Emission-Constrained Dispatch Optimization Using Adaptive Grouped Fish Migration Algorithm in Carbon-Taxed Power Systems

Abstract

With increasing global pressure to decarbonize electricity systems, particularly in regions outside international carbon trading frameworks, it is essential to develop adaptive optimization tools that account for regulatory policies and system-level uncertainty. An emission-constrained power dispatch strategy based on an Adaptive Grouped Fish Migration Optimization (AGFMO) algorithm is proposed. The algorithm incorporates dynamic population grouping, a perturbation-assisted escape strategy from local optima, and a performance-feedback-driven position update rule. These enhancements improve the algorithm’s convergence reliability and global search capacity in complex constrained environments. The proposed method is implemented in Taiwan’s 345 kV transmission system, covering a decadal planning horizon (2023–2033) with scenarios involving varying load demands, wind power integration levels, and carbon tax schemes. Simulation results show that the AGFMO approach achieves greater reductions in total dispatch cost and CO₂ emissions compared with conventional swarm-based techniques, including PSO, GACO, and FMO. Embedding policy parameters directly into the optimization framework enables robustness in real-world grid settings and flexibility for future carbon taxation regimes. The model serves as decision-support tool for emission-sensitive operational planning in power markets with limited access to global carbon trading, contributing to the advanced modeling of control and optimization processes in low-carbon energy systems.

1. Introduction

As climate change mitigation efforts accelerate, carbon trading schemes have become essential instruments for incentivizing low-carbon transitions in energy systems. By assigning economic value to greenhouse gas emissions, these markets encourage utilities and industrial actors to invest in cleaner technologies and reduce their carbon footprint [1,2]. Their widespread adoption has significantly reshaped dispatch strategies and long-term planning in the power sector [3,4].

However, not all regions benefit from participation in such mechanisms. Taiwan, for instance, remains excluded from international carbon markets due to geopolitical constraints [5,6]. This exclusion poses a structural challenge: without access to carbon pricing signals or international offsetting mechanisms, Taiwan must pursue decarbonization through internally driven, policy-compatible dispatch strategies.

Despite recent growth in wind and solar deployment, Taiwan’s power system remains predominantly fossil-fueled, with coal and natural gas accounting for the bulk of generation [7]. The intermittency of renewables adds further complexity, making it difficult to reconcile emission goals with grid reliability [8,9]. Although Taiwan’s renewable energy development includes both solar PV and wind energy, this study focuses primarily on wind power for several reasons. First, Taiwan’s government has designated offshore wind as a core strategy in its long-term decarbonization plan, with substantial capacity expansion planned through to 2030. Second, wind energy exhibits greater resilience during extreme weather events, as evidenced by continued wind turbine operation during Typhoon Danas in 2025, while PV installations suffered widespread damage. Third, due to land and geographical constraints, PV expansion is expected to plateau, making wind energy a more viable long-term lever for carbon reduction. As such, this paper focuses on wind-based dispatch modeling to provide policy-relevant insights and can serve as a reference for similar regions globally.

Existing research offers limited guidance for such non-market contexts. For example, Qiu and Entchev [10] and De Carne et al. [11] focus on renewable variability within market-integrated systems, while Wu et al. [12] and Leal Filho et al. [13] emphasize the role of carbon pricing in reshaping fossil-fuel usage—an option unavailable in Taiwan.

More broadly, the literature assumes the presence of carbon markets, leaving a strategic and methodological gap for regions without such mechanisms. Hameed et al. [14] present optimization models embedded within market structures, and Bechara and Alnouri [15] explore carbon-constrained planning but stop short of addressing renewable-induced dispatch volatility. Han et al. [16] consider demand response and storage, yet do not examine how fossil-fuel-heavy systems maintain dispatch feasibility during energy transitions. Likewise, studies by Russo et al. [17] and Gao et al. [18] address renewable integration in island contexts but do not grapple with the policy isolation that defines Taiwan’s case.

In addition, while optimization techniques such as PSO, NSGA-II, and MOPSO have been applied to emission-constrained dispatch [19,20,21,22,23,24], they often rely on problem structures and parameter regimes that presume market participation. For example, Li et al. [21] and Li and Liu [22] model trading environments or microgrids, but their solutions are poorly suited to national-scale, policy-constrained systems. Liu et al. [23] discuss AI-driven decarbonization without tailoring methods to the technical dynamics of grid-scale fossil–renewable coexistence. Qin et al. [25] offer low-carbon dispatch models but do not tackle algorithmic robustness under volatile load and fuel-price scenarios.

These gaps are addressed through a tailored dispatch optimization method based on an Adaptive Grouped Fish Migration Optimization (AGFMO) algorithm. Designed specifically for systems operating outside of carbon market frameworks, AGFMO integrates genetic recombination strategies with grouped migration dynamics to improve global search capacity and solution adaptability. The proposed method is implemented within a dispatch model that reflects Taiwan’s energy mix, renewable growth projections, and carbon taxation policies.

The contributions of this work are threefold:

(1)

Algorithmic innovation: AGFMO is extended the traditional fish migration model through adaptive dynamic grouping and local disturbance mechanisms, enhancing exploration capacity in constrained, high-renewable scenarios. It addresses the convergence limitations often observed in PSO-based dispatch models.

(2)

Context-specific dispatch framework: The model is explicitly developed for fossil-fuel-dependent grids operating without international market access. It provides actionable strategies for emission-aware dispatch and renewable integration under isolated policy regimes.

(3)

Scalable application for isolated or developing regions: While tailored to Taiwan, the framework is designed to be flexible enough to be adapted to other regions facing similar geopolitical or infrastructural constraints. The framework offers a viable carbon reduction pathway that complements international climate objectives without requiring full market integration.

By addressing both methodological and contextual gaps, this study contributes to the global dialog on low-carbon energy transitions in policy-constrained environments. It offers both a robust optimization tool and a replicable dispatch strategy for achieving meaningful emission reductions in systems that fall outside conventional carbon pricing mechanisms.

Following this introduction, Section 2 uses data mining techniques to develop wind power output forecasts and analyze installed capacity utilization, providing a data-driven foundation for the emission-constrained dispatch model formulated in Section 3. Section 3 defines the optimal power flow framework under carbon emission constraints on Taiwan’s 345 kV system, establishing the context-specific dispatch optimization problem addressed in this work. Section 4 details the design of the proposed AGFMO algorithm and its enhancements over the standard fish migration approach, highlighting how this algorithmic innovation improves global search and convergence for the dispatch optimization. Section 5 presents the simulation studies and results, including scenario analyses under various carbon tax and renewable integration levels. The efficacy of the proposed approach and its advantages over conventional methods are demonstrated by these findings, illustrating the method’s viability and scalability for emission-constrained dispatch planning. Finally, Section 6 provides a summary of key insights and a critical analysis, linking the outcomes back to the contributions and discussing implications for low-carbon power system operation.

