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Ensuring frequency stability in interconnected power systems is challenging due to continuous load variations and fluctuations in tie-line power. This study focuses on designing and optimizing controllers for Automatic Generation Control (AGC) of a two-area power system to achieve zero-frequency deviation under dynamic load conditions. Five different controllers—PID (Proportional-Integral-Derivative), PIDn, FOPID (Fractional Order PID), TID (Tilt-Integral-Derivative), and PIDA—were tested. Their parameters were optimized using seven advanced metaheuristic algorithms: Artificial Rabbit Optimization, Chernobyl Disaster Optimizer (CDO), Modified Chernobyl Disaster Optimizer (mCDO), Golden Jackal Optimization, Honey Badger Algorithm, Mont-Flame Optimization, and Spider Wasp Optimizer. A total of 35 simulation studies were conducted, and performance was evaluated using the Integral of Time-Weighted Absolute Error (ITAE) metric Among the tested controllers, the FOPID-mCDO combination achieved the lowest ITAE value (0.320684), a settling time of 3.6 s, and minimal overshoot (0.0083 Hz) and undershoot (− 0.1480 Hz). Compared to conventional PID controllers, this configuration reduced settling time by 10% and improved frequency stability under dynamic load variations. The proposed mCDO algorithm, which integrates neighborhood–global and wandering search strategies to enhance the exploration–exploitation balance of the original CDO, outperformed the standard CDO by enabling faster convergence and more precise parameter tuning. The findings indicate that the FOPID-mCDO combination is a promising approach for automatic generation control in multi-area power systems.
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1 Introduction
Optimization refers to the process of finding values that provide the best solution in a predefined space. Depending on the structure of the problem and the values being sought, the goal may be to find either the minimum or maximum value. Over time, many optimization algorithms have been developed, and in recent years, the number and variety of these algorithms have rapidly increased. In general, optimization algorithms are divided into two main categories: stochastic and deterministic algorithms, each of which has its own subcategories. Stochastic optimization algorithms are further divided into heuristic and metaheuristic algorithms. Metaheuristic optimization algorithms can be classified into three groups: evolutionary-based, swarm intelligence-based, and physics-chemistry-based algorithms. In addition, other subgroups and approaches can be used to classify these algorithms in various ways [1‐3].
Optimization algorithms are continuously evolving and are widely used, particularly in solving technical problems. These algorithms, inspired by nature and modeling the behaviors, hunting strategies, and lifestyles of living organisms, are extensively applied in fields such as electronics, mechanics, the natural sciences, economics, and strategic planning to find the most economical, fastest, easiest, or highest-value solutions for many uncertain and hard-to-predict problems [4, 5].
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One of the important problems that needs to be solved in control and power systems using optimization algorithms is determining the optimal controller parameters for load/frequency control in multi-area power systems. In multi-area power systems, continuous load variation causes fluctuations in both the voltage amplitude and the frequency. To mitigate these fluctuations, continuous monitoring and regulation are performed through control systems [6].
Power systems generally consist of single-area, two-area, four-area, or even more complex multi-area systems. In the context of a country, power systems are composed of hundreds or even thousands of different types of power generators that work together [7]. In such systems, parameters such as fluctuating power demands, intermittent renewable energy production, and cost-effective power generation are factored into the calculations. When fault conditions are also considered, maintaining system balance, stabilizing voltage amplitude, and keeping the frequency at the desired level becomes a significant challenge. Therefore, simulation studies of multi-area power systems are being conducted, and controllers capable of better managing these systems are being designed [8].
Figure 1 shows a single-line diagram of the simulated system. The system consists of two different power generation units, making it a two-area system. Each power system generates electricity according to its own conditions and power values. The generated voltage is then supply into the transmission line and delivered to consumers via the grid. Additionally, each power unit has its own local loads, and each power system produces voltage within a specified tolerance range. Frequency control is performed between the loads and power plants, as well as between the power plants themselves. In addition, each system is controlled independently. In this study, different types of controllers were examined, and their parameters were optimized using metaheuristic algorithms to enhance system performance.
Modern power systems consist of a mixture of conventional base-load power plants, such as coal, lignite, and nuclear units, as well as discontinuous renewable energy sources including wind, solar, and biomass. The increasing penetration of such variable generation makes it essential to maintain a continuous balance between generation and load in order to preserve system frequency at its nominal value. Any imbalance directly leads to frequency deviations, which may threaten system stability and reliability. Therefore, effective load–frequency control (LFC) is required to ensure stable operation of interconnected power systems. In this study, the LFC problem is addressed by optimally tuning controller parameters to maintain system balance and improve dynamic frequency performance under varying load conditions [7, 8].
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1.1 Literature review
There are numerous studies on TAPS in the literature. Among these, the most relevant works have been carefully examined to better understand the topic. The key aspects generally discussed in these studies include two-area power systems (TAPS), controllers, optimization algorithms, and error indices or test parameters. Additionally, the results obtained and comparisons made within the studies are discussed.
In this study, the parameters of PID controllers used to control a TAPS were tuned using both the Genetic Algorithm (GA) and the harmony search algorithm (HSA). The effectiveness of both algorithms was compared, and the Integral of the Time-Weighted Absolute Error (ITAE) was selected as the criterion for testing the controller parameters. It was found that HSA provided more efficient results [8].
In another study examining both single-area and TAPS, Artificial Neural Network (ANN), PID controllers, and Fuzzy Logic Controllers (FLC) were employed to control each area. The comparison of the controllers was based on overshoot, settling time, and error values, which were presented in tables. It was observed that the ANN produced better results than the other controllers [9]. Similarly, in other studies, frequency control was performed using the FLC alone in one case, and both PI and FLC in another for TAPS, with FLC yielding better results in both cases [10, 11].
In a study involving a TAPS with both photovoltaic and thermal energy generation, the ITAE index was used to tune the PI controller parameters using the RIME algorithm, Black-Widow Optimization Algorithm (BWOA), Salp Swarm Algorithm (SSA), Shuffled Frog Leaping Algorithm (SFLA), Firefly Algorithm (FA), and GA. Overshoot, undershoot, settling time, and error values were compared both graphically and in tables, and it was found that the RIME-tuned PI controller provided the best results for these metrics [12].
Another study optimized and compared PID controllers for frequency regulation in a TAPS using Bull-Lion Optimization (BLO), Gravitational Search Algorithm (GSA), FA, and Particle Swarm Optimization (PSO). The ITAE value was used to compare the optimization algorithms. The study evaluated the performance of PID controllers in terms of settling time, rise time, and overshoot, with BLO outperforming the other techniques by achieving the lowest ITAE value [13].
In a study involving a TAPS with EV batteries, the PID controller parameters (KP, KI, and KD) were optimized using the Honey Badger Algorithm (HBA). The results of the HBA algorithm were compared with those obtained using Ziegler-Nichols and GA. The Integral Absolute Error (IAE) was used as the comparison criterion, and comparisons were made based on minimum, maximum, and average values. It was found that HBA produced better results than the other techniques [14].
In another study, a proportional integral derivative controller with derivative filter (PIn) was used to control a multi-area power system. The controller parameters were optimized using the Sine Cosine Algorithm (SCA), SSA, Symbiotic Organisms Search (SOS), Nelder-Mead Simplex (NMS), Luus-Jaakkola (LJ) algorithms, and the Jaya Optimization Algorithm (JOA). The Integral of The Time Square Error (ITSE) was used to evaluate the algorithms, and JOA provided the best controller parameters according to the ITSE value [15].
In another study, TID and FOTID controllers were employed together for LFC. The controller parameters were tuned using the SSA. Various versions of TID and FOTID controllers were tested across different multi-area power systems, with settling time, undershoot, and overshoot compared. The evaluation was based on the ITAE, IAE, and ITSE criteria [16].
A study on a TAPS using an FOI-PD controller optimized the controller parameters using Improved-Fitness Dependent Optimizer (I-FDO), FA, Fitness Dependent Optimizer (FDO), and Teaching Learning-Based Optimization (TLBO). The algorithms were compared in terms of overshoot, settling time, and undershoot, with I-FDO providing the best results. This study also includes a comprehensive table summarizing different power system types, controllers, and optimization algorithms used [17].
Although these studies have contributed to the field, a systematic comparison of multiple controllers optimized using different metaheuristic algorithms under identical conditions is still lacking. Additionally, newly developed optimization algorithms, such as the Chernobyl Disaster Optimizer (CDO) and its modified version (mCDO), have not been extensively tested for TAPS control. This study aims to fill these research gaps and provide a more robust, comparative analysis of various controllers and optimization techniques.
1.2 Research gap and motivation
The LFC in two-area power systems (TAPS) has been extensively studied using various controllers and optimization techniques. However, systematic comparsions of multiple controllers and optimization algorithms under identical operating conditions are still limited. While the CDO has shown promising results in various optimization problems, its effectiveness in TAPS control remains largely un expplored. In this study, we present a modified variant, mCDO, to enhance convergence speed and parameter tuning for TAPS applications. Dynamic load variations pose an additional challenge, as many studies optimize controllers, which may not refelect real-world power demands. Designing and testing controllers under dynamic load variations is therefore importnat for evaluating robustness and adaptability.
This study aims to address these gaps by evaluating multiple controllers, including PID, PIDn, FOPID, TID, and PIDA-optimized using seven metaheuristic algorithms. The developed mCDO algorithm is systematically analyzed, and the the robustness of the optimized controllers is assessed under dynamic load variations to investigate their practical feasibility in real-world power systems.
1.3 Contributions
This study aims to optimize controller parameters in a TAPS to ensure frequency stability under varying load conditions. The key contributions are summarized as follows:
Development of mCDO
An enhanced version of the CDO algorithm that incorporates an improved neighborhood–global search strategy and a wandering-based exploration mechanism.
These enhancements enable faster convergence, more accurate parameter tuning, and reduced computational cost compared to conventional optimization algorithms.
mCDO mitigates premature convergence and achieves a balanced trade-off between exploration and exploitation, resulting in superior control performance across the tested scenarios.
Comprehensive Performance Evaluation of Multiple Controllers
Conducted a systematic comparative analysis of five controllers (PID, PIDn, FOPID, TID, and PIDA) using consistent performance metrics for LFC.
