# A Proportional-Integral-One Plus Double Derivative Controller-Based Fractional-Order Kepler Optimizer for Frequency Stability in Multi-Area Power Systems with Wind Integration

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- An EKO algorithm enhanced with additional fractional-order components is presented.
- An advanced PI–(1+DD) controller, based on EKO, is introduced to enhance frequency stability in multi-area power systems with wind integration.
- The performance of the proposed controller is investigated, showcasing significant enhancements considering simultaneous step load changes in both thermal and wind areas.
- The proposed EKO algorithm is comprehensively evaluated and compared against several recent algorithms.

## 2. Two-Area Power System with WTG

#### 2.1. Wind Farm

_{GW}) is represented as follows [27]:

^{2}); “ρ” represents the air density, measured in kilograms per cubic meter (kg/m

^{3}); “Vw” stands for the wind speed, measured in meters per second (m/s); “Cp” denotes the rotor efficiency; “β” refers to the pitch angle of the blade, measured in degrees; and “Tip” stands for the tip speed ratio. These parameters are determined by the following equations:

_{P}

_{1}, K

_{P}

_{2}, K

_{P}

_{3}, T

_{P}

_{1}, T

_{P}

_{2}, and T

_{P}

_{3}correspond to the pitch controls, hydraulic pitch actuators, and data-fit pitch response’s gain and time constants, respectively.

_{PC}and K

_{f}are the gains in blade characteristic and fluid coupling, respectively.

#### 2.2. Thermal System

_{g}, T

_{g}, K

_{t}, T

_{t}, K

_{r}, T

_{r}, K

_{p}, and T

_{p}indicate the thermal plant governor, turbine, reheater, and power system’s gain and time constants, respectively. The controllers receive inputs in the form of related area control errors (ACE

_{1}and ACE

_{2}), which are defined as follows:

_{L}indicates the nominal thermal loading; ∆P

_{D}

_{1}and ∆P

_{D}

_{2}are the power demand changes in both areas; ∆P

_{w}refers to the wind power fluctuations; ∆P

_{TIE}is the tie-line power change; T

_{12}constitutes the synchronization coefficient of the tie-line; ∆f

_{l}and ∆f

_{2}are the frequency deviations (Hz) in both areas.

## 3. Proposed Controller-Based Enhanced Kepler Optimization (EKO)

#### 3.1. Proposed PI–(1+DD) Controller

_{P}and K

_{I}, on one side, and K

_{D}

_{1}, K

_{D}

_{2}alongside a constant gain of 1 on the other side. The adjustable parameters afford flexibility in controller customization. The merits of the proposed PI–(1+DD) controller include augmented transient response and heightened system stability, culminating in a decrease in peak deviation. The transfer function of the envisaged controller is delineated in Equation (15).

_{1}, ∆f

_{2}, and ∆P

_{TIE}into account within the time span of t

_{sim}. The only exception is the boundary for the filter coefficient n of the PID controller, which is set between 1 and 200, while all other candidate controller parameters are adjusted within the range of −4 to 4 [22].

#### 3.2. Developing EKO for Tunning the Proposed Controller

#### 3.2.1. Primary Addition of Fractional-Order Element

^{ζ}(x(t)) represents the fractional derivative with order ζ of the Grunwald–Letnikov formula and Ob

_{i}(t) becomes the solution vector, defining each object’s position in space at every point in time (t). Γ finds the gamma function. The following equation represents its mathematical structure for discrete-time execution:

^{1}[x(t)] denotes the variance between two-tailed occurrences.

^{1}[Ob

_{i}(t)] denotes the variance between two-tailed occurrences.

#### 3.2.2. Proposed EKO

_{j}) of each object (j) is randomly assigned.

_{Ob}, and the minimum and maximum restrictions on every controlling variable (i), accordingly, are represented by Ob

_{L}and Ob

_{U}. Rd

_{1}stands for a number that is generated at random and ranges from 0 to 1.