2. Application of Data Mining in Wind Power Output and Installed Capacity Forecasting

2.1. Application of Data Mining in Wind Power Forecasting

Forecasting wind power output is notoriously difficult, largely due to the inherent variability caused by shifting weather patterns, seasonal dynamics, and local geographical features. Taiwan, with its growing portfolio of wind energy installations, faces an urgent need to improve the predictability of wind power output (WPO) and installed capacity (WIC) to manage grid stability and ensure efficient power dispatch [26].

To address this challenge, data mining was adopted—not merely as a statistical tool, but as a means to uncover hidden regularities and inform smarter decision-making. Drawing on several years of historical data in Taiwan, a forecasting model was developed that is designed to support day-ahead power dispatch [27,28].

The analysis revealed patterns that, while not unexpected, are highly actionable; once cleaned and processed, these patterns can reveal insights beyond what raw intuition can provide.

(1)

Data Collection and Cleaning:

The dataset spanned multiple years and included site-level data such as wind speed, power output, temperature, and installed capacity. Basic preprocessing was applied: obvious outliers were removed, and missing values imputed based on local regression. While the dataset was relatively clean, some anomalies were still identified and manually verified.

(2)

Feature Selection and Modeling:

Using multivariate regression analysis, wind speed, ambient temperature, and month of the year were identified as the most predictive variables. The resulting model (see below) offers a practical baseline for regional wind output estimation:

$$P_{wind}={\alpha }_0+{\alpha }_1·Win{d}_{speed}+{\alpha }_2·Temperature+{\alpha }_3·Season+ϵ$$

(1)

where $P_{wind}$ represents the predicted wind power generation, and α₁, α₂, α₃ are regression coefficients. The residual term $ϵ$ captures the deviation between the observed and predicted values, accounting for random fluctuations and other external factors not explicitly modeled. By applying multivariate regression, this model captures the relationships between wind power output and various influencing factors.

(3)

Construction of the wind power forecasting model

Based on the results of the regression analysis, a forecasting model for wind power generation was developed. This model not only reflects patterns in historical data but also dynamically adjusts forecasts using current meteorological data. The forecasting model can be expressed as follows:

$${\widehat{P}}_{wind}=f\left(Win{d}_{speed}, Temperature, Season, Variation\dots \right)$$

(2)

The model is used to predict wind power output under different meteorological conditions, providing a reliable input for optimizing future power dispatch [27,28].

2.2. Relationship Between Data Mining and Installed Capacity

Beyond forecasting wind power generation, this study also investigates how installed wind capacity relates to actual generation efficiency. Understanding how effectively installed capacity is utilized—especially under varying operational and environmental conditions—provides insight into overall system performance. To this end, data mining techniques were applied to uncover trends and correlations that would otherwise remain hidden in the raw data [28].

(1)

Installed capacity analysis:

Data were collected from both onshore and offshore wind farms across Taiwan, including variables such as nameplate capacity, maintenance schedules, downtime records, and actual generation figures. By comparing output under different load conditions, the effective utilization rate of installed capacity was estimated. Notably, offshore installations showed greater variability during seasonal transitions, likely due to harsher environmental exposure [26].

(2)

Modeling the Efficiency Relationship:

To quantify this relationship, a regression-based model was constructed in which generation efficiency was treated as the dependent variable and installed capacity as the key predictor. The model can be expressed as:

$$E{f}_{wind}={\beta }_0+{\beta }_1·S_{installed}+{\beta }_2·T_D+{\beta }_3·M_w+\rho$$

(3)

where $E{f}_{wind}$ represents generation efficiency, $S_{installed}$ denotes the installed capacity, β₀, β₁, β₂, β₃ are regression coefficients. T_D stands for the transmission distance from the wind farm to the nearest grid connection point (km), M_w is the site’s average wind speed (m/s), and $\rho$ is the error term. This model enables the prediction of generation efficiency under varying installed capacity conditions [28]. According to the regression analysis conducted in [28], the estimated values of the coefficients are β₀ = 0.35, β₁ = 0.0024, β₂ = −0.0018, and β₃ = 0.045, which reflect the empirical influence of capacity size, grid distance, and wind conditions on generation performance.

(3)

Integrating forecast and optimization:

The insights obtained through data mining are directly fed into the power dispatch optimization framework. By anticipating shifts in demand and generation potential, resources can be allocated more effectively and dispatch can be scheduled more efficiently. For instance, knowing that offshore capacity drops by 12–15% during typhoon season enables preemptive load rerouting or reserve activation by operators. These predictive components, when integrated into the AGFMO-based optimization (detailed in the next section), enhance not only dispatch precision but also the resilience of the system under carbon emission constraints [29].

By analyzing wind power output in relation to installed capacity through data mining methods, this chapter establishes a data-driven foundation for optimizing power dispatch. The insights gained here—particularly the predictive understanding of utilization rates—will inform the optimization strategy presented in the next section where the design of the AGFMO algorithm and its application to achieving optimal dispatch under carbon emission constraints are presented.

3. Optimal Power Flow (OPF) with Carbon Emission Constraints

To enhance the clarity of the solution framework, an integrated flowchart is provided in Figure 1. This diagram illustrates the main procedural steps of the emission-constrained OPF model, highlighting its integration with the wind utilization prediction module developed in Section 2. Specifically, the forecasted wind availability is treated as a constraint within the OPF formulation. The AGFMO algorithm employs the dispatch of fossil-fuel generators, taking into account both fuel costs and carbon taxation schemes. The resulting outputs include optimal power generation schedules and associated carbon cost metrics. This integration enables a practical low-carbon dispatch strategy suited to policy-constrained energy systems to be developed.

The traditional OPF formulation is extended by incorporating carbon dioxide emission constraints. Using the 345 kV transmission system of the Taiwan Power Company, carbon emission control is simulated under various load conditions for 2023, 2028, and 2033, covering off-peak, average, and peak loads. The objective is to determine whether Taiwan’s carbon emission cap can be met under actual power generation costs, or to estimate the excess emissions if the cap is exceeded. The study also integrates independent power producers (IPPs) for optimal emission control and evaluates the impact of renewable energy investments. To simplify comparing emissions and fuel costs, carbon pricing from international carbon markets is used to convert emissions into the same currency as fuel costs for analysis.

