# Discrete Time Domain Modeling and Control of a Grid-Connected Four-Wire Split-Link Converter

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## Abstract

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## 1. Introduction

## 2. Split-Link Converter Topology and Control

#### 2.1. Problem Statement

- Unequal leakage currents of the capacitors
- Unequal capacitor values
- Unequal time delays during switching
- Asymmetrical charging during transients
- Current measurement errors

#### 2.2. Origin of Neutral-Wire Currents

#### 2.3. Mid-Point Balancing Techniques

## 3. Zero-Sequence Current Injection

#### 3.1. Description of the Technique

#### 3.2. Discrete Time Domain Modeling

#### 3.3. Simulation Results

## 4. Half-Bridge Chopper

#### 4.1. Description of the Technique

#### 4.2. Discrete Time Domain Modeling

#### 4.3. Simulation Results

## 5. Experimental Validation

#### 5.1. No Mid-Point Control

#### 5.2. Zero-Sequence Current Injection

#### 5.3. Half-Bridge Chopper

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Distribution grid simulation results: (

**a**) source currents ${I}_{\mathrm{S}}$, (

**b**) load current ${I}_{\mathrm{L},\mathrm{a}}$, (

**c**) DG-unit currents ${I}_{\mathrm{DG}}$, (

**d**) DG-unit neutral-wire current ${I}_{\mathrm{n}}$, (

**e**) DG-unit dc bus voltage ${V}_{\mathrm{dc}}$. Three-phase currents are denoted in blue, yellow and orange.

**Figure 9.**Zero-Sequence Current Injection step-response (simulation). Circles: simulation model, Crosses: $\mathcal{Z}$ domain model.

**Figure 10.**Simulated effect of an artificial current measurement error on the ZSCI control: (

**a**) Voltage unbalance $\mathsf{\Delta}{V}_{\mathrm{dc}}$, (

**b**) Compensating current ${I}_{\mathrm{comp}}$.

**Figure 14.**Half-Bridge Chopper step-response (simulation). Circles: simulation model, Crosses: $\mathcal{Z}$ domain model.

**Figure 15.**Effect of an artificial current measurement error on the HBC control: (

**a**) Voltage unbalance $\mathsf{\Delta}{V}_{\mathrm{dc}}$, (

**b**) Compensating current ${I}_{\mathrm{comp}}$.

**Figure 16.**Disabling mid-point control at t = 50 ms (experimental): (

**a**) voltage unbalance $\mathsf{\Delta}{V}_{\mathrm{dc}}$, (

**b**) DG-unit three-phase currents ${I}_{\mathrm{DG}}$ in blue, yellow and orange.

**Figure 17.**Effect of a current measurement error on ZSCI on t = 100 ms (experimental): (

**a**) Voltage unbalance $\mathsf{\Delta}{V}_{\mathrm{dc}}$, (

**b**) Compensating current ${I}_{\mathrm{comp}}$.

**Figure 18.**Effect of a current measurement error on HBC on t = 100 ms (experimental): (

**a**) Voltage unbalance $\mathsf{\Delta}{V}_{\mathrm{dc}}$, (

**b**) Compensating current ${I}_{\mathrm{comp}}$.

${\mathit{f}}_{\mathbf{switch}}$ | ${\mathit{C}}_{\mathbf{dc}}$ | ${\mathit{I}}_{\mathbf{ref}}$ | ${\mathit{V}}_{\mathbf{dc}}$ | ${\mathit{L}}_{\mathbf{f}}$ | ${\mathit{C}}_{\mathbf{f}}$ |
---|---|---|---|---|---|

20 kHz | 10 mF | 25 A | 700 V | 2 mH | 5 µF |

${\mathit{R}}_{\mathbf{a},\mathbf{b},\mathbf{c}}$ | ${\mathit{R}}_{\mathbf{n}}$ | L | ${\mathit{R}}_{\mathbf{a},\mathbf{b},\mathbf{c}}$/X | ${\mathit{I}}_{\mathbf{nom}}$ |
---|---|---|---|---|

0.410 Ω/km | 0.713 Ω/km | 0.243 mH/km | 5.37 | 255 A |

${\mathit{\omega}}_{\mathbf{c}}$ | ${\mathit{T}}_{\mathbf{s}}$ | ${\mathit{C}}_{\mathbf{dc}}$ | ${\mathit{I}}_{\mathbf{ref}}$ | ${\mathit{V}}_{\mathbf{dc},\mathbf{ref}}$ | ${\mathit{V}}_{\mathbf{dc}}$ |
---|---|---|---|---|---|

$2\phantom{\rule{3.33333pt}{0ex}}\pi \phantom{\rule{3.33333pt}{0ex}}10\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$ | $50\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{s}$ | $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mF}$ | $24\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ | $600\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ | $400\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

${\mathit{T}}_{\mathbf{s}}$ | ${\mathit{C}}_{\mathbf{dc}}$ | ${\mathit{V}}_{\mathbf{dc}}$ | ${\mathit{L}}_{\mathbf{f}}\phantom{\rule{4pt}{0ex}}\mathbf{and}\phantom{\rule{4pt}{0ex}}{\mathit{L}}_{\mathbf{ch}}$ | ${\mathit{C}}_{\mathbf{f}}$ | ${\mathit{V}}_{\mathbf{grid}}$ |
---|---|---|---|---|---|

50 µs | 1 mF | 400 V | 2.1 mH | 5 µF | 115 V |

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**MDPI and ACS Style**

De Kooning, J.D.M.; Bozalakov, D.; Vandevelde, L.
Discrete Time Domain Modeling and Control of a Grid-Connected Four-Wire Split-Link Converter. *Electronics* **2021**, *10*, 506.
https://doi.org/10.3390/electronics10040506

**AMA Style**

De Kooning JDM, Bozalakov D, Vandevelde L.
Discrete Time Domain Modeling and Control of a Grid-Connected Four-Wire Split-Link Converter. *Electronics*. 2021; 10(4):506.
https://doi.org/10.3390/electronics10040506

**Chicago/Turabian Style**

De Kooning, Jeroen D. M., Dimitar Bozalakov, and Lieven Vandevelde.
2021. "Discrete Time Domain Modeling and Control of a Grid-Connected Four-Wire Split-Link Converter" *Electronics* 10, no. 4: 506.
https://doi.org/10.3390/electronics10040506