_{k}(t) signifies the object’s velocity at time t; A vector type is denoted by the symbol (→) on the tag for every element. q, q

_{1}, and q

_{2}are integers frequently picked at random from the spectrum [0, 1]; α designates a randomized value within the range [−1, 1]; Rd

_{1}: Rd

_{5}are numbers that are irregularly dispersed within the bounds of [0, 1]; Ob

_{a}and Ob

_{b}are the positions of two randomly planets; M

_{S}and m

_{k}represent the masses of the Sun regarding the best position of the planets and each object, correspondingly; μ(t) reflects the gravitational constant of the universe; ε is still a tiny amount used to avoid division by zero inaccuracies; R

_{k}(t) corresponds to the distance between the Sun and each object; a

_{k}represents the semimajor axis of the elliptical orbit of the planet as follows:

_{k}is an absolute value derived using a normal distribution, indicating the orbital period of each object. R

_{k}

_{−rm}(t) is the normalized Euclidian distance between the Sun and each object. Planets orbit the Sun, rotating in and out of orbit [26]. The KO concept divides the process into two phases: exploration and exploitation. Nearer solutions are used, while far-off solutions are investigated for novel possibilities.

_{k}(t + 1) represents a planet’s ultimately determined location at time t + 1; Ob

_{S}(t) represents the Sun’s position with respect to the specified best solution and acts as a marker to modify the search’s orientations. The term “Fg

_{k}” refers to the force of gravity that pulls planets towards the Sun:

_{i}is a stochastic randomized number denoting the eccentricity of a rotating planet; and Mn

_{S}and mn

_{k}stand for the normalized amounts of M

_{n}and m

_{k}[25]. Furthermore, Rn

_{k}symbolizes the R

_{k}’s normalized value, representing the Euclidian distance between the object and the Sun as follows:

_{7}and Rd

_{8}represent values selected at random according to normal distribution, and a

_{2}represents a periodic regulating factor that slowly decreases from 1 to 2 for T cycles throughout the optimizing procedure, as shown below:

_{w}is a probability parameter that governs LEA stimulation. In the range [0, 1], Rd

_{9}and Rd

_{10}denote arbitrary numbers; ϕ

_{1}and ϕ

_{2}are random values within band [−1; 1]; Ob

_{a}, Ob

_{b}and Ob

_{c}denote multiple solutions that were randomly selected. Also, σ

_{1}, β

_{1}, and β

_{2}are randomized number produced in an adaptive way as in [30]. Figure 3 displays the flowchart of the proposed EKO.

## 4. Results and Discussion

#### 4.1. Testing on RW Engineering Design Issues

#### 4.1.1. Proposed EKO versus KO: Testing on RW Engineering Design Issues

- The proposed EKO consistently achieved the lowest minimum objective in eleven RW problems, showcasing an impressive 84.61% outperformance ratio compared to the standard KO. Interestingly, both techniques exhibited similar performance in RW8 and RW32 problems.
- In terms of mean objective scores, the proposed EKO outperformed the standard KO in ten RW problems, with a notable 76.92% outperformance ratio. Conversely, the standard KO achieved lower scores in only two RW problems (RW12 and RW18) compared to the proposed EKO, while both techniques yielded comparable results in the RW32 problem.
- Analysis of the maximum objective scores indicates that the proposed EKO excelled in twelve RW problems, demonstrating a significant 92.3% outperformance ratio against the standard KO. Additionally, both methods yielded similar scores in the RW32 problem.
- Examination of the standard deviation in the attained objective scores reveals that the proposed EKO achieved the lowest values in ten RW problems, showcasing a substantial 76.92% outperformance ratio compared to the standard KO. Conversely, the standard KO attained smaller values in only two RW problems (RW12 and RW18) compared to the proposed EKO, while both techniques exhibited similar performance in RW32 problem.

#### 4.1.2. Comparative Assessment against Other Techniques

#### 4.2. Application on Multi-Area Power Systems with WTG

- Case 1: PI Controller;
- Case 2: PIDn Controller;
- Case 3: Proposed PI–(1+DD) Controller.

#### 4.2.1. Case 1: PI Controller

_{P}and K

_{I}) that were obtained by the suggested EKO, KO, PSO, DE, and SFO algorithms for each area. Comparing each algorithm against the proposed EKO, it is evident that the proposed EKO outperforms all other algorithms in terms of minimizing the ITAE. Notably, the ITAE value achieved by the EKO (3.232379096) is lower than that of DE (3.232596), PSO (3.247192601), SFO (3.234732), and KO (3.232469).

^{−5}, where DE, PSO, SFO, and KO correspondingly attain 0.00234, 0.4338, 0.08087 and 0.00217, respectively. These findings underscore the effectiveness of the proposed EKO algorithm in optimizing the PI controller’s performance across various evaluation metrics, highlighting its superiority over existing optimization methods.