Optimized controller parameters with seven metaheuristic algorithms (ARO, CDO, mCDO, GJO, HBA, MFO, and SWO), with clearly defined algorithm settings and stopping criteria to ensure reproducibility.
Performed a standardized comparison, providing a fair and comprehensive assessment of controller performance under identical operating conditions.
Performance of the FOPID-mCDO Configuration
Among all configurations, the FOPID controller optimized with mCDO achieved the best results, including:
The lowest ITAE value (0.320684)
A 10% reduction in settling time
Minimal overshoot (0.0083 Hz) and undershoot (− 0.1480 Hz)
Demonstrates significantly improved dynamic response compared to conventional PID-based controllers optimized with standard algorithms, offering an effective solution for real-world TAPS applications.
Validation Under Dynamic Load Variations
Tested the FOPID-mCDO framework under continuous and unpredictable load variations to verify robustness and adaptability.
Unlike studies limited to static or predefined load scenarios, this approach demonstrates reliable performance in realistic, dynamically changing power system conditions.
Comparison with Existing Methods
Addresses limitations of prior studies that focus on a single controller, use few metaheuristic algorithms, or consider simplified load scenarios.
Contributions include:
Introduction of mCDO, which outperforms the compaired optimization techniques.
Large-scale comparative analysis across multiple controllers and optimization algorithms.
Evaluation under dynamic, real-world-inspired load conditions.
These contributions demonstrate the effectiveness of the FOPID–mCDO framework as a promising solution for automatic generation control in multi-area power systems.
The organization of the paper is as follows: Sect. 2 presents the mathematical background related to the two-area power system, the controllers, and the mCDO algorithm. In Sect. 3, the simulation studies and results are presented in tables and graphical representations. In Sect. 4, the obtained results are discussed. Finally, Sect. 5, Conclusion, and Future Recommendations, provides suggestions for future research to further improve upon this study.
2 Problem formulation
A detailed schematic of this study is presented in Fig. 2. The figure shows a TAPS, five different controllers used separately to control each area (PID, PIDn, FOPID, TID, PIDA), seven different algorithms (ARO, CDO, mCDO, GJO, HBA, MFO, SWO) used to tune these controllers, and the points where ITAE measurements are taken.
Here, V1 (V) is the voltage of area 1, V2 (V) is the voltage of area 2, δ1 (°) is the phase angle of area 1, δ2 (°) is the phase angle of area 2, Pline12 is the power in the transmission line, and X12 is the impedance of the transmission line.
The phase angle change in each area is given by Eq. (2), where \(f\) (Hz) represents the system frequency.
The power change between the areas is described by Eq. (3), and the synchronizing torque coefficient is defined by Eq. (4). The power change in the transmission line is given by Eq. (5). Frequency variations occur in the system due to instantaneous load changes (ΔPL). Accordingly, the power change and Area Control Error (ACE) are described in Eqs. (6) and (7), where \(B\) is the frequency bias parameter.
$${ACE}_{1}=-B\Delta {f}_{1}-{\Delta P}_{tie}$$
(6)
$${ACE}_{2}=-B\Delta {f}_{2}+{\Delta P}_{tie}$$
(7)
Additionally, R, as shown in Fig. 2, represents the governor speed regulation constant. ΔPref1 and ΔPref2 are the control signal inputs of the PID controllers for areas 1 and 2, and are given by Eqs. (8) and (9). Here, KP, KI, and KD represent the PID controller parameters for proportional, integral, and derivative actions, respectively.
The transfer functions for the connection devices in the TAPS, including the governor, non-reheat turbine, and power system, are given in Eqs. (10)–(12). In these equations, \({T}_{t}\) is the time constant of the non-reheat turbine, \({T}_{g}\) is the time constant of the speed governor, and \({K}_{PS}\) and \({T}_{PS}\) are the gain and time constants of the power system, respectively [7, 18, 19].
In this study, five different controllers were used. Table 1 shows the controllers used in the study, and their corresponding equations are provided in Eqs. (13)–(17). Each controller has a different number of parameters that need to be tuned [20‐24].
FOPID controllers are advanced extensions of conventional PID controllers. By introducing fractional orders into the integral and derivative terms, FOPID controllers enable a more precise and flexible realization of differentiation and integration operations. This additional flexibility allows for improved tuning capability even at lower orders, leading to enhanced control performance. Owing to these characteristics, FOPID controllers generally outperform classical PID controllers in terms of robustness and accuracy. However, the increased number of controller parameters constitutes a potential drawback, as it complicates the tuning process and increases computational complexity.
In this study, controllers were selected based on their anticipated effectiveness in delivering optimal performance. Simpler controllers such as PI and PD, which are derivatives of PID, were not utilized due to their inability to achieve the desired efficiency. Instead, more complex and advanced controllers were preferred to enhance system performance.
To evaluate the effectiveness of PID controllers, several performance indices are utilized, which account for both the error dynamics and the accumulated error over time. Commonly used metrics include the integral of absolute error (IAE), the integral of time-weighted absolute error (ITAE), the integral square error (ISE), and the integral of time square error (ITSE). These indices are essential for analyzing how the control system responds to varying operational conditions and for determining the most efficient control approach. In this study, the ITAE index was selected as the primary performance metric due to its ability to capture both the size and duration of the control error, as represented in Eqs. (18)–(21) [25].
The ITAE criterion evaluates system performance by integrating the absolute value of the error weighted by time. Although alternative error indices can also be employed for performance assessment, the ITAE index is preferred in this study because it simultaneously accounts for both the magnitude of the error and its evolution over time. By penalizing errors that persist for longer durations more heavily, the ITAE metric provides a more effective measure of dynamic system performance, particularly in control applications where fast settling and minimal sustained deviations are desired.
In this study, the ITAE metric was selected owing to its widespread use and acceptance in the control systems literature, despite the availability of alternative evaluation criteria. This widespread use and acceptance of ITAE made it the most appropriate choice for this work. As presented in Eq. (22), the ITAE values for frequency deviations in each power system and the tie-line were aggregated to derive a comparison value for the optimization algorithms. The Objective Function (OF) is formulated as follows:
In this study, the parameters of five different controllers—PID, PIDn, FOPID, TID, and PIDA—were optimized using various algorithms, including ARO, CDO, mCDO, GJO, HBA, MFO, and SWO. Special emphasis was placed on evaluating the performance of the newly developed mCDO algorithm. The effectiveness of the proposed method was rigorously compared against other algorithms to determine its suitability for control parameter optimization in this context.
2.3.1 Overview of the chernobyl disaster optimizer
The Chernobyl Disaster Optimizer (CDO) is a recently introduced metaheuristic algorithm developed by Shehadeh in 2023 [26]. This algorithm falls under the category of physical-based methods. It simulates the catastrophic nuclear event of the Chernobyl disaster in 1986, where increasing pressure and temperature inside reactor No. 4 caused an explosion. This explosion released three types of radiation with varying speeds and masses: alpha, beta, and gamma.
Alpha radiation is the largest and heaviest type, characterized by low speed due to its composition of two protons and two neutrons. Alpha particles carry a positive charge and are highly ionizing [26].
Beta particles have a negative charge, a very small mass, and are moderately ionizing with high speed [26].
Gamma particles are a form of electromagnetic radiation with high frequency and short wavelength, having negligible mass and being weakly ionizing with very high speed [26].
The aforementioned particles, which are harmful to humans, travel from the explosion region to human-inhabited areas, causing damage. This catastrophic event has been simulated as an algorithm, which will now be mathematically presented.
2.3.1.1 Initialization
Let \({X}_{i}=({x}_{i1},{x}_{i2},...,{x}_{id})\) represent a feasible solution in a d-dimensional space, where \({x}_{ij}\in [{lb}_{\text{j}},{ub}_{\text{j}}]\). The corresponding random opposite feasible solution for \({x}_{i,j}\) is given by [26]:
where \({ub}_{j}\) and \({lb}_{\text{j}}\) represent the upper and lower boundaries of the problem variables, and \(rand\) is a uniform random vector ranging from 0 to 1.
2.3.1.2 Updating process
Following a critical reactor event, the uncontrolled release of nuclear energy generates various types of radiation. These radiation particles can travel significant distances from the reactor core, a high-pressure zone where nuclear fission occurs. As they move outward, the pressure they encounter gradually decreases until they reach human-inhabited areas, characterized by lower pressure. This uncontrolled radiation spread poses a substantial threat to human health and can lead to a nuclear catastrophe. The CDO algorithm incorporates a gradient descent factor to simulate this phenomenon, mimicking the movement of these particles toward lower-pressure regions (areas populated by humans), which is mathematically expressed as follows [26]:
where Δ represents the difference between the positions of the particles and the human position, \({\text{X}}_{p}(t)\) denotes the current position of the particles, and \(\text{c}\) is a constant valued at 0.25 for α particles, 0.5 for β particles, and 1 for γ particles Additionally, σ signifies the propagation of particles, which is calculated as follows [26]:
where \(r\) is a uniform random vector ranging from 0 to 1, \({\text{A}}_{h}=\uppi .{r}^{2}\) denotes the area within which humans move, and \(v\) signifies the speed of particles, set at 16,000 for α particles, 270,000 for β particles, and 300,000 for γ particles. Additionally, \(W{S}_{h}\) represents the outdoor walking speed of adults and is calculated as follows [26]:
$$W{S}_{h}=3-1.\left[\frac{3}{T}\right]$$
(27)
where \(T\) is the maximum number of iterations.