3.1. Objective Equation

The following equations were developed to minimize the cost of fuel dispatch in a power system considering carbon dioxide emissions.

$$C_{gen}={\displaystyle \sum }_{k=1}^N{a}_k+b_k{P}_k+c_k{P}_k^2$$

(4)

$$C_{C{O}_2}=\left({\displaystyle \sum }_{k=1}^N{\alpha }_k+{\beta }_k{P}_k+{\gamma }_k{P}_k^2\right)·{\lambda }_{C{O}_2}$$

(5)

$$\mathrm{Min}\left(C_{total}\right)=\min \left(C_{gen}+C_{C{O}_2}\right)$$

(6)

where N represents the total number of generators in the system, while P_k denotes the power output of the k-th generator. The cost structure includes several components: $a_k$ refers to the fixed cost associated with operating the k-th generator, b_k is the linear fuel cost coefficient that indicates how fuel costs increase proportionally with the power output, and c_k captures the quadratic fuel cost coefficient, accounting for the non-linear increase in fuel consumption as the generator output increases. On the emissions side, ${\lambda }_{C{O}_2}$ represents a flat-rate carbon cost per ton of CO₂, which, when combined with generator-specific emission factors, determines the total emission-related cost. This modeling approach reflects a fixed carbon tax scenario, in line with Taiwan’s current situation where no official carbon pricing mechanism has been implemented, and policy development remains in discussion. The inclusion of ${\lambda }_{C{O}_2}$ within the objective function enables direct incorporation of carbon-related penalties into dispatch optimization. Specifically, α_k represents the fixed carbon emissions from the k-th generator, while the linear and quadratic components of carbon emissions are described by β_k and γ_k, respectively, reflecting how emissions vary with power output.

The coefficients a_k, b_k, c_k and α_k, β_k, γ_k are parameterized according to the typical characteristics of fossil-fueled generators in Taiwan. Coal-fired units are modeled with higher baseline cost and emission levels, where a_k = 510, b_k = 14.3, and c_k = 0.0045, while the corresponding emission coefficients are set as α_k = 0.215, β_k = 0.0038, and γ_k = 0.000012. For gas-fired units, the operational cost is relatively lower in fixed and quadratic terms, with a_k = 780, b_k = 19.6, and c_k = 0.0078, and emissions are accordingly reduced with coefficients α_k = 0.172, β_k = 0.0025, and γ_k = 0.000009. Oil-fired units, characterized by their higher carbon intensity and cost, are represented by a_k = 1200, b_k = 36.5, and c_k = 0.0102, alongside emission coefficients of α_k = 0.26, β_k = 0.0045, and γ_k = 0.000015. These parameter values are selected with reference to the Taiwan Power Company’s public disclosures and national emission factor databases [26], ensuring that the dispatch and emission cost modeling is grounded in realistic, region-specific data.

3.1.1. Equality Constraints

The equality constraints ensure that the total power generated by the main generator buses, along with the wind turbine output, must be equal to the total load demand plus transmission losses. The load balance equation can be formulated as follows:

$${\displaystyle \sum }_{k=1}^N{P}_{gen, k}+ {\displaystyle \sum }_{j=1}^y{P}_{wind, j}={\displaystyle \sum }_{i=1}^x{P}_{load, k}+ P_{loss}$$

(7)

where P_gen,k represents the power output of the k-th generator bus, while $P_{wind, j}$ indicates the power produced by the j-th wind turbine bus. The total load at the i-th load bus is denoted by P_load,k and P_loss accounts for the total transmission losses within the system. Additionally, the variables N, y, and x correspond to the total number of generator buses, wind turbine buses, and load buses, respectively.

3.1.2. Inequality Constraints

The inequality constraints reflect the operational limits of the system, including generation capacity bounds, bus voltage ranges, wind power output limits, and overall carbon emission caps. These constraints ensure that the power system operates safely, efficiently, and within the boundaries of environmental regulations. The mathematical formulation of these inequality conditions is as follows:

$$\underset{¯}{P_{gen, k}}\le P_{gen, k}\le \overline{P_{gen, k}}$$

(8)

$$\underset{¯}{U_e}\le U_e\le \overline{U_e}$$

(9)

$$0\le P_{wind}\le \overline{P_{wind}}$$

(10)

$$S_d\le \overline{S_d}$$

(11)

$${\displaystyle \sum }_{k=1}^{N}C{O}_{2,k}\le \overline{C{O}_{2,k}}$$

(12)

The system operates under several constraints. The generation limits for each generator bus, denoted by $\underset{¯}{P_{gen, k}}$ and $\overline{P_{gen, k}}$, set the minimum and maximum allowable power output, while the voltage at each bus e must remain within the bounds $\underset{¯}{U_e}$ and $\overline{U_e}$. The total number of buses in the system is represented by N. For wind turbines, their generation is capped at $\overline{P_{wind}}$. Additionally, transmission line d has a capacity limit defined by $\overline{S_d}$. Regarding environmental considerations, the carbon emissions from the k-th generator bus, denoted as $C{O}_{2,k}$, must not exceed the annual carbon emission limit, $\overline{C{O}_{2,k}}$.

In this study, all wind energy generated is directly dispatched without curtailment. Photovoltaic sources are not considered in the optimization model due to focus on wind-based dispatch, which is consistent with Taiwan’s long-term offshore wind policy emphasis. Battery energy storage systems (BES) are not included in the current formulation, as the study focuses on direct dispatch optimization under renewable uncertainty and fossil balancing, reflecting real-world operational constraints in Taiwan.

3.2. The ECI for Power Flow Solution

The Equivalent Current Injection (ECI) method is employed [30,31] for solving the power flow, as it provides faster convergence compared to the Newton-Raphson method, especially in systems with complex load conditions. Load scenarios are evaluated for short-term (2023), medium-term (2028), and long-term (2033) with the integration of onshore and offshore wind turbines. The focus is on minimizing the power system’s fuel dispatch cost while considering carbon dioxide emissions. The procedure for the equivalent current injection (ECI) method can be described as follows:

Step 1:

The process begins with defining the allowable error threshold, typically set below 10⁻⁶.

Step 2:

Import the system data required for analysis, which includes bus data, transmission line details, as well as predefined voltage settings and real/reactive power values.

Step 3:

Formulate the admittance matrix (Y_Bus) for the power system.

Step 4:

Check for the presence of any voltage-controlled buses (PV Bus) in the system.

If present, proceed to Step 5;

If absent, move to Step 6.

Step 5:

Make necessary adjustments to the Jacobian matrix before moving to Step 6.

Step 6:

Perform LU decomposition on the Jacobian matrix.

Step 7:

Compute the equivalent current injection at load buses and calculate the error values for the real and imaginary components of the generator bus voltage.

Step 8:

Use the computed voltage error values to update the voltage values.

Step 9:

Check if the computed error falls within the acceptable range (error $\le$ tolerance). If so, the process concludes; otherwise, return to Step 5 for further iterations.

