
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39232023
69191
10.1038/s41598-024-69191-z
Article
An innovative bio-inspired Aquila technique for efficient solution of combined power and heat economic dispatch problem
Hakmi Sultan Hassan 1
Moustafa Ghareeb 1
Alnami Hashim 1
Mansour Hany S. E. 2
http://orcid.org/0000-0001-7379-6060
Ginidi Ahmed ahmed.ginidi@eng.suezuni.edu.eg

3
1 https://ror.org/02bjnq803 grid.411831.e 0000 0004 0398 1027 Electrical Engineering Department, College of Engineering, Jazan University, 45142 Jazan, Saudi Arabia
2 https://ror.org/02m82p074 grid.33003.33 0000 0000 9889 5690 Electrical Engineering Department, Suez Canal University, Ismailia, 41522 Egypt
3 https://ror.org/00ndhrx30 grid.430657.3 0000 0004 4699 3087 Department of Electrical Engineering, Faculty of Engineering, Suez University, P.O. Box: 43221, Suez, Egypt
4 9 2024
4 9 2024
2024
14 206388 2 2024
1 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
In the field of power systems, the optimization challenge of combined heat and power units economic dispatch (CHPUED) holds immense importance. This study presents an improved Aquila optimization technique (IAQT) that effectively tackles the CHPUED. The primary objective of the enhanced IAQT model is to minimize the overall cost of power generation in CHP systems while satisfying demand and operational constraints. However, to achieve more accurate cost estimations and avoid suboptimal solutions, it is crucial to consider transmission losses in the optimization model. By incorporating transmission losses, the IAQT algorithm can allocate power generation resources more effectively, leading to improved system efficiency and reduced operational costs. The proposed IAQT algorithm addresses the limitations of the standard AQT and introduces novel features to enhance its search capabilities. One key limitation of the standard AQT is its heavy reliance on the best solution found during optimization. To overcome this drawback, the enhanced IAQT model eliminates the dependency on the best solution and enables a more thorough exploration of the search space. Moreover, the algorithm incorporates specific limitations and constraints for each dimension of the newly generated solutions, ensuring their feasibility and validity. The standard AQT and proposed IAQT are tested on CEC 20 benchmark functions. Moreover, the proposed approach is extensively evaluated through experimentation and testing on various scenarios, including 7–48-unit and large 96-unit systems with/without losses. Furthermore, the overall costs for the 7 unit-system are considered including the reserve constraint. The results exhibit the remarkable performance and efficiency of the enhanced IAQT model, outperforming the standard version and several previously reported results. This validation underscores the significant contribution of the study in addressing the CHPUED and highlights its potential for real-world applications.

Keywords

Aquila optimization
CHP units
Valve point loadings
Improved Aquila technique
Economic dispatch
Subject terms

Electrical and electronic engineering
Energy infrastructure
Mathematics and computing
http://dx.doi.org/10.13039/100009388 Jazan University ISP23-122 Hakmi Sultan Hassan issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Power system operation offers a number of technical and financial benefits by guaranteeing a steady and dependable supply of electricity to fulfil customer demand. This comprises controlling the production, distribution, and transmission of energy as well as upholding grid stability and handling technical problems like voltage regulation and system protection1,2. From a technical standpoint, the main objectives are to maximise power plant efficiency, oversee load dispatch, and put control mechanisms in place to guarantee safe and effective operation. From an economic perspective, power system operation encompasses financial viability and cost issues3. This entails controlling fuel and energy procurement, taking market prices and electricity tariffs into account, and optimising generation schedules and dispatch to reduce operational costs. The combination of energy efficiency and renewable energy sources is also covered by economic views4,5. Therefore, economic research and planning are essential since they take into account several aspects like capital costs, operating expenses, and income creation6. The significance of CHP units (CHPU) in advancing energy solutions aligns closely with the worldwide goals of the sustainable development. One of the most important issues facing power systems is improving economic dispatch in electrical systems, which includes CHPU. Approximately 63% of the global electricity production heavily relies on power plants that burn fossil fuels7. These power plants release substantial amounts of carbon dioxide into the atmosphere, contributing significantly to the overall increase in emissions, accounting for about 46% of the global8. It is widely acknowledged that these emissions are the primary driver of climate change, which poses significant environmental challenges and has far-reaching impacts on ecosystems, weather patterns, and human well-being9. To address these pressing concerns, CHPU, also referred to as cogeneration units, offer a highly effective solution for simultaneous generation of thermal and electrical energy10. Unlike conventional thermal power plants, where a considerable amount of energy is wasted as heat during the conversion of fossil fuels into electricity, CHPU are designed to harness this otherwise wasted heat and utilize it to meet the demand for thermal energy11. This integrated approach significantly increases energy efficiency, reaching approximately 90%, as the same fuel source is utilized for both electricity and heat production12. The benefits of adopting CHPU extend beyond enhanced energy efficiency. They have the potential to reduce overall fuel costs by 10–40% due to the more efficient utilization of primary energy sources13. By minimizing energy wastage, CHPU contribute to cost savings and increased energy affordability. Additionally, the utilization of CHPU can lead to a substantial reduction in environmental pollution, with estimates suggesting a decrease of 13–18% in emissions of greenhouse gases and other harmful pollutants13. This reduction in emissions contributes to mitigating the adverse environmental impacts associated with traditional fossil fuel-based electricity generation.

However, there is growing apprehension about a deficit of fossil fuels due to their gradual decline14,15. Fuel suppliers have imposed additional constraints on fuel supply accords, leading electric power utilities to reschedule power production depending on the availability of fuel16,17. Various factors such as fuel contracts, storage limitations, and fuel delivery quantities can limit the supply and consumption of fuel in a power system18. For example, fuel procurement for electric utilities can occur through long-term commitments, such as take-or-pay contracts, which can span several years and have adjustable base prices. Alternatively, participation in the spot market allows for the purchase of fuel through short-term contracts, ranging from months to days, at competitive prices19.

Because of the economic and physical constraints of fuel transportation systems, fuel contracts often include provisions that outline the minimum and maximum amounts of fuel that can be delivered at any given time19,20. Furthermore, utilities also encounter challenges in managing their fuel inventory due to regulatory or physical limitations on storage capacity. These restrictions impose real constraints on the quantity of fuel that can be stored14. As a result, effectively scheduling energy resources presents a significant challenge in the planning and operation of power systems21. Trefny and Lee have addressed the economic dispatch problem with considering the fuel limits. The suggested approach has been applied to actual control of a power network contributing to short-term fuel managing by automatic generation control22. In21,23, the authors have proposed an approach to determine the optimal scheduling of fuel supply sources. The objective function has been formulated to minimize the total fuel cost. In a previous study24, researchers proposed the use of the nondominated sorting genetic algorithm to address the economic dispatch optimization problem, specifically focusing on the constraints related to fuel usage. Another study25 introduced an enhanced version of the social spider technique, employing strategies such as multi-mating and Gaussian mating radius, to tackle the economic dispatch problem. Furthermore, researchers in26 enhanced the Particle Swarm Optimization (PSO) algorithm by incorporating a double elitist breeding quantum mechanism, aiming to increase the diversity of the swarm and solve the economic dispatch problem. This enhancement was evaluated on widely used test systems comprising 15-unit, 40-unit, and 140-unit setups. In a different approach, researchers in27 developed a collective decision population-based technique inspired by human decision-making processes, specifically designed for the economic dispatch problem. Additionally28, introduced a shrink Gaussian distribution quantum-PSO method for solving the economic dispatch problem iteratively by gradually reducing the Gaussian probability distribution around each particle's learning inclination point.

Indeed, to profitably achieve the operation of CHPU, the optimal economic dispatch problem is adopted. This problem is addressed to find the optimal heat and power units’ schedule. The main aim is to minimize the CHPU fuel cost while sustaining operational constraints29. In30, the authors have proposed the manta ray foraging algorithm to address the problem of optimal scheduling of CHPU. The effects of valve point and wind power have been considered in problem formulation. The suggested approach has been tested on two systems: 5 units and 96 units. Two scenarios of loading have been addressed: peak and daily variation loading. The wind power integration has achieved more economical benefits. The particle swarm optimization (PSO) technique has been proposed to solve the CHPUED31. Time-varying acceleration factors have been applied to succeed in dealing with premature convergence of PSO and improve the result quality. After using the PSO with time-varying acceleration factors, the authors have employed the Monte Carlo approach to solve the uncertainties of load demand and renewable energy sources32. In33, the PSO has also been suggested as an optimization method to study the influence of economic dispatch of coal-fired CHPU on the whole coal consumption of the CHPU. In34, the differential evolution algorithm has been presented to solve the CHPU dynamic economic dispatch problem. In35, the authors have suggested an approach to address the optimal CHPU economic dispatch while considering the security of the electrical system from the voltage stability point of view. In36, the PSO has been proposed to determine the optimum placement and capacity of CHPU systems. The objective function has been designed with the aim of minimizing line losses, enhancing the nodes voltage and system reliability. The suggested approach has been employed to two standard microgrids: 84 and 32-bus networks. In37, the author has addressed the scheduling problem of CHPU of a remote microgrid. The study has been implemented with and without considering the fuel constraints. The microgrid under study has incorporated dispatchable and non-dispatchable units in addition to energy storage systems and plugin electric vehicles. The elephant clan optimizer has been suggested to solve the optimal scheduling problem. In38, the horse herd optimizer has been proposed to address the optimal CHPU scheduling. The problem has been formulated with considering the fuel constraints. The test system has included 13 thermal units, six co-generators, five heat-only units, single equivalent solar system, single equivalent wind generator, and single pumped storage hydro plant. Ramp rate constraints of thermal sources and co-generators have been considered.