As previously mentioned, three types of particles (alpha, beta, and gamma) pose risks to humans following a reactor explosion like that of Chernobyl’s reactor number four. The severity of these risks depends on various factors, including each particle’s speed and ability to penetrate materials. Alpha particles, highly ionizing yet slowest, are easily halted by human skin due to their size. Beta particles travel faster but can still be shielded by common materials such as clothing or aluminum. Gamma rays, however, pose the greatest concern as they travel at nearly the speed of light and penetrate deeply into the body, making them difficult to shield against. This underscores the importance of seeking shelter and evacuation promptly after a nuclear event. The gradient descent factor \(\text{V}\) for alpha, beta, and gamma particles during their impact on humans can be calculated as follows [26]:
where \({\Delta }_{\alpha }\), \({\Delta }_{\beta }\), and \({\Delta }_{\delta }\) represent the estimated distances between the current solution and α, β, and γ particles, respectively. \({\text{X}}_{\alpha }\left(t\right)\), \({\text{X}}_{\beta }\left(t\right)\) and \({\text{X}}_{\delta }\left(t\right)\) denote the positions of α, β, and γ particles at the t-th iteration. Once these approximate distances are determined, the final location of the solution can be calculated using the equation below [26]:
At its core, the CDO algorithm is presented in its most fundamental form to showcase its efficacy in achieving global optimization across diverse application domains. This foundational structure, akin to other optimization algorithms, allows for customized adaptations. By strategically adjusting CDO’s parameters, researchers can potentially enhance its exploration and exploitation capabilities to tackle specific challenges encountered across various disciplines. However, it is essential to recognize that complex optimization problems often exhibit characteristics such as high dimensionality, nonlinearity, multimodality, and non-separability [27]. A singular search strategy, as utilized in the basic CDO algorithm, may encounter challenges in navigating such intricate landscapes effectively.
To address these limitations, the mCDO algorithm is proposed. mCDO integrates two distinct search strategies including neighborhood-global integration strategy and wandering around-based search mechanism., fostering a synergistic combination of localized exploration and broader search capabilities. This dual approach aims to navigate complex optimization landscapes more effectively, thereby facilitating the discovery of superior solutions.
2.4.1 Neighborhood-global integration strategy
In the original CDO algorithm, individuals are updated based on their distance to alpha, beta, and gamma particles. As iterations progress, individuals tend to converge toward optimal solutions, albeit with reduced population diversity. Despite their initial strong exploitation capability, individuals may gradually lose their ability to explore novel solutions. To mitigate this issue, this paper introduces a neighborhood-global integration strategy. By implementing this strategy, the algorithm enhances its performance significantly by harnessing the strengths of Non-neighborhood-based Global Search (NGS) and Neighborhood-based Local Search (NLS) strategies [28]. This dual approach ensures that the algorithm benefits from broad global search capabilities and detailed local exploration, resulting in a more resilient and effective optimization process.
Non-neighborhood-based global search (NGS) mechanism
The NGS principle involves expanding the search beyond local neighborhoods when local solutions fail to exhibit significant improvement over global ones [28]. NGS becomes active when the best local fitness value (\({f}_{local}\)) is greater than or equal to the best global fitness value (\({f}_{global}\)). This condition indicates that the local neighborhood does not offer superior solutions compared to the global best, prompting the algorithm to pursue broader exploration. Upon satisfying \({f}_{local}\ge {f}_{global}\), the algorithm concludes that the local neighborhood’s potential for further exploitation is limited, prompting a shift toward exploring the non-neighborhood region Particle i extends its search toward the globally best particle by randomly selecting k dimensions within the search space and adjusting their values using the following equation [28]:
where \(rand\) is a uniform random vector ranging from 0 to 1, \({\text{X}}_{gbest}\left(t\right)\) denotes the best non-neighbor, while \(\text{A}(\text{t})\) is a transition parameter decreases linearly with an increase in iterations and can be formulated as follows:
$$A(t)=2.02-1.[\frac{1.08}{T}]$$
(32)
where \(T\) is the maximum number of iterations. This strategy promotes extensive exploration of the search space, helping to discover potentially superior solutions that may lie outside the immediate neighborhood.
Neighborhood-based local search (NLS) mechanism
This principle behind NLS is to exploit promising local neighborhoods to enhance solution quality [28]. NLS becomes active when the best fitness value of a randomly selected local neighbor (\({f}_{local}\)) is less than the best global fitness value (\({f}_{global}\)). This condition indicates that the local neighborhood potentially contains superior solutions warranting further exploration. Upon satisfying \({f}_{local}<{f}_{global}\), the algorithm determines that the neighborhood’s quality is sufficient for exploitation. Subsequently, particle i adjusts its position toward a randomly selected neighbor using the following equation [28]:
where \(rand\) is a uniform random vector ranging from 0 to 1, \({\text{X}}_{local}\left(t\right)\) represents a random neighbor, while \(\text{A}(\text{t})\) is a transition parameter expressed as in Eq. (10). This strategy allows the algorithm to intensify its search within areas that have shown promise, thereby improving the chances of finding optimal solutions through local exploitation.
The WAS strategy is integrated to adjust the position of individual i when its fitness fails to improve despite employing either the NLS or NGS strategies [28]. WAS is activated when neither NLS nor NGS yields a fitness improvement for individual i, prompting the algorithm to explore alternative search paths. During WAS, individual i explores around its current position in the search space, potentially discovering new, unexplored regions that may contain superior fitness solutions. This wandering process enables the algorithm to dynamically adapt and diversify its search trajectory, ensuring effective exploration and exploitation of the search space by individual i. The strategy is mathematically formulated as follows: [28]:
where \(rand\) is a uniform random vector ranging from 0 to 1, \({\text{X}}_{gbest}\left(t\right)\) denotes the global best position, while \({r}_{1}\) is a randomly chosen integer values, which is selected from the range of 1 to the total number of search agents, while \(\text{A}(\text{t})\) is a transition parameter expressed as in Eq. (10). This strategy allows the algorithm to intensify its search within areas that have shown promise, thereby improving the chances of finding optimal solutions through local exploitation.
Algorithm 2 presents the pseudocode of the proposed CDO algorithm, and Fig. 3 illustrates its flowchart.
This section presents a detailed experimental analysis of the effectiveness of mCDO. First, mCDO’s performance is compared to a range of well-known and widely used metaheuristic algorithms using the CEC’2020 test suite [29]. Additionally, a Wilcoxon rank-sum test is applied to statistically compare the performance of these algorithms. Finally, mCDO’s convergence behavior and stability are analyzed relative to its competitors.
2.5.1 Benchmark description
To thoroughly assess the performance of the proposed mCDO algorithm against established optimization techniques, a rigorous benchmarking process was conducted. The recently introduced and well-rounded CEC’2020 benchmark suite was selected for this evaluation. This advanced suite features ten computationally challenging optimization problems, encompassing a variety of function types, including unimodal (single peak), multimodal (multiple peaks), hybrid (combinations of types), and composite (constructed from other functions). Notably, all benchmarks are minimization problems, with the search space for each confined to the range of − 100 to 100. Detailed descriptions of the CEC’2020 functions are provided in Table 2. The strength of the CEC’2020 suite lies in its extensive variety, particularly the inclusion of hybrid and composite functions. This rich set of functions allows for a comprehensive assessment of the mCDO algorithm’s capabilities and effectiveness in addressing a wide range of optimization challenges encountered in real-world applications. To further enhance understanding and visualization of these complex functions, three-dimensional representations are provided in Fig. 4.
A comprehensive evaluation of the mCDO algorithm was conducted by comparing its performance against established optimization techniques. This comparison included well-regarded algorithms like the Coati Optimization Algorithm (COA) [30], Seagull Optimization Algorithm (SOA) [31], Whale Optimization Algorithm (WOA) [32], Sinh Cosh Optimizer (SCHO) [33], Teaching–Learning-Based Optimization (TLBO) [34], Particle Swarm Optimization (PSO) [35, 36], Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [37], and Grey Wolf Optimization (GWO) [38, 39]. Table 3 presents the parameter settings used for all algorithms in the benchmark suite, with each configured following the recommendations of its original publication.
Table 3
Parameters of mCDO and its competitors
Algorithm
Parameter
Value
All
Maximum iteration
1000
Number of agents
50
Number of runs
30
COA
–
–
SOA
A
[0,1]
fc
2
WOA
a
[0,1]
SCHO
p, q, α
10, 9, 4.6
β, \(\epsilon\), n
1.55, 0.003, 0.5
ct, u, m
3.6, 0.388, 0.45
TLBO
TF
1.2
PSO
c2
2
c2
2
Vmax
6
CMA-ES
α
2
GWO
a
[2,0]
CDO
\({S}_{\gamma }\)
Rand (1,300,000)
\({S}_{\beta }\)
Rand (1,270,000)
\({S}_{\alpha }\)
Rand (1,16,000)
mCDO (proposed)
A
[2,0.9]
\({S}_{\gamma }\)
Rand (1,300,000)
\({S}_{\beta }\)
Rand (1,270,000)
\({S}_{\alpha }\)
Rand (1,16,000)
The algorithms selected for benchmarking were chosen for their significant impact on optimization, consistently demonstrating strong performance across diverse problems. Classical algorithms are well-established and thoroughly validated, whereas newer algorithms embody innovative, evolving strategies. Moreover, state-of-the-art algorithms, including both competition-winning and top-performing approaches, were included to provide a comprehensive assessment of recent advancements. This selection ensures a balanced evaluation of classical, contemporary, and leading-edge optimization techniques.
To ensure a fair and standardized evaluation, all algorithms were tested on the CEC’2020 benchmark suite with consistent settings: a problem dimension of 20, a population size of 50, and a maximum of 1,000 iterations. To account for potential randomness, each experiment was repeated 30 independent times. The computational environment consisted of MATLAB R2023a running on an Intel(R) Core(TM) i7-9750 CPU @ 3.20 GHz with 32 GB RAM.
2.5.3 Statistical results
Table 4 summarizes the performance of ten algorithms evaluated using the CEC’2020 test suite. The table presents the best, worst, mean, and standard deviation (Std) values obtained by each algorithm across all functions, along with their corresponding rankings. The mean value is the primary determinant of ranking, with standard deviation used as a tiebreaker in cases where multiple algorithms achieve identical means. The final ranking for each algorithm is calculated as the average ranking across all 30 test functions. The last row of Table 4 displays the mean ranking derived from the Friedman rank test [40], providing a comprehensive performance comparison. Values in bold within the table represent the best results achieved among the ten algorithms.
