# A Chaotic Search-Based Hybrid Optimization Technique for Automatic Load Frequency Control of a Renewable Energy Integrated Power System

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## Abstract

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## 1. Introduction

_{2}/H∞ robust control, model predictive control, linear matrix control, and fractional order (FO) control [8] have been applied by the researchers to solve the ALFC problem under dynamic conditions of a power system. For most control applications, the PID controller is extensively used due to its simplicity and easy implementation. However, with traditional tuning methods, optimizing the gains of the PID controller is one of the most important tasks to achieve improved dynamic and steady-state responses of the control system. This can be resolved by optimizing the control gains of a controller with various intelligent techniques such as fuzzy [9], artificial neural network (ANN) [10], and adaptive neuro-fuzzy inference system (ANFIS) [11]. However, these control schemes necessitate a significant amount of time to formulate the rules and train the network [12]. As a result, to overcome the problems of these control methods, many researchers have used various heuristic-based stochastic optimization techniques such as genetic algorithm (GA), gravitational search algorithm (GSA), firefly algorithm (FA), artificial bee colony, (ABC), particle swarm optimization (PSO), Cuckoo search (CS), Grey Wolf Optimization (GWO), teaching learning-based optimization (TLBO), dragon fly optimization (DFO), differential evaluation (DE), etc., to tune the control gains of the PID controller for the ALFC application [13,14]. Also, the performance of these algorithms may degrade for the power system operating with a different generation system coupled with its mechanical devices that required retuning of the controller gains as the configuration of the system operation changes. Hence, hybrid techniques are essential in tuning the gains of the controller, as they have a better exploration and exploitation search capability compared to other specified methods.

- A chaotic-based (1D mapping sequence) hybrid SSO-GSA optimization technique is implemented to optimize the parameters of the PID controller for ALFC of HPS.
- A dynamic condition of comprehensive analysis is carried out for the proposed CSSO-GSA technique under different combinations of power sources interconnected into the TAIPS.
- Sensitivity analysis is carried out under load disturbance and varying (wind and solar) power conditions in real time in order to study the robustness of the proposed CSSO-GSA technique.
- A stability analysis was performed to explore the frequency stability of the HPS model.
- A comparative analysis with a literature study is conducted to validate the performance of the proposed CSSO-GSA control technique and to exhibit its global convergence ability.

## 2. Two-Area Interconnected Power System (TAIPS) Model

#### 2.1. RE Sources

#### 2.2. Bio Power Sources

#### 2.3. Energy Storage

_{2}) and store it. In the case of power fluctuations in RE sources or low power yields (due to low wind and solar power), the stored H

_{2}can be used by the FC to generate power. The first-order transfer function for non-linear models of AE and FC is defined in the following Equations (5) and (6), respectively [35].

## 3. Control System

_{P}, K

_{I}, and K

_{D}are the controller’s proportional, integral, and derivative gains. The closed loop feedback control with tuning parameters of controllers for each area of the system is shown in Figure 1. Considering the given set value (Y

_{ref}), actual plant output (Y

_{plant}), and error signal e (t) of a closed loop feedback control system, the u (t) of controller output in each area can be expressed as follows [15]:

_{1}and ACE

_{2}are the errors of the control areas 1 and 2, respectively, and K

_{P1}/K

_{P2}, K

_{I1}/K

_{I2}, and K

_{d1}/K

_{d2}are the proportional, integral, and derivative control gains of the PID controller in areas 1 and 2, respectively. In this study, the integral time absolute error (ITAE) as given in Equation (10) is chosen as an objective function to optimize the control gains (K

_{P}, K

_{I}, and K

_{d}) of the PID controller by minimizing the ACE of the system. The ITAE objective function (J) is defined as,

#### 3.1. Control Techniques

#### 3.1.1. Sperm Swarm Optimization (SSO)

_{Sbest}(current best), and X

_{Sgbest}(global best) [39].

- Initial velocity of sperm: The sperm swarm takes a random position and its velocity in that position is determined by the pH value. The initial velocity of movement of sperm can be expressed as,

_{1}is the random number of pH value (varying between 7 and 14).