In this paper, an enhanced version of the AQT is proposed which seeks to overcome certain limitations associated with the standard AQT and introduces novel features to enhance its searching capabilities. The primary drawback of the standard AQT lies in its heavy reliance on the best solution found during the optimization process. To address this limitation, the proposed enhanced IAQT model eliminates the dependency on the best solution, thereby allowing for a more comprehensive exploration of the search space. Furthermore, the algorithm incorporates limitations and constraints specific to each dimension of the newly generated solutions, ensuring the feasibility and validity of the solutions. Fitness evaluation is performed, and improved solutions are selectively retained based on their objective values. The proposed enhanced IAQT model is developed for the CHPUED to minimize the overall cost of power generation while satisfying the demand and operational constraints. However, neglecting transmission losses in the optimization model can lead to suboptimal solutions and inaccurate cost estimations. The contribution of this article can be summarized as follows:An improved version of the Aquila Optimization Technique (IAQT) is proposed to overcome certain limitations associated, enhance its searching capabilities,

The proposed IAQT and the standard AQT are assessed on the CHPUED problem effectively.

Through extensive experimentation and testing on various scenarios with/without considering transmission losses including the 7- unit, 48-unit and large 96-unit systems,

The overall costs for the 7 unit-system are considered including the reserve constraint.

The standard AQT and proposed IAQT are tested on Congress on Evolutionary Computation 2020 (CEC 20) benchmark functions and compared with the reported algorithms.

It is demonstrated that the proposed IAQT outperforms the standard version and several reported results with remarkable performance and efficiency.

The paper is divided into five sections that include its main content. In "Proposed IAQT for CHPUED", the improved version of the Aquila Optimization Technique is mathematically displayed. In "Mathematical model of CHPUED formulation", the mathematical formulation of the CHPUED model is developed. A thorough examination and explanation of the numerical outcomes for each of these six scenarios and CEC 20 benchmark functions can be found in "Simulation results". Moreover, "Conclusions" offers the final conclusions.

Proposed IAQT for CHPUED

The Aquila Technique (AQT) represents a meta-heuristic optimization method inspired by the ferocious aquila bird of prey. The aquila's ability to catch a wide variety of animals is attributed to its remarkable agility, quickness, powerful feet, and breadth. Accordingly, many hunting tactics rely on the estimation of the natural tendencies of the prey. To rapidly and expertly adjust its hunting methods, the AQT takes into account the surrounding aspects of the hunting setting in addition to the Aquila's diverse talents39. To initiate the optimization process, the AQT produces an initial set of feasible solutions inside the defined searching space. Since the ensuing solutions are created at random, a large range of parameter options are covered. This starting population serves as the foundation for additional algorithmic iterations in the search for the optimal solution40:1 Qukjt=Rd1×UPPj-LOWj+LOWj;k=1,2,3,⋯,NQandj=1,2,3,⋯,Dm

In the AQT, each solution, denoted as Qukj, represents a candidate solution with a dimension (Dm). The variable "t" represents the current iteration of the algorithm, while LOWj and UPPj represent the least and highest limits of the search space for each dimension. Rd1 indicates a random vector following uniform distribution while NQ Refers to the number of Aquila solutions in each iteration. The hunting features of the Aquila are modeled in the AQT using the Aquila's quest for target approach, which consists of two stages. Each stage comprises two distinct processes, designed to mimic the Aquila's hunting behaviors.

Stage 1: exploration

In this stage, two diversified processes are involved: contour flight and expanded exploration41. This stage is carefully formulated to simulate the Aquila's hunting strategies and is expressed as follows:2 QuN=QuBt×1-ttM+∑i=1QuQuitNQu-QuBt×r2QuBt×LevyDm+Qu1t+Z×r3×cosθ-sinθifr4<12Else

where, QuN is the new generated candidate Aquila solution while the subscript N refers to “new”. The maximum number of iterations, denoted as tM, determines the overall duration of the optimization process. The term 1-ttM is employed to regulate the exploration stage which is gradually reducing the exploration as the algorithm progresses. The Levy flight distribution function, LevyDm governs the random movement of the solutions within the search space, mimicking the flight patterns observed in nature. The best solution with the minimum fitness score is denoted as QuB. The AQT uses three randomized numbers, r2,r3 and r4,, which are created inside the range of [0,1], to add even more randomization. These arbitrary values enhance the algorithm's decision-making process at each iteration and enable the system to behave in an adaptive manner. Qu1 represents a randomly selected solution from the current population. It introduces diversity and exploration by considering different candidate solutions during the optimization process.

Additionally, two adaptive parameters, namely Z and θ, play a crucial role in the algorithm. These parameters are evaluated based on specific criteria or conditions determined by the problem at hand. The precise evaluation process for Z and θ depends on the context and requirements of the optimization task.3 Z=D1×U+Z1

4 θ=1.5×π-D1×W

The variable D1 represents integer numbers ranging from 1 to Dm. Two small, fixed values, U and W, are incorporated into the algorithm. (U = 0.00565 and W = 0.005). Z1 takes on values between 1 and 20, determining the number of search cycles.

Equation (2) in the algorithm describes the exploration behavior of the Aquila. When a random number, r4, is less than 0.5, the Aquila engages in expanded exploration, as explained in the first part of the equation. On the other hand, if r4 is greater than or equal to 0.5, the Aquila follows the contour flight strategy, as indicated in the second part of the equation.

It is observed that the exploration behavior of the Aquila is influenced by the optimal solution found so far, denoted as (QuB), as shown in Eq. (2). This dependence can help the algorithm perform better in smaller search regions, but it can also cause performance issues in larger solution spaces. This dependence reduces the algorithm's ability to explore new areas by raising the possibility of becoming stuck in local optima.

An improved AQT (IAQT) is presented in order to overcome this restriction and improve the AQT's exploratory capabilities. It includes many randomly distributed diverse replies from the whole population, as expressed in the following formula. With this change, the search procedure should be more varied and the possibility of becoming stuck in a local optima reduced, allowing for a more thorough investigation of the solution space:5 QuN=Qu1t+r2×Qu2t-Qu3tQu1t+LevyDm×Qu2t-Qu3tifr4<12Else

The proposed model addresses a drawback of the standard AQT by alleviating its reliance on the best solution. This improvement is achieved by eliminating any term or factor related to the best solution in the algorithm's formulation.

Stage 2: exploitation

This stage introduces two additional processes to enhance the exploitation capabilities of the AQT. These processes are known as expanded exploitation and narrowed exploitation. The exploitation stage is designed as follows:6 QuN=α×QuBt-∑i=1QuQuitNQu-r4+r5×UPP-LOW+LOW×δQuBt×t2×z9-1(1-tM)2-Qu(t)×2r6-r7+2-2ttM×LevyDm+2r7-r8ifz4<12Else

where, α and δ are specified fixed parameters at 0.1.

As depicted in Eq. (6), the Aquila exhibits reduced movement as it identifies its target prey. Once the target is chosen, the Aquila engages in a gradual and thorough attack using expanded exploitation, as described in the first part of the equation. This strategy is employed when a randomly generated value (z4) is smaller than 0.5. On the other hand, when z4 is greater than or equal to 0.5, the Aquila adopts stalking behavior, pushing the prey across the ground. This narrowed exploitation step is particularly useful when dealing with larger prey.

Additionally, every aspect of the recently developed solution needs to be assessed in light of the particular restrictions or limitations given by the current issue. The viability and legitimacy of every dimension in the solution are governed by these constraints.7 QuNj=UPPjLOWjifQuNj>UPPjifQuNj<LOWjj=1,2,3,⋯,Dm

Furthermore, the fitness function of each newly created solution is assessed. If the objective value of the newly created solution (OQuN)) is lower than that of the previous solution (OQu), the old solution is replaced with the new one. This replacement mechanism ensures that only improved solutions are retained and incorporated into the algorithm's progress.8 QuN=QuNQuifOQuN<OQuElse

A flowchart of the proposed IQAT is developed in Fig. 1.Figure 1 Flowchart of IQAT.