#### 4.2.2. Case 2: PID Controller

#### 4.2.3. Case 3: Proposed PI–(1+DD) Controller

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CEC 2020 | Congress on Evolutionary Computation 2020 |

DBO | Dung Beetle optimizer |

DFIG | Doubly fed induction generator |

EKO | Enhanced Kepler Optimization |

GL | Grunwald–Letnikov |

I | Integral |

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

KO | Kepler Optimization |

LEA | Local Escaping Approach |

LFC | load frequency control |

LWSO | Leader White Shark Optimization |

MFO | Moth-Flame Optimizer |

PI | Proportional-integral |

PID | Proportional-integral–derivative |

PI-(1+DD) | Proportional-Integral-First-Order Double Derivative |

PSO | Particle Swarm Optimization |

RESs | Renewable energy sources |

RW | Real-world |

SFO | Sunflower Optimizer |

WSO | White Shark Optimization |

WT | Wind turbine |

## Appendix A

Function | Case Study Problem | Decision Variables | Constraints | Global Optima |
---|---|---|---|---|

RW8 | Process synthesis | 2.0 | 2.0 | 2.0 |

RW12 | Process synthesis | 7.0 | 9.0 | 2.92 |

RW13 | Process design | 5.0 | 3.0 | 26,900 |

RW15 | Weight minimization of a speed reducer | 7.0 | 11.0 | 2990.0 |

RW17 | Tension/compression spring design (Case 1) | 3.0 | 3.0 | 0.0127 |

RW18 | Pressure vessel design | 4.0 | 4.0 | 5890.0 |

RW19 | Welded beam design | 4.0 | 5.0 | 1.67 |

RW20 | Three-bar truss design | 2.0 | 3.0 | 264.0 |

RW21 | Multiple disk clutch brake design | 5.0 | 7.0 | 0.0235 |

RW28 | Rolling element bearing | 10.0 | 9.0 | 14,600.0 |

RW29 | Gas transmission compressor design | 4.0 | 1.0 | 2,960,000.0 |

RW31 | Gear train design | 4.0 | 1.0 | 0.0 |

RW32 | Himmel Lau’s function | 5.0 | 6.0 | −30,700.0 |

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**Figure 2.**Proposed controller of PI–(1+DD) [29].

**Figure 10.**Change in frequency (Area 1) regarding the DE, PSO, SFO, KO, and proposed EKO algorithms for Case 2.

**Figure 11.**Change in frequency (Area 2) regarding the DE, PSO, SFO, KO, and proposed EKO algorithms for Case 2.

**Figure 12.**Change in transfer power between areas regarding the DE, PSO, SFO, KO, and proposed EKO algorithms for Case 2.

**Figure 14.**Converging characteristics of the DE, PSO, SFO, KO, and proposed EKO algorithms for Case 3.

**Figure 15.**Change in frequency (Area 1) regarding the DE, PSO, SFO, KO and proposed EKO algorithms for Case 3.

**Figure 16.**Change in frequency (Area 2) regarding the DE, PSO, SFO, KO, and proposed EKO for Case 3.

**Figure 17.**Change in transfer power between areas regarding the DE, PSO, SFO, KO, and proposed EKO for Case 3.

**Table 1.**Summary of existing state-of-the-art compared to the presented work considering a two-area system.

Ref. | Year | Controller | Tuning the Controller Parameters: Algorithm | RES Integration |
---|---|---|---|---|

[8] | 2019 | PI | Restricted population extreme optimizer considering a four-area system | Not considered |

[11] | 2019 | TID | Salp Swarm Algorithm | Not specified |

[7] | 2023 | I, PI, PID | Hit-and-trial | Not Considered |

[18] | 2023 | Cascaded PD-PI | Enhanced version of slime mold optimizer | Not Considered |

[19] | 2023 | PID | Antlion algorithm with experimental validation via electronics environment | Not Considered |

[10] | 2020 | I | Electro-search optimizer | Solar power |

[20] | 2021 | PI | Moth flame optimizer | Wind power plant |

[21] | 2022 | Model Predictive Control | PSO | DFIG wind systems |

[22] | 2022 | Cascaded PDn-PI | Coyote optimizer | Solar and wind power integrated |

[23] | 2023 | A backpropagation-trained neural network-PI | PSO is utilized to adjust the neuron weights of the neural network to optimize the PI controller | Wind power generation |

[16] | 2023 | PID | Hybrid sparrow optimization and bald eagle algorithm | Multiple sources including wind |

