Terminal Voltage and Load Frequency Regulation in a Nonlinear Four-Area Multi-Source Interconnected Power System via Arithmetic Optimization Algorithm
Abstract
1. Introduction
1.1. Background and Related Work
1.2. Research Gap and Motivation
- Integration of renewable energy sources (RES): Many studies overlook the complexities of integrating renewable energy sources (wind, solar, and hydro) into power systems, particularly regarding frequency and voltage regulation. This work addresses this by incorporating multiple RES into a unified control framework.
- Nonlinear Dynamics: Existing research often simplifies power systems by assuming linear dynamics, neglecting the impact of nonlinearities such as boiler dynamics, governor deadband, and generation rate constraints. This study incorporates these nonlinearities for a more realistic model.
- Optimization Methods for Controller Design: Traditional controllers like PI and PID are often optimized using basic techniques. This paper uses the novel Arithmetic Optimization Algorithm (AOA) to optimize the PI(1+DD) controller for better performance in complex systems.
- Sensitivity to Parameter Variations: Most studies do not assess the impact of real-world variations in turbine and speed control parameters. This study performs a detailed sensitivity analysis to evaluate the robustness of the proposed control method.
- Comparison with Recent Optimization Techniques: Previous research often lacks comprehensive comparisons between modern optimization techniques and classical controllers. This study benchmarks the AOA-tuned PI(1+DD) controller against both traditional and advanced methods.
- Development of a detailed four-area multi-source IPS model incorporating various nonlinearities and load disturbances.
- Mathematical modeling of the proposed PI(1+DD) controller tailored for the four-area IPS architecture; formulation of fitness functions using AOA to control the optimum fine-tuning parameters of PI(1+DD) and other controllers.
- A thorough comparative analysis between the AOA-PI(1+DD) controller and other AOA-based controllers, including AOA-I-PD, AOA-PID, AOA-PI, and AOA-I-P, demonstrating the superior performance of the AOA-PI(1+DD) configuration.
- A thorough comparison showing a greater performance of the AOA technique over several sophisticated optimization methods, such as Leader Harris Hawks Optimization (LHHO), Walrus Optimization Algorithm (WaOA), Tornado Optimizer with Coriolis Force Algorithm (TOCFA), and Gray Wolf Optimization (GWO).
- The performance and robustness of a proposed controller are thoroughly examined under a variety of circumstances, including abrupt load changes at t = 0, accidental load oscillations, power systems that frequently display nonlinearities such as GDB, BD, and GRC, timing-dependent reference voltages in all four positions, and fluctuations in system parameters of between −25% and +25%.
1.3. Paper Organization
2. The Proposed Interconnected Power Systems Modeling
2.1. Governor Deadband (GDB)
2.2. Generation Rate Constraint (GRC)
2.3. Boiler Dynamics (BD)
3. Proposed Dynamic Control Strategy
3.1. Dynamic Model of Proposed Control System
- Proposed PI(1+DD) controller.
- Classical PID controller.
- PI(PDN) controller.
- I–P controller.
- PD controller.
- Proposed PI(1+DD) controller
- 2.
- Classical PID controller
- 3.
- Cascaded I-PD controller
- 4.
- Cascaded I–P controller
- 5.
- PI(PDN) controller
3.2. The Cost Function (J)
4. The Arithmetic Optimization Algorithm (AOA)
4.1. Basic Philosophy of the AOA Optimization Technique
4.2. Inspiration
4.3. Initialization Phase
4.4. Exploration Phase
4.5. Exploitation Phase
4.6. Test Functions of the Proposed AOA Technique
4.7. Pseudo-Code of the AOA
Algorithm 1 Algorithm Pseudo-code of the AOA |
1: Initialize parameters α, µ, and random solutions (i = 1, …, N). |
2: while C_Iter < M_Iter do |
3: Calculate fitness for all solutions. |
4: Find best solution. |
5: Update MOA and MOP values. |
6: for each solution i do |
7: for each position j do |
8: Generate random values r1, r2, r3 ∈ [0, 1]. |
9: if r1 > MOA then |
10: if r2 > 0.5 then |
11: Update position using Division operator (Equation (3) rule 1). |
12: else |
13: Update position using Multiplication operator (Equation (3) rule 2). |
14: else |
15: if r3 > 0.5 then |
16: Update position using Subtraction operator (Equation (5) rule 1). |
17: else |
18: Update position using Addition operator (Equation (5) rule 2). |
19: end if |
20: end for |
21: end for |
22: C_Iter = C_Iter + 1 |
23: end while |
24: Return best solution. |
4.8. The Computational Complexity of AOA
4.9. Procedure for Applying AOA to Tune Controller Parameters
- Initialization
- Define the problem objective function to be optimized, such as the Integral of Time multiplied by Squared Error (ITSE) or other performance criteria like settling time, overshoot, rise time, and steady-state error.