In regard of Fig. 1, the main steps of the standard AQT can be further explained as follows:

Step 1, initialization: The AQT starts by generating an initial population of potential solutions within the predefined search space. Each solution is represented by a set of parameters that define its position in the search space.

Step 2, exploration: During the exploration phase, the AQT simulates the Aquila's behavior of searching for prey by flying over a large area (contour flight) and by performing random yet directed movements (expanded exploration) within the search space. This phase is crucial for covering a wide range of the solution space to avoid local optima. Step 3, exploitation: The exploitation phase focuses on refining the search around the most promising areas identified during the exploration phase. The AQT mimics the Aquila's precise and targeted attacks on prey, using strategies like expanded exploitation and narrowed exploitation to converge towards the best solution.

Step 4, adaptive parameters: The AQT uses adaptive parameters such as Z and θ to modify the behavior of the algorithm during the optimization process. These parameters are adjusted based on the problem's specific characteristics and the progress of the search.

Step 5, solution update: The AQT updates the solutions in the population based on the hunting strategies and the random components. The fitness of each solution is evaluated, and the algorithm retains only the improved solutions.

Step 6, termination condition: The AQT continues to iterate until a termination condition is met, which could be a maximum number of iterations, a desired fitness threshold, or a time limit.

A myriad of advantages of the IAQT are noticed which includes a good balance between exploration and exploitation that is ensured by the IAQT's hunting strategies, which is essential for efficient optimization. Moreover, the use of adaptive parameters allows the IAQT to be flexible and suitable for a wide range of optimization problems. Furthermore, the randomization and diverse exploration strategies help the IAQT to avoid getting trapped in local optima, leading to better global search capabilities.

Mathematical model of CHPUED formulation

The primary goal of CHPUED is to minimize the overall expense of heat and electricity generation. A visual representation of the economic CHPUED challenge, including several limitations. The Fuel Cost (FC) function can be defined as all of the expenses of the power–heat amalgamation, power-only units (POU), and heat-only units (HOU)42. This is expressed through the following equation:9 MinCF=∑x=1NpCFn1x(Pxp)+∑y=1NhCFn2y(Hyh)+∑z=1NcphCFn3z(Pzcph,Hzcph)$/h

where CFn1x(Pxp) is the xth POUs’ cost; Np is the power-only units’ number; CFn2y(Hyh) is the yth heat-only units’ cost; Nh is the heat-only units’ number; CFn3z(Pzcph,Hzcph) is the zth CHP units’ cost; NCHP is the CHP units’ number.

The three cost functions CFn1x(Pxp),CFn2y(Hyh) and CFn3z(Pzcph,Hzcph) can be mathematically represented as follows:

(1) CFn1x of xth power units10 CFn1x(Pxp)=ϕ1x(Pxp)2+ϕ2xPxp+ϕ3x+λxsin(ρx(Pxmin-Pxp))$/h

where the parameters (ϕ1x,ϕ2x,andϕ3x) depict the coefficients for xth power-only units’ cost.

The valve-point impacts can be modeled by the sinusoidal term (λxsin(ρx(Pxmin-Pxp))) in Eq. (10) which illustrates clearly non-convexity and the non-differentiability of CHPUED43,44. The coefficients for modeling valve point impacts are denoted by λx, and ρx45. The increased consumption is represented by the valve point effect, which superimposes the unit consumption curve with the pulsating effect caused by rapid valve opens. Figure 2 illustrates the FC taking into account the valve point impact with αx = 0.0002, γx = 580, βx = 8.3, ρx = 0.04, Pxpmin = 50, Pxpmax = 650, and λx = 350. This figure clearly highlights the valve point effect. The figure below makes it evident that, for the majority of the generated power, the valve point effect results in higher unit production costs than the quadratic cost formulation. Therefore, it can be concluded that depending solely on the simplified cost equation—which ignores the valve point effect—will not result in the best solution for the system's actual operation.Figure 2 Valve point impact on fuel cost46.

(2) CFn2y of yth HOUs11 CFn2y(Hyh)=λ1y(Hyh)2+λ2yHyh+λ3y$/h

where the parameters (λ1y,λ2y,andλ3y) characterize the coefficients for yth heat-only units’ cost.

(3) CFn3z of zth CHPUs12 CFn3z(Pzcph,Hzcph)=β1z(Pzcph)2+β2zPzcph+β3z+β4z(Hzcph)2+β5zHzcph+β6zHzcphPzcph$/h

where the parameters (β1z,β2z,β3z,β4z,β5z and β6z) characterize the zth CHP unit’s cost coefficients.

Constraints

When minimizing the given cost function, the following constraints are considered.

Bounds of POUs’ capacity13 Pxmin≤Pxp≤Pxmaxx=1,...,Np,

Equation (14) can be used to determine how power generation and demand are balanced47:

Power balance constraint14 ∑x=1NpPxp+∑z=1NcphPzcph=PD+Ploss

where PD give explanation for the power demand, while Ploss demonstrates the network losses.

Heat balance constraint15 ∑z=1NcphHzcph+∑y=1NhHyh=HD,

where HD establishes thermal demand48.

CHPU capacity limits16 Pzcphl(Hzcph)≤Pzcph≤Pzcphu(Hzchp)z=1,...,Ncph,

17 Hzcphl(Pzcph)≤Hzcph≤Hzcphu(Pzcph)z=1,...,Ncph,

HOUs’ production limits18 Hyl≤Hyh≤Hyuy=1,...,Nh,

where the power and heat unit limits are expressed by lower “l” and upper “u” as superscripts.

Simulation results

To validate the effectiveness of the proposed IAQT, a comparative analysis was conducted using data collected for the Economic Dispatch (ED) with Combined Power and Heat Units (CHPUs), and it was contrasted with the standard AQT. Three commonly employed test systems, namely the 7-unit, 48-unit and 96-unit systems, were employed to assess the performance of both techniques, each differing in configuration and scalability. In the first test system, denoted as the 7-unit system, there are two CHPUs, four POUs, and one HOU. The system parameters, encompassing loss coefficients, fuel prices, and CHPU limits, are defined in49. The power and heat loading levels for this system are 600 MW and 150 MWth, respectively, as outlined in49. As detailed in45, the second test system comprises 48 units, consisting of 10 HOUs, 12 CHPUs, and 26 POUs. The corresponding load demands for power and heat in this system are 4700 MW and 2500 MWth, respectively. As illustrated in42, the third test system comprises 96 units, consisting of 20 HOUs, 24 CHPUs, and 52 POUs. The corresponding load demands for power and heat in this system are 9400 MW and 5000 MWth, respectively. The standard AQT and the proposed IAQT involve 100 individuals, and the number of iterations for the first and second systems are 300 and 3000, respectively. For the purpose of the study, four scenarios are investigated, and they are outlined as follows:

Scenario 1: Fuel costs minimization for the 7 unit-system without consideration of losses.

Scenario 2: Fuel costs minimization for the 7 unit-system with consideration of power losses.

Scenario 3: Fuel costs minimization for the 48 unit-system without consideration of losses.

Scenario 4: Fuel costs minimization for the 48 unit-system with consideration of power losses.

Scenario 5: Fuel costs minimization for the large 96 unit-system without consideration of losses.

Scenario 6: Fuel costs minimization for the large 7 unit-system considering the reserve constraint.

Testing the standard AQT and the proposed IAQT on latest standard test function CEC20 test functions.

Implementation for Scenario 1

To minimize fuel expenses without incorporating losses, the ED with CHPUs can be resolved using the proposed IAQT and the standard AQT. Table 1 illustrates the best parameters for the operation of the HOUs, CHPUs, and POUs obtained using the standard AQT in this case as well as the proposed IAQT.Table 1 Optimal operational values and best costs of of the standard AQT and the proposed IAQT for Scenario 1.

	Outputs	AQT	IAQT	
POUs	P1	74.80225	44.8751	
P2	91.42406	98.54052	
P3	175	112.6739	
P4	124.9242	209.8174	
CHP 1	P5	93.77399	94.09275	
H5	40.07545	40.00024	
CHP 2	P6	31.82824	27.71361	
H6	74.45646	74.99711	
HOU	H7	43.7153	47.28927	
Costs ($/h)	10,220.62	10,091.93	

Based on the outcomes in Table 1, the proposed IAQT has the lowest possible fuel costs of 10,091.93 $/h which reveals impressive outcomes. To achieve this result, the proposed IAQT attains the operational values are 40.00024, 74.99711 and 47.28927 MWth for the heat units and 44.8751, 98.54052, 112.6739, 209.8174, 94.09275 and 27.71361 MW for the power outputs. In contrast, the standard AQT produces fuel costs of 10,220.62 $/hr which is worse than the proposed IAQT. Additionally, the obtained expenses of the proposed IAQT and the standard AQT of Scenario 1 are provided in Fig. 3 for each simulated run. As can be demonstrated, in every simulated run, the proposed IAQT outperforms the standard AQT. The proportion of improvement significantly varies from 1.2587% to 3.8270%.Figure 3 Achieved costs for all runs of the standard AQT and the proposed IAQT for Scenario 1.