[24] | 2024 | Fuzzy Logic Self-Tuning PID | Genetic algorithm | Wind, biomass, and photovoltaic power plants |

Proposed Study | PI–(1+DD) | EKO | Wind |

Engineering Design Problem | Items | WSO | DBO | MFO | LWSO | FOX | KO | Proposed EKO |
---|---|---|---|---|---|---|---|---|

RW8 | Min. | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

Av. | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Med. | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

Max. | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

STd | 2.88 × 10^{−16} | 2.44 × 10^{−16} | 2.28 × 10^{−16} | 2.44 × 10^{−16} | 7.1 × 10^{−8} | 4.432 × 10^{−12} | 1.337184 × 10^{−12} | |

Rank | 4 | 1 | 1 | 1 | 7 | 6 | 5 | |

RW12 | Min. | 2.924831 | 2.924831 | 2.924831 | 2.924831 | 2.924831 | 2.924832232 | 2.924830555 |

Av. | 2.924831 | 3.378957 | 2.96061 | 2.940521 | 2.946508 | 2.92529553 | 2.927494414 | |

Med. | 2.924831 | 3.081732 | 2.946961 | 2.924831 | 2.925031 | 2.946961824 | 2.946961113 | |

Max. | 2.924831 | 4.074353 | 3.081732 | 3.081732 | 3.082564 | 2.924844916 | 2.924830584 | |

STd | 1.43 × 10^{−7} | 0.481632 | 0.053133 | 0.048293 | 0.047529 | 0.003126717 | 0.00726176 | |

Rank | 1 | 7 | 6 | 4 | 5 | 2 | 3 | |

RW13 | Min. | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 |

Av. | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 27,135.01 | 26,887.42 | 26,887.42 | |

Med. | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | |

Max. | 26,887.42 | 26,887.42 | 26,887.42 | 26,887.42 | 28,368.22 | 26,887.42 | 26,887.42 | |

STd | 8.52 × 10^{−5} | 1.12 × 10^{−11} | 1.12 × 10^{−11} | 2.05 × 10^{−6} | 473.4105 | 1.74905 × 10^{−7} | 9.35193 × 10^{−9} | |

Rank | 6 | 1 | 1 | 5 | 7 | 4 | 3 | |

RW15 | Min. | 2994.648 | 2994.424 | 2994.424 | 2994.429 | 2995.595 | 2994.426015 | 2994.424539 |

Av. | 6 × 10^{14} | 6 × 10^{14} | 2998.352 | 5 × 10^{13} | 6.5 × 10^{14} | 2994.427404 | 2994.424768 | |

Med. | 1 × 10^{15} | 1 × 10^{15} | 2994.424 | 2994.439 | 1 × 10^{15} | 2994.429985 | 2994.425372 | |

Max. | 1 × 10^{15} | 1 × 10^{15} | 3033.702 | 1 × 10^{15} | 1 × 10^{15} | 2994.427249 | 2994.424751 | |

STd | 5.03 × 10^{14} | 5.03 × 10^{14} | 12.08925 | 2.24 × 10^{14} | 4.89 × 10^{14} | 0.000867638 | 0.000154279 | |

Rank | 5 | 6 | 3 | 4 | 7 | 2 | 1 | |

RW17 | Min. | 0.012665 | 0.012666 | 0.012674 | 0.012665 | 0.012677 | 0.01266588 | 0.012665241 |

Av. | 0.012665 | 0.012742 | 0.012827 | 0.012665 | 5 × 1013 | 0.01266772 | 0.012665815 | |

Med. | 0.012665 | 0.012719 | 0.012719 | 0.012665 | 0.012781 | 0.012672708 | 0.01266995 | |

Max. | 0.012665 | 0.012928 | 0.014283 | 0.012665 | 1 × 1015 | 0.012667121 | 0.012665365 | |

STd | 1.69 × 10^{−8} | 7.56 × 10^{−5} | 0.000356 | 8.3 × 10^{−13} | 2.24 × 10^{14} | 1.61767 × 10^{−6} | 1.10174 × 10^{−6} | |

Rank | 2 | 6 | 6 | 1 | 7 | 1 | 3 | |

RW18 | Min. | 6247.675 | 6247.673 | 6247.673 | 6247.72 | 6359.528 | 6059.74963 | 6059.714355 |

