# Design of Robust Integral Terminal Sliding Mode Controllers with Exponential Reaching Laws for Solar PV and BESS-Based DC Microgrids with Uncertainties

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background and Motivation

#### 1.2. Literature Review

#### 1.3. Contribution

- The solar PV unit and the BESS work in conjunction to ensure generation-load power balance while keeping the DC-bus voltage within an acceptable operating range.
- To improve the sliding phase’s steady-state tracking performance, a modified integral terminal sliding surface (ITSS) is proposed.
- A newly modified reaching law is proposed, which can confirm the finite-time convergence of the ITSS in a faster way as compared to the conventional reaching law.
- The control law is derived by satisfying the Lyapunov stability theorem. Therefore, the proposed controller surpasses existing controllers in terms of overall stability when paired with an energy management system. The BESS unit also aids the solar PV unit in fulfilling the DCMG’s goal by supplying power to DC loads independently.
- Extensive simulation results are used to illustrate the usefulness of the proposed controller under fluctuating of solar irradiation and load demand. In addition, the simulation results are compared to those of previously designed controllers.

#### 1.4. The Organization of this Paper

## 2. DCMG Configuration

- A solar PV unit based on maximum power point tracking (MPPT) that is linked to the DC-bus through a DDBC;
- A BESS, which is used to smooth the power flow within the DCMG. To control the charging and discharging current of the BESS, a BDDC is used in between it and the common DC-bus;
- DC loads, which are connected to the DC-bus directly; and
- Finally, it has a 120 V DC-bus, to which all components are connected either directly or through a power electronics converter.

## 3. DCMG Operational Modes

**Mode I**(${P}_{pv}>{P}_{load}$ and $SoC<So{C}_{max}$)

**Mode II**(${P}_{pv}<{P}_{load}$ and $SoC>So{C}_{min}$)

**Mode III**(${P}_{pv}>{P}_{load}$ and $SoC>So{C}_{max}$)

**Mode IV**(${P}_{pv}<{P}_{load}$ and $SoC<So{C}_{min}$)

## 4. Modeling of Each Microgrid Component

#### 4.1. DDBC Model for the Solar PV

#### 4.2. BDDC Model for the BESS

## 5. Proposed Controller Design

#### 5.1. Controller Design for the DDBC

#### 5.1.1. Design Steps

#### 5.1.2. Stability Analysis

#### 5.1.3. Reachability Analysis

**Proof.**

#### 5.2. Controller Design for the BDDC

## 6. Simulation Results and Discussion

## 7. Conclusions

- The simulation results clearly indicate that the designed controller can provide faster transient and dynamic stability under a variety of DCMG operational conditions as compared to the existing controllers.
- Even with substantial changes in source and load power, the developed controller can follow the reference value of the DC-bus voltage with less overshoot and shorter settling time.
- In certain additional ways, the proposed ITSMC outperforms its competitors in terms of implementation robustness, reduced complexity, and chattering-free outputs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACMG | AC microgrid |

BESS | Battery energy storage system |

BDDC | Bidirectional DC-DC converter |

BSC | Nonlinear backstepping controller |

DAB | Dual active bridge |

DPG | Diesel power generator |

DDBC | DC-DC boost converter |

DCMG | DC microgrid |

FLC | Fuzzy logic controller |

FBLC | Nonlinear feedback linearization control |

ITSMC | Integral terminal sliding mode controller |

ITSS | Modified integral terminal sliding surface |

MGs | Microgrids |

MERL | Modified exponential reaching law |

MPC | Model predictive controller |

MPPT | Maximum power point tracking |

RES | Renewable energy source |

SoC | State of charge |

SMC | Nonlinear sliding mode control |

SPV | Solar photovoltaic |

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**Figure 6.**Dynamic response of (