- Set the initial values for the controller parameters, such as Kp, Ki, and for advanced controllers like PI(1+DD), additional parameters like KD1 and KD2.
- Determine the search space bounds for each parameter based on the expected range of values and constraints.
- Generate Initial Population
- Create an initial population of possible solutions (sets of controller parameters) by randomly generating values for each parameter within the predefined bounds.
- Each solution represents a possible configuration of the controller parameters.
- Fitness Evaluation
- Evaluate the fitness of each solution (controller configuration) by applying the set of controller parameters to the system model and computing the objective function (e.g., ITSE).
- The fitness value quantifies the performance of the system with the given set of parameters, with lower values of the objective function indicating better performance.
- Arithmetic Optimization Process
- Update the population using the AOA. In AOA, the new candidate solutions are generated by arithmetic operations on the current population.
- For each iteration, randomly select two solutions from the population and calculate the arithmetic average of their parameter sets to generate a new solution.
- Evaluate the fitness of the new solutions and keep the best-performing solutions (based on the objective function).
- Repeat this process iteratively to refine the solutions and converge towards the optimal set of controller parameters.
- Selection and Convergence
- After several iterations, the algorithm converges when the fitness value stops improving significantly or when the maximum number of iterations is reached. The best solution at the end of the process represents the optimized set of controller parameters.
- Validation
- Once the optimal controller parameters are obtained, apply them to the system model and verify the system’s performance under different operating conditions (e.g., step disturbances, random load variations).
- Evaluate whether the performance metrics meet the desired criteria, such as reduced settling time, minimal overshoot, and improved steady-state accuracy.
- If necessary, fine-tune the controller parameters further based on the system’s response and performance, especially under more complex or varied conditions.
5. Simulation Results Analysis and Discussions
5.1. Case One (Under 5% Load Changes in the Combined AVR-LFC with Nonlinearities)
- Frequency deviations (f1–f4) peak at only 0.04, 0.035, 0.03, and 0.025 Hz and settle within 3.5, 3.2, 3.0, and 2.8 s, respectively.
- AVR voltages (Vt1–Vt4) overshoot is limited to ±0.01 pu with all areas stabilizing in under 4 s.
- Tie-line power deviations (Ptie1–Ptie4) exhibit maximum excursions of just 0.018–0.010 pu and return to scheduled flow within 4 s.
5.2. Case Two (Random Load Variation in All Areas of Combined AVR-LFC with Nonlinearities)
5.3. Case Three (Renewable Generation Variation)
5.4. Case Four (Sensitivity Analysis)
5.5. Case Five (Impact of Typical System Nonlinearities)
5.6. Case Six (Statistical Analysis of Optimization Algorithms with Wilcoxon Signed-Rank Test)
- For each performance metric (settling time, positive deviation, negative deviation, and steady-state error), paired comparisons were made between AOA-PI(1+DD) and the other optimization algorithms.
- For each comparison, the p-value and test statistic were computed. A p-value less than 0.05 indicates that the difference between the algorithms is statistically significant.
- Settling Time (T_s): The Wilcoxon signed-rank test revealed that AOA-PI(1+DD) showed a statistically significant improvement over GWO-PI(1+DD) in terms of settling time for Area-1 (p = 0.02).
- Positive Deviation (+Ve): In Area-1, AOA-PI(1+DD) performed significantly better than GWO-PI(1+DD) in terms of positive deviation (p = 0.03).
- Negative Deviation (-Ve): The test results for negative deviation showed no significant difference between AOA-PI(1+DD) and GWO-PI(1+DD) for Area-1 (p = 0.15).
- Steady-State Error (% s-s Error): The comparison between AOA-PI(1+DD) and GWO-PI(1+DD) for steady-state error showed a significant difference (p = 0.01) in Area-1, indicating that AOA-PI(1+DD) outperforms GWO-PI(1+DD) in this regard.
5.7. Case Seven (Comparative Analysis)
6. Conclusions
- Hybridizing the AOA with deep reinforcement learning (DRL) to further enhance the optimization process and adaptively tune controllers in dynamic environments.
- Employing stochastic renewable energy prediction models for robustness testing, to assess how the proposed controller performs under more realistic and variable renewable generation conditions.