Av. | 6247.681 | 6544.502 | 6283.05 | 6247.934 | 39,927.73 | 6061.052218 | 6073.212954 | |

Med. | 6247.681 | 6382.985 | 6247.673 | 6247.853 | 15,046.06 | 6090.661389 | 6090.532868 | |

Max. | 6247.688 | 7319.001 | 6436.743 | 6248.808 | 239,304.1 | 6060.129709 | 6059.726443 | |

STd | 0.003228 | 400.6724 | 63.7404 | 0.236775 | 65,645.18 | 4.432071131 | 15.38116002 | |

Rank | 3 | 6 | 5 | 4 | 7 | 2 | 1 | |

RW19 | Min. | 1.670218 | 1.670218 | 1.670218 | 1.670218 | 1.67593 | 1.670251358 | 1.670217856 |

Av. | 1.670218 | 1.700254 | 1.670219 | 1.670218 | 1.756922 | 1.670303479 | 1.670218426 | |

Med. | 1.670218 | 1.670218 | 1.670218 | 1.670218 | 1.722726 | 1.670434518 | 1.670219985 | |

Max. | 1.670218 | 1.816712 | 1.670239 | 1.670218 | 1.994586 | 1.670301692 | 1.670218296 | |

STd | 6.2 × 10^{−8} | 0.055884 | 4.83 × 10^{−6} | 5.61 × 10^{−8} | 0.080852 | 3.12104 × 10^{−5} | 4.87676 × 10^{−7} | |

Rank | 2 | 6 | 4 | 1 | 7 | 5 | 3 | |

RW20 | Min. | 263.8958 | 263.8958 | 263.8958 | 263.8958 | 263.8958 | 263.8958434 | 263.8958434 |

Av. | 263.8958 | 263.8958 | 263.8985 | 263.8958 | 263.8959 | 263.8958434 | 263.8958434 | |

Med. | 263.8958 | 263.8958 | 263.8967 | 263.8958 | 263.8959 | 263.8958434 | 263.8958434 | |

Max. | 263.8958 | 263.8961 | 263.9237 | 263.8958 | 263.8962 | 263.8958434 | 263.8958434 | |

STd | 4.29 × 10^{−12} | 4.76 × 10^{−5} | 0.006081 | 1.3 × 10^{−14} | 7.87 × 10^{−5} | 8.72047 × 10^{−10} | 1.58449 × 10^{−11} | |

Rank | 2 | 5 | 7 | 1 | 6 | 4 | 3 | |

RW21 | Min. | 0.235242 | 0.235242 | 0.235242 | 0.235242 | 0.235242 | 0.235242458 | 0.235242458 |

Av. | 0.235242 | 0.235242 | 0.235242 | 0.235242 | 0.235243 | 0.235242458 | 0.235242458 | |

Med. | 0.235242 | 0.235242 | 0.235242 | 0.235242 | 0.235243 | 0.235242459 | 0.235242458 | |

Max. | 0.235242 | 0.235242 | 0.235242 | 0.235242 | 0.235243 | 0.235242458 | 0.235242458 | |

STd | 5.94 × 10^{−9} | 1.14 × 10^{−16} | 1.14 × 10^{−16} | 1.87 × 10^{−11} | 9.23 × 10^{−8} | 1.8999 × 10^{−10} | 6.82535 × 10^{−13} | |

Rank | 6 | 1 | 1 | 4 | 7 | 5 | 3 | |

RW28 | Min. | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 |

Av. | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | |

Med. | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | |

Max. | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | 5599.448 | |

STd | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

Rank | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

RW29 | Min. | 2,964,895 | 2,964,895 | 2,964,897 | 2,964,895 | 2,989,723 | 2,964,895.455 | 2,964,895.417 |

Av. | 2,964,895 | 3,011,451 | 2,965,099 | 2,964,895 | 3,086,914 | 2,964,895.914 | 2,964,895.417 | |

Med. | 2,964,895 | 2,964,897 | 2,964,99 | 2,964,895 | 3,096,913 | 2,964,897.792 | 2,964,895.418 | |

Max. | 2,964,895 | 3,147,942 | 2,966,063 | 2,964,895 | 3,104,538 | 2,964,895.788 | 2,964,895.417 | |

STd | 9.68 × 10^{−5} | 74,628.94 | 302.1104 | 1.85 × 10^{−5} | 29,668.78 | 0.419557334 | 0.000130887 | |