- Personal sperm best solution: It is the best solution that the sperm has achieved thus far. The rule for personal best solution can be represented as,

_{2}is the random number of pH value (varying between 7 and 14), Temp_rand

_{1}is the temperature metrics (varying between 35.1 and 38.5 degree Celsius), X

_{i}is the current location of sperm “i” at iteration “t”.

- Global best solution: It is determined on the basis of the sperm’s data that is closest to the goal at the moment (this sperm will be winner in the end). The mathematical rule for the global best solution is stated as,

_{3}is the random number of pH value (varying between 7 and 14), Temp_rand

_{2}is the temperature metrics (varying between 35.1 and 38.5 degrees Celsius), X

_{i}is the current location of sperm “i” at iteration “t”.

_{i}(t)) can be written as [19,39],

_{i}(t) = X

_{i}(t) + V

_{i}(t)

#### 3.1.2. Gravitational Search Algorithm (GSA)

_{mK}is the Euclidian distance between objects m and k.

_{m}denotes the random number (0, 1). The acceleration of mth agent is given by,

_{m}(t) is the inertia mass of mth agent.

_{m}is a random number with the range of (0, 1).

#### 3.1.3. Hybrid SSO-GSA Technique

_{i}(t) is the velocity of the agent, pH_rand

_{1}and pH_rand

_{2}are the metrics of pH, Temp_rand

_{1}is the metric of temperature, rand (n) is the random constant (varying between 0 and 1), and ac

_{i}(t) is the acceleration of agent “i” at iteration “t”. X

_{Sgbest}is the sperm global best value, and X

_{i}(t) is the current location of swarm. For this proposed control algorithm, the pH range of 7 to 14 was chosen because it is particularly suited for sperm motility and is regarded as very alkaline and non-toxic to sperm. The temperature inside the vagina ranged from 35.1 to 37.4 degrees Celsius, according to the study of several research works. This temperature can increase to 38.5 degrees Celsius in some cases due to vaginal blood pressure circulation. Based on this information, the temperature range was determined to be between 35.1 and 38.5 degrees Celsius [19,22]

_{i}(t + 1) = X

_{i}(t) + V

_{i}(t + 1)

_{i}(i = 1, 2, 3,……, N),

- Estimate the gravitational force (using Equation (17)),
- Estimate the gravitational constant (using Equation (18)),
- Estimate the resultant forces (using Equation (19)),
- Estimation of acceleration of sperm (using Equation (20)),

_{i}(t) (using Equation (23)),

_{i}, (t) (using Equation (24)).

#### 3.1.4. Chaotic-Based Hybrid SSO-GSA

_{i}(t) is the velocity of agent, pH_rand

_{1}and pH_rand

_{2}are the metrics of pH (varying between 7 and 14), Temp_rand

_{1}is the metric of temperature (varying between 35.1 and 38.5 degrees Celsius), and ac

_{i}(t) is the acceleration of agent “i” at iteration “t”. X

_{Sgbest}is the sperm global best value, and X

_{i}(t) is the current location of swarm. SSO (velocity damping factor, D–rand (0, 1), temperature–rand (35.5, 38.5), pH–rand (7, 14), swarm size–30, and total number of iterations–100) are the parameters assumed for tuning the gains of the following: the PID controller, SSO-GSA and CSSO-GSA (velocity damping factor, D–rand (0,1); temperature–rand (35.5, 38.5); pH–rand (7, 14); constant, α = 20; initial gravitational constant, G

_{0}= 1; total number of agents = 30; and number of iterations = 100). The tuned control gains of the PID controller by the proposed chaotic-based SSO-GSA technique and other techniques (SSO, GSA, and SSO-GSA) for different conditions (Case 1(a): integration of all power sources in HPS; Case 1(b): integration of thermal power and bio energy sources in HPS; Case 1(c): integration of thermal power and RE sources in HPS) are illustrated in Table 1, as below:

## 4. Results and Discussion

#### 4.1. Time Domain Analysis of HPS with Integration of Different Power Sources

_{1}) in area 1, frequency deviations (Δf

_{2}) in area 2, and tie line power variations (ΔP

_{tie}) of HPS, are illustrated in Table 2, Table 3 and Table 4, respectively. From the results of the Case 1(a) analysis, it is inferred that the steady-state performance indices were substantially improved with the chaotic-based hybrid tuned controller than with other techniques (hybrid, SSO, and GSA). Furthermore, the tie-line power tends to be more stable and smooth with the proffered chaotic-based hybrid strategy than other methods. In addition, the results of transient indices show that the proffered method (chaotic hybrid) exhibits minimal ST (s) with a reduced CE, RT, and |P-M| than other methods (hybrid, SSO, and GSA).

