# Nonlinear Hierarchical Easy-to-Implement Control for DC MicroGrids

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. DC MicroGrid Model

#### 1.1. PV System Modeling

#### 1.1.1. Solar Array

#### 1.1.2. DC/DC Boost Converter

#### 1.2. Storage System

#### 1.2.1. Battery Model

#### 1.2.2. Supercapacitor

#### 1.2.3. Bidirectional Boost Converters

#### 1.3. Interconnected Model

## 2. DC MicroGrid Control Strategy

#### 2.1. High Level Controller for PV Array Source

#### 2.2. High Level Controller for Storage System

## 3. Local Control Level

#### 3.1. Storage System Control

#### 3.1.1. Battery Control Law

#### 3.1.2. Supercapacitor Control Law

#### 3.2. PV System Control

## 4. Stability Study of the Interconnected System

**Lemma**

**1.**

**Proof.**

## 5. Simulation Results

^{2}for $t<$ 12.4 s and from $t>$ 24.8 s it is set to 300 W/m

^{2}. One can notice very small DC bus voltage variations following the transients, quickly recovering and stabilizing at the rated (1000 V DC) in about 10 ms. The maximum DC bus voltage error equals 4 V = 0.4%, well inside the $\pm 5\%$ desired operation region.

#### Comparison with PI Control

^{2}for $t<$ 12.4 s and $t>$ 24.8 s, and is equal to 300 W/m

^{2}otherwise. There, one sees that the DC bus voltage varies briefly following the transients, recovering and stabilizing at the rated DC bus voltage of 1000 V DC in about 1 s. The maximum DC bus voltage error is equal to 58 V. The battery’s reference and actual waveforms are shown on the fourth screen from the top. The reference signal varies slowly and the actual waveform follows it with a small switching frequency ripple.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${I}_{PV}$ | Output current of solar cell |

${P}_{PV}$ | Output power of solar cell |

${V}_{PV}$ | Terminal voltage of PV cell |

${I}_{bat}$ | Output current of the battery |

${V}_{bat}$ | Output voltage of the battery |

${P}_{bat}$ | Output power of the battery |

${I}_{sc}$ | Output current of the super capacitor |

${V}_{sc}$ | Output voltage of the super capacitor |

${P}_{sc}$ | Output power of the super capacitor |

${P}_{DC}$ | Output power of the DC microgrid |

T | Cell’s reference Temperature |

G | Solar irradiation |

$SOC$ | State of charge of battery |

${R}_{01}$, ${R}_{02}$ | internal resistances of DC/DC converter for the PV |

${R}_{1}$, ${R}_{2}$ | resistances of DC/DC converter for the PV |

${L}_{1}$ | inductance for the boost converter for the PV |

${C}_{1}$${C}_{2}$ | capacitance of DC/DC converter for the PV |

${V}_{C1}$, ${V}_{C2}$ | Voltage of capacitance ${C}_{1}$ and ${C}_{2}$ of DC/DC converter for the PV |

${i}_{L1}$ | current of inductance ${L}_{1}$ for the boost converter for the PV |

${R}_{03}$, ${R}_{04}$ | internal resistances of DC/DC converter for the battery |

${R}_{3}$, ${R}_{4}$ | resistances of DC/DC converter for the battery |

${L}_{2}$ | inductance for the boost converter for the battery |

${C}_{3}$${C}_{4}$ | capacitance of DC/DC converter for the battery |

${V}_{C3}$, ${V}_{C4}$ | Voltage of capacitance ${C}_{3}$ and ${C}_{4}$ of DC/DC converter for the battery |

${i}_{L2}$ | current of inductance ${L}_{1}$ for the boost converter for the battery |

${R}_{05}$, ${R}_{06}$ | internal resistances of DC/DC converter for the super capacitor |