A detailed derivation of the ECI method can be found in [32]. This method’s ability to handle complex and dynamic load conditions, while minimizing fuel and carbon costs, highlights its suitability for modern power systems with integrated renewable energy sources.

4. Adaptive Grouped Fish Migration Optimization (AGFMO) with Carbon Emission Constraints

4.1. AGFMO

The Fish Migration Optimization (FMO) algorithm draws inspiration from the natural behaviors of fish, particularly their migratory patterns in search of food, breeding grounds, and favorable environmental conditions [32]. These biologically driven behaviors are abstracted into a computational framework capable of addressing complex numerical optimization problems.

Building upon the standard FMO, an improved variant, termed Adaptive Grouped Fish Migration Optimization (AGFMO), is proposed. The core enhancement lies in the integration of an adaptive dynamic population grouping strategy (ADGPS), which facilitates a more effective balance between global exploration and local exploitation throughout the optimization process. Specifically, AGFMO partitions the fish population into subgroups based on their fitness values, allowing different search behaviors to emerge dynamically during the algorithm’s evolution. This design enables the algorithm to more efficiently navigate multi-objective and constrained optimization landscapes—such as power dispatch problems under carbon emission limits.

The original FMO relies on a simplified bioenergetic model to describe energy consumption during fish migration. This is represented mathematically as:

$$E=a·t+a·U_s^x$$

(13)

where E represents the energy consumption, t is the time spent, U_s is the swimming speed of the fish, and a, b, and x are constants related to metabolic rate, energy use scaling, and swimming speed, respectively. In this study, the parameter values were adopted from existing literature on fish bioenergetics modeling, with a = 0.6, b = 1.2, and x = 2.0, reflecting typical metabolic scaling observed in similar agent-based or population-based models [32]. These constants ensure that the model realistically captures the trade-off between exploration and exploitation in the fish migration process, as adapted in the original FMO framework.

In AGFMO, the population of fish is dynamically divided into subgroups based on their fitness values. This strategy allows the algorithm to switch between global exploration and local exploitation, increasing the chances of finding a global optimum while still refining local solutions.

The steps of the AGFMO algorithm with ADGPS are as follows:

(1)

Initialization: The initial fish population is generated by randomly assigning positions and velocities within predefined bounds. Each fish’s position P_i represents a candidate solution to the optimization problem (e.g., generator dispatch levels), and is initialized as:

$$P_i=P_{min}+rand()·\left(P_{max}-P_{min}\right)$$

(14)

where P_min and P_max represent the lower and upper bounds of the search space.

(2)

Fitness function: Each fish is evaluated using a fitness function that accounts for both economic and environmental objectives. Specifically:

$$\mathrm{Fitness}\left(P_i\right)={\omega }_1{C}_{gen}\left(P_i\right)+{\omega }_2{C}_{CO2}\left(P_i\right)$$

(15)

where C_gen(P_i) is the power generation cost of solution P_i, C_CO2(P_i) is the carbon emission cost of the same solution, ω₁ and ω₂ are weight factors representing the relative importance of generation cost and carbon emissions, respectively.

(3)

Swim process: In each iteration, every fish evaluates multiple candidate positions within its movement range before selecting the best one. The swimming distance is governed by its velocity Us and a time t parameter:

$$d=U_s·t$$

(16)

The energy consumption during swimming is given by:

$$E=a·U_s^x$$

(17)

After evaluating all candidates, the fish moves to the position with the highest fitness. The parameters for the energy model are adopted based on standard bioenergetic formulations, where a = 0.6 and x = 2.0, consistent with metabolic scaling observed in previous fish migration optimization literature [32]. The swimming time t is normalized to 1 for simplicity, allowing U_s to directly represent the movement range in each iteration.

(4)

Dynamic grouping: The population is dynamically divided into two groups based on fitness:

(1)

Local search group: Fish with higher fitness values perform fine-tuned local exploitation around known optima.

(2)

Global search group: Lower-performing individuals explore under-sampled regions of the search space to maintain population diversity and avoid premature convergence.

(5)

Migration process: Fish that fail to improve their fitness through local search are reallocated to new global positions, a process aimed at escaping local optima. Their updated position is calculated using a velocity-based rule inspired by social best guidance:

$$P_i\left(t+1\right)=P_i\left(t\right)+U_i\left(t\right)$$

(18)

where $P_i\left(t\right)$ is the current position, and $U_i\left(t\right)$ is the updated velocity based on the fish’s best position and the global best position found so far.

Cooperative evolution: At each generation, cooperative crossover allows individuals from different groups to exchange information. The offspring position is generated using:

$$P_{new}=\beta ·P_{partent1}+\left(1-\beta \right)·P_{partent2}$$

(19)

where β is a parameter controlling the contribution of each parent, and P_parent1 and P_parent2 are selected from different groups.

4.2. AGFMO for Solving the Power Dispatch Problem with Carbon Emission Constraints

As formulated in Section 3, the carbon-constrained optimal power flow (OPF) problem aims to minimize the total generation cost and associated CO₂ emissions while adhering to a set of operational and environmental constraints. These include generator output limits, transmission line capacities, and regulatory caps on carbon emissions. Equation (4) defines the multi-objective cost function, integrating both economic and environmental components.

To solve this non-linear and high-dimensional optimization problem, the AGFMO algorithm is applied. AGFMO is well-suited for the carbon-constrained dispatch setting, as it combines adaptive subgrouping, cooperative search dynamics, and bio-inspired movement strategies to balance convergence and diversity.

Within the AGFMO framework, each fish is used to represent a candidate dispatch vector P = {P₁, P₂, …, P_n}, corresponding to the power outputs of all generating units. The fitness function evaluates each fish based on a weighted sum of fuel cost and CO₂ emissions:

$$\min \left({\omega }_1·C_{gen}·P_i\left(t\right)+{\omega }_2·C_{C{O}_2}·P_i\left(t\right)\right)$$

(20)

where ω₁ and ω₂ are weights that reflect the importance of generation cost and carbon emissions, respectively.

The Adaptive Grouped Fish Migration Optimization (AGFMO) algorithm addresses the constrained optimization problem by iteratively guiding a population of candidate solutions—represented as individual “fishes”—toward the optimal power dispatch configuration. The algorithm proceeds through the following steps:

• Step 1: Initialization

(1)

Population Initialization: Each fish is randomly initialized within permissible generation bounds, where its position P_i corresponds to a specific generation schedule across all units.

(2)

Velocity Initialization: Each fish is also assigned an initial velocity vector U_i, representing its search direction in the solution space.

• Step 2: Fitness Evaluation

Each fish is evaluated based on a composite fitness function that integrates both fuel generation cost and carbon emission penalties.