Table 4
Comparison of mCDO and its competitors in solving the CEC’2020 functions
F
index
COA
SOA
WOA
SCHO
TLBO
PSO
CMA-ES
GWO
CDO
mCDO
1
Best
1.560E + 10
1.718E + 09
9.412E + 06
9.616E + 08
1.536E + 09
1.009E + 02
6.883E + 03
2.098E + 04
2.107E + 10
1.003E + 02
Mean
2.644E + 10
4.064E + 09
3.416E + 07
3.767E + 09
5.137E + 09
1.391E + 03
1.360E + 10
1.413E + 08
2.137E + 10
6.333E + 02
Worst
3.627E + 10
7.874E + 09
9.717E + 07
9.906E + 09
9.037E + 09
5.367E + 03
2.595E + 10
1.257E + 09
2.163E + 10
2.317E + 03
Std
5.450E + 09
1.589E + 09
1.915E + 07
2.669E + 09
2.105E + 09
1.420E + 03
5.995E + 09
2.818E + 08
1.489E + 08
5.850E + 02
Rank
10
6
4
5
7
2
8
3
9
1
2
Best
4.699E + 03
1.948E + 03
2.867E + 03
2.458E + 03
1.952E + 03
1.660E + 03
5.203E + 03
1.474E + 03
3.937E + 03
1.108E + 03
Mean
5.455E + 03
3.415E + 03
3.892E + 03
3.288E + 03
2.731E + 03
2.831E + 03
5.672E + 03
2.824E + 03
4.902E + 03
1.292E + 03
Worst
6.224E + 03
5.277E + 03
5.270E + 03
4.045E + 03
3.906E + 03
3.969E + 03
6.144E + 03
5.560E + 03
5.356E + 03
1.699E + 03
Std
3.285E + 02
7.002E + 02
6.171E + 02
4.162E + 02
4.380E + 02
4.162E + 02
2.301E + 02
1.226E + 03
3.184E + 02
1.657E + 02
Rank
9
6
7
5
2
4
10
3
8
1
3
Best
9.347E + 02
7.973E + 02
8.153E + 02
8.196E + 02
7.829E + 02
7.442E + 02
7.926E + 02
7.413E + 02
9.655E + 02
7.231E + 02
Mean
1.009E + 03
8.701E + 02
9.262E + 02
8.710E + 02
8.425E + 02
7.646E + 02
8.110E + 02
7.728E + 02
9.908E + 02
7.291E + 02
Worst
1.052E + 03
9.364E + 02
1.037E + 03
9.643E + 02
9.866E + 02
8.022E + 02
8.238E + 02
8.375E + 02
1.020E + 03
7.401E + 02
Std
3.112E + 01
2.883E + 01
5.866E + 01
3.594E + 01
4.905E + 01
1.480E + 01
6.343E + 00
2.958E + 01
1.455E + 01
4.168E + 00
Rank
10
7
8
6
5
2
4
3
9
1
4
Best
1.900E + 03
1.900E + 03
1.900E + 03
1.900E + 03
1.906E + 03
1.901E + 03
1.903E + 03
1.900E + 03
1.900E + 03
1.900E + 03
Mean
1.900E + 03
1.900E + 03
1.900E + 03
1.900E + 03
1.932E + 03
1.902E + 03
1.907E + 03
1.902E + 03
1.900E + 03
1.900E + 03
Worst
1.900E + 03
1.900E + 03
1.902E + 03
1.900E + 03
2.022E + 03
1.904E + 03
1.909E + 03
1.906E + 03
1.900E + 03
1.900E + 03
Std
0.000E + 00
2.060E-03
4.531E-01
0.000E + 00
2.562E + 01
7.901E-01
1.689E + 00
1.540E + 00
0.000E + 00
0.000E + 00
Rank
1
5
6
1
10
8
9
7
1
1
5
Best
8.626E + 05
7.203E + 04
1.726E + 05
4.965E + 04
4.755E + 04
3.465E + 04
5.136E + 05
2.448E + 04
3.560E + 05
3.500E + 03
Mean
3.681E + 06
5.405E + 05
1.594E + 06
8.019E + 05
1.144E + 06
1.239E + 05
1.130E + 07
2.883E + 05
9.401E + 05
1.240E + 04
Worst
1.106E + 07
1.503E + 06
8.898E + 06
2.726E + 06
4.117E + 06
3.313E + 05
3.955E + 07
1.591E + 06
1.789E + 06
4.318E + 04
Std
2.418E + 06
3.862E + 05
1.780E + 06
7.477E + 05
1.006E + 06
7.838E + 04
1.115E + 07
3.151E + 05
3.088E + 05
8.939E + 03
Rank
9
4
8
5
6
2
10
3
7
1
6
Best
2.676E + 03
1.781E + 03
1.930E + 03
1.760E + 03
1.724E + 03
1.645E + 03
2.336E + 03
1.656E + 03
2.810E + 03
1.600E + 03
Mean
3.160E + 03
2.068E + 03
2.407E + 03
2.112E + 03
2.033E + 03
2.084E + 03
2.870E + 03
1.863E + 03
3.115E + 03
1.601E + 03
Worst
3.608E + 03
2.589E + 03
2.929E + 03
2.636E + 03
2.601E + 03
2.511E + 03
3.239E + 03
2.201E + 03
3.504E + 03
1.602E + 03
Std
2.357E + 02
1.840E + 02
2.646E + 02
2.256E + 02
2.052E + 02
2.496E + 02
2.222E + 02
1.407E + 02
1.828E + 02
4.975E-01
Rank
10
4
7
6
3
5
8
2
9
1
7
Best
1.098E + 05
1.968E + 04
6.995E + 04
8.026E + 03
9.210E + 03
4.391E + 03
1.872E + 05
9.835E + 03
4.535E + 05
2.123E + 03
Mean
1.725E + 06
1.637E + 05
7.017E + 05
4.420E + 05
4.172E + 05
8.231E + 04
5.519E + 06
1.409E + 05
1.874E + 07
3.388E + 03
Worst
7.533E + 06
4.738E + 05
2.821E + 06
1.870E + 06
1.950E + 06
3.709E + 05
1.755E + 07
4.261E + 05
7.771E + 07
6.242E + 03
Std
1.763E + 06
1.374E + 05
6.388E + 05
5.314E + 05
4.635E + 05
8.394E + 04
4.563E + 06
9.497E + 04
2.209E + 07
1.167E + 03
Rank
8
4
7
5
6
2
9
3
10
1
8
Best
4.183E + 03
2.435E + 03
2.315E + 03
2.341E + 03
2.489E + 03
2.300E + 03
2.300E + 03
2.306E + 03
4.215E + 03
2.300E + 03
Mean
5.594E + 03
5.574E + 03
4.025E + 03
3.580E + 03
2.787E + 03
2.786E + 03
6.514E + 03
3.725E + 03
5.455E + 03
2.301E + 03
Worst
6.656E + 03
6.923E + 03
6.313E + 03
5.607E + 03
3.359E + 03
5.400E + 03
7.209E + 03
7.158E + 03
7.499E + 03
2.303E + 03
Std
7.652E + 02
8.882E + 02
1.652E + 03
1.240E + 03
2.459E + 02
1.016E + 03
1.148E + 03
1.778E + 03
1.357E + 03
8.152E-01
Rank
8
9
6
5
3
2
10
4
7
1
9
Best
3.133E + 03
2.826E + 03
2.868E + 03
2.957E + 03
2.889E + 03
2.500E + 03
2.968E + 03
2.812E + 03
3.317E + 03
2.811E + 03
Mean
3.312E + 03
2.874E + 03
3.008E + 03
3.045E + 03
2.961E + 03
2.989E + 03
3.011E + 03
2.859E + 03
3.360E + 03
2.828E + 03
Worst
3.625E + 03
2.933E + 03
3.166E + 03
3.213E + 03
3.111E + 03
3.163E + 03
3.049E + 03
2.919E + 03
3.404E + 03
2.849E + 03
Std
1.152E + 02
2.402E + 01
6.755E + 01
6.278E + 01
6.302E + 01
1.457E + 02
2.408E + 01
3.745E + 01
2.074E + 01
9.285E + 00
Rank
9
3
5
8
4
6
7
2
10
1
10
Best
3.799E + 03
2.954E + 03
2.921E + 03
2.951E + 03
2.995E + 03
2.902E + 03
3.222E + 03
2.914E + 03
5.916E + 03
2.910E + 03
Mean
5.153E + 03
3.083E + 03
3.033E + 03
3.106E + 03
3.212E + 03
2.966E + 03
3.948E + 03
2.962E + 03
5.939E + 03
2.953E + 03
Worst
6.854E + 03
3.402E + 03
3.176E + 03
3.681E + 03
4.050E + 03
3.003E + 03
4.996E + 03
3.020E + 03
5.970E + 03
3.006E + 03
Std
7.410E + 02
9.037E + 01
4.609E + 01
1.616E + 02
2.314E + 02
2.671E + 01
4.626E + 02
3.289E + 01
1.229E + 01
3.483E + 01
Rank
9
5
4
6
7
3
8
2
10
1
Mean rank
8.300
5.300
6.200
5.200
5.300
3.600
8.300
3.200
8.000
1.000
Rank
9
5
7
4
5
3
9
2
8
1
Bold values represent the best results
Table 4 demonstrably showcases mCDO’s superiority over the nine compared algorithms across the CEC’2020 benchmark suite. This exceptional performance is evident across all function classes, including unimodal (F1), multimodal (F2-F4), hybrid (F5-F7), and composite (F8-F10) functions. The overall ranking of the ten algorithms, from most effective to least effective, is as follows: mCDO, GWO, PSO, SCHO, SOA, TLBO (with SOA and TLBO achieving identical rankings), WOA, CDO, COA, and CMA-ES (where COA and CMA-ES also share a ranking). Following the calculation of average rankings, mCDO convincingly secures the top position with a value of 1.00. These compelling results lead to a definitive conclusion: mCDO exhibits the most distinguished relative performance among the evaluated algorithms.
Figure 5 presents a radar chart of each algorithm across the CEC’2020 benchmark functions. The radar chart is a valuable tool for providing a comprehensive comparison of the algorithms’ performance across multiple functions, allowing us to quickly identify which algorithm delivers the best balance of accuracy and efficiency across the benchmark functions in a single plot. In this visualization, a smaller enclosed area corresponds to superior overall performance, as it indicates that the algorithm has consistently performed well across all evaluation functions. As shown in the figure, mCDO exhibits the most compact area, indicating its position as the top-performing algorithm within the test suite.
