# Proportional-Integral-Derivative Controller Based-Artificial Rabbits Algorithm for Load Frequency Control in Multi-Area Power Systems

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(This article belongs to the Section Engineering)

## Abstract

**:**

## 1. Introduction

- ▪
- In contrast to the scenario employed by the other methods, a unique model is utilized in the ARA.
- ▪
- Given the distinctive characteristics of the ARA, this paper’s focus is on optimizing the PID controller settings for the LFC problems.
- ▪
- The simulation techniques make use of a two-area non-reheat thermal MAPS.
- ▪
- The suggested ARA is used in comparison to the particle swarm optimization (PSO), differential evolution (DE), JAYA optimizer, and self-adaptive multi-population elitist (SAMPE) JAYA optimizers in three distinct test situations with different sets of disturbances.
- ▪
- The outcomes produced by the ARA-based PID controller design are evaluated against a number of published methods.
- ▪
- These simulated results demonstrate that the developed PID controller relying on the ARA is efficient and excellent at managing load frequency management in multiple-area power grids.
- ▪
- It is reliable and produces superior outcomes when compared to other indices and instances.

## 2. Problem Formulation

#### 2.1. MAPS Model

_{1}and u

_{2}), the power change in the tie-line (∆P

_{TIE}), and the power change in the demands (∆P

_{D}

_{1}and ∆P

_{D}

_{2}). The outputs are the area control errors (ACE

_{1}and ACE

_{2}), and the deviations in system frequencies (∆f

_{1}and ∆f

_{2}) [15].

_{PIDn}) is mathematically denoted as:

_{1}and ACE

_{2}) that come from:

#### 2.2. Objective Function

- The frequency variation should recover to zero once the load is altered.
- The integral of the frequency error must have the lowest feasible value.
- The control loop needs to be sufficiently stable.
- Each region shall carry out its load under normal circumstances, and after a load disruption, the power exchange between areas should be quickly restored to its planned value. A time-domain goal function is modified using integral criteria to determine the best PID controller gains as follows:

_{2}may be easily advanced to take into account reducing the peak-overshoots of the frequency variations for both regions and in the tie-line power transfer. This evolution of the fitness form benefits from obtaining a sufficient damping ratio to provide a certain level of stability [15]. The limits of the controller parameter settings are the problem limitations. As a result, the design issue might be described as the subsequent optimization issue problem.

For PID controller: Kp

_{min}≤ Kp ≤ Kp

_{max},

Ki

_{min}≤ Ki ≤ Ki

_{max},

Kd

_{min}≤ Kd ≤ Kd

_{max},

n

_{min}≤ n ≤ n

_{max}

_{1}or J

_{2}. Every controller parameter’s lowest and maximum values are denoted by the subscripts “min” and “max.” The relative quantities are determined to be 0 and 3, and the border of the filter factor n is selected to be between 0 and 500 [15].

## 3. Mathematical Model of the Proposed Artificial Rabbits Algorithm (ARA)

_{i}indicates the position of the rabbit, lb and ub refer to the lower and upper limits of the considered variables, n and dim are, respectively, the population size and the number of control variables of the problem.

#### 3.1. Detour Foraging

_{i}and Y

_{i}are the new and old positions of rabbit (i); SND is governed by the standard normal distribution; L is the traveling distance, which reflects the movement rate; round and randperm are functions for rounding the value into the closest integer and randomizing permutation of the integers from 1 to dim; $v$

_{1}, $v$

_{2}and $v$

_{3}indicate three randomized values within range [0, 1], and T

_{max}indicates the highest number of iterations.

#### 3.2. Randomized Hiding

_{i,j}is the jth burrow of the rabbit (i); H is the concealing parameter and gradually decreases from 1 to 1/T

_{max}with a random perturbation over the course of iterations; $v$

_{4}is a randomized value within range [0, 1]. Based on this characteristic, those burrows are first created in a rabbit’s larger neighborhood. This neighborhood shrinks as the iterations grow more numerous.

#### 3.3. Energy Shrink (Switch from Exploration to Exploitation)

## 4. Simulation Results and Discussions

- Case study 1: Step load perturbation in area 1 only.
- Case study 2: Step load perturbation in area 2 only.
- Case study 3: Step load perturbation in area 1 and area 2.

#### 4.1. Simulation of Case Study 1

_{p}, K

_{i}, K

_{d}, and n throughout every area. In comparison to PSO, DE, JAYA, and SAMPE-JAYA, which had minimum ITAE values of 0.0769, 0.0781, 0.077, and 0.0769, respectively, the proposed ARA achieved a minimum value of 0.0754. As shown, the proposed ARA can acquire significant improvements in the ITAE value of 1.949, 3.455, 2.077, and 1.949 %, respectively, compared to PSO, DE, JAYA and SAMPE-JAYA.