Table 2 outlines the robustness metrics for Scenario 1 with comparing the standard AQT and the proposed IAQT. The metrics include minimum, mean, maximum, and standard deviation values, where they are derived from the outcomes illustrated in Fig. 3. The outcomes demonstrate that the proposed IAQT exhibits superior resilience performance compared to the standard AQT. Specifically, the proposed IAQT achieves the lowest values for minimum, mean, maximum, and standard deviation (STD) at 10,091.93, 10,092.02, 10,092.2, and 0.069043 $/hr, respectively, leading to notable enhancements of 1.25, 2.30, 3.82, and 99.91%, respectively.Table 2 Robustness metrics of of the standard AQT and the proposed IAQT for Scenario 1.

Costs ($/h)	AQT	IAQT	Improvement %	
Minimum	10,220.62	10,091.93	1.259109	
Mean	10,330.27	10,092.02	2.306314	
Maximum	10,493.64	10,092.2	3.825512	
STD	74.86093	0.069043	99.90777	

Furthermore, Fig. 4 presents the convergence rates of the proposed IAQT and the standard AQT in relation to their best run, worst run, and the average across all simulated runs. It is evident that the suggested IAQT exhibits superior convergence characteristics during its development, particularly in the reduction of fuel costs over the course of iterations. The standard AQT remained confined to a local optimal zone and did not achieve lower fitness values during the 300 iterations. Figure 5 further illustrates the discrepancies among the best run, worst run, and average of all runs for both the IAQT and the standard AQT in Scenario 1, highlighting a substantial improvement of approximately 5.45%, 5.32%, and 7.07% for the average, best, and worst scenarios, respectively, occurring after around 45% of the total iteration count.Figure 4 Convergence rates of the standard AQT and the proposed IAQT for Scenario 1.

Figure 5 Proportion difference for the best, average and worst of all runs for the standard AQT and the proposed IAQT for Scenario 1.

Implementation for Scenario 2

To minimize fuel expenses with incorporating losses, the ED with CHPUs can be resolved using the proposed IAQT and the standard AQT. Table 3 illustrates the best parameters for the operation of the HOUs, CHPUs, and POUs obtained using the standard AQT in this case as well as the proposed IAQT. Based on this data, the proposed IAQT has the lowest possible fuel costs of 10,094.43 $/h which reveals impressive outcomes. To achieve this result, the proposed IAQT attains the operational values are 40.00001, 74.99037 and 46.99536 MWth for the heat units and 45.7441, 98.53964, 112.674, 209.8162, 94.041 and 28.01427 MW for the power outputs. In contrast, the standard AQT produces fuel costs of 10,217.5 $/h which is worse than the proposed IAQT.Table 3 Optimal operational values and best costs of of the standard AQT and the proposed IAQT for Scenario 2.

	Outputs	AQT	IAQT	
POUs	P1	69.06792	45.7441	
P2	85.67367	98.53964	
P3	108.7214	112.674	
P4	205.5838	209.8162	
CHP 1	P5	91.3615	94.041	
H5	40.3895	40.00001	
CHP 2	P6	43.90357	28.01427	
H6	69.72355	74.99037	
HOU	H7	36.37288	46.99536	
Costs ($/h)	10,217.5	10,094.43	

Additionally, the obtained expenditures of the proposed IAQT and the standard AQT of Scenario 2 are provided in Fig. 6 for each simulated run. As can be demonstrated, in every simulated run, the proposed IAQT outperforms the standard AQT. The proportion of improvement significantly varies from 1.2019 to 4.8985%.Figure 6 Achieved costs for all runs of the standard AQT and the proposed IAQT for Scenario 2.

Table 4 outlines the robustness metrics for Scenario 2 by comparing the standard AQT and the proposed IAQT. The metrics include minimum, mean, maximum, and standard deviation values, where they are derived from the outcomes illustrated in Fig. 6. The outcomes demonstrate that the proposed IAQT exhibits superior resilience performance compared to the standard AQT. Specifically, the proposed IAQT achieves the lowest values for minimum, mean, maximum, and standard deviation (STD) at 10,094.43, 10,094.55, 10,094.87, and 0.082304 $/h, respectively, leading to notable enhancements of 1.20, 2.56, 4.89, and 99.92%, respectively.Table 4 Robustness metrics of of the standard AQT and the proposed IAQT for Scenario 2.

Costs ($/h)	AQT	IAQT	Improvement %	
Minimum	10,217.5	10,094.43	1.204476	
Mean	10,360.2	10,094.55	2.564126	
Maximum	10,614.44	10,094.87	4.894992	
Standard deviation	101.8008	0.082304	99.91915	

Furthermore, analogies are drawn with the outcomes of strong optimization techniques applied to address the ED problems in the literature. Table 5 exhibits the evaluation of the AQT and the proposed IAQT with published techniques which are Artificial Ecosystem Algorithm (AEA)50, Modified AEA (MAEA)50, CPSO51, differential evolution DE51,52, PSO31, LCA53, RCGA54, PSO with time-varying acceleration coefficients TVAC-PSO32, ECSA55, WVO-PSO56, IGA57, WVO56, PSO32, and TVAC- BCO54. It is evident that the proposed IAQT offers superior performance characteristics than the alternatives.Table 5 Comparative results for Scenario 2 for 7-unit system.

Optimizer	Costs ($/h)	
IAQT	10,094.43	
AQT	10,217.5	
MAEA50	10,095.02453	
AEO50	10,095.11736	
LCA53	12,451.4000	
CPSO51	10,325.3000	
IGA57	10,107.9071	
TVAC-PSO31	10,100.3000	
ECSA55	10,121.9466	
PSO32	10,178.4311	
RCGA54	10,667.0000	
TVAC-PSO32	10,244.0200	
DE52	10,317.0000	
BCO54	10,317.0000	
WVO-PSO56	10,372.0000	
DE51	10,317.0000	
WVO56	10,317.0000	

Furthermore, Fig. 7 presents the convergence rates of the proposed IAQT and the standard AQT in relation to their best run, worst run, and the average across all simulated runs. It is evident that the suggested IAQT exhibits superior convergence characteristics during its development, particularly in the reduction of fuel costs over the course of iterations. The standard AQT remained confined to a local optimal zone and did not achieve lower fitness values during the 300 iterations. Figure 8 further illustrates the discrepancies among the best run, worst run, and average of all runs for both the IAQT and the standard AQT in Scenario 2, highlighting a substantial improvement of approximately 5.47%, 6.21%, and 5.44% for the average, best, and worst scenarios, respectively, occurring after around 47% of the total iteration count.Figure 7 Convergence rates of the standard AQT and the IAQT for Scenario 2.

Figure 8 Proportion difference for the best, average and worst of all runs for the standard AQT and the proposed IAQT for Scenario 2.

Implementation for Scenario 3

To minimize fuel expenses without incorporating losses, the ED with CHPUs can be resolved using the proposed IAQT and the standard AQT. Table 6 illustrates the best parameters for the operation of the HOUs, CHPUs, and POUs obtained using the standard AQT in this case as well as the proposed IAQT. As shown in Table 6, the proposed IAQT has the lowest possible fuel costs of 116,645.4 $/h which reveals impressive outcomes. In contrast, the standard AQT produces fuel costs of 123,063.4 $/h which is worse than the proposed IAQT. The totals for Sum (H) and Sum (P) successfully meet the specified heat and power requirements of 2500 MWth and 4700 MW, respectively, as depicted in Table 1. Moreover, all outcomes fall within the feasible range, with numerous individual results precisely positioned at either the lower or upper bounds.Table 6 Optimal operational values and best costs of the standard AQT and the proposed IAQT for Scenario 3.