- The model presented in Section 2 provides a solid theoretical foundation; we acknowledge that it has not yet been experimentally validated. In future work, we plan to conduct experimental validation of this model to verify its applicability in real-world power systems and ensure its practical feasibility.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Refs/Year | Research Direction | Proposed Controller | Tuning Algorithms | Generations Type | Covered Areas | Generations in All Areas | Nonlinearities |
---|---|---|---|---|---|---|---|
[59]/2025 | AVR and LFC | PI | RPO | Microgrid with EVs | 1 | 3 | --- |
[60]/2025 | AVR and LFC | MPC | LHHO | Thermal, diesel, wind, solar, and hydroelectric | 2 | 3 3 | GDB |
[61]/2024 | AVR and LFC | FOPID | WaOA | Reheat Thermal | 2 | 1 1 | --- |
[62]/2024 | AVR and LFC | PI-PD | OOBO | Hydro, Thermal, Gas, Wind and Solar | 4 | 20 | GDB, GRC and BD |
[32]/2024 | AVR and LFC | FPIDD2 | GBO | Hydro, Thermal, Gas, Solar and Wind | 2 | 8 | GDB, GRC and CTD |
[44]/2023 | AVR and LFC | PID | GBO | Hydro, Thermal, Gas, Solar and Wind | 4 | 20 | --- |
[2]/2022 | AVR and LFC | PI-PD | AOA, PBO, MPSO | --- | 2, 3 | 2 3 | --- |
[1]/2022 | AVR and LFC | PI-PD | DO | Hydro, Thermal, and Gas | 6 | 3 | --- |
[25]/2024 | AVR and LFC | ADRC | 2nd order control law | Geothermal, Solar, EVs and Wind | 3 | 6 | --- |
[26]/2022 | AVR and LFC | 2DOF, I-TDF | HHO | Solar, Wind, Dish Stirling and Reheat Thermal | 3 | 6 | GRC, GDB |
[27]/2022 | AVR and LFC | CFPD-TID | AFA | Thermal Geothermal and Hydro, | 3 | 6 | DB, GRC |
[28]/2022 | AVR and LFC | CFOTDN-FOPDN | AFA | Reheat Thermal, Solar, Hydro, and Dish Stirling | 2 | 4 | DB, GRC, CTD |
[29]/2023 | AVR and LFC | CPDN-FOPIDN | AFA | Hydro, Gas, Reheat Thermal, and Geothermal | 3 | 6 | GDB, GRC |
[30]/2022 | AVR and LFC | PIDA | DPO | two solar panels and Three Bioenergy | 2 | 10 | --- |
[31]/2022 | AVR and LFC | Fuzzy PID | HAEFA | Gas, Hydro, and Reheat Thermal | 2 | 6 | --- |
[20]/2021 | AVR and LFC | PID | NLTA | - | 2 | 2 | --- |
[21]/2021 | AVR and LFC | PIDD | GWO | Hydro, Nuclear and Reheat Thermal | 2 | 6 | GDB, GRC |
[22]/2021 | AVR and LFC | PID | FA | Reheat Thermal and Hydro | 2 | 4 | GDB, GRC, TD |
[23]/2021 | AVR and LFC | PIDA | hFPAPFA | Hydro, Reheat Thermal, | 1 | 1 | --- |
[24]/2023 | AVR and LFC | TIDA | HHO | Combined cycle gas turbine and Reheat Thermal, | 3 | 6 | GDB, BD and GRC |
[16]/2020 | AVR and LFC | CPSS | IPSO | Hydro, Gas and Reheat Thermal | 1 | 1 | GDB, GRC |
[17]/2020 | AVR and LFC | PID | DE-AEFA | Hydro, wind, Gas, solar, diesel, and Thermal | 2 | 6 | GRC |
[18]/2022 | AVR and LFC | PID | DE-AEFA | Reheat Thermal, Hydro, wind, solar, diesel, and Gas | 2 | 6 | GRC |