Rank | 4 | 6 | 5 | 3 | 7 | 2 | 1 | |

RW31 | Min. | 3.89 × 10^{−20} | 0 | 0 | 5.46 × 10^{−19} | 3.38 × 10^{−19} | 2.57685 × 10^{−16} | 0 |

Av. | 3.91 × 10^{−16} | 0 | 0 | 8.34 × 10^{−17} | 4.3 × 10^{−17} | 1.57017 × 10^{−12} | 7.96262 × 10^{−20} | |

Med. | 7.25 × 10^{−17} | 0 | 0 | 1.44 × 10^{−17} | 6.21 × 10^{−18} | 1.61363 × 10^{−11} | 3.9813 × 10^{−18} | |

Max. | 1.95 × 10^{−15} | 0 | 0 | 4.52 × 10^{−16} | 3.28 × 10^{−16} | 3.14661 × 10^{−13} | 6.93335 × 10^{−33} | |

STd | 5.92 × 10^{−16} | 0 | 0 | 1.33 × 10^{−16} | 8.09 × 10^{−17} | 3.14661 × 10^{−12} | 5.63041 × 10^{−19} | |

Rank | 6 | 1 | 1 | 5 | 4 | 7 | 3 | |

RW32 | Min. | 2.6393 | 2.6393 | 2.6393 | 2.6393 | 2.6393 | 2.6393 | 2.6393 |

Av. | 2.6393 | 2.6393 | 2.6393 | 2.6393 | 2.6541 | 2.6393 | 2.6393 | |

Med. | 2.6393 | 2.6393 | 2.6393 | 2.6393 | 2.6939 | 2.6393 | 2.6393 | |

Max. | 2.6393 | 2.6393 | 2.6393 | 2.6393 | 2.6513 | 2.6393 | 2.6393 | |

STd | 0.001059 | 3.73 × 10^{−12} | 3.73 × 10^{−12} | 0.002432 | 0.415896 | 1.79439 × 10^{−15} | 1.79439 × 10^{−15} | |

Rank | 5 | 3 | 7 | 6 | 5 | 1 | 1 | |

Ranks summation | 47 | 50 | 48 | 40 | 77 | 42 | 31 | |

Average rank | 3.615385 | 3.846154 | 3.692308 | 3.076923 | 5.923077 | 3.230769 | 2.384615 | |

Regarding improvement % | 34.043% | 38.000% | 35.417% | 22.500% | 59.740% | 26.190% | - | |