Outputs	AQT	IAQT	Outputs	AQT	IAQT	Outputs	AQT	IAQT	
P1	628.3189	448.8118	P 22	64.2718	109.8756	H 30	75.96489	90.35739	
P 2	170.5926	226.3723	P 23	81.53478	77.60479	H 31	40.06825	40.01261	
P3	163.2366	300.5324	P 24	93.61713	77.54376	H 32	31.63817	25.30517	
P 4	161.2947	159.7351	P 25	98.21684	92.49187	H 33	141.3112	106.63	
P 5	132.3101	110.0442	P 26	88.42704	92.40986	H 34	82.81921	80.37727	
P 6	74.64129	159.7393	P 27	94.33445	89.66306	H 35	111.573	109.0663	
P 7	113.3066	110.3156	P 28	45.44266	45.65637	H 36	88.51348	79.98218	
P 8	117.9868	159.8288	P 29	90.30556	147.7092	H 37	37.34161	40.61791	
P 9	134.7935	110.0201	P 30	41.35367	57.78963	H 38	34.63534	22.58507	
P 10	90.84718	77.42627	P 31	10.70067	10.02868	H 39	326.3527	426.318	
P 11	73.57869	77.67754	P 32	60.75232	46.67157	H 40	58.03857	59.99985	
P 12	106.278	92.60267	P 33	148.5804	84.26043	H 41	59.9357	59.99983	
P 13	65.843	92.46824	P 34	50.67257	46.22846	H 42	119.8106	119.9996	
P 14	521.7812	359.0482	P 35	93.50224	88.60213	H 43	51.85454	119.999	
P 15	315.8886	226.3983	P 36	57.45095	45.77075	H 44	600.5088	426.9685	
P 16	125.346	225.2167	P 37	10.87183	11.44218	H 45	59.92403	59.99972	
P 17	110.1976	109.8696	P 38	71.52524	40.68624	H 46	58.39993	59.99972	
P 18	71.142	159.739	H 27	110.9511	109.6617	H 47	119.37	119.9999	
P 19	105.2037	109.8798	H 28	79.11116	79.88342	H 48	119.946	119.9998	
P 20	121.073	109.9613	H 29	91.93172	142.2369	Costs ($/h)	123,063.4	116,645.4	
P 21	94.77948	109.8784							

Additionally, the obtained expenditures of the proposed IAQT and the standard AQT of Scenario 3 are provided in Fig. 9 for each simulated run. As can be demonstrated, in every simulated run, the proposed IAQT outperforms the standard AQT. The proportion of improvement significantly varies from 4.2891 to 18.9476%.Figure 9 Achieved costs for all runs of the standard AQT and the proposed IAQT for Scenario 3.

Table 7 outlines the robustness metrics for Scenario 3 with comparing the standard AQT and the proposed IAQT. The metrics include minimum, mean, maximum, and standard deviation values, where they are derived from the outcomes illustrated in Fig. 9. The outcomes demonstrate that the proposed IAQT exhibits superior resilience performance compared to the standard AQT. Specifically, the proposed IAQT achieves the lowest values for minimum, mean, maximum, and standard deviation (STD) at 116,645.4, 117,426.9, 118,444.9, and 359.2806 $/hr, respectively, leading to notable enhancements of 5.21, 7.82, 17.69, and 91.52%, respectively.Table 7 Robustness metrics of of the standard AQT and the proposed IAQT for Scenario 3.

Costs ($/h)	AQT	IAQT	Improvement %	
Minimum	123,063.4	116,645.4	5.215182	
Mean	127,396.8	117,426.9	7.825825	
Maximum	143,913.8	118,444.9	17.69732	
Standard deviation	4240.754	359.2806	91.52791	

An exhaustive comparison is offered in Table 8 between the proposed IAQT and various other methodologies outlined in the literature which are multi-verse algorithm (MVA)58, salp swarm algorithm (SSA)59, MVO58, jellyfish search optimizer (JFSO)60, supply–demand optimization (SDO)61, the gravitational search algorithm (GSA)62, differential evolution (DE)58, CPSO31, PSO with time-varying acceleration coefficients (PSO-TVAC)31, marine predator algorithm (MPA)46, civilized swarm optimization (CSO) incorporated Powell’s pattern search (PPS)63, crow search algorithm (CSA)58, gray wolf optimization (GWO)58, the GSO-based algorithm with ranger operators and modified scrounger (MGSO)64, and manta ray foraging algorithm (MRFA)58. The comprehensive analysis of cost and performance in Table 8 unambiguously establishes the superior performance of the proposed IAQT over other optimization techniques. Specifically tailored for Combined Power and Heat Units Economic Dispatch (CHPUED) optimization, the IAQT emerges as the optimal choice, delivering the most favorable outcomes in comparison to alternative methods. This table unequivocally illustrates that, among a spectrum of optimizers, the proposed IAQT excels in both performance and cost-effectiveness. Additionally, this comparison solidly confirms the efficacy and superiority of the proposed IAQT when applied in conjunction with CHPUED.Table 8 Comparative results for Scenario 3 for 48-unit system.

Optimizer	Best costs ($/h)	Mean costs ($/h)	Worst costs ($/h)	
IAQT	116,645.4	117,426.9	118,444.9	
AQT	123,063.4	127,396.8	143,913.8	
KOA48	116,650.0870	117,104.5447	117,915.5359	
AEA50	118,881.4473	120,045.6955	124,396.4722	
MAEA50	116,897.8879	118,004.3493	119,424.0332	
JFSO60	117,365.09	–	–	
GSA62	119,775.9	–	–	
MRFO58	117,336.9	117,875.4	118,217.5	
GWO58	122,583.3	–	–	
CPSO32	120,918.9	–	–	
MPA46	116,860.6	–	–	
TVAC-PSO32	118,962.5	–	–	
MGSO64	117,366.09	–	–	
MVO58	117,657.9	118,724	119,249.3	
CSA58	122,953.5	–	–	
DE58	120,482.7	–	–	
SSA58	120,174.1	121,110.2	122,636.8	
CSO and PPS63	117,367.09	–	–	

Furthermore, Fig. 10 presents the convergence rates of the proposed IAQT and the standard AQT in relation to their best run, worst run, and the average across all simulated runs. It is evident that the proposed IAQT exhibits superior convergence characteristics during its development, particularly in the reduction of fuel costs over the course of iterations. The standard AQT remained confined to a local optimal zone and did not achieve lower fitness values during the 300 iterations. Figure 11 further illustrates the discrepancies among the best run, worst run, and average of all runs for both the IAQT and the standard AQT in Scenario 2, highlighting a substantial improvement of approximately 43.52%, 22.37%, and 70.49% for the average, best, and worst scenarios, respectively.Figure 10 Convergence rates of the standard AQT and the proposed IAQT for Scenario 3.

Figure 11 Proportion difference for the best, average and worst of all runs for the standard AQT and the proposed IAQT for Scenario 3.

Implementation for Scenario 4

To minimize fuel expenses incorporating losses, the ED with CHPUs can be resolved using the proposed IAQT and the standard AQT. Table 9 illustrates the best parameters for the operation of the HOUs, CHPUs, and POUs obtained using the standard AQT in this case as well as the proposed IAQT. Based on this data, the proposed IAQT has the lowest possible fuel costs of 117,330.4 $/h which reveals impressive outcomes. In contrast, the standard AQT produces fuel costs of 122,705.3 $/h which is worse than the proposed IAQT. The totals for Sum (H) and Sum (P) successfully meet the specified heat and power requirements of 2500 MWth and 4700 MW, respectively, as depicted in Table 9. Moreover, all outcomes fall within the feasible range, with numerous individual results precisely positioned at either the lower or upper bounds.Table 9 Optimal operational values and best costs of of the standard AQT and the proposed IAQT for Scenario 4.

Outputs	AQT	IAQT	Outputs	AQT	IAQT	
P1	628.5506	448.8292	P 32	36.23425	52.71409	
P 2	223.7313	301.4361	P 33	145.6543	92.03028	
P3	289.4614	224.7431	P 34	58.48826	40.66324	
P 4	60.26776	109.8776	P 35	85.51618	101.1052	
P 5	72.58179	109.8783	P 36	61.60169	40.93777	
P 6	60.31301	109.9222	P 37	14.25958	11.64105	
P 7	121.7469	159.7351	P 38	47.5988	41.74357	
P 8	63.54701	109.9738	H 27	124.7783	115.3052	
P 9	117.2921	109.8686	H 28	99.69397	76.43838	
P 10	116.3564	77.52365	H 29	122.0261	120.9325	
P 11	71.39263	77.40128	H 30	78.29069	80.92044	
P 12	88.6331	92.40274	H 31	38.02201	41.63405	
P 13	93.59475	92.40015	H 32	20.5546	28.05084	
P 14	458.6817	538.5604	H 33	140.0497	110.99	
P 15	246.8761	300.9239	H 34	90.4389	75.5717	
P 16	173.2352	150.2945	H 35	107.0429	116.0828	
P 17	76.2886	159.8111	H 36	93.50389	75.80997	
P 18	61.87222	109.8746	H 37	40.31566	40.70363	
P 19	115.0649	109.8765	H 38	25.57153	23.06509	
P 20	157.7332	159.7504	H 39	336.1001	436.7963	
P 21	95.62523	159.7599	H 40	59.87565	59.99769	
P 22	171.6541	109.9225	H 41	59.78425	59.9993	
P 23	75.80825	77.45171	H 42	108.7756	119.9951	
P 24	110.0402	77.41419	H 43	120	119.9967	
P 25	113.6744	55.5928	H 44	511.3411	437.7162	
P 26	120	92.48794	H 45	60	59.9998	
P 27	118.1172	99.72298	H 46	58.97297	59.99794	
P 28	69.33218	41.66635	H 47	116.7606	119.9985	
P 29	111.79	109.7457	H 48	88.10146	119.9979	
P 30	74.57252	46.85794	Sum (H)	2500	2500	
P 31	11.06765	13.81297	Sum (P)	4700	4700	
			Costs ($/h)	122,705.3	117,330.4	

Additionally, the obtained expenditures of the proposed IAQT and the standard AQT of Scenario 4 are provided in Fig. 12 for each simulated run. As can be demonstrated, in every simulated run, the proposed IAQT outperforms the standard AQT. The proportion of improvement significantly varies from 2.7485 to 14.4476%.Figure 12 Achieved costs for all runs of the standard AQT and the proposed IAQT for Scenario 4.