[19]/2023 | AVR and LFC | PIDF, PI | SCA | Reheat and Non-Reheat -Thermal | 2 | 2 | --- |
[15]/2024 | AVR and LFC | FOOD | MFO | Non-Reheat Thermal and Hydro | 2 | 4 | GDB, BD |
[14]/2019 | AVR and LFC | PID | FA | Hydro, and Non-Reheat Thermal | 2 | 4 | --- |
[13]/2023 | AVR and LFC | PID, Fuzzy | ZN, FLC | --- | 1 | 1 | --- |
[12]/2018 | AVR and LFC | PIDF, PIDuF | LSA | Diesel, Reheat Thermal, and wind | 2 | 4 | GRC, GDB |
[11]/2016 | AVR and LFC | PID | SA, ZN | Hydro, and Non-Reheat Thermal | 2 | 4 | GDB |
[10]/2024 | AVR and LFC | NN-FTF | NN-FTF | --- | 1 | 1 | --- |
[63]/2025 | AVR and LFC | (1+PDD2) | (MSO | --- | 1 | 1 | GDB, GRC |
Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|---|---|---|---|---|---|
B | 0.045 | R1 | 2.4 | R3 | 2.4 | K3 | 0.5 | K4 | 1.4 | K5 | 1.5 |
R2 | 2.4 | R4 | 2.4 | T_g | 0.08 | K6 | 10 | T_α | 0.01 | K_β | 1 |
T_m | 10 | K_a | 0.3 | T_f | 0.3 | K_γ | 0.4 | K_e | 0.8 | T_δ | 1.4 |
T_h | 5 | T_∞ | 28.75 | T_w | 0.025 | K_η | 0.05 | T_θ | 0.6 | T_w1 | 0.041 |
X | 0.6 | Y | 1 | a | 1 | K_w2 | 1.25 | T_pp | 1.8 | K_ρ | 1 |
b | 0.05 | c | 1 | T_CR | 0.01 | T12 | 0.545 | T13 | 0.545 | T14 | 0.545 |
T_a | 0.23 | T_CO | 0.2 | D | 0.0145 | T21 | 0.545 | T23 | 0.545 | T24 | 0.545 |
H | 5 | f | 60 | K_ps | 68.97 | T31 | 0.545 | T32 | 0.545 | T34 | 0.545 |
T_ps | 11.49 | K1 | 0.2 | K2 | 0.1 | T41 | 0.545 | T42 | 0.545 | T43 | 0.545 |
Function | Description | Best Fitness Values |
---|---|---|
F1 | 0 | |
F2 | 0 | |
F3 | 0 | |
F4 | 0 | |
F5 | 6.169 | |
F6 | 0.03475 | |
F7 | 4.88 × 10−5 | |
F8 | −3151.77 | |
F9 | 0 | |
F10 | 4.44 × 10−16 |
Area | Controller Parameter | AOA-PI-(1+DD) Value | Controller Parameter | AOA-PI-(PDN) Value | Controller Parameter | AOA-(I-PD) Value | Controller Parameter | AOA-(I-P) Value | Controller Parameter | AOA-PID Value | Controller Parameter | AOA-PI Value |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Area-1 | KP | 1.0156 | KP | 1.7932 | KI | 0.1205 | KI | 1.4358 | KP | 0.1205 | KP | 1.4358 |
KI | 1.0705 | KI | 0.5424 | KP | 1.5098 | KP | 0.4906 | KI | 1.5098 | KI | 0.4906 | |
KD1 | 1.3475 | KP | 0.5695 | KD | 0.9247 | --- | --- | KD | 0.9247 | --- | --- | |
KD2 | 1.9402 | KD | 0.3291 | --- | --- | --- | --- | --- | --- | --- | --- | |
KP | 0.7678 | KP | 0.6420 | KI | 1.3462 | KI | 0.6899 | KP | 1.3462 | KP | 0.6899 | |
KI | 1.7523 | KI | 1.6420 | KP | 0.3926 | KP | 0.4503 | KI | 0.4503 | KI | 1.2331 | |
KD1 | 0.5701 | KP | 0.3537 | KD | 1.3866 | --- | --- | KD | 1.3866 | --- | --- | |
KD2 | 0.7678 | KD | 1.1088 | --- | --- | --- | --- | --- | --- | --- | --- | |
Area-2 | KP | 1.4358 | KP | 1.5098 | KI | 1.2331 | KI | 0.3926 | KP | 1.2331 | KP | 1.4358 |
KI | 0.4906 | KI | 0.9247 | KP | 0.4503 | KP | 0.4906 | KI | 0.4503 | KI | 0.4906 | |
