# Parameter Estimation of a Grid-Tied Inverter Using In Situ Pseudo-Random Perturbation Sources

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Grid-Tied Inverter

#### 2.1. Inverter Topology

#### 2.2. Mathematical Analysis of Inverter

## 3. Simulated Small-Signal Analysis of Inverter

## 4. Parameter Estimation

#### 4.1. Frequency-Domain Sensitivity Analysis of Filter and Controller Parameters

#### 4.2. Parameter Estimation Methodology

#### 4.3. Parameter Estimation Results

## 5. Experimental Comparison of Pseudo-Random Perturbation

#### 5.1. Pseudo-Random Perturbation Sources

#### 5.2. Description of the Practical Inverter under Investigation

#### 5.3. PRIS Perturbation

#### 5.4. PRBS Perturbation

#### 5.5. Critical Comparison of Pseudo-Random Perturbation Strategies

#### 5.6. Estimation of an Analytical Transfer Function of the Inverter Output Impedance from Measurement Data

^{th}order transfer function fits the measured ${Z}_{o}\left(f\right)$ most accurately. The approximated transfer function of the output impedance, ${Z}_{o,approx}\left(s\right)$, is:

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Magnitude and phase frequency response of the estimated inverter output impedance compared to the analytical inverter output impedance.

**Figure 6.**The magnitude and phase response of the analytical output impedance, ${Z}_{o}\left(f\right)$, as a function of (

**a**) ${k}_{i}$, (

**b**) ${k}_{p}$, (

**c**) ${\omega}_{pr}$, (

**d**) ${\omega}_{g}$, (

**e**) ${C}_{f}$, (

**f**) ${L}_{f}$, and (

**g**) ${L}_{g}$.

**Figure 9.**Comparison of the magnitude and phase response of the output impedance of the model after parameter estimation is performed, ${Z}_{o,M}\left(f\right)$, and the target system ${Z}_{o,S}\left(f\right)$.

**Figure 13.**The output voltage, ${v}_{o,n}\left(t\right)$, and current, ${i}_{o,n}\left(t\right)$, under normal operating conditions.

**Figure 16.**Small-signal perturbations of the current, (${i}_{o,p}\left(t\right)-{i}_{o,n}\left(t\right)$), and the voltage, (${v}_{o,p}\left(t\right)-{v}_{o,n}\left(t\right)$), after PRIS perturbation.

**Figure 17.**Magnitude and phase response of ${Z}_{o}\left(f\right)$ of the practical inverter under investigation using PRIS perturbation.

**Figure 19.**Small-signal perturbations of the current, (${i}_{o,p}\left(t\right)-{i}_{o,n}\left(t\right)$), and the voltage, (${v}_{o,p}\left(t\right)-{v}_{o,n}\left(t\right)$), after PRBS perturbation.

**Figure 20.**Magnitude and phase response of ${Z}_{o}\left(f\right)$ of the practical inverter under investigation using PRBS perturbation.

**Figure 21.**Magnitude squared coherence, ${C}_{({i}_{o,p}\left(t\right)-{i}_{o,n}\left(t\right)),({v}_{o,p}\left(t\right)-{v}_{o,n}\left(t\right))}\left(f\right)$, using PRIS and PRBS perturbations.

**Figure 22.**Magnitude and phase response of the analytical Laplace-domain approximation of the measured output impedance, ${Z}_{o,approx}\left(f\right)$, compared to ${Z}_{o}\left(f\right)$.

Parameter | ${\mathit{k}}_{\mathit{p}}$ | ${\mathit{k}}_{\mathit{i}}$ | ${\mathit{\omega}}_{\mathbf{pr}}$ | ${\mathit{\omega}}_{\mathit{g}}$ | C_{f}[$\mathsf{\mu}\mathbf{F}$] | L_{f}[mH] | L_{g}[$\mathsf{\mu}\mathbf{H}$] |
---|---|---|---|---|---|---|---|

Value | 5.4 | 400 | 1 | 314.16 | 5.3 | 18 | 9 |

Coefficient | Value |
---|---|

${b}_{0}$ | ${C}_{f}{L}_{f}{L}_{g}$ |

${b}_{1}$ | ${C}_{f}{L}_{g}({k}_{p}+2{L}_{f}{\omega}_{pr})$ |

${b}_{2}$ | ${L}_{f}+{L}_{g}+2{C}_{f}{L}_{g}{k}_{i}{\omega}_{pr}+2{C}_{f}{L}_{g}{k}_{p}{\omega}_{pr}+2{C}_{f}{L}_{g}{\omega}_{g}^{2}$ |

${b}_{3}$ | ${k}_{p}+2{L}_{f}{\omega}_{pr}+{C}_{f}{L}_{g}{k}_{p}{\omega}_{g}^{2}$ |

${b}_{4}$ | $2{k}_{i}{\omega}_{pr}+{L}_{f}{\omega}_{g}^{2}+{L}_{g}{\omega}_{g}^{2}$ |

${b}_{5}$ | ${k}_{p}{\omega}_{g}^{2}$ |

${a}_{0}$ | 0 |

${a}_{1}$ | ${C}_{f}{L}_{f}$ |

${a}_{2}$ | ${C}_{f}{k}_{p}+2{C}_{f}{L}_{f}{\omega}_{pr}$ |

${a}_{3}$ | $1+2{C}_{f}{k}_{i}{\omega}_{pr}+2{C}_{f}{k}_{p}{\omega}_{pr}+{C}_{f}{L}_{f}{\omega}_{g}^{2}$ |

${a}_{4}$ | $2{\omega}_{pr}+{C}_{f}{k}_{p}{\omega}_{g}^{2}$ |

${a}_{5}$ | ${\omega}_{g}^{2}$ |

**Table 3.**Summary of the effect of the controller and filter parameters on the resonant frequency points of the output impedance.