#### 4.2. Sensitivity Analysis

_{1}and Δf

_{2}) and tie-line power (ΔP

_{tie}) of the interconnected HPS network for these uncertain load changes are shown in the Figure 10. From the results, it is inferred that the deviations in frequency and tie-line power were settled faster with the proposed chaotic-hybrid method of tuning the controller. As per the load profile (Figure 9), when there is an increment of load (at 20 s), the results of power deviation (ΔP

_{tie}) as shown in Figure 10 clearly indicate that the proposed chaotic-hybrid method offers reduced power devaition (ΔP

_{tie}= −5.24 × 10

^{−4}p.u) than with the hybrid method (ΔP

_{tie}= −5.58 × 10

^{−4}p.u). At same time, reduced frequency deviations in area 1 and 2 (Δf

_{1}= −1.86 × 10

^{−3}Hz and Δf

_{2}= −5.55 × 10

^{−3}Hz) were observed with the proposed chaotic-hybrid method rather than with hybrid control method (Δf

_{1}= −1.99 × 10

^{−3}Hz and Δf

_{2}= −6.29 × 10

^{−3}Hz). When there is a decrement of load (at 40 s), reduced power deviation (ΔP

_{tie}= 7.21 × 10

^{−4}p.u) and frequency deviations (Δf

_{1}= 2.63 × 10

^{−3}Hz and Δf

_{2}= 7.97 × 10

^{−3}Hz) were observed with the chaotic-hybrid method rather than the power deviation (ΔP

_{tie}= 8.03 × 10

^{−4}p.u) and frequency deviations (Δf

_{1}= 2.91 × 10

^{−3}Hz and Δf

_{2}= 9.25 × 10

^{−3}Hz) of the hybrid control method. Thus, the proposed technique had a more robust and faster response time to converge into a steady-state under consecutive load disturbances in HPS.

_{tie}= −7.18 × 10

^{−4}p.u) and frequency deviations (Δf

_{1}= −1.93 × 10

^{−3}Hz and Δf

_{2}= −3.02 × 10

^{−3}Hz) were observed with the chaotic-hybrid method rather than with power deviation (ΔP

_{tie}= −8.40 × 10

^{−4}p.u) and frequency deviations (Δf

_{1}= −2.27 × 10

^{−3}Hz and Δf

_{2}= −3.52 × 10

^{−3}Hz) of the hybrid control method. Thus, from the results, it is seen that the proposed technique provides a better and smoother response with reduced undershoot and overshoot than in the hybrid technique, during the case of real time solar power variation.

_{tie}= 3.12 × 10

^{−3}p.u) and frequency deviations (Δf

_{1}= 7.85 × 10

^{−3}Hz and Δf

_{2}= 10.95 × 10

^{−3}Hz) with the chaotic-hybrid method were found to be lesser than power (ΔP

_{tie}= 3.48 × 10

^{−3}p.u) and frequency deviations (Δf

_{1}= 8.44 × 10

^{−3}Hz and Δf

_{2}= 11.57 × 10

^{−3}Hz) of the hybrid method. Thus, from the results of varying wind power analysis, it is inferred that the proposed method offers reduced frequency oscillations and steady-state error, as compared to the hybrid method.

#### 4.3. Convergence Performance

#### 4.4. Stability Analysis

_{ref}is the plant output reference and f

_{P}is the actual plant output.

#### 4.5. Comparative Analysis

_{1}and Δf

_{2}) and tie-line power variation (Δp

_{tie}) of the HPS model is portrayed in Figure 18. It is seen from the response that the proposed method quickly settles to a steady-state compared to the PSO, WHO, MFO, and SSO method of tuning the controller. Further, the controller technique has been examined in terms of % improvement in ST and ITAE, as given in Table 11. It is seen that the proffered chaotic-hybrid method outperforms significantly compared to other techniques presented.