Table 10 outlines the robustness metrics for Scenario 4 with comparing the standard AQT and the proposed IAQT. The metrics include minimum, mean, maximum, and standard deviation values, where they are derived from the outcomes illustrated in Fig. 12. The outcomes demonstrate that the proposed IAQT exhibits superior resilience performance compared to the standard AQT. Specifically, the proposed IAQT achieves the lowest values for minimum, mean, maximum, and standard deviation (STD) at 117,330.4, 118,494.9, 119,569.7, and 505.6826 $/h, respectively, leading to notable enhancements of 4.38, 8.07, 13.02, and 85.76%, respectively.Table 10 Robustness metrics of the standard AQT and the proposed IAQT for Scenario 4.

Costs ($/h)	AQT	IAQT	Improvement %	
Minimum	122,705.3	117,330.4	4.380306	
Mean	128,909.2	118,494.9	8.078783	
Maximum	137,483.6	119,569.7	13.0298	
Standard deviation	3551.368	505.6826	85.76091	

Furthermore, Fig. 13 presents the convergence rates of the proposed IAQT and the standard AQT in relation to their best run, worst run, and the average across all simulated runs. It is evident that the proposed IAQT exhibits superior convergence characteristics during its development, particularly in the reduction of fuel costs over the course of iterations. The standard AQT remained confined to a local optimal zone and did not achieve lower fitness values during the 300 iterations. Figure 14 further illustrates the discrepancies among the best run, worst run, and average of all runs for both the IAQT and the standard AQT in Scenario 2, highlighting a substantial improvement of approximately 34.39%, 20.38%, and 48.83% for the average, best, and worst scenarios, respectively.Figure 13 Convergence rates of the standard AQT and the proposed IAQT for Scenario 4.

Figure 14 Proportion difference for the best, average and worst of all runs for the standard AQT and the proposed IAQT for Scenario 4.

Implementation for Scenario 5

For the 96-unit system, the ED with CHPUs can be resolved using the proposed IAQT and the standard AQT. Table 11 illustrates the best parameters for the operation of the HOUs, CHPUs, and POUs obtained using the standard AQT in this case as well as the proposed IAQT. Based on this data, the proposed IAQT has the lowest possible fuel costs of 234,235.3629 $/h which reveals impressive outcomes. In contrast, the standard AQT produces fuel costs of 261,738.5541 $/h which is worse than the proposed IAQT. The totals for Sum (H) and Sum (P) successfully meet the specified heat and power requirements of 5000 MWth and 9400 MW, respectively, as depicted in Table 11. Moreover, all outcomes fall within the feasible range, with numerous individual results precisely positioned at either the lower or upper bounds.Table 11 Optimal operational values and best costs of of the standard AQT and the proposed IAQT for Scenario 5.

Outputs	AQT	IAQT	Outputs	AQT	IAQT	Outputs	AQT	IAQT	
P1	269.8027904	628.3358084	P42	296.7251853	224.3793524	H59	56.33210891	108.716696	
P2	124.6304446	224.2564724	P43	133.5875813	152.5182534	H60	82.79570134	77.52649438	
P3	206.0717024	75.84570582	P44	162.3557357	159.9417068	H61	93.80441084	125.7111519	
P4	154.1792675	109.96158	P45	88.5124899	159.1270209	H62	101.2080418	103.1870812	
P5	147.1766478	159.8302781	P46	150.1204357	109.9162008	H63	42.81305178	42.98155821	
P6	110.2101914	159.5599187	P47	140.7657046	110.0895514	H64	41.94007158	25.55230397	
P7	108.4818619	159.9547539	P48	68.75636792	110.0516859	H65	118.0118719	120.0263696	
P8	101.3883831	159.6094466	P49	115.5602954	77.38467291	H66	123.0734979	89.03079335	
P9	77.41564302	109.9471805	P50	115.9332929	76.7005516	H67	121.408225	125.8936096	
P10	104.776965	77.12104042	P51	97.61079166	92.56064225	H68	108.8684309	88.86667794	
P11	119.3586111	77.28510616	P52	63.57765887	55.28644453	H69	46.44309593	44.14895982	
P12	60.3514913	93.11052475	P53	196.2124051	86.84856452	H70	25.79143594	24.57959973	
P13	104.9492797	92.30523819	P54	91.33057632	50.12376293	H71	117.9348589	109.7190083	
P14	333.5758087	448.6882271	P55	157.9463826	108.0341686	H72	67.07680432	91.40534829	
P15	209.3233296	298.9658918	P56	59.16502781	49.54193224	H73	101.7504944	116.2696287	
P16	315.8797514	149.653607	P57	24.09905762	17.47833903	H74	115.9094219	85.90862156	
P17	148.9107529	110.1950724	P58	71.07247328	60.21078326	H75	47.69231095	40.75440268	
P18	155.8394852	109.901037	P59	161.5949957	88.0895374	H76	37.53109429	22.25004185	
P19	115.7278541	159.6712335	P60	55.17286246	42.97584678	H77	397.030675	408.8258408	
P20	128.102418	159.4614243	P61	87.19283438	118.3837066	H78	60	59.55449589	
P21	147.6929282	159.7724604	P62	76.92586899	72.72990462	H79	31.73167267	59.92226706	
P22	117.285228	109.6830028	P63	32.30835879	16.97737299	H80	109.9412553	119.7820758	
P23	72.69901883	76.67369722	P64	83.34456745	47.39192594	H81	75.20312592	119.4902789	
P24	120	77.37025472	P65	139.5649132	108.1786706	H82	315.9646772	410.5132607	
P25	106.7486801	92.40023686	P66	98.58503589	56.32836462	H83	55.59933235	59.90329696	
P26	109.8284522	92.40533998	P67	111.2149988	120.4183964	H84	59.89084663	59.98329135	
P27	353.7323898	538.6766055	P68	80.70040967	56.17427211	H85	97.04397873	119.7559527	
P28	72.26383903	149.1886108	P69	32.1944743	19.70419591	H86	119.3313247	119.7846381	
P29	256.416649	224.4423619	P70	55.4623506	45.19351866	H87	430.5212066	418.888056	
P30	165.6231211	160.2308987	P71	122.3838357	89.80822248	H88	36.65338276	59.98385552	
P31	113.9348551	109.9000384	P72	42.81513036	59.09536496	H89	59.64914916	59.88964855	
P32	177.5011989	109.8978483	P73	216.0412145	101.7771566	H90	116.9314593	119.9386431	
P33	152.6946089	109.8213928	P74	88.55870203	52.65503113	H91	51.8368352	119.9316551	
P34	110.7278808	109.8652055	P75	41.9930951	11.84885727	H92	688.95848	412.4206874	
P35	124.571397	109.7034497	P76	73.81645771	40.1734606	H93	59.80789812	59.85275237	
P36	86.67274467	77.2919024	H53	165.8289891	107.8829103	H94	60	59.99068893	
P37	65.75834254	77.70569872	H54	93.66219985	83.70269287	H95	114.3707701	119.9676006	
P38	68.39939211	92.37277469	H55	79.89824033	119.8417727	H96	114.3026727	119.9155401	
P39	94.73266745	92.19800881	H56	74.67721713	83.15707272	Sum (P)	9400	9400	
P40	87.93587855	269.2784531	H57	44.40098142	43.17005978	Sum (H)	5000	5000	
P41	65.42648087	149.3647729	H58	36.3787012	31.42261869	Costs ($/h)	261,738.5541	234,235.3629	

Additionally, the obtained expenditures of the proposed IAQT and the standard AQT of Scenario 5 are provided in Fig. 15 for each simulated run. As can be demonstrated, in every simulated run, the proposed IAQT outperforms the standard AQT. The proportion of improvement significantly varies from 9.504061 to 38.47635%.Figure 15 Achieved costs for all runs of the standard AQT and the proposed IAQT for Scenario 5.