To further evaluate the accuracy of mCDO relative to competing algorithms, a comparative analysis of Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) was conducted across various test functions. Both RMSE and MAE serve as quantitative measures of the discrepancy between the obtained solutions and the known theoretical optima. Lower values of these metrics indicate superior algorithm performance, characterized by reduced error and enhanced stability. As illustrated in Table 5, mCDO consistently achieves lower RMSE and MAE values across most test functions. This compelling evidence suggests that mCDO exhibits a high degree of precision, with its solutions demonstrably closer to the theoretical optima. Consequently, mCDO establishes itself as the most accurate algorithm among the evaluated methods.
Table 5
RMSE and MAE results of mCDO and its competitors on CEC’2020 functions
F
Index
COA
SOA
WOA
SCHO
TLBO
PSO
CMA-ES
GWO
CDO
mCDO
1
MAE
2.644E + 10
4.064E + 09
3.416E + 07
3.767E + 09
5.137E + 09
1.291E + 03
1.360E + 10
1.413E + 08
2.137E + 10
5.333E + 02
RMSE
2.698E + 10
4.354E + 09
3.900E + 07
4.591E + 09
5.539E + 09
1.901E + 03
1.482E + 10
3.110E + 08
2.137E + 10
7.844E + 02
2
MAE
4.355E + 03
2.315E + 03
2.792E + 03
2.188E + 03
1.631E + 03
1.731E + 03
4.572E + 03
1.724E + 03
3.802E + 03
1.918E + 02
RMSE
4.367E + 03
2.415E + 03
2.857E + 03
2.226E + 03
1.687E + 03
1.779E + 03
4.578E + 03
2.103E + 03
3.815E + 03
2.516E + 02
3
MAE
3.088E + 02
1.701E + 02
2.262E + 02
1.710E + 02
1.425E + 02
6.459E + 01
1.110E + 02
7.277E + 01
2.908E + 02
2.912E + 01
RMSE
3.103E + 02
1.724E + 02
2.334E + 02
1.746E + 02
1.504E + 02
6.621E + 01
1.112E + 02
7.837E + 01
2.912E + 02
2.941E + 01
4
MAE
0.000E + 00
3.761E-04
1.478E-01
0.000E + 00
3.222E + 01
2.259E + 00
6.650E + 00
1.600E + 00
0.000E + 00
0.000E + 00
RMSE
0.000E + 00
2.060E-03
4.694E-01
0.000E + 00
4.089E + 01
2.389E + 00
6.854E + 00
2.203E + 00
0.000E + 00
0.000E + 00
5
MAE
3.679E + 06
5.388E + 05
1.592E + 06
8.002E + 05
1.142E + 06
1.222E + 05
1.130E + 07
2.866E + 05
9.384E + 05
1.070E + 04
RMSE
4.380E + 06
6.592E + 05
2.366E + 06
1.087E + 06
1.511E + 06
1.444E + 05
1.574E + 07
4.220E + 05
9.863E + 05
1.385E + 04
6
MAE
1.560E + 03
4.684E + 02
8.072E + 02
5.122E + 02
4.333E + 02
4.837E + 02
1.270E + 03
2.632E + 02
1.515E + 03
1.229E + 00
RMSE
1.577E + 03
5.021E + 02
8.481E + 02
5.582E + 02
4.779E + 02
5.424E + 02
1.289E + 03
2.973E + 02
1.526E + 03
1.323E + 00
7
MAE
1.723E + 06
1.616E + 05
6.996E + 05
4.399E + 05
4.151E + 05
8.021E + 04
5.517E + 06
1.388E + 05
1.874E + 07
1.288E + 03
RMSE
2.444E + 06
2.107E + 05
9.402E + 05
6.830E + 05
6.164E + 05
1.151E + 05
7.111E + 06
1.673E + 05
2.869E + 07
1.725E + 03
8
MAE
3.394E + 03
3.374E + 03
1.825E + 03
1.380E + 03
5.871E + 02
5.863E + 02
4.314E + 03
1.525E + 03
3.255E + 03
1.007E + 02
RMSE
3.476E + 03
3.485E + 03
2.443E + 03
1.841E + 03
6.349E + 02
1.158E + 03
4.459E + 03
2.320E + 03
3.518E + 03
1.007E + 02
9
MAE
9.122E + 02
4.738E + 02
6.078E + 02
6.453E + 02
5.614E + 02
5.889E + 02
6.107E + 02
4.588E + 02
9.601E + 02
4.280E + 02
RMSE
9.192E + 02
4.743E + 02
6.114E + 02
6.482E + 02
5.649E + 02
6.061E + 02
6.112E + 02
4.603E + 02
9.603E + 02
4.281E + 02
10
MAE
2.653E + 03
5.832E + 02
5.333E + 02
6.059E + 02
7.122E + 02
4.664E + 02
1.448E + 03
4.616E + 02
3.439E + 03
4.534E + 02
RMSE
2.751E + 03
5.899E + 02
5.352E + 02
6.264E + 02
7.476E + 02
4.671E + 02
1.518E + 03
4.627E + 02
3.439E + 03
4.547E + 02
Bold values represent the best results
2.5.4 Wilcoxon rank sum test
This subsection employs the nonparametric Wilcoxon rank-sum test [41] to statistically evaluate the significance of performance differences between mCDO and the compared algorithms. The test is conducted at a significance level of α = 0.05. The resulting p-values are presented in Table 6, with values exceeding 0.05 underlined. Notably, ‘NaN’ denotes instances where the algorithms’ performance is too similar to establish a statistically significant distinction. As evident from Table 6, the sparsity of ‘NaN’ markers and underlined values signifies a low degree of similarity between mCDO and its competitors regarding the optimization results obtained from the benchmark functions. By consolidating the analyses from prior subsections, it can be concluded that mCDO exhibits the most exceptional overall performance among the evaluated metaheuristic algorithms, including the original CDO version.
Table 6
p-value of Wilcoxon sum test between mCDO and its competitors
F
COA
SOA
WOA
SCHO
TLBO
PSO
CMA-ES
GWO
CDO
1
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
5.555E-02
1.510E-11
3.020E-11
3.020E-11
2
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.669E-11
1.510E-11
6.066E-11
3.020E-11
3
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
3.020E-11
3.020E-11
4
NaN
3.337E-01
2.096E-02
NaN
6.059E-13
6.059E-13
1.000E + 00
1.212E-12
NaN
5
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
2.252E-11
1.510E-11
4.077E-11
3.020E-11
6
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
3.020E-11
3.020E-11
7
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.510E-11
1.305E-10
1.510E-11
3.020E-11
3.020E-11
8
9.779E-12
9.779E-12
9.779E-12
9.779E-12
9.779E-12
2.450E-03
9.779E-12
1.956E-11
1.956E-11
9
1.510E-11
1.737E-10
1.510E-11
1.510E-11
1.510E-11
4.242E-09
1.510E-11
2.891E-03
3.020E-11
10
1.510E-11
8.474E-10
3.361E-10
2.786E-10
2.747E-11
9.049E-02
7.472E-02
1.494E-01
3.020E-11
Values over 0.05 are underlined
2.5.5 Convergence analysis
To facilitate a clearer understanding of convergence behavior, we analyze the convergence curves of each algorithm on the CEC’2020 benchmark functions. Figure 6 illustrates the convergence curves of mCDO and its competitors, providing insights into their exploration and exploitation patterns. CDO exhibits rapid initial improvement but quickly stagnates, indicating strong exploitation but limited exploration, which leads to premature convergence on suboptimal solutions for F1, F2, F3, and F5–F9. Other algorithms, including GWO, CMA-ES, PSO, TLBO, SCHO, WOA, SOA, and COA, demonstrate slower yet steadier convergence, reflecting a gradual shift from exploration in the early iterations to exploitation in later stages. In contrast, mCDO achieves fast convergence while maintaining sustained exploration, effectively avoiding local optima and approaching the global optimum. This balance allows mCDO to consistently outperform the competing algorithms in both convergence speed and solution quality across the benchmark suite.
The boxplot [42] is utilized to provide a discrete analysis of the algorithm’s optimization performance, effectively highlighting the algorithm’s stability throughout the search process. The median value within each boxplot represents the average performance across 30 optimization runs, with lower median values indicating superior convergence accuracy. The upper and lower quartile values depict the variability observed in the optimization results. Outliers, represented by individual data points situated beyond the whiskers, signify instances of exceptional instability during the optimization process. As illustrated in Fig. 7, mCDO consistently achieves the lowest median values across functions F3, F4, F6, F7, F8, and F9. Notably, these boxplots exhibit minimal data fluctuation, further emphasizing mCDO’s exceptional stability and superiority within the improved version.
To assess the computational complexity of mCDO relative to key competing algorithms under the CEC2020 framework, we measured their performance on test problems with dimensions of 10 and 20. Three runtime metrics were considered: \({T}_{0}\), \({T}_{1}\), and \({T}_{2}\). Specifically, \({T}_{0}\) represents the time required solely by the CEC’2020 evaluation system. \({T}_{1}\) denotes the time taken to perform 1,000 iterations of the D-dimensional reference function F1. Finally, \({T}_{2}\) corresponds to the average runtime of 1,000 iterations of the same function, computed as the arithmetic mean over five separate measurements of \({T}_{2}\), i.e., \(\overline{{{\varvec{T}}}_{2}}\)= mean (\({T}_{2}\)).
The computational overhead of each algorithm was evaluated using the metrics \({T}_{0}\), \({T}_{1}\), \({T}_{2}\), and the derived measure (\(\overline{{{\varvec{T}}}_{2}}\)− \({T}_{1}\))/\({T}_{0}\). To ensure fair and standardized comparisons, all algorithms were executed under identical conditions: processing a single entity per iteration, without parallel computing or vectorized optimizations. All experiments were implemented in MATLAB following a consistent programming methodology. The baseline execution time \({T}_{0}\) was determined according to the formula presented below [29]:
Table 7 presents the results of the time complexity comparison among the algorithms. As reported in the table, mCDO exhibits higher computational complexity, reflecting greater per-iteration overhead due to its enhanced mechanisms for balancing exploration and exploitation. Overall, while mCDO may incur a higher computational cost in specific scenarios than simpler algorithms, its superior solution quality, as demonstrated by faster convergence and higher accuracy across the CEC’2020 benchmarks, makes it a worthwhile choice for complex optimization tasks where precision outweighs runtime efficiency.