## 5. Conclusions

_{1}and Δf

_{2}) and tie-line power variation (ΔP

_{tie}) under differnet conditions (1(a), 1(b), and 1(c)). From the results of case studies, it can be concluded that the profferred chaotic-hybrid tuned PID controller offers excellent performance in terms of enhancing the steady-state indices and minimizing the indices of ST, CE, RT, and |P-M|. Furthermore, a sensitivity analysis was carried out to validate the self adaptiveness and robustness of the proposed controller under various conditions such as load variation and uncertain conditions of wind and solar power sources. From the results of sensitivity analysis, it is inferred that the proposed chaotic-hybrid tuned PID controller outperforms the hybrid technique in terms of controlling frequency and power oscillations during the case of consecutive load disturbances and varying wind/solar power in real time conditions. Moreover, from the analysis of convergence performance, it can be seen that the proposed technique offers excellent performance, as it converges faster than the other algorithms (SSO, GSA, and hybrid SSO-GSA). In addition, results from the stability and comparative analysis clearly indicate that the proposed controller is more stable and exhibits minimum ST and ITAE in terms of frequency deviations than other methods. When compared to the existing control approach, the proposed control strategy achieves a significant improvement in ST and IATE by 60.204% and 40.055% in area 1 and 57.856% and 39.820% in area 2 of frequency control. The development of an improved hybrid control mechanism for optimizing frequency and voltage regulation in multi-area power systems is the future scope of this research endeavor.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

CSSO | Chaotic sperm swarm optimization |

GSA | Gravitational search algorithm |

HPS | Hybrid power system |

RE | Renewable energy |

ALFC | Automatic load frequency control |

AGC | Automatic generation control |

GDB | Governor dead band |

GRC | Generator rate constant |

PID | Proportional integral derivative |

TAIPS | Two-area interconnected power system |

STPS | Solar thermal power source |

WTGS | Wind turbine generation source |

K_{S} & K_{T} | Gain constants of solar collector and turbine |

T_{S} & T_{T} | Time constants of solar collector and turbine |

K_{WTG} & T_{WTG} | Gain and time constants of wind turbine |

K_{BT} & T_{BT} | Bio-turbine gain and time constants |

TCR & T_{BG} | Gas and combustion delay constants |

X_{C} & Y_{C} | Lead and lag time constants |

K_{BA} & T_{BA} | Gain and delay constants of valve actuator |

K_{AE} and T_{AE} | Gain and time constants of aqua electrolyzer |

1-Kn | Fraction of wind and solar power |

K_{FC} & T_{FC} | Gain and time constants of ruel cell |

ACE | Area control error |

D & Vi | Damping factor and velocity of sperm “I” |

X_{Sbest} | Personal best value of sperm |

X_{gbest} | Global best value of sperm |

M_{ak} | Active gravitational mass of object k |

M_{pm} | Passive gravitational mass of object m |

G (t) | Gravitational constant at time |

GO & -de | Initial value & descending coefficient |

X_{K} & X_{K+1} | Iterative sequences (current and next) |

r | Bifurcation parameter of sine map |

chaos (n) | Sine function of chaotic map |

P_{Tg} | Total power generation of sources |

P_{bd} & P_{bg} | Power generation of bio-diesel and bio-gas units |

P_{wind} & P_{solar} | Power generation of wind turbine and solar thermal |

P_{AE} | Power absorption from aqua electrolyzer |

P_{FC} | Power generation of fuel cell |

ITAE & IAE | Integral time absolute error and integral absolute error |

ITSE & ISE | Integral time square error and integral square error |

CE & ST | Control effort and settling time |

Δf_{1} & Δf_{2} | Frequency deviations in area 1 and 2 |

ΔP_{tie} | Intertie power variation |

CLTF | Closed loop transfer function |

## Appendix A

#### Appendix A.1. Thermal Reheat Power Block

_{12}= T

_{21}= 0.08674 p.u.MW/rad.Hz (Synchronizing coefficients), K

_{pi}= 200 Hz/(p.u. MW) (gain of power block), T

_{pi}= 20 s (time constant), K

_{ri}= 0.5 (reheat gain), T

_{ri}= 10 s (time constant), T

_{gi}= 0.08 s (Governor time constant), T

_{ti}= 0.3 s (Steam turbine time constant), B

_{i}= 0.425 p.u. MW/Hz (frequency bias), R

_{i}= 2.4 p.u. MW/Hz (speed regulation of Governor).