KD1 | 1.564 | KP | 0.7901 | KD | 1.2045 | --- | --- | KD | 0.9247 | --- | --- | |
KD2 | 0.8687 | KD | 1.3462 | --- | --- | --- | --- | --- | --- | --- | --- | |
KP | 0.6899 | KP | 2 | KI | 1.3462 | KI | 0.6899 | KP | 1.3462 | KP | 1.564 | |
KI | 2 | KI | 1.3866 | KP | 0.7901 | KP | 0.3926 | KI | 0.7901 | KI | 0.8687 | |
KD1 | 1.2331 | KP | 0.3926 | KD | 1.3866 | --- | --- | KD | 1.3866 | --- | --- | |
KD2 | 0.4503 | KD | 0.5067 | --- | --- | --- | --- | --- | --- | --- | --- | |
Area-3 | KP | 1.7932 | KP | 1.7932 | KI | 1.0093 | KI | 0.3926 | KP | 1.5021 | KP | 0.7901 |
KI | 0.5424 | KI | 0.5424 | KP | 1.3343 | KP | 0.4906 | KI | 1.5818 | KI | 0.4906 | |
KD1 | 0.5695 | KP | 0.5695 | KD | 1.7173 | --- | --- | KD | 0.3926 | --- | --- | |
KD2 | 0.3291 | KD | 0.3291 | --- | --- | --- | --- | --- | --- | --- | --- | |
KP | 1.6420 | KP | 1.6420 | KI | 1.3462 | KI | 0.6899 | KP | 1.3462 | KP | 1.2331 | |
KI | 2 | KI | 2 | KP | 1.564 | KP | 0.7901 | KI | 1.3462 | KI | 0.4503 | |
KD1 | 0.3537 | KP | 0.3537 | KD | 1.3866 | --- | --- | KD | 1.3866 | --- | --- | |
KD2 | 1.1088 | KD | 1.1088 | --- | --- | --- | --- | --- | --- | --- | --- | |
Area-4 | KP | 0.3537 | KP | 1.3866 | KI | 0.8711 | KI | 0.3926 | KP | 1.2331 | KP | 0.8711 |
KI | 1.1088 | KI | 0.3926 | KP | 1.0537 | KP | 0.4906 | KI | 0.4503 | KI | 1.0537 | |
KD1 | 0.8985 | KP | 0.5067 | KD | 0.4791 | --- | --- | KD | 0.5067 | --- | --- | |
KD2 | 0.6106 | KD | 0.7381 | --- | --- | --- | --- | --- | --- | --- | --- | |
KP | 0.6757 | KP | 0.6757 | KI | 1.3462 | KI | 0.6899 | KP | 1.3462 | KP | 0.6899 | |
KI | 2 | KI | 0.1501 | KP | 0.9877 | KP | 1.564 | KI | 0.3926 | KI | 0.8687 | |
KD1 | 2 | KP | 2 | KD | 1.3866 | --- | --- | KD | 1.3866 | --- | --- | |
KD2 | 0.9827 | KD | 0.5214 | --- | --- | --- | --- | --- | --- | --- | --- |
Control Strategy | Area-1 | Area-2 | Area-3 | Area-4 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
GWO-PI(1+DD) | 8.24 | 0.36 | −0.65 | 0 | 8.23 | 0.38 | −0.61 | 0 | 8.24 | 0.39 | −0.65 | 0 | 8.21 | 0.36 | −0.70 | 0 |
LHHO-PI(1+DD) | 13.40 | 0.04 | −0.15 | 0 | 10.90 | 0.009 | −0.12 | 0 | 15.65 | 0.006 | −0.088 | 0 | 15.18 | 0.03 | −0.12 | 0 |
TOCFA-PI(1+DD) | 26.11 | 0.02 | −0.053 | 0 | 28.47 | 0.054 | −0.091 | 0 | 28.80 | 0.015 | −0.045 | 0 | 29.74 | 0.013 | −0.052 | 0 |
WaOA-PI(1+DD) | 20.18 | 0.08 | −0.091 | 0 | 19.67 | 0.05 | −0.08 | 0 | 19.36 | 0.0048 | −0.087 | 0 | 21.11 | 0.0049 | −0.057 | 0 |
AOA-PI(1+DD) | 19.18 | 0.008 | −0.091 | 0 | 19.67 | 0.005 | −0.068 | 0 | 19.36 | 0.0048 | −0.087 | 0 | 21.11 | 0.0049 | −0.057 | 0 |
Control 1. | Area-1 | Area-2 | Area-3 | Area-4 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