Table 7
Time complexity outcomes
Dimension
Algorithm
\({T}_{0}\)
\({T}_{1}\)
\(\overline{{T}_{2}}\)
(\(\overline{{T}_{2}}\)−\({T}_{1}\))/\({T}_{0}\)
10
COA
0.2157
0.5368
0.4602
0.3551
SOA
0.2157
0.3449
0.3390
0.0275
WOA
0.2157
0.2640
0.2673
0.0154
SCHO
0.2157
0.6886
0.6139
0.3461
TLBO
0.2157
0.7295
0.7763
0.2170
PSO
0.2157
0.3598
0.3230
0.1707
CMA-ES
0.2157
3.8523
5.2971
6.6987
GWO
0.2157
0.4780
0.6660
0.8715
CDO
0.2157
0.4969
0.6565
0.7400
mCDO
0.2157
12.9023
14.9113
9.3141
20
COA
0.2157
0.6016
0.6492
0.2208
SOA
0.2157
0.4896
0.6901
0.9294
WOA
0.2157
0.3061
0.4176
0.5168
SCHO
0.2157
0.9412
1.1555
0.9934
TLBO
0.2157
0.8066
1.1338
1.5172
PSO
0.2157
0.4376
0.6714
1.0840
CMA-ES
0.2157
4.3789
5.6578
5.9292
GWO
0.2157
0.8882
0.7754
0.5229
CDO
0.2157
0.9210
0.8025
0.5492
mCDO
0.2157
17.3355
17.3857
0.2327
Our in-depth analysis reveals that mCDO exhibits the most consistent performance across multiple trials. This stability is attributed to the synergistic integration of two key strategies: the neighborhood-global integration strategy and the WAS strategy. These strategies maintain a high degree of population diversity, fostering robust exploration capabilities across the entire problem space. Consequently, mCDO demonstrates exceptional efficacy in searching for the optimal solution, effectively mitigating the risk of premature convergence on suboptimal solutions. Moreover, these enhancements significantly reduce the stochasticity associated with mCDO’s results, enabling it to consistently converge on the global optimum solution.
3 Simulation results for AGC of a two-area power system
Initially, the optimization of controller parameters was performed under steady conditions within a predefined search space. Subsequently, different load conditions were applied, and the system’s response to these variations in the controller parameters was analyzed.
3.1 Parameter configuration
In order to maintain the stability of the TAPS and minimize frequency deviations, five different controllers—PID, PIDn, FOPID, TID, and PIDA—were individually tested to identify the controller that provided the most optimal response. The parameters of these controllers were tuned using several optimization algorithms, including ARO, CDO, mCDO, GJO, HBA, MFO, and SWO. In the initial phase of the simulations, a 10% load was applied to the first area, deliberately disturbing the system’s balance. After identifying the best-performing controller parameters using the most effective algorithm, the system was further examined under wider load variation scenarios.
For the optimization process, ARO, CDO, mCDO, GJO, HBA, MFO, and SWO algorithms were employed, as they were expected to yield superior results. Each algorithm was executed with 40 agents over 50 iterations, resulting in a total of 2000 simulation runs to achieve the final outcomes.
The specific data used for TAPS in the simulation study are presented in Table 8 [19].
Table 8
Values used in the simulation
Variable
Value
Unit
Tg
0.08
s
Tt
0.3
s
T12
0.545
p.u. MW/rad
Tps
20
s
Kps
120
Hz/p.u
B
0.425
p.u. MW/Hz
R
2.4
Hz/p.u
f
60
Hz
In order to disturb the balance of the TAPS, a 10% load disturbance was applied to the first area. All simulations were carried out under these conditions, and the controller parameters were determined accordingly.
In Table 9, the range of variation for the optimized controller parameters is presented.
Table 9
Change in the intervals of the optimized controllers for the optimization algorithms
Variable
Lower
Upper
KP
0.01
10
KI
0.01
10
KD
0.01
10
Kt
0.01
10
Ka
0.01
10
n
1
1000
λ
0.01
3
μ
0.01
3
nTID
0.01
3
Frequency range
0.001
1000
Approximation order
5
5
The ranges provided here were determined based on the results obtained from preliminary trials. Additionally, the controller spaces utilized in the reviewed literature were also taken into consideration during this process.
3.2 Results
As a result of the following steps, the values presented in Tables 10 and 11 were obtained.
Table 10
All of simulation results for PID, PIDn and FOPID
Controller
Optimization algorithms
KP
KI
KD
Kt
Ka
nTID
n
λ
μ
ITAE
Solver
PID
Artificial Rabbit Optimization
3.1447
9.9998
2.1990
–
–
–
–
–
–
0.3284
Fixed step (automatic solver)
CDO
3.06467
10
2.17763
–
–
–
–
–
–
0.3284
Fixed step (automatic solver)
mCDO
3.0396
10
2.1685
–
–
–
–
–
–
0.3284
Fixed step (automatic solver)
Golden Jackal Optimization
3.0385
10
2.1619
–
–
–
–
–
–
0.3284
Fixed step (automatic solver)
Honey Badger Algorithm
3.0396
10
2.1685
–
–
–
–
–
–
0.3284
Fixed step (automatic solver)
Moth-Flame Optimization
3.0396
10
2.1685
–
–
–
–
–
–
0.3284
Fixed step (automatic solver)
Spider Wasp Optimizer
6.7356
9.4533
3.9927
–
–
–
–
–
–
0.4397
Fixed step (automatic solver)
PIDn
Artificial Rabbit Optimization
3.0631
9.9980
2.1508
–
–
–
717.0123
–
–
0.3285
Fixed step (automatic solver)
CDO
2.790926
10
2.143538
–
–
–
422.1989
–
–
0.3286
Fixed step (automatic solver)
mCDO
3.0419
10
2.1688
–
–
–
999.7138
–
–
0.3284
Fixed step (automatic solver)
Golden Jackal Optimization
3.0619
10
2.1658
–
–
–
431.9013
–
–
0.3284
Fixed step (automatic solver)
Honey Badger Algorithm
3.0416
10
2.1688
–
–
–
1000
–
–
0.3284
Fixed step (automatic solver)
Moth-Flame Optimization
3.0388
10
2.1680
–
–
–
1000
–
–
0.3284
Fixed step (automatic solver)
Spider Wasp Optimizer
2.6304
9.995
1.8967
–
–
–
899.7993
–
–
0.3318
Fixed step (automatic solver)
FOPID
Artificial Rabbit Optimization
5.2270
9.8864
2.3555
–
–
–
–
1.0126
1.2412
0.337
ode1be
CDO
5.4218
10
4.3574
–
–
–
–
0.993418
1.01475
0.3963
ode1be
mCDO
3.4218
9.9999
1.8422
–
–
–
–
1.0036
1.1629
0.320684
ode1be
Golden Jackal Optimization
4.9057
10
2.0549
–
–
–
–
1.0105
1.2370
0.3267
ode1be
Honey Badger Algorithm
10
10
10
–
–
–
–
1.0096
1.1835
0.469
ode1be
Moth-Flame Optimization
3.41
10
1.8281
–
–
–
–
1.0036
1.1666
0.320696
ode1be
Spider Wasp Optimizer
2.0388
5.8632
2.5429
–
–
–
–
1.0391
1.3298
0.7229
ode1be
Table 11
All of simulation results for TID and PIDA
Controller
Optimization algorithms
KP
KI
KD
Kt
Ka
nTID
n
λ
μ
ITAE
Solver
TID
Artificial Rabbit Optimization
–
9.9996
2.6046
2.7232
–
2.9148
–
–
–
0.3578
ode1be
CDO
–
10
2.56096
2.44201
–
3
–
–
–
0.3556
ode1be
mCDO
–
9.9999
2.5530
2.4111
–
3
–
–
–
0.3556
ode1be
Golden Jackal Optimization
–
10
2.5532
2.4113
–
3
–
–
–
0.3556
ode1be
Honey Badger Algorithm
–
10
5.0306
10
–
1.7847
–
–
–
0.4053
ode1be
Moth-Flame Optimization
–
10
2.5523
2.4059
–
3
–
–
–
0.3556
ode1be
Spider Wasp Optimizer
–
9.8938
3.3081
6.7312
–
2.7893
–
–
–
0.4107
ode1be
PIDA
Artificial Rabbit Optimization
3.11
9.9074
2.2260
–
0.0246
–
–
–
–
0.3331
Fixed step (automatic solver)
CDO
2.26209
10
2.02215
–
0.0685261
–
–
–
–
0.3301
Fixed step (automatic solver)
mCDO
3.0164
10
2.1663
–
0.01
–
–
–
–
0.3285
Fixed step (automatic solver)
Golden Jackal Optimization
3.0158
10
2.1440
–
0.0157
–
–
–
–
0.3286
Fixed step (automatic solver)
Honey Badger Algorithm
3.0165
10
2.1663
–
0.01
–
–
–
–
0.3285
Fixed step (automatic solver)
Moth-Flame Optimization
3.0189
10
2.1676
–
0.01
–
–
–
–
0.3285
Fixed step (automatic solver)
Spider Wasp Optimizer
7.9823
9.9781
3.1989
–
0.52597
–
–
–
–
0.441
Fixed step (automatic solver)
Certain controllers and optimization algorithms inherently exhibit numerical sensitivities to variations in simulation step size. Such variations may lead to numerical instabilities, particularly division-by-zero issues in derivative-based controllers. To ensure numerical stability and reliable simulation outcomes, an appropriate adjustment of the solver configuration is therefore required.
Accordingly, a fixed-step solver was employed in some simulation scenarios, whereas the ode1be solver was adopted in others to enhance numerical robustness. MATLAB/Simulink solvers operating in automatic mode are internally optimized to achieve the fastest stable execution. However, reducing the step size to suppress numerical singularities results in a disproportionate increase in computational cost. In this context, the use of the ode1be solver served exclusively as a practical measure to maintain numerical stability rather than as a methodological or performance-oriented choice.