• Step 3: Dynamic grouping

Using the ADGPS, the population is split into two subgroups based on fitness:

(1)

Local search group: Fish with better fitness values perform local exploitation, refining the solutions in their immediate neighborhood.

(2)

Global search group: Fish with lower fitness values explore new regions of the search space, ensuring diversity and preventing premature convergence.

• Step 4: Local search (swimming process)

For fish in the local search group, a local search is applied within a smaller range. Each fish evaluates potential new positions by updating its position and velocity based on its current best-known position and the global best solution.

The velocity update is influenced by the fish’s personal best position P_best and the global best position P_{global_best}:

$$U_i\left(t+1\right)=\omega ·U_i\left(t\right)+c_1·r_1\left(P_{best}-P_i\left(t\right)\right)+c_2·r_2\left(P_{global\_best}-P_i\left(t\right)\right)$$

(21)

where $\omega$ is the inertia weight, c₁ and c₂ are cognitive and social coefficients, r₁ and r₂ are random numbers between 0 and 1.

• Step 5: Global search (migration process)

For fish in the global search group, a migration step is performed to explore the global search space.

• Step 6: Cooperative evolution

Periodically, the fish in the local and global groups exchange information through cooperative evolution. This exchange enables the global group to benefit from the knowledge of the local group and vice versa.

• Step 7: Update fitness and selection

New candidate positions are evaluated. If a fish’s new position yields improved fitness, it is retained; otherwise, the fish reverts to its previous best-known solution.

• Step 8: Termination

The algorithm halts when either the maximum number of iterations is reached or when improvements in the global best solution fall below a predefined threshold over successive generations.

• Step 9: Return the Optimal Dispatch Solution

Upon convergence, the fish with the best fitness value is taken to represent the optimal dispatch configuration that minimizes both cost and emissions while satisfying all system constraints.

The following Algorithm 1 outlines the AGFMO pseudocode for solving the power dispatch problem:

Algorithm 1. Pseudocode of the AGFMO algorithm.

Initialize parameters: Set population size N, maximum iterations T, velocity limits, and generator output bounds. Initialize fish population: P ← Random positions Pi within bounds for each fish; U ← Random initial velocities Ui for each fish; Evaluate initial fitness: For each fish Pi in population:    Compute fitness: (Pi)$={\omega }_1·C_{gen}·P_i\left(t\right)+{\omega }_2·C_{C{O}_2}·P_i\left(t\right)$    Store personal best position: Pbest_i ← Pi Set global best: Gbest ← best(Pbest_i) While iteration < T do:    Sort fish by fitness F(Pi)    Divide population into two groups:    Local search group: top-performing fish;    Global search group: remaining fish;    For each fish in local search group:    Update velocity;    Update position;    Apply bounds to Pi;    Re-evaluate fitness F(Pi);    If F(Pi) < F(Pbest_i):       Pbest_i = Pi    For each fish in global search group:    Migrate to a new random position within bounds;    Re-evaluate fitness F(Pi);    If F(Pi) < F(Pbest_i):       Pbest_i = Pi    Apply cooperative evolution across groups;    Update fish positions and velocities across all dimensions;    If stopping criteria met (e.g., max iterations or no fitness improvement):       Break loop;    Return:    Return best-found solution $P_{best}$

5. Case Analysis and Discussion

The Taiwan 345 kV power system, shown in Figure 1, was analyzed using the ECI method for power flow and the AGFMO algorithm to optimize system parameters, minimizing carbon emissions and operational costs. This approach effectively handles the complexity of the 345 kV system. Each test case considered varying load conditions and renewable energy integration, providing insights into system performance. Tests were conducted using MATLAB 2016b on a PC with a 2.9-GHz Intel processor and 16 GB memory.

5.1. The Model of Taipower’s 345 kV High-Voltage Transmission System

The Taipower 345 kV system is the core network used for Taiwan’s power transmission [26], as shown in Figure 2. It uses high-voltage transmission lines to deliver electricity generated by nuclear, thermal, hydroelectric power plants, and renewable energy sources such as wind and solar power to substations across Taiwan. The electricity is then stepped down and supplied to users. This system encompasses key components such as power plants, transmission lines, and substations, ensuring stable and efficient power transmission. It also incorporates redundancy designs and protection measures to handle faults and ensure the safety and reliability of the power grid.

The system’s total load for each year in the cases is referenced from the study [33]. In 2023, the peak load was 33,927 MW, the annual average load was 24,816 MW, and the off-peak load was 15,706 MW. For 2028 and 2033, the load growth estimates are based on Taipower’s projections for the next decade [34]. In 2028, the peak load is expected to reach 40,263 MW, with an annual average load of 29,451 MW and an off-peak load of 18,337 MW. By 2033, the peak load is forecasted to rise to 47,033 MW, with an annual average load of 34,403 MW and an off-peak load of 20,465 MW.

As highlighted in related studies [35], the proportion of different power generation capacities in Taiwan correlates with load growth forecasts. In 2023, the total installed capacity was approximately 38,446 MW, with small hydropower and solar power plants excluded due to their negligible impact on overall capacity. This adjustment, consistent with common modeling practices, slightly reduces the total compared to the Taiwan Power Company’s official data but maintains the key trends needed for macro-level analysis.

Looking ahead, Taiwan Power Company’s 2022 development plan projects a steady increase in generation capacities over the next decade [34]. The projected expansions include renewable energy, conventional hydropower, thermal power, and nuclear power, reflecting the Taiwan’s strategic focus on diversifying its energy sources and increasing capacity to meet future load demands. By aligning with the latest expected commercial operation years, this study provides a forward-looking analysis that captures dynamic shifts in Taiwan’s power system and offers valuable insights for future energy planning.

5.2. Utilizing Data Mining for the Assessment of Wind Power Output and Installed Capacity

Using data mining and multivariate regression analysis, a cyclical pattern in load fluctuations was identified, with low-load periods in February, average loads in May and October, and peak loads in July (as shown in

Table 1. Comprehensive statistics of wind power generation in Taiwan’s power system.

Metric	Year Peak	Year Average	Year Low
Rep. month	7	10	2
Wind power generation (kWh)	23,167,537	8,265,104	100,165,267
Operating hours (H)	108,579.55	104,333.1	105,411.15
Downtime (H)	10,489.66	14,706.91	5948.85
Avg. hourly generation (kWh)	31,139.52	111,377.83	134,630.74
Fault-free avg. hourly generation (kWh)	33,536.25	120,778.67	151,114.34
Onshore net gen./capacity (%)	11.66	42.04	52.97
Offshore net gen./capacity (%)	13.82	61.55	65.83
Monthly availability (%)	91.22	87.53	94.7

), reflecting seasonal power demand shifts. As Taiwan plans to expand offshore wind power, onshore and offshore wind generation are evaluated separately. Onshore data cover Taiwan’s main island, while offshore data are based on estimates from islands like Penghu and Kinmen [36]. A regression model was used to calculate average hourly power generation per kilowatt of installed capacity under different load conditions. During peak loads, generation averages 11.67 kWh per kW, 42.05 kWh during average loads, and 52.97 kWh during low loads. Lower generation rates are typically associated with malfunctions or maintenance [36]. Wind power analysis under seasonal and monthly load changes provides insights into wind output under varying conditions.

