# Frequency Fluctuation Mitigation in a Single-Area Power System Using LQR-Based Proportional Damping Compensator

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## Abstract

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## 1. Introduction

- A new LQR-based proportional damping compensator is proposed to mitigate the frequency fluctuation of a single-area power system;
- A hybrid multiprocessor-based processor-in-loop (PIL) technique is introduced in the paper to validate the performance of the proposed control strategy; and
- The performance of the proposed controller is evaluated both in simulation and experimental environments in terms of frequency deviation, settling time, and steady-state error for various load conditions.

## 2. Modeling of the Single-Area Power System

_{g}), governors are utilized in power systems to detect frequency biases caused by fluctuations in load and eradicate them.

_{L}, to the frequency fluctuation, Δf

_{d}, are provided below.

_{T}= 0.5 s and τ

_{g}= 0.2 s, respectively. The governor inertia constant (H) is set to 5 s and the frequency sensitive load coefficient (D) is set to 0.8. Load change varies from 0.2 p.u. to 1 p.u. throughout the simulation study.

## 3. Design of the Proposed Controller

#### 3.1. Design of the LQR

#### 3.2. Design of the Proportional Damping Compensator

_{i}), capacitor (C

_{i}), and inductor (L

_{i}). To analyze the effects of the resistor, capacitor, and inductor, three compensators are considered with different values for R

_{i}, C

_{i}, and L

_{i}[28].

#### 3.3. Integrating the LQR with Proportional Damping Compensator

## 4. Simulation Results Analysis

#### 4.1. Response of the Single-Area Power System with LQR

#### 4.2. Response of the Single-Area Power System with LQR and Damping Compensator

#### 4.3. Response of the Single-Area Power System with the Proposed LQR-based Proportional Damping Compensator

## 5. Experimental Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

τ_{g} | is the time constant of the governor; |

τ_{T} | is the time constant of the turbine; |

ΔP_{ref} | is the reference set power; |

Δf_{d} | is the frequency deviation; |

ΔP_{L} | is the load variation; |

H | is the governor inertia constant; |

D | is the frequency sensitive load coefficient; |

R | is the speed regulation of the governor; |

Abbreviations | |

LFC | Load frequency control |

AGC | Automatic generation control |

PIL | Processor-in-loop |

LQR | Linear quadratic regulator |

SG | Synchronous generator |

PID | Proportional–integral–derivative |

MPC | Model predictive control |

SMC | Sliding mode control |

ANN | Artificial neural networks |

EID | Equivalent input disturbances |

ADRC | Active disturbance rejection control |

SOPTD | Second order plus time delay |

FOPID | Fractional-order proportional–integral–derivative |

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**Figure 3.**Modeling of proposed LQR-based proportional damping compensator for a single-area power system.

**Figure 5.**Block representation of the proposed LQR-based proportional damping compensator for single-area power system.

**Figure 6.**Root locus of a single-area power system with the proposed LQR-based proportional damping compensator.

**Figure 7.**Frequency fluctuation of single-area power system with LQR for different values of gain of the controller.

**Figure 9.**Frequency deviation comparison between compensator-3 and the proposed controller for a single-area power system.

**Figure 10.**Comparative frequency deviation analysis for different controllers of a single-area power system.

**Figure 11.**Frequency deviation analysis for different loads on a single-area power system with the proposed controller.

**Figure 12.**Frequency deviation analysis for dynamic load variation on a single-area power system with the proposed controller.

**Figure 14.**Experiment frequency fluctuation response against load variation for a single-area power system.

**Figure 15.**Experimental frequency fluctuation response of different existing LQR-integrated compensators for a single-area power system.

**Figure 16.**Experimental frequency fluctuation response: open loop, PID, LQR, and the proposed controller.

**Figure 17.**Experimental frequency deviation of the proposed controller against multiple load variation in a single-area power system.

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**MDPI and ACS Style**

Das, P.; Biswas, S.P.; Mondal, S.; Islam, M.R.
Frequency Fluctuation Mitigation in a Single-Area Power System Using LQR-Based Proportional Damping Compensator. *Energies* **2023**, *16*, 4804.
https://doi.org/10.3390/en16124804

**AMA Style**

Das P, Biswas SP, Mondal S, Islam MR.
Frequency Fluctuation Mitigation in a Single-Area Power System Using LQR-Based Proportional Damping Compensator. *Energies*. 2023; 16(12):4804.
https://doi.org/10.3390/en16124804

**Chicago/Turabian Style**

Das, Pranta, Shuvra Prokash Biswas, Sudipto Mondal, and Md Rabiul Islam.
2023. "Frequency Fluctuation Mitigation in a Single-Area Power System Using LQR-Based Proportional Damping Compensator" *Energies* 16, no. 12: 4804.
https://doi.org/10.3390/en16124804