# Optimal Adaptive Gain LQR-Based Energy Management Strategy for Battery–Supercapacitor Hybrid Power System

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## Abstract

**:**

## 1. Introduction

- Robust control ensures a good energy quality provided to the load side and an extended Hybrid Energy Storage System (HESS) component lifetime;
- Combination of different methods to achieve an optimal performance.

## 2. HPS Topology Structure and Modeling

_{batt}is the battery voltage, E

_{0}is the battery constant voltage (V), K is the polarization constant (V/Ah), Q is the battery capacity (Ah), i* is the filtered battery current (A), i

_{t}is the actual battery charge (Ah), A

_{b}is the exponential zone amplitude (V), B is the exponential zone time constant inverse (Ah

^{(−1)}), and R

_{b}is the battery internal resistance (Ω). As for the polarization resistance Pol

_{res}, which is only present when charging the battery, it is expressed as follows:

_{SC}is given as follows:

_{T}is the total electric charge (Coulombs), C

_{T}is the supercapacitor module capacitance, R

_{SC}is the supercapacitor module resistance (Ω), and i

_{SC}is the supercapacitor module current (A). It is dependent on the supercapacitor’s state of charge. The supercapacitor block is given in Figure 5.

## 3. The Proposed EMS

_{bus}is the bus energy, and P

_{load}, P

_{SC}, and P

_{batt}are the load, supercapacitor, and battery powers, respectively. The supercapacitor power reference can be obtained as

**J**:

^{(−1)}B

^{T}. S is the gain matrix defined as

#### 3.1. LQR Controller

#### 3.2. SSA Optimizer

_{2}and c

_{3}are random variables [0,1]. ub and lb are the upper and lower search space limits. The follower’s movement can be modeled as

## 4. Simulation Results

_{1}and R

_{2}are the internal converter resistors (Ω); L

_{1}and L

_{2}are the converter inductors (mH); V

_{SC}ref is the SC voltage reference (V); ${V}_{bus}^{ref}$ is the (V); C

_{SC}and C

_{bus}are the SC and the bus capacitance values, respectively (F); C

_{batt}is the battery capacity (Ah).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. LQR Matrices

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**Figure 2.**Hybrid power system (HPS) topologies: (

**a**) passive topological structure; (

**b**) semi-active topology; (

**c**) another type of semi-active topology; (

**d**) fully active topology.

**Figure 3.**Studied system: (

**a**) overview of the system; (

**b**) electrical circuit of the considered system.

Parameters | Value |
---|---|

R_{1}, R_{2} (Ω) | 0.1 |

L_{1}, L_{2} (mH) | 2 |

${V}_{SC}^{ref}$ (V) | 200 |

${V}_{bus}^{ref}$ (V) | 400 |

C_{SC} (F) | 120 |

C_{bus} (µF) | 2000 |

C_{batt} (Ah) | 1500 |

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**MDPI and ACS Style**

Ferahtia, S.; Djeroui, A.; Mesbahi, T.; Houari, A.; Zeghlache, S.; Rezk, H.; Paul, T. Optimal Adaptive Gain LQR-Based Energy Management Strategy for Battery–Supercapacitor Hybrid Power System. *Energies* **2021**, *14*, 1660.
https://doi.org/10.3390/en14061660

**AMA Style**

Ferahtia S, Djeroui A, Mesbahi T, Houari A, Zeghlache S, Rezk H, Paul T. Optimal Adaptive Gain LQR-Based Energy Management Strategy for Battery–Supercapacitor Hybrid Power System. *Energies*. 2021; 14(6):1660.
https://doi.org/10.3390/en14061660

**Chicago/Turabian Style**

Ferahtia, Seydali, Ali Djeroui, Tedjani Mesbahi, Azeddine Houari, Samir Zeghlache, Hegazy Rezk, and Théophile Paul. 2021. "Optimal Adaptive Gain LQR-Based Energy Management Strategy for Battery–Supercapacitor Hybrid Power System" *Energies* 14, no. 6: 1660.
https://doi.org/10.3390/en14061660