In addition,

Table 2. Yearly wind power generation evaluation results.

Year	Onshore Wind (MWh)	Offshore Wind (MWh)	Total Wind (MWh)
2023	1,468,670.42	0	1,468,670.42
2028	2,317,986.21	1,088,886.33	3,406,872.54
2033	3,090,528.53	2,750,002.32	5,840,530.85

shows a significant increase in both onshore and offshore wind power generation from 2023 to 2033. Offshore wind, which has no contribution in 2023, is projected to rise significantly by 2033, highlighting its growing importance in Taiwan’s energy mix. This approach not only explains and predicts complex systems, as shown in the tables, but also lays a solid foundation for future power scheduling and wind energy development.

5.3. Minimization of Fuel Dispatch Costs and Emissions Under Carbon Tax Scenarios

The AGFMO algorithm was used to obtain the results presented in

Table 3. Minimization of fuel dispatch cost under carbon emission pricing in 2023.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	137,341,336	150,594,559	172,301,044
	Avg (65%)	73,819,293	90,000,017	104,131,988
	Low (15%)	41,299,501	46,504,288	52,340,511
Total cost (NTD)		713,713,208,005	831,270,998,730	962,388,054,661
Avg cost (NTD/MWh)		3233.01	3770.75	4247.43
Carbon emissions (tons)	Peak (20%)	19,001	18,521	18,759
	Avg (65%)	13,950	14,121	13,487
	Low (15%)	4694	4613	4673
Total emissions (tons)		122,079,250	117,758,501	115,991,850
Emissions (tons/MWh)		0.53	0.52	0.51

,

Table 4. Minimization of fuel dispatch cost under carbon emission pricing in 2028.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	167,182,804	183,251,821	211,410,310
	Avg (65%)	82,240,770	91,441,180	102,483,167
	Low (15%)	1,566,809,501,147	1,720,687,815,156	1,870,560,887,881
Total cost (NTD)		5847.21	6402.41	7021.08
Avg cost (NTD/MWh)		21,791	21,751	21,897
Carbon emissions (tons)	Peak (20%)	17,451	17,007	15,907
	Avg (65%)	7081	7351	6752
	Low (15%)	148,012,721	145,670,911	143,819,537
Total emissions (tons)		0.55	0.54	0.53
Emissions (tons/MWh)		292,001,901	314,174,027	345,627,781

and

Table 5. Minimization of fuel dispatch cost under carbon emission pricing in 2033.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	333,370,429	361,553,285	405,567,495
	Avg (65%)	158,657,560	145,566,522	160,612,670
	Low (15%)	3,179,699,580,483	3,311,754,575,311	3,588,500,497,124
Total cost (NTD)		9801.21	10,528.36	11,507.33
Avg cost (NTD/MWh)		29,720	29,598	29,512
Carbon emissions (tons)	Peak (20%)	24,702	24,501	23,052
	Avg (65%)	10,021	8587	8981
	Low (15%)	204,311,588	201,302,527	193,521,851
Total emissions (tons)		0.64	0.64	0.62
Emissions (tons/MWh)		560,526,821	582,511,836	605,521,781

for dispatch cost and CO₂ emissions under varying carbon tax levels. Each table corresponds to a different planning year—2023, 2028, and 2033—evaluated using the same optimization model described in Section 3, with updated load and capacity data. The findings reveal a clear pattern: while increasing carbon taxes substantially raises dispatch costs, the corresponding reductions in emissions are relatively modest.

In 2023 (Table 3), with a carbon tax of 500 NTD/ton, the dispatch cost during the peak period is 137,341,336 NTD, and annual CO₂ emissions total 122,079,250 tons. When the tax increases to 2500 NTD/ton, the dispatch cost escalates dramatically to 962,388,054,661 NTD, while emissions drop only slightly to 115,991,850 tons. This suggests a non-linear relationship where emission reductions taper off despite a steep rise in cost.

A similar trend is observed in 2028 (Table 4). At 500 NTD/ton, dispatch costs are 1.57 trillion NTD, with emissions at 148 million tons. At 2500 NTD/ton, costs rise to nearly 1.87 trillion NTD, yet emissions only decline to 143.8 million tons. These results imply that the marginal effectiveness of carbon taxes in reducing emissions diminishes over time as the energy system matures.

The trend continues in 2033 (Table 5). Even with a high carbon tax of 2500 NTD/ton, emissions fall only slightly—from 204.3 million tons to 193.5 million tons—while dispatch costs climb sharply to 3.59 trillion NTD. This indicates that without structural changes in fuel usage, carbon taxes alone yield limited emission reduction benefits, despite increasing economic costs.

Table 6. Analysis of carbon tax and fuel mix by year.

Year	Carbon Tax (NTD)	Total Generation (MW)	Coal Generation (MW)	Coal Mix (%)	Oil Generation (MW)	Oil Mix (%)	Gas Generation (MW)	Gas Mix (%)
2023	500	17,811.20	11,081.11	62.21	450.21	2.52	6279.88	35.25
	1500	18,001.05	10,840.53	58.84	251.81	1.39	7160.52	39.77
	2500	17,765.50	10,237.036	57.50	377.03	2.12	7171.88	40.36
2028	500	22,362.27	12,957.07	57.82	709.39	3.17	8721.69	39.00
	1500	22,262.65	12,545.19	56.23	731.19	3.28	9011.33	40.47
	2500	22,350.35	10,956.77	48.92	1157.14	5.17	10258.32	45.89
2033	500	27,793.71	19,440.05	69.80%	511.44	1.84	7881.06	28.35
	1500	27,454.14	18,955.15	68.90%	376.90	1.37	8159.95	29.72
	2500	27,473.60	17,146.30	62.28%	758.88	2.76	7171.88	34.95

illustrates changes in the fuel mix under different carbon tax levels. In 2023, coal accounted for 62.21% and gas 35.25% of total electricity generation under a 500 NTD/ton tax. With a tax of 2500 NTD/ton, coal’s share declines marginally to 57.50%, while gas rises to 40.36%. However, even in 2033 under the same high tax rate, coal still makes up 62.28% of generation—indicating little progress toward decarbonization. These results suggest that without complementary policy interventions, carbon pricing alone is insufficient to significantly reduce coal dependence.