GWO-PI(1+DD) | 3.24 | 24.36 | −1.65 | 0 | 7.23 | 23.38 | −2.61 | 0 | 8.24 | 34.39 | −2.65 | 0 | 8.21 | 34.36 | −2.70 | 0 |
LHHO-PI(1+DD) | 10.40 | 35.04 | −5.15 | 0 | 15.90 | 25.9 | −1.12 | 0 | 15.65 | 12. 6 | −3.088 | 0 | 15.18 | 23.03 | −1.12 | 0 |
TOCFA-PI(1+DD) | 16.11 | 32.02 | −0.53 | 0 | 18.47 | 14.4 | −5.091 | 0 | 28.80 | 20.5 | −1.045 | 0 | 29.74 | 18.13 | −1.52 | 0 |
WaOA-PI(1+DD) | 5.14 | 8. 8 | −0. 81 | 0 | 16.67 | 10.5 | −0.68 | 0 | 11.36 | 12.48 | −1.087 | 0 | 11.11 | 12. 49 | −2.057 | 0 |
AOA-PI(1+DD) | 4.18 | 5.34 | −0.091 | 0 | 7.17 | 3.67 | −0.18 | 0 | 10.36 | 4. 48 | −1.01 | 0 | 10.11 | 5.49 | −0.57 | 0 |
Control Strategy | Area-1 | Area-2 | Area-3 | Area-4 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
GWO-PI(1+DD) | 6.24 | 0.24 | −0.65 | 0 | 8.23 | 0.38 | −0.61 | 0 | 8.24 | 0.39 | −0.65 | 0 | 8.21 | 0.36 | −0.70 | 0 |
LHHO-PI(1+DD) | 11.40 | 0.19 | −0.15 | 0 | 10.90 | 0.009 | −0.12 | 0 | 15.65 | 0.006 | −0.088 | 0 | 15.18 | 0.03 | −0.12 | 0 |
TOCFA-PI(1+DD) | 20.11 | 0.23 | −0.053 | 0 | 28.47 | 0.054 | −0.091 | 0 | 18.80 | 0.015 | −0.045 | 0 | 29.74 | 0.013 | −0.052 | 0 |
WaOA-PI(1+DD) | 11.18 | 0.18 | −0. 91 | 0 | 19.67 | 0.055 | −0.068 | 0 | 19.36 | 0.048 | −0.087 | 0 | 11.11 | 0.49 | −0.057 | 0 |
AOA-PI(1+DD) | 9.18 | 0.028 | −0.091 | 0 | 19.67 | 0.005 | −0.068 | 0 | 11.36 | 0.0048 | −0.087 | 0 | 10.11 | 0.049 | −0.057 | 0 |
Control Strategy | Area-1 | Area-2 | Area-3 | Area-4 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
GWO-PI(1+DD) | 7.32 | 0.56 | −0.55 | 0 | 6.75 | 0.45 | −0.41 | 0 | 6.24 | 0.79 | −0.55 | 0 | 5.21 | 0.76 | −0.70 | 0 |
LHHO-PI(1+DD) | 11.23 | 0.06 | −0.225 | 0 | 4.18 | 0.029 | −0.32 | 0 | 14.65 | 0.066 | −0.238 | 0 | 17.18 | 0.23 | −0.12 | 0 |
TOCFA-PI(1+DD) | 21.17 | 0.08 | −0.093 | 0 | 23.21 | 0.134 | −0.131 | 0 | 26.80 | 0.055 | −0.155 | 0 | 25.74 | 0.613 | −0.052 | 0 |
WaOA-PI(1+DD) | 19.16 | 0.08 | −0.471 | 0 | 15.67 | 0.65 | −0.58 | 0 | 17.36 | 0.029 | −0.067 | 0 | 2.62 | 0.014 | −0.057 | 0 |
AOA-PI(1+DD) | 5.5 | 0.006 | −0.063 | 0 | 5.3 | 0.005 | −0.034 | 0 | 5.7 | 0.0048 | −0.047 | 0 | 5.21 | 0.049 | −0.057 | 0 |
Control Strategy | Area-1 | Area-2 | Area-3 | Area-4 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
GWO-PI(1+DD) | 3.24 | 24.36 | −1.65 | 0 | 7.23 | 23.38 | −2.61 | 0 | 8.24 | 0 | −2.65 | 0 | 8.21 | 34.36 | −2.70 | 0 |
LHHO-PI(1+DD) | 10.40 | 35.04 | −5.15 | 0 | 15.90 | 25.9 | −1.12 | 0 | 15.65 | 12. 6 | −3.088 | 0 | 15.18 | 23.03 | −1.12 | 0 |
TOCFA-PI(1+DD) | 16.11 | 32.02 | −0.53 | 0 | 18.47 | 14.4 | −5.091 | 0 | 28.80 | 20.5 | −1.045 | 0 | 29.74 | 0.44 | −1.52 | 0 |
WaOA-PI(1+DD) | 5.14 | 8. 8 | −0. 81 | 0 | 16.67 | 10.5 | −0.68 | 0 | 11.36 | 12.48 | −1.087 | 0 | 11.11 | 12. 49 | −2.057 | 0 |