Consequently, the solver selection does not introduce any performance bias or comparative advantage among the evaluated controllers. This conclusion is further supported by the results presented in Table 11, where the PIDA controllers operating under fixed-step conditions exhibit superior performance compared to the TID controller implemented using the ode1be solver.
The parameters of each controller were optimized using ARO, CDO, mCDO, GJO, HBA, MFO, and SWO algorithms.
The optimization was carried out based on the ITAE value, with the controller parameters corresponding to the smallest ITAE value selected, as this represents the lowest error.
Convergence curves for the seven algorithms applied to each controller were plotted to facilitate comparison.
Finally, the optimized controller parameters were used to simulate the TAPS, and frequency response graphs for each area, along with the tie-line, were plotted and analyzed.
Figures 8, 9, 10, 11 and 12 display the convergence curves obtained from optimizing each controller with the respective algorithms. Additionally, the detailed characteristics of these convergence curves are shown below each figure.
Figure 8 illustrates the convergence curves of the optimization algorithms applied to the PID controller. While SWO exhibited the poorest performance, all other algorithms identified controller parameters that resulted in the same ITAE value after the 26th iteration. This is also reflected in Table 10. The best PID controller parameters found by SWO yielded an ITAE value of 0.4397, whereas all other algorithms achieved an ITAE value of 0.3284. Thus, with the exception of SWO, the other algorithms can be effectively used for tuning PID controller parameters. Notably, mCDO and HBA reached the optimal values most rapidly, converging by the 7th iteration.
The results for the PIDn controller are shown in Fig. 9. Here, mCDO, GJO, HBA, and MFO achieved the lowest ITAE value of 0.3284. mCDO remained stable after the 9th iteration.
Figure 10 presents the convergence curves for the optimization of the FOPID controller parameters, highlighting the comparative performances of seven optimization algorithms: mCDO, ARO, CDO, GJO, HBA, MFO, and SWO. The most notable observation from the convergence curves is that mCDO consistently outperforms the other algorithms in terms of achieving the lowest ITAE value.
The mCDO algorithm reaches the optimal solution by the 16th iteration, which demonstrates its efficiency in converging quickly to the best result. This rapid convergence is a significant advantage in real-time applications, where computational efficiency is crucial. Furthermore, the mCDO algorithm achieves an ITAE value of 0.320684, which is the lowest among all the algorithms tested. The corresponding optimal controller parameters are determined as KP = 3.4218, KI = 9.9999, KD = 1.8422, l = 1.0036, and m = 1.1629.
In contrast, the MFO algorithm, while yielding competitive results, converges more slowly compared to mCDO and does not reach as low an ITAE value. The slower convergence is evident in the curve, where MFO shows a plateau before it eventually reaches a similar performance. Additionally, other algorithms such as CDO and GJO demonstrate moderate convergence behavior, stabilizing around higher ITAE values, which are not as optimal as mCDO.
However, the SWO algorithm exhibits the poorest performance, failing to converge to a satisfactory solution within the 50 iterations. This is reflected in its significantly higher ITAE value, and the fact that it does not show meaningful improvements after early iterations.
The results confirm the robustness and effectiveness of the mCDO algorithm in tuning the FOPID controller for optimal performance in multi-area power systems. Compared to other well-known optimization techniques, mCDO not only reaches a more accurate solution but does so with a faster convergence rate, making it a highly suitable candidate for real-time control applications where computational efficiency is critical.
The results of the TID controller optimization are presented in Fig. 11. The CDO, mCDO, GJO, and MFO algorithms yielded the same ITAE value of 0.3556. As depicted in the figure, mCDO converged and stabilized after the 8th iteration. The poorest performance was observed with the SWO algorithm.
Upon further analysis of Fig. 11, several important observations can be made regarding the performance of the algorithms used to optimize the TID controller parameters:
Initial Convergence Rates: Most algorithms begin with a high fitness value (around 0.55 to 0.60), but within the first 5 iterations, significant convergence is observed, particularly in algorithms like GJO, MFO, and mCDO. These algorithms exhibit a rapid decrease in fitness value, indicating their efficiency in quickly identifying optimal parameter sets. In contrast, SWO shows very slow progress and struggles to improve the fitness value throughout the iterations, demonstrating poor performance relative to other algorithms.
Stabilization: The mCDO algorithm reaches a stable fitness value of 0.3556 by the 8th iteration, indicating that it has effectively found the optimal parameters for the TID controller. Similarly, CDO, GJO, and MFO converge to the same optimal fitness value, indicating robust performance in identifying the best parameter set.
Comparison of Algorithm Performance: While mCDO, CDO, GJO, and MFO deliver the lowest ITAE value and show strong convergence and stability, SWO exhibits the worst performance, maintaining a fitness value around 0.44 throughout the optimization process. This suggests that SWO is not suitable for optimizing the TID controller parameters in this context.
Performance of ARO and HBA: ARO and HBA make significant improvements in the early iterations but fail to reach the same optimal level as mCDO, CDO, GJO, and MFO, stabilizing at higher fitness values.
Zoomed-In Section (Lower Graph): The zoomed-in portion of the graph further demonstrates that after the 5th iteration, mCDO, CDO, GJO, and MFO maintain their optimal fitness values at approximately 0.3556, highlighting their consistent and reliable performance.
In conclusion, the convergence curves in Fig. 11 demonstrate the superior performance of mCDO, GJO, MFO, and CDO in optimizing the TID controller parameters. These algorithms exhibit fast and stable convergence, making them the most effective in achieving the lowest ITAE values. On the other hand, the SWO algorithm shows a clear inability to effectively optimize the TID controller parameters, as evidenced by its slow progress and poor final result.
Figure 12 displays the optimization results for the PIDA controller, where the mCDO, HBA, and MFO algorithms all achieved the same optimal ITAE value of 0.3285. This outcome is further corroborated by Table 10, which lists identical ITAE values for these three algorithms.
A closer examination of the convergence curves reveals that HBA stabilized at the 17th iteration, demonstrating its ability to quickly converge on the optimal solution. This early convergence suggests that HBA is particularly efficient at identifying the best parameters with fewer iterations, making it highly computationally efficient.
In contrast, mCDO required more iterations to stabilize, with convergence occurring at the 23rd iteration. Although it took longer than HBA, mCDO showed steady progress toward the optimal solution. This suggests that mCDO might be more thorough in its search, exploring a broader range of possible solutions before selecting the best set of parameters.
Meanwhile, MFO followed a slightly different pattern, showing minor fluctuations early in the optimization process before settling at the same ITAE value. While MFO ultimately achieved the desired outcome, its more variable early behavior indicates that it may take more time to converge in certain cases, as it initially explores a wider parameter space.
On the other hand, SWO and ARO performed significantly worse, with SWO showing particularly slow convergence and a final ITAE value that was far from optimal. These results indicate that SWO and ARO are not as effective for optimizing the PIDA controller in this scenario, as they struggled to match the performance of mCDO, HBA, and MFO.
Overall, the convergence curves shown in Fig. 12 clearly demonstrate that HBA, mCDO, and MFO are the most effective algorithms for optimizing the PIDA controller, each reaching the same optimal ITAE value but with varying speeds and behaviors during the convergence process. The superior performance of HBA, particularly its quick stabilization, stands in contrast to the slower and less efficient results of SWO and ARO, which failed to converge effectively.
Following the observation that the mCDO algorithm consistently provided the best optimization performance among all tested algorithms, the frequency responses of the system using the PID, PIDn, FOPID, TID, and PIDA controllers, optimized by mCDO, were analyzed in detail. Figures 13, 14, 15, 16 and 17 illustrate the dynamic frequency responses in each area (Df1 and Df2) and the tie-line (Dptie).
Fig. 13
Response of the system with the PID controller tuned using mCDO
In Fig. 13, the frequency response for the system under the control of the mCDO-optimized PID controller shows an initial sharp frequency deviation, with Δf1 and Δf2 dropping to approximately -0.15 Hz before stabilizing. with the tie-line (Δptie) exhibiting a similar trend, indicating that while the PID controller manages to stabilize the system, it introduces moderate undershoot and oscillations during the transient phase.
Figure 14 presents the frequency responses when using the PIDn controller optimized by mCDO. In this case, the system exhibits less oscillatory behavior compared to the standard PID controller. The frequency deviations in both areas stabilize more quickly and with reduced overshoot, suggesting that the PIDn controller offers improved damping characteristics and a more controlled response under load disturbances.
In Fig. 15, the frequency responses for the FOPID controller optimized by mCDO demonstrate further improvements. The fractional-order controller results in a smoother system response, with minimal overshoot and quicker convergence to the steady state. The tie-line frequency deviation (Δptie) also shows better performance compared to the previous controllers, confirming that the FOPID controller provides superior dynamic response in managing frequency deviations.
The TID controller, as shown in Fig. 16, performs similarly to the FOPID controller, with rapid stabilization and minimal oscillations observed in both areas and the tie-line. The system returns to steady-state conditions efficiently, highlighting the TID controller’s capability to handle dynamic fluctuations with high precision.
Finally, Fig. 17 illustrates the frequency responses for the PIDA controller optimized by mCDO. The system experiences an initial drop in frequency, similar to the other controllers, but quickly stabilizes within a few seconds. Both areas and the tie-line show minimal deviations post-stabilization, indicating that the PIDA controller is well-suited for maintaining system stability in multi-area power systems.
In conclusion, the results from Figs. 13, 14, 15, 16 and 17 confirm that the mCDO algorithm is highly effective in optimizing various controllers. Among the controllers tested, the FOPID and TID controllers consistently exhibit the best dynamic performance, with rapid stabilization, minimal overshoot, and a smoother transition to steady-state. This highlights their potential for application in systems requiring precise frequency control and fast recovery from disturbances.
Table 12 summarizes the results extracted from the frequency response graphs shown in Figs. 13, 14, 15, 16 and 17. The table presents the undershoot (Us), overshoot (Os), and settling time (Ts) values derived from the frequency deviations in the two areas (Δf1 and Δf2) and the tie-line (Δptie). These values provide a quantitative comparison of the performance of the various controllers (PID, PIDn, FOPID, TID, and PIDA) optimized by the mCDO algorithm.