Although higher carbon taxes shift generation marginally toward gas, low-cost coal remains dominant. For example, in 2023, the AGFMO-optimized dispatch results in 11,081.11 MW of coal generation and 6279.88 MW of gas. At 2500 NTD/ton, coal generation falls to 10,237.04 MW and gas increases to 7171.88 MW. Despite these changes, the system remains constrained by its reliance on coal, limiting the effectiveness of carbon pricing in driving down emissions.

Overall, while the AGFMO algorithm effectively minimizes costs under given constraints, the results highlight a critical insight: carbon taxes alone do not lead to significant emission reductions unless accompanied by broader structural reforms in the energy system. Long-term decarbonization will require complementary policies beyond price signals—such as fuel switching incentives, renewable energy mandates, or stricter emissions caps.

5.4. Balancing Cost and Carbon: Dispatch Optimization Under Emission Limits

The results point to an increasingly evident trend in energy system planning: carbon emission constraints are becoming central to dispatch decisions—not just technically, but politically and ethically as well. The AGFMO algorithm, in its dynamic OPF form, attempts to reconcile these layers of complexity. Yet, as seen in the 2023 simulation (

Table 7. Results of 2023 cost minimization analysis with carbon emission constraints.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	141,838,709	156,131,118	175,651,393
	Avg (65%)	76,247,689	88,157,397	104,580,690
	Low (15%)	42,287,502.8	48,534,629	52,611,982
Total cost (NTD)		734,161,332,165	851,213,119,090	980,934,218,911
Avg cost (NTD/MWh)		3157.36	3734.95	4313.99
Carbon emissions (tons)	Peak (20%)	19,431	18,217	19,097
	Avg (65%)	14,549	14,425	13,310
	Low (15%)	4931	4917	4831
Total emissions (tons)		122,008,876	119,537,619	116,607,114
Emissions (tons/MWh)		0.53	0.52	0.51

), the presence of a carbon cap did not really change the outcome. Why? Because emissions were already within policy limits that year, meaning cost minimization and compliance essentially coincided.

That alignment does not hold in 2028 (

Table 8. Results of 2028 cost minimization analysis with carbon emission constraints.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	296,721,83	310,584,094	345,069,350
	Avg (65%)	179,164,910	192,945,169	202,292,615
	Low (15%)	94,600,393	96,617,308	103,659,348
Total cost (NTD)		1,654,241,780,591	1,738,198,483,069	1,897,727,342,301
Avg cost (NTD/MWh)		6038.39	6460.29	7076.44
Carbon emissions (tons)	Peak (20%)	22,614	22,419	23,202
	Avg (65%)	15,526	15,712	15,578
	Low (15%)	6406	6850	6735
Total emissions (tons)		139,023,194	138,564,127	139,328,479
Emissions (tons/MWh)		0.51	0.52	0.51

) or 2033 (

Table 9. Results of 2033 cost minimization analysis with carbon emission constraints.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	629,145,597	646,260,891	688,809,335
	Avg (65%)	524,092,942	547,841,323	520,853,527
	Low (15%)	207,220,315	212,196,309	216,637,323
Total cost (NTD)		4,325,948,476,247	4,439,729,682,009	4,473,557,069,706
Avg cost (NTD/MWh)		13,348.43	13,962.43	14,090.23
Carbon emissions (tons)	Peak (20%)	26,075	26,086	26,681
	Avg (65%)	14,687	14,878	14,740
	Low (15%)	7095	7041	7034
Total emissions (tons)		141,207,615	140,531,437	140,919,598
Emissions (tons/MWh)		0.44	0.44	0.43

). The rising load demand starts to push the system up against those same carbon thresholds. For example, Taiwan’s 2035 target is 139.1 million tons of CO₂—a 25–28% reduction from 2005 levels. In 2028, even with a steep carbon tax of 2500 NTD/ton, emissions still clock in at 139,328,479 tons. Close, but not quite. By 2033 (Table 9), the goal is still exceeded unless carbon abatement mechanisms are significantly tightened. It is a clear tension: reducing emissions while keeping systems cost-efficient becomes harder as demand rises.

And it is not just about emissions. Table 7 shows that dispatch costs, surprisingly, stay in a fairly tight range across tax levels. This raises a different kind of question: if cost does not vary much, why is carbon not falling faster? The answer seems to lie in structural inertia—namely, coal. Even with tax pressure, it does not exit the mix easily.

The AGFMO model adapts dispatch in response to cost and carbon signals. In 2023, under a 500 NTD/ton tax, emissions hit 122 million tons, and the dispatch cost lands at 734.2 billion NTD. At higher tax levels, emissions nudge downward, but not dramatically. The signal is there, but the system is not pivoting fast enough.

This all points to a broader insight: carbon pricing helps, but it will not carry the whole decarbonization load. Without complementary policy moves—fuel switching mandates, capacity caps, or technology subsidies—progress stalls. Carbon taxes can shape the direction, but structural reform is what gets you there.

As shown in

Table 10. Cost minimization under wind power increment and carbon emission constraints in 2023.

Metric		Carbon Tax (NTD)
		500	1500	2500
Dispatch cost (NTD)	Peak (20%)	620,480,292	611,651,084	688,949,738
	Avg (65%)	494,575,870	494,948,891	475,215,728
	Low (15%)	203,415,678	204,225,477	218,363,863
Total cost (NTD)		4,139,911,890,922	4,160,944,758,703	4,213,821,810,374
Avg cost (NTD/MWh)		12,640.65	12,916.99	13,269.20
Carbon emissions (tons)	Peak (20%)	25,877	25,906	27,117
	Avg (65%)	14,311	14,455	14,185
	Low (15%)	7705	7602	7513
Total emissions (tons)		138,466,684	138,690,758	138,102,927
Emissions (tons/MWh)		0.43	0.43	0.42

, in 2023, without the addition of new renewable energy, the system’s carbon emissions reached approximately 140,919,598 tons, exceeding the policy limit of 139,000,000 tons. By integrating 2500 MW of wind power, emissions were reduced to 139,466,684 tons, successfully meeting the emissions target.

Under a carbon tax of 500 NTD/ton, the total dispatch cost was reduced to 4,139,911,890,922 NTD, with an average generation cost of 12,640.65 NTD/MWh. When the carbon tax increased to 1500 NTD/ton, the total dispatch cost rose slightly to 4,160,944,758,703 NTD, with an average generation cost of 12,916.99 NTD/MWh. At the highest carbon tax of 2500 NTD/ton, the dispatch cost increased further to 4,213,821,810,374 NTD, while emissions remained controlled at 138,102,927 tons, staying within the target.