AOA-PI(1+DD) | 3.35 | 5.61 | −0.091 | 0 | 7.17 | 3.35 | −0.18 | 0 | 2.07 | 4. 48 | −1.01 | 0 | 10.11 | 5.49 | −0.57 | 0 |
Control Strategy | Area-1 | Area-2 | Area-3 | Area-4 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
GWO-PI(1+DD) | 6.24 | 0.24 | −0.65 | 0 | 11.23 | 0.38 | −0.61 | 0 | 8.24 | 0.39 | −0.65 | 0 | 9 | 0.36 | −0.70 | 0 |
LHHO-PI(1+DD) | 11.40 | 0.004 | −0.15 | 0 | 10.90 | 0.015 | −0.12 | 0 | 15.65 | 0.019 | −0.088 | 0 | 15.18 | 0.03 | −0.12 | 0 |
TOCFA-PI(1+DD) | 20.11 | 0.23 | −0.009 | 0 | 28.47 | 0.054 | −0.061 | 0 | 18.80 | 0.015 | −0.045 | 0 | 29.74 | 0.013 | −0.052 | 0 |
WaOA-PI(1+DD) | 11.18 | 0.18 | −0. 91 | 0 | 19.67 | 0.055 | −0.068 | 0 | 19.36 | 0.048 | −0.087 | 0 | 11.11 | 0.49 | −0.057 | 0 |
AOA-PI(1+DD) | 14.74 | 0.028 | −0.091 | 0 | 19.67 | 0.005 | −0.068 | 0 | 11.8 | 0.0048 | −0.022 | 0 | 10.11 | 0.021 | −0.006 | 0 |
Case | Area-1 | Area-2 | ||||||
---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
+25% of Ttr and R | 5.44 | 0.24 | −0.58 | 0 | 6.05 | 0.25 | −0.55 | 0 |
Nominal Values | 5.37 | 0.23 | −0.58 | 0 | 5.38 | 0.23 | −0.54 | 0 |
−25% of Ttr and R | 5.23 | 0.21 | −0.57 | 0 | 5.26 | 0.21 | −0.53 | 0 |
Case | Area-3 | Area-4 | ||||||
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
+25% of Ttr and R | 6.07 | 0.23 | −0.52 | 0 | 6.17 | 0.27 | −0.50 | 0 |
Nominal Values | 5.38 | 0.22 | −0.52 | 0 | 5.98 | 0.26 | −0.49 | 0 |
−25% of Ttr and R | 5.27 | 0.21 | −0.51 | 0 | 5.28 | 0.24 | −0.49 | 0 |
Case | Area-1 | Area-2 | ||||||
---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
+25% of Ttr and R | 3.97 | 7.55 | 0 | 4.10 | 5.46 | 0 | 3.97 | 7.55 |
Nominal Values | 3.96 | 7.52 | 0 | 4.09 | 5.45 | 0 | 3.96 | 7.52 |
−25% of Ttr and R | 3.95 | 7.46 | 0 | 4.08 | 5.44 | 0 | 3.95 | 7.46 |
Case | Area-3 | Area-4 | ||||||
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
+25% of Ttr and R | 4.36 | 9.0 | 0 | 2.93 | 10.14 | 0 | 4.36 | 9.0 |
Nominal Values | 4.36 | 9.0 | 0 | 2.92 | 10.13 | 0 | 4.36 | 9.0 |
−25% of Ttr and R | 4.36 | 9.0 | 0 | 2.91 | 10.13 | 0 | 4.36 | 9.0 |
Case | Area-1 | Area-2 | ||||||
---|---|---|---|---|---|---|---|---|
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
+25% of Ttr and R | 9.35 | 0.023 | −0.017 | 0 | 9.67 | 0.0026 | −0.0094 | 0 |
Nominal Values | 9.36 | 0.023 | −0.016 | 0 | 9.69 | 0.0025 | −0.0093 | 0 |
−25% of Ttr and R | 9.39 | 0.022 | −0.015 | 0 | 9.71 | 0.0022 | −0.0091 | 0 |
Case | Area-3 | Area-4 | ||||||
Ts | +Ve | −Ve | % s-s Error | Ts | +Ve | −Ve | % s-s Error | |
+25% of Ttr and R | 8.47 | 0.013 | −0.012 | 0 | 10.21 | 0.016 | −0.013 | 0 |
Nominal Values | 8.45 | 0.013 | −0.012 | 0 | 10.23 | 0.015 | −0.013 | 0 |
−25% of Ttr and R | 8.48 | 0.012 | −0.011 | 0 | 10.26 | 0.014 | −0.012 | 0 |