**a**) $SoC$ of a BESS and the control signal for (

**b**) a solar PV system and (

**c**) a BESS system.

Parameter of the BESS (Lithium Ion Battery) | |

Parameters | Values |

Terminal voltage of the BESS, ${v}_{b}$ | 48 V |

Rated capacity of the BESS, ${Q}_{b}$ | 200 Ah |

Resistance of the converter, ${r}_{b}$ | 0.025 $\Omega $ |

Inductance of the converter, ${L}_{b}$ | 0.3 mH |

Parameter of the solar PV unit | |

Generated power at MPP, ${P}_{pv}$ | 10 kW |

Resistance of the converter, ${r}_{pv}$ | 0.05 $\Omega $ |

Inductance of the converter, ${L}_{pv}$ | 0.352 mH |

Input capacitance of solar PV unit, ${C}_{pv}$ | 2200 $\mathsf{\mu}$F |

Parameter of the common DC-bus | |

Rated common DC-bus voltage, ${V}_{dc}$ | 120 V |

Common DC-bus capacitance, ${C}_{dc}$ | 1 mF |

Transient Time (s) | Settling Time (ms) | Overshoot/Undershoot (%) | ||||
---|---|---|---|---|---|---|

Proposed | Existing 1 | Existing 2 | Proposed | Existing 1 | Existing 2 | |

2.5 | 120 | 200 | 160 | 17.35 | 24.24 | 24.24 |

4.5 | 130 | 200 | 160 | 1.55 | 3.91 | 23 |

8 | 130 | 200 | 160 | 0.30 | 1.06 | 23 |

Transient Time (s) | Settling Time (ms) | Overshoot/Undershoot (%) | ||||
---|---|---|---|---|---|---|

Proposed | Existing 1 | Existing 2 | Proposed | Existing 1 | Existing 2 | |

4.5 | 15 | 205 | 170 | 0 | 16.37 | 15 |

8 | 15 | 230 | 170 | 0 | 58.46 | 35 |

Transient Time (s) | Settling Time (ms) | Overshoot/Undershoot (%) | ||||
---|---|---|---|---|---|---|

Proposed | Existing 1 | Existing 2 | Proposed | Existing 1 | Existing 2 | |

2.5 | 20 | 210 | 180 | 33.67 | 33.67 | 33.67 |

4.5 | 15 | 205 | 180 | 30 | 72.34 | 72.34 |

8 | 5 | 300 | 180 | 0 | 27.28 | 25 |

Transient Time (s) | Settling Time (ms) | Overshoot/Undershoot (%) | ||||
---|---|---|---|---|---|---|

Proposed | Existing 1 | Existing 2 | Proposed | Existing 1 | Existing 2 | |

2.5 | 3 | 300 | 200 | 0 | 0.166 | 0.166 |

4.5 | 3 | 300 | 200 | 0 | 0.166 | 0.166 |

8 | 2 | 300 | 200 | 0 | 0.375 | 0.375 |

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## Share and Cite

**MDPI and ACS Style**

Yeasmin, S.; Roy, T.K.; Ghosh, S.K.
Design of Robust Integral Terminal Sliding Mode Controllers with Exponential Reaching Laws for Solar PV and BESS-Based DC Microgrids with Uncertainties. *Sustainability* **2022**, *14*, 7802.
https://doi.org/10.3390/su14137802

**AMA Style**

Yeasmin S, Roy TK, Ghosh SK.
Design of Robust Integral Terminal Sliding Mode Controllers with Exponential Reaching Laws for Solar PV and BESS-Based DC Microgrids with Uncertainties. *Sustainability*. 2022; 14(13):7802.
https://doi.org/10.3390/su14137802

**Chicago/Turabian Style**

Yeasmin, Sabrina, Tushar Kanti Roy, and Subarto Kumar Ghosh.
2022. "Design of Robust Integral Terminal Sliding Mode Controllers with Exponential Reaching Laws for Solar PV and BESS-Based DC Microgrids with Uncertainties" *Sustainability* 14, no. 13: 7802.
https://doi.org/10.3390/su14137802