The integration of wind power led to significant cost reductions across different carbon tax levels. For a 500 NTD/ton tax, generation costs were reduced by 186,036,585,325 NTD; for a 1500 NTD/ton tax, the reduction was 278,784,923,306 NTD; and under a 2500 NTD/ton tax, the cost was reduced by 259,735,259,332 NTD compared to scenarios without renewable energy integration. To maintain cost efficiency, the optimal purchase price for wind energy should not exceed 16,172 NTD/MWh at a 500 NTD/ton carbon tax, 17,987.9 NTD/MWh at a 1500 NTD/ton tax, and 22,789 NTD/MWh at a 2500 NTD/ton tax. Ensuring that renewable energy purchase prices remain below these levels would prevent excessive increases in generation costs, making wind power integration both environmentally and economically viable.

This analysis highlights the importance of integrating renewable energy, such as wind power, to meet emissions targets while keeping generation costs manageable. Carbon pricing alone is insufficient to achieve these goals, underscoring the need for renewable energy expansion to ensure long-term sustainability.

5.5. Performance Evaluation of AGFMO

To achieve the most precise cost analysis and evaluation, as illustrated in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10, the AGFMO algorithm is utilized. Compared to other algorithms, AGFMO demonstrates superior accuracy in minimizing dispatch costs across a variety of constrained scenarios, making it the optimal solution for handling complex optimization tasks under multiple constraints.

To further demonstrate AGFMO’s outstanding performance, this section evaluates its application under wind power increments and carbon tax constraints. Figure 3, Figure 4 and Figure 5 illustrate the convergence of dispatch costs under carbon taxes of 500, 1500, and 2500 NTD, with an annual average rate of 65%. The results indicate that AGFMO is highly effective in managing multi-constraint optimization problems, particularly over long-term iterations where its convergence continues to improve. In contrast, other algorithms like PSO [23], genetic ant colony optimization (GACO) [37] and FMO [32], although showing faster initial convergence, tend to plateau early and exhibit limited effectiveness in cost minimization under stricter conditions.

The quantitative comparison in

Table 11. Comparative performance of AGFMO and FMO under different carbon tax scenarios (65% annual average wind utilization).

Carbon Tax (NTD)	Algorithm	Total Cost (NTD)	Iterations to Converge	Success Rate	Avg. Euclidean Distance
500	FMO	495,583,410	83	88.9%	0.136
	AGFMO	494,575,870	93	96.7%	0.182
1500	FMO	495,001,915	236	86.7%	0.129
	AGFMO	494,948,891	81	93.3%	0.176
2500	FMO	475,443,854	116	90.0%	0.141
	AGFMO	475,215,728	110	96.7%	0.185

Note: Success rate is calculated based on 30 independent runs for each scenario, with success defined as achieving a final cost within 0.1% of the best-known solution.

provides further evidence for these observations. Across all carbon tax scenarios, AGFMO consistently achieves lower total dispatch costs, converges in fewer iterations or with more stable patterns, maintains a higher success rate, and preserves greater population diversity (measured by average Euclidean distance) compared to the original FMO. These metrics were obtained from 30 independent simulation runs for each scenario, with success defined as achieving a final cost within 0.1% of the best-known solution. This confirms that the adaptive grouping strategy in AGFMO not only improves convergence speed but also enhances solution robustness and stability under varying policy constraints.

To further validate the general optimization capability of AGFMO beyond the power dispatch context, the algorithm was also evaluated on selected CEC2020 benchmark functions under a 15-dimensional (15D) search space. Due to space constraints, three representative functions—F1 (unimodal), F5 (hybrid), and F9 (composition)—were selected to cover diverse optimization landscapes. The results, summarized in

Table 12. AGFMO performance on selected CEC2020 benchmark functions (15D).

Function	Ideal Value	Mean Fitness Value	STD	Relative Error
F1	100	101.6114	0.02	1.6114%
F5	1700	1755.845	25.48	3.2856%
F9	2400	2516.537	10.63	4.8557%

, show that AGFMO achieves mean fitness values close to the ideal optima, with low standard deviations and small relative errors, indicating high stability and accuracy. In particular, the algorithm demonstrates consistent convergence behavior across unimodal and multimodal functions, as illustrated in Figure 6 for the F9 case. These findings confirm that AGFMO’s adaptive grouping mechanism is effective not only in complex, real-world dispatch problems but also in generalized benchmark scenarios.

AGFMO’s superior performance can be attributed to its adaptive mechanism, which allows it to flexibly explore the solution space and reduce the likelihood of becoming trapped in local optima. This adaptability is especially valuable when managing complex energy systems, where balancing fluctuating renewable energy sources with strict carbon emission limits is crucial. The results not only demonstrate AGFMO’s capability to optimize dispatch costs but also highlight its robustness and reliability in addressing real-world energy challenges.

In summary, AGFMO provides both the highest precision in cost evaluation and long-term stability, making it an ideal tool for future energy systems that must meet increasing environmental and regulatory demands.

6. Conclusions and Critical Analysis

This study introduced an Adaptive Grouped Fish Migration Optimization (AGFMO) algorithm to address the complex challenge of optimizing power dispatch in carbon-constrained environments, using Taiwan as a representative case. In the absence of access to international carbon trading mechanisms, the results show that meaningful emissions reductions are still achievable through internal, algorithm-driven dispatch strategies. By incorporating projected renewable energy development and fluctuating carbon tax scenarios, the proposed framework offers a realistic pathway toward meeting Taiwan’s national targets by 2033.

From a methodological standpoint, the AGFMO algorithm extends conventional metaheuristics by incorporating adaptive grouping, recombination, and disturbance strategies that enhance global search behavior and mitigate premature convergence. These features have proven critical in navigating the non-linear, multi-constraint dispatch landscape associated with high renewable penetration and emission limits. The comparative performance gains over traditional methods underscore AGFMO’s potential as a flexible optimization engine for energy systems facing structural constraints.

On the practical front, the findings carry broader implications beyond the Taiwan case. Many regions—whether politically isolated or institutionally underdeveloped—face similar challenges in balancing fossil-fuel dependence with decarbonization mandates. The AGFMO-based framework offers a scalable, market-independent solution for dispatch planning, capable of adapting to local resource availability, policy settings, and operational uncertainties. Importantly, it offers a viable strategy for aligning domestic energy planning with broader climate goals, even in the absence of external market incentives.

In summary, advanced algorithmic design can play a pivotal role in enabling low-carbon transitions within constrained environments. AGFMO not only delivers technical improvements over existing optimization methods but also addresses a critical policy gap by enabling emission reductions in systems excluded from global trading regimes. Its adaptability and robustness position it as a strong candidate for application across a wider range of energy planning contexts in the era of deep decarbonization.