#### Appendix A.2. Load and System

_{i}= 8.33 × 10

^{−3}p.u. MW/Hz (Step load perturbation with frequency), H

_{i}= 5 s (System inertia).

#### Appendix A.3. Solar Thermal Power Block

_{S}= 1.8 and T

_{S}= 1.8 s (Solar collector gain and time constant), K

_{T}= 1 and T

_{T}= 0.3 s (Solar thermal steam turbine gain and time constant).

#### Appendix A.4. Wind Turbine Block

_{WTG}= 1, T

_{WTG}= 1.5 s (Wind turbine gain and time constant).

#### Appendix A.5. Aqua Electrolyser and Fuel Cell Block

_{AE}= 0.002 and T

_{AE}= 0.5 s (electrolyser gain and time constant), K

_{FC}= 0.01 and T

_{FC}= 4 s (Fuel cell gain and time constant), K

_{n}= 0.6 (renewable power sharing constant).

#### Appendix A.6. Bio-Gas Power Block

_{BT}= 1 and T

_{BT}= 0.2 s (bio-turbine gain and time constant), T

_{CR}= 0.01 s and T

_{BG}= 0.23 s (bio-gas and combustion delay constants), b

_{B}= 0.5 (constant of valve actuator), X

_{C}= 0.6 s and Y

_{C}= 1 s (lead and lag time constants).

#### Appendix A.7. Bio-Diesel Power Block

_{VA}= 1 and T

_{VA}= 0.5 s (valve gain and valve actuator delay constant), K

_{BE}= 1 and T

_{BE}= 0.5 s (engine gain and time constant).

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**Figure 2.**Renewable energy sources: (

**a**) bio-diesel generation; (

**b**) bio-gas generation; (

**c**) solar thermal and wind power generation.

**Figure 4.**Sperm swarm with global best (winner) [22].

Case Conditions | Control Gains | SSO | GSA | SSO-GSA | Ch SSO-GSA |
---|---|---|---|---|---|

1(a) | K_{p} | −9.912 | −6.057 | −7.00 | −6.998 |

K_{i} | −17.00 | −15.360 | −17.00 | −16.999 | |

K_{d} | −4.739 | −4.145 | −3.844 | −3.677 | |

1(b) | K_{p} | −7.822 | −6.154 | −6.942 | −7.00 |

K_{i} | −13.344 | −15.230 | −16.951 | −16.985 | |

K_{d} | −2.70 | −4.264 | −3.409 | −2.877 | |

1(c) | K_{p} | −10.638 | −7.821 | −6.993 | −6.993 |

K_{i} | −16.515 | −9.129 | −14.458 | −16.726 | |

K_{d} | −3.207 | −3.001 | −2.709 | −2.702 |

Case 1(a) | Δf_{1} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| | IAE ×10 ^{−2} | ITAE ×10 ^{−2} | ISE ×10 ^{−4} | ISTE ×10 ^{−4} | CE | |

SSO | 8.9611 | 1.5021 | 0.0117 | 1.91 | 5.87 | 1.703 | 1.466 | 0.4298 |

GSA | 7.3772 | 1.6212 | 0.0122 | 2.229 | 5.94 | 1.717 | 2.062 | 0.4737 |

Hybrid | 5.0012 | 1.4721 | 0.0116 | 1.908 | 4.967 | 1.361 | 1.492 | 0.4297 |

Chaotic-Hybrid | 4.9149 | 1.2282 | 0.009 | 1.906 | 4.961 | 1.361 | 1.365 | 0.4293 |

Case 1(a) | Δf_{2} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| | IAE ×10 ^{−2} | ITAE ×10 ^{−2} | ISE ×10 ^{−4} | ISTE ×10 ^{−4} | CE | |

