# Nonlinear Modelling and Control of a Power Smoothing System for a Novel Wave Energy Converter Prototype

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## Abstract

**:**

## 1. Motivation and Problem Statement

## 2. Electrical System of the Wave Energy Converter

#### 2.1. Generator Units

#### 2.2. Supercapacitor Bank

#### 2.3. DC/DC Converter

## 3. Control of the DC-Link

## 4. Stability Analysis

- (i)
- all coefficients ${a}_{i}$ are positive (i.e., ${a}_{i}>0$ for $i\in \{1,2,\dots \}$) and
- (ii)
- all leading principal minors ${D}_{i}$ of the Hurwitz matrix are positive.

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\mathbb{N},\mathbb{R}$ | natural and real numbers |

$\mathit{x}\text{}:={({x}_{1},\dots ,{x}_{n})}^{\top}\in {\mathbb{R}}^{n}$ | column vector, $n\phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}\mathbb{N}$ (where means “is defined as” and ${}^{\top}$ means “transposed”) |

$\parallel \mathit{x}\parallel \text{}:=\sqrt{{\mathit{x}}^{\top}\mathit{x}}$ | Eucilidean norm of $\mathit{x}$ |

$\mathit{A}\in {\mathbb{R}}^{n\times m}$ | real matrix with n rows and m columns, n,$m\in \mathbb{N}$ |

${\mathit{O}}_{n\times m}\in {\mathbb{R}}^{n\times m}$ | zero matrix |

${\mathit{I}}_{n}\text{}:=diag(1,\dots ,1)\in {\mathbb{R}}^{n\times n}$ | identity matrix |

${x}_{\mathrm{ref}}$ | reference value of e.g., voltage, current and power |

${x}_{\mathrm{nom}}$ | nominal value of e.g., voltage, current and power |

${x}_{\mathrm{max}}$ | maximum value of e.g., voltage, current and power |

${x}_{\mathrm{min}}$ | minimum value of e.g., voltage, current and power |

$\overline{x}$ | average value of e.g., voltage, current and power |

${u}_{\mathrm{dc}}$ | DC-link voltage |

${u}_{\mathrm{bat}},{i}_{\mathrm{bat}}$ | battery voltage and current |

${i}_{\mathrm{w}},{p}_{\mathrm{w}}$ | current and power from the generator units |

${i}_{\mathrm{s}},{p}_{\mathrm{s}}$ | current and power flowing into the DC/DC converter |

${\xi}_{\mathrm{s},\mathrm{ref}}$ | quantity, representing the integral action of the PI-controller |

C | capacitance of the supercapacitor bank |

${R}_{\mathrm{esr}}$ | equivalent series resistors (ESR) of the supercapacitor bank |

${u}_{\mathrm{c}},{i}_{\mathrm{c}}$ | voltage and current of the capacitance |

${u}_{\mathrm{esr}},{p}_{\mathrm{esr}}$ | voltage and power of the ESR |

${E}_{\mathrm{c}}$ | stored energy in the supercapacitor bank |

${\tau}_{\mathrm{s}}$ | time constant of the current dynamics of the DC/DC converter |

${\tau}_{\mathrm{w}}$ | time constant of the low pass filter to filter ${p}_{\mathrm{w}}$ |

${V}_{\mathrm{r}},{\tau}_{\mathrm{r}}$ | parameter of the PI-controller |

${i}_{\mathrm{res}},{V}_{\mathrm{res}},{f}_{\mathrm{res}}$ | restoring current, factor and frequency |

${\mathit{x}}^{\star}\text{}:={({u}_{\mathrm{c}}^{\star},{i}_{\mathrm{s}}^{\star},{\xi}_{\mathrm{s},\mathrm{ref}}^{\star},{u}_{\mathrm{c},\mathrm{ref}}^{\star},{\overline{i}}_{\mathrm{w}}^{\star})}^{\top}$ | quantity vector of the operation point |

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**Figure 2.**Simplified schematic of the electrical system of the wave energy converter and illustration of the power flow, where ${p}_{\mathrm{w}}$ is the fluctuating power from the WEC, ${p}_{\mathrm{c}}$ is the power flowing in and out of the supercapacitors and ${p}_{\mathrm{s}}$ is the smoothed power flow to the battery.