#### 4.2. Simulation of Case Study 2

_{p}, K

_{i}, K

_{d}, and n throughout every area. As shown, the proposed ARA achieved a minimum value of 0.0754. On the other side, PSO, DE, JAYA, and SAMPE-JAYA records minimum ITAE values of 0.0816, 0.082, 0.078, and 0.077, respectively. As shown, the proposed ARA can acquire significant improvements in the ITAE value of 7.587, 8.038, 3.322, and 2.066%, respectively, compared to PSO, DE, JAYA, and SAMPE-JAYA.

#### 4.3. Simulation of Case Study 3

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

ACE | Area control error |

R | Governor speed droop characteristics |

B | Frequency bias factor |

u | Governor control inputs |

T_{g} | Governor time constants (seconds) |

∆P_{g} | Changes in valve position of the governor (per unit (p.u.)) |

T_{t} | Turbine time constants (seconds) |

∆P_{t} | Power changes in turbine output (p.u.) |

k_{p} | Gain |

T_{p} | Power system time constants (seconds) |

∆P_{D} | Power demand changes |

∆P_{TIE} | Tie-line power change (p.u.), |

T_{12} | Synchronization coefficient between areas 1 and 2 |

∆f | Power system frequency change (Hz), |

P_{Rg} | MW capacity of area g (g = 1, 2) |

a_{12} | Constant |

K_{p}, K_{i} and K_{d} | Gains of PID controller of proportional, integral and derivative, respectively |

ITAE | Integral time-multiplied absolute value of the error |

t_{sim} | Simulation time |

J | Objective function to be considered |

## Appendix A

_{R}= 2000 MW (rating), P

_{L}= 1000 MW (nominal loading); f = 60 Hz; R

_{1}= R

_{2}= 2.4 Hz/pu; B

_{1}= B

_{2}= 0.045 pu MW/Hz; T

_{g}

_{1}= T

_{g}

_{2}= 0.08 s; T

_{t}

_{1}= T

_{t}

_{2}= 0.3 s; K

_{P}

_{1}= K

_{P}

_{2}= 120 Hz/pu MW; T

_{P}

_{1}= T

_{P}

_{2}= 20s; T

_{12}= 0.545 pu; a

_{12}= −1.

## Appendix B

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**Figure 4.**Dynamic responses for SAMPE-JAYA, JAYA, DE, PSO and the proposed ARA under case study 1. (

**a**) Change in frequency in area 1. (

**b**) Change in frequency in area 2. (

**c**) Tie-line power deviation.

**Figure 5.**Statistical indices for SAMPE-JAYA, JAYA, DE, PSO and the proposed ARA under case study 1.

**Figure 6.**Dynamic responses for SAMPE-JAYA, JAYA, DE, PSO and the proposed ARA under case study 2. (

**a**) Change in frequency in area 1. (

**b**) Change in frequency in area 2. (

**c**) Tie-line power deviation.

**Figure 7.**Statistical indices for SAMPE-JAYA, JAYA, DE, PSO and the proposed ARA under case study 2.

**Figure 8.**Dynamic responses for SAMPE-JAYA, JAYA, DE, PSO and the proposed ARA under case study 3. (

**a**) Change in frequency in area 1. (

**b**) Change in frequency in area 2. (

**c**) Tie-line power deviation.

**Figure 9.**Statistical indices for SAMPE-JAYA, JAYA, DE, PSO and the proposed ARA under case study 3.

Algorithm | SAMPE-JAYA | JAYA | DE | PSO | Proposed ARA | |
---|---|---|---|---|---|---|

Controller parameters | K_{P1} | 1.8066 | 1.8394 | 1.7101 | 1.9602 | 1.875133 |

K_{i1} | 2.9895 | 3 | 3 | 3 | 2.997462 | |

K_{d1} | 0.5654 | 0.5806 | 0.5284 | 0.6083 | 0.578319 | |

n_{1} | 88.111 | 72.985 | 372.86 | 385.58 | 115.132 | |

K_{P2} | 2.1364 | 1.4843 | 2.8899 | 2.9756 | 2.979045 | |

K_{i2} | 0.4187 | 0.4306 | 1.0332 | 1.5561 | 0.729651 | |

K_{d2} | 1.7534 | 1.0095 | 1.9478 | 2.6638 | 1.050161 | |

n_{2} | 146.04 | 425.37 | 332.02 | 497.92 | 15.40281 | |

ITAE Value | 0.0769 | 0.077 | 0.0781 | 0.0769 | 0.075401 | |

ITAE Improvement % compared to the proposed ARA | 1.949 | 2.077 | 3.455 | 1.949 | - |

Controller | Optimization Technique | Reference | Settling Times(s) | Objective Value | ||
---|---|---|---|---|---|---|