SSO | 7.7883 | 1.1822 | 0.0125 | 2.098 | 5.727 | 1.51 | 1.516 | 0.4231 |

GSA | 7.3174 | 1.2758 | 0.0128 | 2.011 | 5.097 | 1.52 | 1.526 | 0.4674 |

Hybrid | 4.9754 | 1.1583 | 0.0126 | 2.011 | 5.097 | 1.52 | 1.52 | 0.4244 |

Chaotic-Hybrid | 4.9081 | 1.1469 | 0.0109 | 2.007 | 5.087 | 1.215 | 1.465 | 0.4219 |

Case 1(a) | ΔP_{tie} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| ×10 ^{−4} | IAE ×10 ^{−4} | ITAE ×10 ^{−3} | ISE ×10 ^{−7} | ISTE ×10 ^{−7} | CE | |

SSO | 7.6649 | 0.1035 | 3.926 | 5.695 | 2.145 | 1.4 | 1.201 | 0.0117 |

GSA | 4.9898 | 0.1208 | 4.1778 | 6.724 | 1.619 | 1.66 | 1.69 | 0.0131 |

Hybrid | 3.9631 | 0.1059 | 3.8427 | 5.588 | 1.245 | 1.31 | 1.19 | 0.0119 |

Chaotic-Hybrid | 3.8716 | 0.1034 | 3.8235 | 5.643 | 1.237 | 1.31 | 1.17 | 0.0112 |

Case 1(b) | Δf_{1} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| | IAE ×10 ^{−2} | ITAE ×10 ^{−2} | ISE ×10 ^{−4} | ISTE ×10 ^{−4} | CE | |

SSO | 11.8398 | 1.7903 | 0.018 | 3.714 | 12.09 | 3.894 | 4.911 | 0.7275 |

GSA | 11.1111 | 1.907 | 0.0179 | 3.556 | 11.05 | 3.88 | 5.201 | 0.6362 |

HSSO-GSA | 10.0584 | 1.4721 | 0.0116 | 2.966 | 9.047 | 3.071 | 3.565 | 0.5712 |

Chaotic-Hybrid | 9.7448 | 1.2282 | 0.0115 | 2.951 | 9.006 | 3.069 | 3.371 | 0.57 |

Case 1(b) | Δf_{2} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| | IAE ×10 ^{−2} | ITAE ×10 ^{−2} | ISE ×10 ^{−4} | ISTE ×10 ^{−4} | CE | |

SSO | 11.7259 | 1.7495 | 0.0184 | 3.862 | 12.35 | 4.207 | 5.363 | 0.721 |

GSA | 10.9367 | 1.5025 | 0.0185 | 3.697 | 11.37 | 4.172 | 5.746 | 0.6306 |

Hybrid | 9.7646 | 1.3517 | 0.0185 | 3.098 | 9.254 | 3.343 | 3.963 | 0.5652 |

Chaotic-Hybrid | 9.2949 | 1.3488 | 0.0184 | 3.017 | 9.225 | 3.323 | 3.768 | 0.5636 |

Case 1(b) | ΔP_{tie} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| ×10 ^{−4} | IAE ×10 ^{−4} | ITAE ×10 ^{−3} | ISE ×10 ^{−7} | ISTE ×10 ^{−7} | CE | |

SSO | 8.7086 | 10.88 | 5.938 | 9.416 | 2.639 | 3.355 | 3.211 | 0.0185 |

GSA | 7.4632 | 9.3586 | 5.7349 | 10.07 | 2.724 | 3.505 | 3.788 | 0.0171 |

Hybrid | 4.2268 | 0.295 | 5.2203 | 8.294 | 2.209 | 2.769 | 2.531 | 0.0156 |

Chaotic-Hybrid | 3.8279 | 0.2947 | 3.8235 | 7.823 | 2.109 | 2.724 | 2.292 | 0.0154 |

Case 1(c) | Δf_{1} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−5} | |P-M| | IAE ×10 ^{−3} | ITAE ×10 ^{−2} | ISE ×10 ^{−5} | ISTE ×10 ^{−6} | CE | |

