit{\zeta}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}})$ | Damping of Modes 7 and 8— Experimental Rig $({\mathit{\zeta}}_{\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}})$ |
---|---|---|

14 | 0.226 | 0.232 |

15 | 0.220 | 0.197 |

16 | 0.215 | 0.151 |

17 | 0.211 | 0.105 |

**Table 9.**Comparison between the damping of modes 7 and 8 given by the proposed model and the those found in the experimental system, for SCR = 1.8577 and PLL bandwidth equal to 40.723 Hz.

Reference Value in the Converter [A] $({\mathit{I}}_{1\mathit{d}}^{*\mathit{c}})$ | Damping of Modes 7 and 8—Proposed Model $({\mathit{\zeta}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}})$ | Damping of Modes 7 and 8—Experimental Rig $({\mathit{\zeta}}_{\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}})$ |
---|---|---|

14 | 0.183 | 0.145 |

15 | 0.168 | 0.118 |

16 | 0.153 | 0.090 |

17 | 0.137 | 0.068 |

**Table 10.**Comparison between the damping of modes 7 and 8 given by the proposed model and the those found in the experimental system, for SCR = 2.1652 and PLL bandwidth equal to 51.514 Hz.

Reference Value in the Converter [A] $({\mathit{I}}_{1\mathit{d}}^{*\mathit{c}})$ | Damping of Modes 7 and 8—Proposed Model $({\mathit{\zeta}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}})$ | Damping of Modes 7 and 8—Experimental Rig $({\mathit{\zeta}}_{\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}})$ |
---|---|---|

14 | 0.163 | 0.178 |

15 | 0.143 | 0.140 |

16 | 0.123 | 0.084 |

17 | 0.102 | 0.050 |

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**MDPI and ACS Style**

Burgos-Mellado, C.; Costabeber, A.; Sumner, M.; Cárdenas-Dobson, R.; Sáez, D.
Small-Signal Modelling and Stability Assessment of Phase-Locked Loops in Weak Grids. *Energies* **2019**, *12*, 1227.
https://doi.org/10.3390/en12071227

**AMA Style**

Burgos-Mellado C, Costabeber A, Sumner M, Cárdenas-Dobson R, Sáez D.
Small-Signal Modelling and Stability Assessment of Phase-Locked Loops in Weak Grids. *Energies*. 2019; 12(7):1227.
https://doi.org/10.3390/en12071227

**Chicago/Turabian Style**

Burgos-Mellado, Claudio, Alessandro Costabeber, Mark Sumner, Roberto Cárdenas-Dobson, and Doris Sáez.
2019. "Small-Signal Modelling and Stability Assessment of Phase-Locked Loops in Weak Grids" *Energies* 12, no. 7: 1227.
https://doi.org/10.3390/en12071227