∆P_{TIE} | ∆F_{2} | ∆F_{1} | ITAE | |||

PI | Conventional | [12] | 28.27 | 45.01 | 45 | 3.5795 |

PI | GA | [12] | 9.37 | 11.39 | 10.59 | 2.7475 |

PI | BFOA | [12] | 6.35 | 7.09 | 5.52 | 1.8379 |

PI | DE | [7] | 5.75 | 8.16 | 8.96 | 0.9911 |

PI | PSO | [8] | 5.0 | 7.82 | 7.37 | 1.2142 |

PI | BFOA-PSO | [8] | 5.73 | 7.65 | 7.39 | 1.1865 |

PI | FA | [12] | 5.62 | 7.22 | 7.11 | 0.8695 |

PID | FA | [12] | 4.78 | 5.49 | 4.25 | 0.4714 |

PID | Proposed ARA | Presented | 3.059294 | 2.901341 | 2.195834 | 0.075401 |

Algorithm | SAMPE-JAYA | JAYA | DE | PSO | Proposed ARA | |
---|---|---|---|---|---|---|

Controller parameters | K_{P1} | 2.7421 | 2.017 | 2.5822 | 2.2948 | 2.794512 |

K_{i1} | 0.3762 | 1.9978 | 0.4092 | 1.1983 | 0.509808 | |

K_{d1} | 3 | 2.403 | 1.2792 | 1.0794 | 0.995084 | |

n_{1} | 307.57 | 57.928 | 409.97 | 500 | 15.03582 | |

K_{P2} | 1.9881 | 1.9102 | 2.4546 | 2.2328 | 1.865161 | |

K_{i2} | 2.9963 | 3 | 3 | 2.9705 | 2.997782 | |

K_{d2} | 0.5997 | 0.6106 | 0.6618 | 0.6757 | 0.576527 | |

n_{2} | 500 | 372.48 | 410.78 | 136.77 | 166.8088 | |

ITAE Value | 0.077 | 0.078 | 0.082 | 0.0816 | 0.075409 | |

ITAE Improvement % compared to the proposed ARA | 2.066566 | 3.322123 | 8.038117 | 7.587323 | - |

Algorithm | SAMPE-JAYA | JAYA | DE | PSO | Proposed ARA | |
---|---|---|---|---|---|---|

Controller parameters | K_{P1} | 1.8022 | 2.1068 | 1.5717 | 1.7764 | 1.820495 |

K_{i1} | 2.6685 | 2.9333 | 2.7589 | 3 | 2.994891 | |

K_{d1} | 0.6436 | 1.4944 | 0.3809 | 0.9385 | 0.577278 | |

n_{1} | 138.3 | 85.067 | 147.82 | 392.86 | 67.01898 | |

K_{P2} | 1.7795 | 1.1472 | 2.3342 | 1.562 | 1.565437 | |

K_{i2} | 2.859 | 2.9487 | 2.9998 | 2.5835 | 2.989529 | |

K_{d2} | 0.4627 | 0.4789 | 0.9962 | 0.6575 | 0.462035 | |

n_{2} | 340.75 | 489.58 | 332.55 | 500 | 448.9027 | |

ITAE Value | 0.1726 | 0.2272 | 0.2021 | 0.2354 | 0.146308 | |

ITAE Improvement % compared to the proposed ARA | 17.97 | 55.29 | 38.13 | 60.89 | - |

Change | SAMPE-JAYA | JAYA | DE | PSO | Proposed ARA |
---|---|---|---|---|---|

∆F_{1} | 2.4343 | 3.6834 | 3.0722 | 2.2596 | 2.2967 |

∆F_{2} | 1.8693 | 1.6293 | 2.0993 | 3.1665 | 0.9804 |

∆P_{TIE} | 3.1625 | 3.2571 | 3.4722 | 3.5186 | 2.2305 |

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## Share and Cite

**MDPI and ACS Style**

El-Sehiemy, R.; Shaheen, A.; Ginidi, A.; Al-Gahtani, S.F.
Proportional-Integral-Derivative Controller Based-Artificial Rabbits Algorithm for Load Frequency Control in Multi-Area Power Systems. *Fractal Fract.* **2023**, *7*, 97.
https://doi.org/10.3390/fractalfract7010097

**AMA Style**

El-Sehiemy R, Shaheen A, Ginidi A, Al-Gahtani SF.
Proportional-Integral-Derivative Controller Based-Artificial Rabbits Algorithm for Load Frequency Control in Multi-Area Power Systems. *Fractal and Fractional*. 2023; 7(1):97.
https://doi.org/10.3390/fractalfract7010097

**Chicago/Turabian Style**

El-Sehiemy, Ragab, Abdullah Shaheen, Ahmed Ginidi, and Saad F. Al-Gahtani.
2023. "Proportional-Integral-Derivative Controller Based-Artificial Rabbits Algorithm for Load Frequency Control in Multi-Area Power Systems" *Fractal and Fractional* 7, no. 1: 97.
https://doi.org/10.3390/fractalfract7010097