**Figure 3.**CAD figure of the supercapacitor bank with a photograph of one supercapacitor module of the prototype.

**Figure 5.**(

**a**) Picture of the DC/DC converter VarioString VS-70 and (

**b**) LLC resonant topology of the VarioString VS-70 DC/DC converter with a split resonant capacitor technique.

**Figure 7.**Numerical stability analysis of the coefficients ${a}_{0}$ to ${a}_{3}$ and the determinant ${D}_{3}$ over the input operational current ${i}_{\mathrm{w}}^{\star}$ and for two different reference voltages ${u}_{\mathrm{c},\mathrm{ref}}^{\star}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{u}_{\mathrm{dc},\mathrm{min}}$ [] and ${u}_{\mathrm{c},\mathrm{ref}}^{\star}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{u}_{\mathrm{dc},\mathrm{max}}$ [], where ${u}_{\mathrm{c},\mathrm{ref}}^{\star}={u}_{\mathrm{dc},\mathrm{min}}$ accounts for the worst-case.

**Figure 8.**Comparison of the measured [] and simulated [] quantities of the DC-link control, with the rod position ${h}_{\mathrm{w}}$ at the top, followed by the current ${i}_{\mathrm{w}}$ and the smoothed current ${i}_{\mathrm{s}}$, the DC-link voltage ${u}_{\mathrm{dc}}$ and the power of the generator units ${p}_{\mathrm{w}}$ and the smoothed power ${p}_{\mathrm{s}}$. The measured current ${i}_{\mathrm{w}}$ and the power of the generator units ${p}_{\mathrm{w}}$ are used as inputs for the simulation [].

**Table 1.**Eletrical parameters of the used supercapacitor cell, produced by Maxwell Technologies [34].

Electrical Parameter | Symbol | Value |
---|---|---|

Rated Capacitance | ${C}_{i}$ | $350\phantom{\rule{3.33333pt}{0ex}}\mathrm{F}$ |

Equivalent Series Resistor | ${R}_{\mathrm{esr},i}$ | $3.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathsf{\Omega}$ |

Rated Voltage | ${U}_{\mathrm{nom},i}$ | $2.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

Absolute Maximum Voltage | ${U}_{\mathrm{max},i}$ | $2.85\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

Absolute Maximum Current | ${I}_{\mathrm{max},i}$ | $170\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ |

Maximum Leakage Current | ${I}_{\mathrm{leak},i}$ | $0.30\phantom{\rule{3.33333pt}{0ex}}\mathrm{mA}$ |

Electrical Parameter | Symbol | Value |
---|---|---|

Rated Capacitance | C | $2.19\phantom{\rule{3.33333pt}{0ex}}\mathrm{F}$ |

Equivalent Series Resistor | ${R}_{\mathrm{esr}}$ | $512\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathsf{\Omega}$ |

Rated Voltage | ${U}_{\mathrm{nom}}$ | $432\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

Absolute Maximum Voltage | ${U}_{\mathrm{max}}$ | $456\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

Absolute Maximum Current | ${I}_{\mathrm{max}}$ | $170\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ |

**Table 3.**Eletrical parameters of the VarioString VS-70 [42].

Electrical Parameter | Symbol | Value |
---|---|---|

Maximum Input Power | ${P}_{\mathrm{s},\mathrm{max}}$ | $4.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{kW}$ |

Maximum Input Voltage | ${U}_{\mathrm{s},\mathrm{max}}$ | $600\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

Maximum Input Current | ${I}_{\mathrm{s},\mathrm{max}}$ | $13\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ |

Nominal Battery Voltages | ${U}_{\mathrm{bat},\mathrm{nom}}$ | $48\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}$ |

Maximum Battery Current | ${I}_{\mathrm{bat},\mathrm{max}}$ | $70\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}$ |

Maximum Efficiency | $\eta $ | >98% |

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## Share and Cite

**MDPI and ACS Style**

Krüner, S.; Hackl, C.M.
Nonlinear Modelling and Control of a Power Smoothing System for a Novel Wave Energy Converter Prototype. *Sustainability* **2022**, *14*, 13708.
https://doi.org/10.3390/su142113708

**AMA Style**

Krüner S, Hackl CM.
Nonlinear Modelling and Control of a Power Smoothing System for a Novel Wave Energy Converter Prototype. *Sustainability*. 2022; 14(21):13708.
https://doi.org/10.3390/su142113708

**Chicago/Turabian Style**

Krüner, Simon, and Christoph M. Hackl.
2022. "Nonlinear Modelling and Control of a Power Smoothing System for a Novel Wave Energy Converter Prototype" *Sustainability* 14, no. 21: 13708.
https://doi.org/10.3390/su142113708