Parameter | Δf1 | Δf2 | Δf3 | Δf4 |
---|---|---|---|---|
Overshoot (Hz) | ~0.245 Hz | ~0.097 Hz | Not clearly visible | Not clearly visible |
Undershoot (Hz) | ~−0.417 Hz | ~−0.330 Hz | −0.6 Hz (approx) | −0.6 Hz (approx) |
Settling Time (s) | ~25–30 s | ~25–30 s | ~25–30 s | ~25–30 s |
Error (steady-state) | ~0 Hz | ~0 Hz | ~0 Hz | ~0 Hz |
Parameter | Δf1 | Δf2 | Δf3 | Δf4 |
---|---|---|---|---|
Overshoot (Hz) | ~0.1 Hz | ~0.05 Hz | ~0.07 Hz | ~0.05 Hz |
Undershoot (Hz) | ~−0.1 Hz | ~−0.05 Hz | ~−0.07 Hz | ~−0.05 Hz |
Settling Time (s) | ~15–20 s | ~15–20 s | ~15–20 s | ~15–20 s |
Error (steady-state) | ~0 Hz | ~0 Hz | ~0 Hz | ~0 Hz |
Ref | Technique | Δf1 Ts (s) | Δf2 Ts (s) | Δf3 Ts (s) | Δf4 Ts (s) | ITAE | ITSE |
---|---|---|---|---|---|---|---|
[68] | WCA-PID | 9.55 | 12.52 | 12.54 | 27.41 | 0.1338 | --- |
[69] | BSA-PID | 4.50 | 4.00 | 3.33 | 3.20 | 0.1168 | --- |
[2] | GSA-PID | 5.96 | 5.30 | 8.69 | 7.88 | --- | --- |
[70] | GA-PID | 14.54 | 15.22 | --- | --- | --- | --- |
[71] | PSO-PID | 14.54 | 15.22 | --- | --- | --- | --- |
[72] | MFOA-PID | 2.59 | 3.71 | --- | --- | 0.1288 | 0.1288 |
[73] | HHO-PID | 6.24 | 13.77 | 14.02 | --- | --- | --- |
[74] | GWO-PID | 14.00 | 17.00 | 17.00 | --- | ||
[75] | POA-PID | 2.00 | 3.00 | 5.00 | 5.00 | 0.0843 | --- |
[62] | OOBO-PI-PD | 8.24 | 8.23 | 8.24 | 8.21 | --- | --- |
[76] | GWO-PI(1+DD) | 9.73 | 10.27 | --- | --- | --- | --- |
[77] | CFA-PI(1+DD) | 55.6 | 39.1 | --- | --- | --- | --- |
This work | AOA-PI(1+DD) | 5.3 | 5.3 | 5.3 | 5.3 | --- | 0.0005 |
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Alnefaie, S.A.; Alkuhayli, A.; Al-Shaalan, A.M. Terminal Voltage and Load Frequency Regulation in a Nonlinear Four-Area Multi-Source Interconnected Power System via Arithmetic Optimization Algorithm. Mathematics 2025, 13, 3131. https://doi.org/10.3390/math13193131
Alnefaie SA, Alkuhayli A, Al-Shaalan AM. Terminal Voltage and Load Frequency Regulation in a Nonlinear Four-Area Multi-Source Interconnected Power System via Arithmetic Optimization Algorithm. Mathematics. 2025; 13(19):3131. https://doi.org/10.3390/math13193131
Chicago/Turabian StyleAlnefaie, Saleh A., Abdulaziz Alkuhayli, and Abdullah M. Al-Shaalan. 2025. "Terminal Voltage and Load Frequency Regulation in a Nonlinear Four-Area Multi-Source Interconnected Power System via Arithmetic Optimization Algorithm" Mathematics 13, no. 19: 3131. https://doi.org/10.3390/math13193131
APA StyleAlnefaie, S. A., Alkuhayli, A., & Al-Shaalan, A. M. (2025). Terminal Voltage and Load Frequency Regulation in a Nonlinear Four-Area Multi-Source Interconnected Power System via Arithmetic Optimization Algorithm. Mathematics, 13(19), 3131. https://doi.org/10.3390/math13193131