SSO | 3.2277 | 1.1733 | 0.0079 | 4.582 | 6.411 | 2.12 | 6.93 | 0.0765 |

GSA | 3.4342 | 1.2096 | 0.0082 | 5.579 | 7.41 | 2.64 | 8.18 | 0.1096 |

Hybrid | 2.4341 | 1.79 | 0.0086 | 4.949 | 2.69 | 2.04 | 6.55 | 0.074 |

Chaotic-Hybrid | 2.2529 | 1.006 | 0.0086 | 4.357 | 2.587 | 2.03 | 5.96 | 0.065 |

Case 1(c) | Δf_{2} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−6} | |P-M| | IAE ×10 ^{−3} | ITAE ×10 ^{−2} | ISE ×10 ^{−5} | ISTE ×10 ^{−6} | CE | |

SSO | 2.9461 | 2.4659 | 0.0109 | 4.582 | 6.411 | 2.82 | 8.93 | 0.0768 |

GSA | 3.1682 | 3.6062 | 0.0108 | 5.579 | 7.041 | 2.64 | 8.58 | 0.1124 |

Hybrid | 2.4166 | 4.6685 | 0.0106 | 4.9403 | 2.638 | 2.78 | 8.52 | 0.0767 |

Chaotic-Hybrid | 2.1222 | 1.4119 | 0.0106 | 4.4899 | 2.627 | 2.79 | 8.48 | 0.0682 |

Case 1(c) | ΔP_{tie} | |||||||
---|---|---|---|---|---|---|---|---|

ST (s) | RT (s) ×10 ^{−4} | |P-M| ×10 ^{−4} | IAE ×10 ^{−4} | ITAE ×10 ^{−4} | ISE ×10 ^{−8} | ISTE ×10 ^{−8} | CE | |

SSO | 6.0225 | 1.4713 | 0.3852 | 3.59 | 3.37 | 1.93 | 7.91 | 0.0026 |

GSA | 5.9878 | 1.88883 | 0.3852 | 2.33 | 5.69 | 2.37 | 0.67 | 0.0047 |

Hybrid | 3.3736 | 1.46258 | 0.3827 | 2.32 | 2.895 | 1.80 | 1.88 | 0.0032 |

Chaotic-Hybrid | 2.8889 | 1.226 | 0.3827 | 2.3 | 2.595 | 1.6 | 4.91 | 0.0024 |

Optimization Method | Signal | ST (s) | Improved (%) in ST | ITAE (×10 ^{−2}) | Improved (%) in ITAE |
---|---|---|---|---|---|

PSO | Δf_{1} | 12.3503 | - | 8.276 | - |

Δf_{2} | 11.6461 | - | 8.453 | - | |

WHO | Δf_{1} | 8.4912 | 31.247 | 5.494 | 33.615 |

Δf_{2} | 4.7195 | 59.476 | 5.626 | 33.443 | |

MFO | Δf_{1} | 8.625 | 30.163 | 5.475 | 33.844 |

Δf_{2} | 4.7481 | 59.23 | 5.601 | 33.739 | |

SSO | Δf_{1} | 8.9611 | 27.442 | 5.87 | 29.072 |

Δf_{2} | 7.7883 | 33.125 | 5.727 | 32.248 | |

Chaotic-Hybrid | Δf_{1} | 4.9149 | 60.204 | 4.961 | 40.055 |

Δf_{2} | 4.9081 | 57.856 | 5.087 | 39.820 |

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## Share and Cite

**MDPI and ACS Style**

Sundararaju, N.; Vinayagam, A.; Veerasamy, V.; Subramaniam, G.
A Chaotic Search-Based Hybrid Optimization Technique for Automatic Load Frequency Control of a Renewable Energy Integrated Power System. *Sustainability* **2022**, *14*, 5668.
https://doi.org/10.3390/su14095668

**AMA Style**

Sundararaju N, Vinayagam A, Veerasamy V, Subramaniam G.
A Chaotic Search-Based Hybrid Optimization Technique for Automatic Load Frequency Control of a Renewable Energy Integrated Power System. *Sustainability*. 2022; 14(9):5668.
https://doi.org/10.3390/su14095668

**Chicago/Turabian Style**

Sundararaju, Nandakumar, Arangarajan Vinayagam, Veerapandiyan Veerasamy, and Gunasekaran Subramaniam.
2022. "A Chaotic Search-Based Hybrid Optimization Technique for Automatic Load Frequency Control of a Renewable Energy Integrated Power System" *Sustainability* 14, no. 9: 5668.
https://doi.org/10.3390/su14095668