50 Hz | 58 Hz | 515 Hz | 23 kHz | |||||
---|---|---|---|---|---|---|---|---|

Parameter | Damping | Shifting | Damping | Shifting | Damping | Shifting | Damping | Shifting |

${k}_{i}$ | x | x | ||||||

${k}_{p}$ | x | x | ||||||

${\omega}_{pr}$ | x | x | ||||||

${\omega}_{g}$ | x | x | ||||||

${C}_{f}$ | x | x | x | |||||

${L}_{f}$ | x | x | ||||||

${L}_{g}$ | x | x |

GA Parameters | Parameter Value |
---|---|

Population size | 70 |

Number of real variables | 7 |

Crossover fraction | 0.8 |

Mutation fraction | 0.01 |

Migration fraction | 0.2 |

Seed for random number generator | 0 |

Maximum iterations | 700 |

Population size | 200 |

Parallel computing enabled | Yes |

Particle Swarm Parameters | Parameter Value |
---|---|

Maximum inertia weight | 0.1 |

Minimum inertia weight | 1.1 |

Maximum iterations | 1400 |

Minimum adaptive neighborhood size | 0.25 |

Particle velocity adjustment weight | 1.49 |

Neighbourhood velocity adjustment weight | 1.49 |

Swarm size | 100 |

Step | ${\mathit{k}}_{\mathit{p}}$ | k_{p}Error [%] | ${\mathit{k}}_{\mathit{i}}$ | k_{i}Error [%] | ${\mathit{\omega}}_{\mathbf{pr}}$ | ω_{pr}Error [%] | ${\mathit{\omega}}_{\mathit{g}}$ | ω_{g}Error [%] | C_{f}[$\mathsf{\mu}$F] | C_{f}Error [%] | L_{f}[mH] | L_{f}Error [%] | L_{g}[$\mathsf{\mu}$H] | L_{g}Error [%] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 4.76 | 11.89 | 560.60 | 40.15 | 0.60 | 40.29 | 313.35 | 0.26 | 6.27 | 18.27 | 15.20 | 15.54 | 7.61 | 15.44 |

2 | 5.40 | 0.02 | 400.08 | 0.02 | 0.99 | 0.03 | 314.16 | 0.00 | 5.3 | 0.02 | 0.018 | 0.01 | 9 | 0.05 |

**Table 7.**Specifications for the Sunny Boy 1600TL [61].

Specification | Value |
---|---|

Maximum DC Voltage | 600V |

Nominal DC voltage | 400 V |

Minimum DC voltage | 125 V |

Nominal AC power | 1600 W |

Nominal AC voltage | $220,230,240{V}_{RMS}$ |

Maximum output current | $11{A}_{RMS}$ |

Power factor | 1 |

Maximum efficiency | $96.0\%$ |

Topology | Transformerless |

Setup | PRBS Order | ${\mathit{f}}_{\mathit{clk}}$ | ${\mathit{V}}_{\mathit{d}}$ | ${\mathit{R}}_{\mathit{PRIS}}$ | ${\mathit{L}}_{\mathit{PRIS}}$ | ${\mathit{C}}_{\mathit{PRIS}}$ |
---|---|---|---|---|---|---|

1 | 15 | 30 kHz | 160 ${V}_{DC}$ | 100 $\mathrm{\Omega}$ | 2.2 mH | 100 nF |

2 | 12 | 6 kHz | 160 ${V}_{DC}$ | 100 $\mathrm{\Omega}$ | 1 mH | 1 $\mathsf{\mu}$F |

Setup | PRBS Order | ${\mathit{f}}_{\mathit{clk}}$ | ${\mathit{V}}_{\mathit{d}}$ |
---|---|---|---|

1 | 14 | 16 kHz | 5 ${V}_{DC}$ |

2 | 12 | 3 kHz | 5 ${V}_{DC}$ |

PRIS Source | PRBS Source |
---|---|

The series RLC circuit that is incorporated in the PRIS source allows for increased protection of the H-bridge and DC source, while also providing an additional means of controlling the time-and frequency-domain characteristics of the PRIS | Fewer components are used to construct the PRBS source |

Can be connected in parallel with the system under test thus Can be used to characterize an inverter and the grid simultaneously | Can be connected in series with the system under test and based on practical measurements that are conducted in this work, it produces better perturbation |

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## Share and Cite

**MDPI and ACS Style**

Gerber, I.P.; Mwaniki, F.M.; Vermeulen, H.J.
Parameter Estimation of a Grid-Tied Inverter Using In Situ Pseudo-Random Perturbation Sources. *Energies* **2023**, *16*, 1414.
https://doi.org/10.3390/en16031414

**AMA Style**

Gerber IP, Mwaniki FM, Vermeulen HJ.
Parameter Estimation of a Grid-Tied Inverter Using In Situ Pseudo-Random Perturbation Sources. *Energies*. 2023; 16(3):1414.
https://doi.org/10.3390/en16031414

**Chicago/Turabian Style**

Gerber, Ian Paul, Fredrick Mukundi Mwaniki, and Hendrik Johannes Vermeulen.
2023. "Parameter Estimation of a Grid-Tied Inverter Using In Situ Pseudo-Random Perturbation Sources" *Energies* 16, no. 3: 1414.
https://doi.org/10.3390/en16031414