${R}_{5}$, ${R}_{6}$ | resistances of DC/DC converter for the super capacitor |

${L}_{3}$ | inductance for the boost converter for the super capacitor |

${C}_{5}$${C}_{6}$ | capacitance of DC/DC converter for the super capacitor |

${V}_{C5}$, ${V}_{C6}$ | Voltage of capacitance ${C}_{5}$ and ${C}_{6}$ of DC/DC converter for the super capacitor |

${i}_{L3}$ | current of inductance ${L}_{1}$ for the boost converter for the super capacitor |

${C}_{dc}$ | DC-link capacitance of DC micro grid |

f | Frequency of the AC grid |

${f}_{c}$ | cutoff frequency for the filter |

${K}_{1}$… ${K}_{10}$ | positive tuning gains parameters |

${u}_{1}$, ${u}_{2}$ and ${u}_{3}$ | Duties cycle of DC/DC converter |

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**Figure 7.**Power reference components (

**top**graph: ${i}_{L2}^{\ast}$,

**bottom**graph: ${i}_{L3}^{\ast}$).

**Figure 14.**Comparison of the ${V}_{DC}$ dynamics by applying nonlinear control (in blue) and classical PI control (in red).

${R}_{01}$ | $0.001\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${R}_{1}$ | $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${C}_{1}$ | 0.1 F |

${R}_{02}$ | $0.002\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${R}_{2}$ | $0.001\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${C}_{2}$ | 0.01 F |

${R}_{03}$ | $0.001\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${R}_{3}$ | $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${C}_{3}$ | 0.1 F |

${R}_{04}$ | $0.0015\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${R}_{4}$ | $0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${C}_{4}$ | 0.01 F |

${R}_{05}$ | $0.001\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${R}_{5}$ | $0.4\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${C}_{5}$ | 0.1 F |

${R}_{06}$ | $0.0015\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${R}_{6}$ | $0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\Omega}$ | ${C}_{6}$ | 0.01 F |

${L}_{1}$ | 0.0033 H | ${L}_{2}$ | 0.0033 H | ${L}_{3}$ | 0.0033 H |

${V}_{PV}$ | 1000 V | ${P}_{n}$ | 1 MW | ${V}_{DC}$ | 1000 V |

${V}_{SC}$ | 500 V | ${V}_{BAT}$ | 500 V | ${V}_{DC}$ | 1000 V |

Steps | Peak | Peak | Settling | Settling |
---|---|---|---|---|

Overshoot | Overshoot | Time | Time | |

Linear | Nonlinear | Linear | Nonlinear | |

Control ${\mathit{V}}_{\mathbf{DC}}$ [V] | Control ${\mathit{V}}_{\mathbf{DC}}$ [V] | Control [s] | Control [s] | |

$t=5$ s | $-58$ | $-3$ | $1.2$ | $0.01$ |

$t=12.4$ s | $-45$ | $-3$ | $0.1$ | $0.01$ |

$t=24.8$ s | 48 | 4 | $0.15$ | $0.01$ |

$t=42.8$ s | 58 | 4 | 1 | $0.02$ |

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**MDPI and ACS Style**

Siad, S.B.; Iovine, A.; Damm, G.; Galai-Dol, L.; Netto, M.
Nonlinear Hierarchical Easy-to-Implement Control for DC MicroGrids. *Energies* **2022**, *15*, 969.
https://doi.org/10.3390/en15030969

**AMA Style**

Siad SB, Iovine A, Damm G, Galai-Dol L, Netto M.
Nonlinear Hierarchical Easy-to-Implement Control for DC MicroGrids. *Energies*. 2022; 15(3):969.
https://doi.org/10.3390/en15030969

**Chicago/Turabian Style**

Siad, Sabah B., Alessio Iovine, Gilney Damm, Lilia Galai-Dol, and Mariana Netto.
2022. "Nonlinear Hierarchical Easy-to-Implement Control for DC MicroGrids" *Energies* 15, no. 3: 969.
https://doi.org/10.3390/en15030969