Table 12 outlines the robustness metrics for Scenario 5 with comparing the standard AQT and the proposed IAQT. The metrics include minimum, mean, maximum, and standard deviation values. The outcomes demonstrate that the proposed IAQT exhibits superior resilience performance compared to the standard AQT. Specifically, the proposed IAQT achieves the lowest values for minimum, mean, maximum, and standard deviation (STD) at 234,235.4, 236,198.4, 237,322.7, and 745.4072 $/h, respectively, leading to notable enhancements of 10.51, 18.88, 38.48, and 97.50%, respectively.Table 12 Robustness metrics of the standard AQT and the proposed IAQT for Scenario 4.

Costs ($/h)	AQT	IAQT	Improvement %	
Minimum	261,738.6	234,235.4	10.50789	
Mean	291,160.8	236,198.4	18.877	
Maximum	385,742.2	237,322.7	38.47635	
Standard deviation	29,859.68	745.4072	97.50363	

Calculation time per iteration for the scenarios

To make sure that the IAQT is significantly more significant than the AQT, the calculation time per iteration in seconds is calculated for each of the five scenarios as demonstrated in Table 13. It can be noticed from the table that the IAQT demonstrates a marked improvement in computational efficiency over the AQT, which implies that it can solve all scenarios faster, making it more suitable for real-time applications and scenarios requiring quick solutions. The improvements in IAQT could be attributed to algorithmic optimizations such as more efficient exploration and exploitation strategies, better convergence mechanisms, or enhanced handling of candidate solutions.Table 13 Calculation time in seconds of the AQT and IAQT for the scenarios under investigation.

Applied algorithms	Calculation time (seconds per iteration)	
AQT	IAQT	
Scenario 1	0.788107	0.346567	
Scenario 2	0.815123	0.362167	
Scenario 3	0.82657	0.369483	
Scenario 4	0.844019	0.386	
Scenario 5	0.84612	0.403	

Implementation for Scenario 6

In this scenario, the model of CHP economic dispatch is upgraded to include further constraints regarding the power reserve in the system. This upgraded model not only targets the minimization of the overall fuel production costs, but also guarantees getting specified reserve of the power generation in the whole system that enables it to face any possible contingency. Therefore, a minimum power reserve should be maintained for each power plant as follows:19 Pxmin≤Pxp≤Pxmaxx=1,...,Np,

20 wherePxR=PxR,pmax-Pxpx=1,...,Np

where PxR manifests the margin reserve for each power plant x, PxR,pmax illustrates the minimum % reserve for each power plant x which is set to 10%.

Secondly, a minimum power reserve should be maintained for the whole system as follows:21 ∑xpPxR∑xpPxR,pmax≥TPxR,pmin

where TPxR,pmin illustrates the minimum % reserve for the whole system which is 30%.

In this model, several additional limitations are considered for preparing specified reserves for each generating station as modeled in Eq. (19). In addition, the whole reserve in the system is to be kept as modelled in Eq. (21). To analyze this model, the proposed IAQT is applied in comparison to the standard AQT considering the 7-unit system. Table 14 illustrates the best parameters for the operation of the HOUs, CHPUs, and POUs obtained using both methods. In this regard, diverse reserve percentages are addressed with 10%, 20%, 30% and 35% reserve.Table 14 Optimal operational values and best costs of of the standard AQT and the proposed IAQT for different cases of Scenario 6.

Reserve	10%	20%	30%	35%	
	Outputs	AQT	IAQT	AQT	IAQT	AQT	IAQT	AQT	IAQT	
POUs	P1	66.87072361	44.90882174	57.66132732	55.01179891	37.14533684	52.47666686	47.97255323	48.74274233	
P2	69.72024827	98.54142459	86.79311612	98.53057271	80.3402516	87.41544863	81.22115222	81.17133294	
P3	121.8566131	112.6730069	120.7684066	112.5648454	113.9874525	122.4819127	110.0451012	113.7328694	
P4	210.2607943	209.8121358	198.3303368	199.972335	174.6570411	174.9888084	162.2506917	162.4896704	
CHP 1	P5	90.39666723	94.06298795	93.75448765	93.91706933	148.7309515	122.5951421	151.4407975	153.8455236	
H5	40.89495345	40.00162304	42.69232553	40.00337866	45.13896652	40.04202131	47.06970407	40.01786129	
CHP 2	P6	49.58468995	27.88643243	30.08566871	28.74607583	2.806034017	1.122330558	0.775098833	0.923104986	
H6	62.07131143	74.99792867	56.41969947	74.99729556	61.7696403	74.62449268	24.91833725	74.90431817	
HOU	H7	38.34399862	47.1156389	63.49463183	46.2566286	85.42432568	74.25317676	124.3065639	74.17257684	
Costs ($/h)	10,317.81589	10,092.00289	10,406.78807	10,156.24198	11,682.76872	10,910.46709	12,080.61572	11,569.64787	

Based on the outcomes in Table 14, when the reserve is 10%, 20%, 30%, and 35%, the proposed IAQT provides the fuel costs of 10,092.00289, 10,156.24198, 10,910.46709, 11,569.64787 $/h, respectively, which reveals impressive outcomes, while the standard AQT provides the fuel costs of 10,317.81589, 10,406.78807, 11,682.76872, 12,080.61572 $/h, respectively. The proportion of improvement significantly varies from 1.259121% to 6.641654%. Moreover, the best costs of the standard AQT and the proposed IAQT versus the reserve percentage for the different reserve percentage are depicted in Fig. 16. It can be noticed from this figure that when the reserve percentage represents 10%, the fuel cost is as the same as the usual case (without any reserve percentage). However, the fuel costs are dramatically increased after 20% to reach to a high fuel cost at 35% which show the importance of the reserve percentage constraint.Figure 16 The best costs of the standard AQT and the proposed IAQT versus the reserve percentage for different cases of Scenario 6.

Furthermore, Fig. 17 presents the convergence rates of the proposed IAQT and the standard AQT in relation to their best run across all simulated runs for four cases including 10, 20, 30, 35% reserve. It is evident that the proposed IAQT exhibits superior convergence characteristics during its development, particularly in the reduction of fuel costs over the course of iterations. The standard AQT remained confined to a local optimal zone and did not achieve lower fitness values during the 300 iterations.Figure 17 Convergence rates of the standard AQT and the proposed IAQT for different cases of Scenario 6.

Testing on real world (RW) engineering design issues

To evaluate the effectiveness of the proposed IAQT in tackling limited, non-convex optimization challenges, computational analyses of RW problems were performed. For testing, nine optimization issues were taken from CEC 2020, namely from the domains of chemical and mechanical engineering. Based on a reference65, the constraint functions' upper and lower limit violations were calculated.

The firm penalty method has been implemented to control the constraints in the case studies which were provided. For each of the nine benchmarks, the population size was fixed at 200, and each case study was allowed a maximum of 500 iterations. Important information about the inequality constraints and benchmark functions used in the study is taken into account66. Table 15 showcases the RW optimization engineering benchmark cases featured in the CEC 2020 competition. The associated average converging properties are displayed in Fig. 18 after the IAQT and AQT are applied for fifty distinct runs.Table 15 RW benchmarks comprised in the CEC 2020.

Case study problem	Function	Constraints	Decision variables	
Process synthesis	RW8	2.0	2.0	
Process synthesis	RW12	9.0	7.0	
Process design	RW13	3.0	5.0	
Weight minimization of a speed reducer	RW15	11.0	7.0	
Tension/compression spring design (case 1)	RW17	3.0	3.0	
Pressure vessel design	RW18	4.0	4.0	
Welded beam design	RW19	5.0	4.0	
Three-bar truss design	RW20	3.0	2.0	
Multiple disk clutch brake design	RW21	7.0	5.0	

Figure 18 Convergence curves of the IAQT and AQT for RW engineering problems.

The results attained were contrasted with those of various optimization solvers, such as including White Shark Optimization WSO67, Dung Beetle optimizer (DBO)68, Leader WSO (LWSO)66, Kepler Optimization (KO)69, and Fox Algorithm70. For every one of the nine RW case studies, Table 16 presents a comparison between the IAQT's efficiency and that of competing optimization techniques. For each of the fifty distinct runs that the algorithms generated, the best (Min), worst (Max), median (Med), average (Av), and standard deviation (STd) of the fitness scores achieved are indicated in this table. Moreover, rank describes the sequence in which each algorithm performs, with the best method recording the highest order. The least values of the Min, Med, Max, Av, and STd metrics are used to determine the ranking.Table 16 Statistical analysis for the RW engineering benchmarks.