Table 12
Optimal controller parameters obtained with mCDO and comparison performance in terms of TS, US and OS
Controller
US (undershoot)
OS (overshoot)
TS (settling time)
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
PID
−0.1515
−0.1050
−0.0363
0.0088
0.00006
0.0001
4
5
5
PIDn
−0.1520
−0.1040
−0.0363
0.0088
0
0
4
5
5
FOPID
-0.1480
-0.1030
-0.0363
0.0083
0
0
3.6
4.8
4.4
TID
−0.1500
−0.1045
−0.0370
0.0128
0.00108
0
3.7
5.1
4.9
PIDA
−0.1510
−0.1040
−0.0363
0.00885
0
0
4
5
5
From the data presented in Table 12, it is evident that the mCDO-optimized FOPID controller consistently delivers superior performance in the TAPS. Specifically, the FOPID controller exhibits the lowest overshoot (0.0083 for Δf1) and undershoot values (− 0.1480 for Δf1), as well as the shortest settling time (3.6 s for Δf1). These values clearly indicate that the FOPID controller provides better dynamic performance in terms of minimizing frequency deviations and rapidly restoring system stability after disturbances.
In contrast, while the PID and PIDn controllers also manage to stabilize the system, their performance is inferior to that of the FOPID controller. The PID and PIDn controllers exhibit similar settling times, with both requiring 4–5 s to stabilize the system. However, the overshoot and undershoot values for these controllers are slightly higher than those for the FOPID controller, suggesting that they are less effective in mitigating transient frequency deviations.
The TID controller performs similarly to the FOPID controller in terms of settling time, with a slightly longer stabilization period (3.7–5.1 s). However, its overshoot (0.0128 for Δf1) and undershoot values (− 0.1500 for Δf1) are higher, indicating that the TID controller introduces more oscillatory behavior during the transient period.
Finally, the PIDA controller also demonstrates strong performance, with its overshoot and undershoot values comparable to those of the PIDn controller. However, it does not outperform the FOPID controller in any of the evaluated metrics, suggesting that the FOPID controller is the optimal choice for frequency control in TAPS when optimized by the mCDO algorithm.
In conclusion, the data in Table 12, combined with the graphical results from Figs. 13, 14, 15, 16 and 17, confirm that the FOPID controller optimized by mCDO provides the best overall performance, offering the fastest stabilization time and the lowest overshoot and undershoot values. This highlights the effectiveness of the FOPID controller for achieving precise and rapid frequency control in multi-area power systems.
When analyzing the undershoot (Us) values, it is evident that the FOPID controller provides the best performance, particularly for Δf1 and Δf2. Specifically, the undershoot values are Δf1 = −0.1480 and Δf2 = −0.1030, which are the lowest across all controllers. The tie-line frequency deviation Δftie = − 0.0363 remains consistent with the values obtained for the PID, PIDn, and PIDA controllers, suggesting similar behavior in stabilizing the tie-line frequency.
In terms of overshoot (Os), the FOPID controller also demonstrates superior control performance. With an overshoot value of Δf1 = 0.0083, it records the lowest overshoot among the controllers tested. Additionally, both Δf2 and Δftie maintain overshoot values of 0, indicating that the FOPID controller is highly effective at minimizing transient peaks during the frequency recovery process.
When evaluating the settling time (Ts), the FOPID controller again proves to be the most efficient. The settling times for the frequency deviations are Δf1 = 3.6 s, Δf2 = 4.8 s, and Δftie = 4.4 s, representing the fastest stabilization times among all the controllers. This rapid convergence to steady-state conditions highlights the FOPID controller’s effectiveness in mitigating frequency deviations and quickly restoring system equilibrium.
Thus, the FOPID controller, when optimized by the mCDO algorithm, outperforms the other controllers in terms of undershoot, overshoot, and settling time. This makes it the most suitable controller for achieving precise and efficient frequency control in a TAPS.
Figure 18 presents the combined results obtained from all controllers.
Fig. 18
Comparison of the results obtained from all five controllers
Finally, a robustness analysis of the system was conducted using a FOPID controller optimized by mCDO under continuous load variation conditions. As shown in Fig. 19, high-power loads were repeatedly connected and disconnected in both Area 1 and Area 2 to simulate dynamic operating conditions. The system’s response to these disturbances, in terms of frequency deviations, is illustrated in Fig. 20.
During this test, continuous load changes were applied to both areas, and the frequency deviations at the tie-line and in each area were monitored. As depicted in Fig. 20, the system initially exhibited frequency disturbances due to the load changes. However, the frequency deviations in both areas quickly converged back to zero, indicating that the system stabilized within a short period.
The system response for each variation interval under varying load conditions is detailed comprehensively in Tables 13, 14, 15, and 16.
Table 13
Load variation conditions with mCDO and performance in terms of TS, US and OS (between t = 1 s and 5 s)
Controller
US (undershoot)
OS (overshoot)
TS (settling time)
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
FOPID
−0.3
−0.205
−0.07275
0.0165
0
0
4.6
3.3
4.5
Table 14
Load variation conditions with mCDO and performance in terms of TS, US and OS (between t = 5 s and 15 s)
Controller
US (undershoot)
OS (overshoot)
TS (settling time)
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
FOPID
−0.72
−1.042
0
0
0.058
0.254
8.8
8.8
9
Table 15
Load variation conditions with mCDO and performance in terms of TS, US and OS (between t = 15 s and 20 s)
Controller
US (undershoot)
OS (overshoot)
TS (settling time)
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
FOPID
−0.593
−1.042
−0.145
0.033
0
0
18.6
17.1
18.1
Table 16
Load variation conditions with mCDO and performance in terms of TS, US and OS (between t = 20 s and 25 s)
Controller
US (undershoot)
OS (overshoot)
TS (settling time)
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
Δf1
Δf2
Δftie
FOPID
0
0
−0.0728
0.206
0.296
0
22.3
22.3
23.6
Here, the settling times (Ts) represent the first instance where the frequency deviation reduces to zero within the specified time intervals. The system’s rapid recovery and stabilization demonstrate the effectiveness of the FOPID controller in mitigating disturbances. Despite the high-power load variations, the frequency deviations are minimal, and the system maintains stability without significant oscillations. This highlights the robustness of the FOPID controller, which provides a fast and reliable response to load changes, ensuring system balance. The results confirm that the system is highly resilient and capable of handling continuous load changes with minimal impact on overall performance.
4 Discussion
In this study, load frequency control parameters in a TAPS were adjusted using a FOPID controller optimized by the mCDO algorithm. The effectiveness of mCDO was compared with other optimization algorithms, including ARO, CDO, GJO, HBA, MFO, and SWO. The performance of the FOPID controller was also evaluated against PID, PIDn, TID, and PIDA controllers. Based on the simulation results, the following conclusions were drawn:
mCDO either provided the best results or achieved outcomes comparable to the other algorithms for tuning controller parameters. This indicates that mCDO performs well and can be reliably used for controller parameter optimization.
Although CDO and mCDO are fundamentally the same algorithm, mCDO outperforms CDO. The modifications made in mCDO enhance its performance.
The performance of the optimization algorithms was evaluated using the ITAE (Integral of Time-weighted Absolute Error) index. Other error performance indices were also examined, but the results remained consistent. As ITAE is widely used in the literature, the final results were presented based on this index.
In multi-area power systems, maintaining frequency stability during load variations requires minimizing harmonics and ensuring that the system’s response occurs quickly and with minimal amplitude deviation.
The simulation results for the FOPID controller showed that the overshoot (Os) and undershoot (Us) values were significantly low, and the settling time (Ts) was remarkably short.
The robustness analysis demonstrated that the FOPID controller provided a rapid response, allowing the system to return to equilibrium in a short period.
When reviewing similar studies in the literature, it was observed that the Os and Us values showed partial improvement in Δf1 and Δf2, while a significant improvement was achieved in the tie-line frequency (Δftie). The settling time also improved. Furthermore, the steady-state error was zero across all controllers [19].
In conclusion, these findings indicate that load frequency control in multi-area power systems can be effectively and efficiently achieved by using a FOPID controller optimized with the mCDO algorithm.
5 Conclusion and future recommendations
This paper has studied the frequency stability in a two-area interconnected power system under dynamic load variations. Five controllers—PID, PIDn, FOPID, TID, and PIDA—were designed and optimized using seven advanced metaheuristic algorithms, including ARO, CDO, mCDO, GJO, HBA, MFO, and SWO. A total of 35 simulation scenarios were evaluated using the Integral of Time-Weighted Absolute Error (ITAE) metric, along with system settling time, overshoot, and undershoot.
The results demonstrate that the FOPID controller optimized with the mCDO algorithm achieved the best performance, attaining the lowest ITAE (0.320684), minimal overshoot (0.0083 Hz) and undershoot (− 0.1480 Hz), and a settling time of 3.6 s. Compared to conventional PID controllers, this configuration reduced settling time by 10% and significantly enhanced frequency stability under dynamic load conditions. The mCDO algorithm, by integrating neighborhood–global and wandering search strategies, effectively improved the exploration–exploitation balance of the original CDO, leading to faster convergence and more precise parameter tuning. These findings indicate that the FOPID-mCDO combination is a highly promising approach for automatic generation control in multi-area power systems.
Future Recommendations:
Further improvements may be achieved by exploring hybrid or intelligent control systems.
Optimization of controller parameters remains crucial, and new algorithms should be tested under consistent conditions for reliable performance evaluation.
Increasing the number of agents or iterations in optimization algorithms can enhance performance but requires careful monitoring of convergence behavior and simulation time.
Future research should investigate alternative algorithm-controller combinations to achieve even better system performance under varying load scenarios.
Declarations
Conflict of interest
The authors declare no competing interests.
Ethical approval
The author declares that this article complies the ethical standard.
Declaration of generative AI and AI-assisted technologies in the writing process
While preparing this work, the authors used ChatGPT to improve language and readability. After using this tool, the authors reviewed and edited the content as needed and took full responsibility for the publication’s content.
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