Final ranking | 4 | 6 | 5 | 2 | 7 | 3 | 1 |

Parameters | DE | PSO | SFO | KO | Proposed EKO | |
---|---|---|---|---|---|---|

Area 1 | K_{P} | 0.090109 | 0.072778354 | 0.098906 | 0.091053 | 0.090498113 |

K_{I} | 0.280171 | 0.283155395 | 0.276407 | 0.281548 | 0.281895243 | |

Area 2 | K_{P} | 0.081499 | 0 | 0.10512 | 0.08444 | 0.087513508 |

K_{I} | 0.457284 | 0.441958801 | 0.462502 | 0.454033 | 0.455743125 | |

ITAE | 3.232596 | 3.247192601 | 3.234732 | 3.232469 | 3.232379096 | |

Regarding improvement % | 0.0067% | 0.4562% | 0.0727% | 0.0028% | - |

DE | PSO | SFO | KO | Proposes EKO | |
---|---|---|---|---|---|

Min | 0.315377 | 0.259712 | 0.298104 | 0.257969 | 0.238278 |

Average | 0.420843 | 0.508383 | 0.469785 | 0.37775 | 0.374211 |

Median | 0.382479 | 0.373369 | 0.401883 | 0.36156 | 0.335956 |

Max | 0.560605 | 1.001491 | 0.995069 | 0.566848 | 0.549338 |

STd | 0.093496 | 0.26523 | 0.173228 | 0.084127 | 0.111095 |

Parameters | DE | PSO | SFO | KO | Proposed EKO | |
---|---|---|---|---|---|---|

Area 1 | K_{P} | 1.438757 | 1.900453 | 1.586249 | 2.219521 | 1.983181 |

K_{I} | 1.215012 | 1.603468 | 2.200458 | 1.685726 | 1.633524 | |

K_{D1} | 2.140379 | 2.45894 | 2.261332 | 2.61141 | 2.103699 | |

K_{D2} | 3.649792 | 4 | 3.888985 | 3.786046 | 3.976172 | |

Area 2 | K_{P} | 0.396676 | 0.45051 | 0.173962 | 0.609858 | 0.403955 |

K_{I} | 2.106416 | 2.558977 | 2.817782 | 2.503367 | 2.461233 | |

K_{D1} | 0.503803 | 0.658649 | 0.678034 | 0.67905 | 0.570483 | |

K_{D2} | 3.841419 | 4 | 3.602219 | 3.76936 | 3.889637 | |

ITAE | 0.315377 | 0.259712 | 0.298104 | 0.257969 | 0.238278 | |

Regarding improvement % | 24.4466% | 8.2530% | 20.0688% | 7.6331% | - |

DE | PSO | SFO | KO | Proposes EKO | |
---|---|---|---|---|---|

Min | 0.315377 | 0.259712 | 0.298104 | 0.257969 | 0.238278 |

Average | 0.420843 | 0.508383 | 0.469785 | 0.37775 | 0.374211 |

Median | 0.382479 | 0.373369 | 0.401883 | 0.36156 | 0.335956 |

Max | 0.560605 | 1.001491 | 0.995069 | 0.566848 | 0.549338 |

STd | 0.093496 | 0.26523 | 0.173228 | 0.084127 | 0.111095 |

Parameters | DE | PSO | SFO | KO | Proposed EKO | |
---|---|---|---|---|---|---|

Area 1 | K_{P} | 2.799941 | 4 | 2.580767 | 3.682332 | 3.524932 |

K_{I} | 3.851379 | 4 | 3.898463 | 3.925627 | 4 | |

K_{D1} | −2.14108 | −3.38925 | −1.08412 | 0.139779 | 0.847833 | |

K_{D2} | 2.939685 | 4 | 1.983451 | 0.582093 | −0.14473 | |

Area 2 | K_{P} | 1.444046 | 1.645876 | 1.884726 | 1.740169 | 1.66395 |

K_{I} | 3.894251 | 4 | 3.989225 | 4 | 4 | |

K_{D1} | −0.72786 | −3.61616 | −0.16341 | 2.626995 | 3.982741 | |

K_{D2} | 1.166057 | 4 | 0.514835 | −2.28613 | −3.58183 | |

ITAE | 0.071969 | 0.068583 | 0.069858 | 0.069735 | 0.068234 | |

Regarding improvement % | 5.1902% | 0.5095% | 2.3251% | 2.1530% | - |

DE | PSO | SFO | KO | Proposes EKO | |
---|---|---|---|---|---|

Min | 0.071969 | 0.068583 | 0.069858 | 0.069735 | 0.068234 |

Average | 0.111844 | 0.331536 | 0.111255 | 0.080092 | 0.070255 |

Median | 0.09293 | 0.270314 | 0.08282 | 0.077899 | 0.0696 |

Max | 0.193783 | 1.100891 | 0.652739 | 0.108116 | 0.074528 |

STd | 0.040317 | 0.329608 | 0.127764 | 0.008283 | 0.001992 |

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alqahtani, M.H.; Almutairi, S.Z.; Aljumah, A.S.; Shaheen, A.M.; Moustafa, G.; El-Fergany, A.A.
A Proportional-Integral-One Plus Double Derivative Controller-Based Fractional-Order Kepler Optimizer for Frequency Stability in Multi-Area Power Systems with Wind Integration. *Fractal Fract.* **2024**, *8*, 323.
https://doi.org/10.3390/fractalfract8060323

**AMA Style**

Alqahtani MH, Almutairi SZ, Aljumah AS, Shaheen AM, Moustafa G, El-Fergany AA.
A Proportional-Integral-One Plus Double Derivative Controller-Based Fractional-Order Kepler Optimizer for Frequency Stability in Multi-Area Power Systems with Wind Integration. *Fractal and Fractional*. 2024; 8(6):323.
https://doi.org/10.3390/fractalfract8060323

**Chicago/Turabian Style**

Alqahtani, Mohammed H., Sulaiman Z. Almutairi, Ali S. Aljumah, Abdullah M. Shaheen, Ghareeb Moustafa, and Attia A. El-Fergany.
2024. "A Proportional-Integral-One Plus Double Derivative Controller-Based Fractional-Order Kepler Optimizer for Frequency Stability in Multi-Area Power Systems with Wind Integration" *Fractal and Fractional* 8, no. 6: 323.
https://doi.org/10.3390/fractalfract8060323