Engineering design Problem	Items	FOX	DBO	LWSO	KO	WSO	AQT	IAQT	
RW8	Min	2	2	2	2	2	2.000000189	2	
Max	2	2	2	2	2	2.000230784	2	
Med	2	2	2	2	2	2.000028542	2	
Av	2	2	2	2	2	2.00004541	2	
STd	7.10E−08	2.44E−16	2.44E−16	4.43E−12	2.88E−16	5.06245E−05	0	
Rank	6	2	2	5	4	7	1	
RW12	Min	2.924831	2.924831	2.924831	2.924832232	2.924831	2.92762277	2.924830554	
Max	3.082564	4.074353	3.081732	2.924844916	2.924831	4.056558343	2.924830554	
Med	2.925031	3.081732	2.924831	2.946961824	2.924831	2.947332457	2.924830554	
Av	2.946508	3.378957	2.940521	2.92529553	2.924831	3.055730707	2.924830554	
STd	0.047529	0.481632	0.048293	0.003126717	1.43E-07	0.301437711	4.87817E-13	
Rank	5	6	4	3	2	7	1	
RW13	Min	26,887.42	26,887.42	26,887.42	26,887.42	26,887.42	26,887.42569	26,887.42221	
Av	27,135.01	26,887.42	26,887.42	26,887.42	26,887.42	26,889.06527	26,887.42221	
Med	26,887.42	26,887.42	26,887.42	26,887.42	26,887.42	26,888.77662	26,887.42221	
Max	28,368.22	26,887.42	26,887.42	26,887.42	26,887.42	26,893.79546	26,887.42221	
STd	473.4105	1.12E−11	2.05E−06	1.75E−07	8.52E−05	1.516588701	1.83746E−11	
Rank	7	1	4	3	5	6	2	
RW15	Min	2995.595	2994.424	2994.429	2994.426015	2994.648	3010.217827	2994.42503	
Max	1.00E+15	1.00E+15	1.00E+15	2994.427249	1.00E+15	3526.398188	2994.430811	
Med	1.00E+15	1.00E+15	2994.439	2994.429985	1.00E+15	3080.767639	2994.426278	
Av	6.50E+14	6.00E+14	5.00E+13	2994.427404	6.00E+14	3102.697049	2994.426395	
STd	4.89E+14	5.03E+14	2.24E+14	0.000867638	5.03E+14	89.32178353	0.001044252	
Rank	6	5	3	2	4	7	1	
RW17	Min	0.012677	0.012666	0.012665	0.01266588	0.012665	0.012926363	0.012665233	
Max	1.00E+15	0.012928	0.012665	0.012667121	0.012665	0.019892019	0.012665233	
Med	0.012781	0.012719	0.012665	0.012672708	0.012665	0.015054181	0.012665233	
Av	5.00E+13	0.012742	0.012665	0.01266772	0.012665	0.015489395	0.012665233	
STd	2.24E+14	7.56E−05	8.30E−13	1.62E−06	1.69E−08	0.001766053	4.26757E−11	
Rank	6	5	2	2	4	7	1	
RW18	Min	6359.528	6247.673	6247.72	6059.74963	6247.675	6061.404777	6059.714335	
Max	239,304.10	7319.001	6248.808	6060.129709	6247.688	7426.06085	6059.714335	
Med	15,046.06	6382.985	6247.853	6090.661389	6247.681	6143.355423	6059.714335	
Av	39,927.73	6544.502	6247.934	6061.052218	6247.681	6357.091738	6059.714335	
STd	65,645.18	400.6724	0.236775	4.432071131	0.003228	395.8513983	9.18728E−13	
Rank	7	5	4	2	3	6	1	
RW19	Min	1.67593	1.670218	1.670218	1.670251358	1.670218	1.709024647	1.670217726	
Max	1.994586	1.816712	1.670218	1.670301692	1.670218	2.011737284	1.670217726	
Med	1.722726	1.670218	1.670218	1.670434518	1.670218	1.84393659	1.670217726	
Av	1.756922	1.700254	1.670218	1.670303479	1.670218	1.835601164	1.670217726	
STd	0.080852	0.055884	5.61E-08	3.12E-05	6.20E-08	0.074246485	3.04283E-11	
Rank	6	5	2	4	3	7	1	
RW20	Min	263.8958	263.8958	263.8958	263.8958434	263.8958	263.8985956	263.8958434	
Max	263.8962	263.8961	263.8958	263.8958434	263.8958	264.1978412	263.8958434	
Med	263.8959	263.8958	263.8958	263.8958434	263.8958	263.990995	263.8958434	
Av	263.8959	263.8958	263.8958	263.8958434	263.8958	264.0097935	263.8958434	
STd	7.87E−05	4.76E−05	1.30E−14	8.72E−10	4.29E−12	0.0788497	2.86067E−13	
Rank	6	5	1	4	3	7	2	
RW21	Min	0.235242	0.235242	0.235242	0.235242458	0.235242	0.235253231	0.235242458	
Max	0.235243	0.235242	0.235242	0.235242458	0.235242	0.236360109	0.235242458	
Med	0.235243	0.235242	0.235242	0.235242459	0.235242	0.235579216	0.235242458	
Av	0.235243	0.235242	0.235242	0.235242458	0.235242	0.235624358	0.235242458	
STd	9.23E−08	1.14E−16	1.87E−11	1.90E−10	5.94E−09	0.000245927	3.11969E−12	
Rank	6	1	3	4	5	7	2	
Ranks summation	55	35	25	29	33	61	12	
Average rank	6.1111	3.8889	2.7778	3.2222	3.6667	6.7778	1.3333	
Regarding improvement %	78.18%	65.72%	52.00%	58.62%	63.64%	80.33%	–	
Final ranking	6	5	2	3	4	7	1	

To assess the comparison between the AQT and the proposed IAQT. Table 16 demonstrates that the IAQT outperforms the AQT by 0.52% in achieving the lowest minimal goal in nine RW issues on a regular basis. With a noteworthy 4.4% outperformance ratio, the IAQT exceeds the AQT in terms of mean objective ratings in the nine RW problems. The examination of the maximum objective scores reveals that the IAQT outperforms the AQT in nine RW issues, with a noteworthy 16.13% outperformance ratio. The IAQT outperforms the AQT by a significant 99.5%, as demonstrated by the standard deviation of achieved objective scores.

Moreover, the average rank provides an overview of how effectively methods perform in relation to each other across the assessed criteria. Better performance is indicated by a lower average rank. With an average rating of 1.3333, the IAQT shows dominance and efficacy across all parameters taken into consideration. With a final ranking, the IAQT takes first place, demonstrating its overall superiority over the other algorithms that were assessed. By evaluating the average rank, the IAQT outperforms each algorithm in certain noteworthy ways. In comparison to FOX, the IAQT exhibits the most improvement, measuring 78.18% better. Furthermore, it attains noteworthy enhancements of 65.72%, 58.62%, and 63.64% in relation to DBO, KO, and WSO, in that order. The IAQT demonstrates a significant 80.33% improvement over the AQT.

Conclusions

In this paper, the proposed an improved version of the Aquila Optimization Technique (IAQT) is developed for the CHPUs economic dispatch to minimize the overall cost of power generation while satisfying the demand and operational constraints. The algorithm incorporates limitations and constraints specific to each dimension of the newly generated solutions, ensuring the feasibility and validity of the solutions. Fitness evaluation is performed, and improved solutions are selectively retained based on their objective values. The improved version of the Aquila Optimization Technique (IAQT) is proposed to overcome certain limitations associated, enhance its searching capabilities. The proposed IAQT and the standard AQT are assessed on the including 7–48-unit and large 96-unit systems of CHP economic dispatch test systems effectively. Through extensive experimentation and testing on various scenarios with/without considering transmission losses. The standard AQT and proposed IAQT are tested on CEC 20 benchmark functions. Furthermore, the overall costs for the 7 unit-system are considered including the reserve constraint. It is demonstrated that the proposed IAQT outperforms the standard version and several reported results with remarkable performance and efficiency. The IAQT demonstrates heightened resilience compared to the standard AQT throughout achieving superior performance in terms of acquiring the minimum, maximum, standard deviation, and mean. Furthermore, the proposed IAQT not only excels in these statistical measures but also exhibits enhanced convergence characteristics during its evolution. This is particularly evident in its ability to consistently reduce fuel expenditures throughout the iterations’ duration.

Acknowledgements

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number ISP23-122.

Author contributions

S.H.H.: Resources; Methodology; Project administration; Supervision; Roles and Writing original draft. G.M. :Data curation; H.A..: Formal analysis; Methodology; Project administration; Roles and Writing original draft. H.S. E. M: Data curation; Formal analysis; Validation; Roles/Writing original draft. A.G.: Validation; Methodology; Software; Project administration; Writing—review and editing.

Data availability

Data analyzed or generated during study are included in this article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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