An improved weighted average algorithm with Cloud-Based Risk-Conscious stochastic model for building energy optimization

Abstract

This paper presents a new approach toward building energy optimization through proposing a cloud theory-based stochastic model that considers the risk that arises due to uncertainty in building cooling and heating efficiency. The main aim of this study is minimizing the annual energy consumption (AEC) in a reduced-order office building by addressing the intrinsic variability of the environmental parameters. One of the key contributions of this work is developing an Improved Weighted Average Algorithm (IWAA), introducing a Dynamic Weight Update Mechanism to further balance exploration and exploitation throughout optimization. The proposed method is evaluated under three situations: (1) CEC-2022 benchmark functions, (2) minimized office building energy optimization excluding risk in uncertain parameters, and (3) stochastic building energy optimization model including risk in uncertain cooling and heating efficiencies. The optimization model is also applied to three different weather conditions to highlight its applicability in varying environmental conditions. The results demonstrate that the IWAA is learned much more effectively than the typical algorithms, such as WAA, PSO, and WOA, by providing more stable and consistent results for lower AEC values. Furthermore, injecting uncertainty into the optimization problem with the help of the cloud theory framework is identified as the most significant factor in getting more realistic and credible energy forecasting. The findings illustrate the strength of the proposed IWAA, which achieves better optimization performance with its incorporation of uncertainties and balancing of exploration-exploitation trade-offs. The model is a strong candidate for real-world energy optimization problems, with potential benefits for the design of sustainable energy-efficient buildings.

Introduction

The accelerating challenges of climate change and the global pursuit of sustainable development have positioned energy efficiency as a central priority in engineering and policy initiatives¹. To meet international objectives such as those outlined in the Paris Climate Agreement, the design of zero-emission and low-energy buildings has become increasingly essential². Within this context, Building Energy Optimization (BEO) plays a critical role by aiming to minimize energy consumption while maintaining indoor comfort and system performance³. However, the optimization of building energy systems remains highly complex due to numerous interacting variables, non-linear dynamics, and operational constraints that often render traditional deterministic methods inadequate. Consequently, metaheuristic algorithms have gained prominence because of their flexibility and ability to explore vast, high-dimensional search spaces effectively⁴. However, conventional metaheuristics frequently encounter issues with premature convergence and entrapment in local optima that can lower their robustness and generalization capability in complex real-world applications. Recent research has thus focused on developing improved or hybrid metaheuristic algorithms employing dynamic weight adaptation, hybrid search mechanisms, or exploration-exploitation balancing techniques to improve their stability and efficiency in optimization. In addition, uncertainty in some key parameters–like heating and cooling efficiencies, weather variability, and fluctuating energy demands–introduces further complexity that is often ignored by deterministic models^5–7. If these uncertainties are neglected, there can easily be significant deviations in the energy predictions and suboptimal operational performance. Therefore, stochastic modeling approaches that are able to quantify and incorporate risk will be indispensable if reliability and realism in building energy optimization are to be achieved. With the integration of algorithm improvement and stochastic risk modeling, the optimization frameworks will be able to obtain more resilient and trustworthy solutions that are close to the actual building performance under uncertain environmental conditions.

Recent advances in building energy optimization (BEO) have increasingly applied hybrid and intelligent computational techniques to enhance efficiency, comfort, and sustainability. For instance, a hybrid metaheuristic derived from the Pelican Optimization Algorithm (POA) and Single Candidate Optimizer (SCO) was presented to effectively combine the global search of POA with the local search capability of SCO, achieving balanced optimization performance⁸. Similarly, Memetic Algorithms (MAs) were also tailored to multi-objective optimization for smart buildings with dynamic adaptation to evolving environmental and energy needs⁹. In another study, artificial neural networks were coupled with extrapolation techniques for cooling/heating demand prediction, which was also optimized using Particle Swarm Optimization (PSO) and Biogeography-Based Optimization for improved forecast accuracy¹⁰. In¹¹, the knowledge-informed, performance-based generative design (PGD) optimization framework for sustainable building developments that can overcome time-related limitations of traditional methods by informing the PGD process with a knowledge graph is presented. The implementation of the framework in the design case study optimized modular layout both to decrease cooling energy consumption and increase daylight utilization. In¹², the Grey Wolf Optimizer (GWO) is employed to reduce the annual energy consumption of an office building in Seattle’s climate. The study not only applies GWO to energy minimization but also thoroughly analyzes the influence of different algorithm parameters on its optimization capability. Through the investigation into GWO’s sensitivity to different configurations, the study highlights the necessity for proper parameter tuning in order to enhance the algorithm’s efficiency and solution precision in building energy applications. In¹³, one improved transient search optimization algorithm based on a Rosenbrock’s direct rotation technique has been used to solve the building energy optimization for two simple and detailed buildings with the purpose of optimizing the consumption of building energy. In¹⁴, a knowledge-informed performance-based generative design optimization framework for sustainable buildings is presented, with the aim of mitigating the time issue by integrating a knowledge graph into PGD. An application of the proposed framework has been demonstrated in the design project related to optimizing module layout with the aim of minimizing cooling energy and maximizing daylighting. A MILP method was implemented for optimizing MPC for HVAC systems in multi-family houses with multiple objective functions being taken into consideration¹⁵. Besides, some new metaheuristics were proposed, such as a Golden Sine-enhanced Exponential Distribution Optimizer (EDO), which incorporated search capability and power in energy and hybrid system optimization under uncertain surface area and load conditions¹⁶.

In continuation of the reviewed deterministic studies, a number of recent works have extended BEO toward stochastic frameworks for explicitly addressing the influence brought about by uncertainties within design, environmental, and operational parameters. In¹⁷, a stochastic adjustment framework for shading devices was developed that considered the impact of uncertainty in window blind usage on the overall building energy consumption. Therein, a stochastic shading model for common office buildings was developed and its parameters optimized by way of hyperparameter tuning through a machine learning algorithm with ensuing improved prediction accuracy when compared to static control strategies. Similarly, in⁶, a stochastic optimization approach was proposed that aimed at improving the energy performance of buildings under holistic uncertainty conditions by explicitly including physical and design uncertainties such as variations in thickness, density, and thermal conductivity of envelope components including walls, roofs, and floors. The results verified that stochastic formulations provided more realistic and robust energy predictions than those of deterministic methods. In¹⁸, uncertainty in office-building energy consumption was investigated in an EnergyPlus-based Monte Carlo simulation framework. The occupancy schedules were generated using a stochastic process based on the Markov chain method while setpoint schedules were determined using a PMV-based decision model. In that study, the intrinsic variability in occupancy behavior was effectively modeled and the resultant impact on energy performance was quantified. In¹⁶, an enhanced Exponential Distribution Optimizer (EDO) was presented for solving building energy optimization problems under uncertainty in conditioned surface area and temperature control parameters. The proposed method adopted Monte Carlo simulation to evaluate the propagation of risk to ensure robustness of solutions in energy performance estimation. Furthermore, in¹⁹, an intelligent hybrid computational framework involving Monte Carlo analysis, EnergyPlus simulation, and ANN was developed to enhance the prediction of both energy consumption and thermal comfort level. As highlighted therein, this model showed the benefit of combining stochastic sampling with data-driven learning for improved generalization under uncertain conditions. Finally²⁰, performed a meta-model-based sensitivity analysis using a suite of advanced statistical tools, such as validation tests, Morris’ screening method, multivariate adaptive regression splines (MARS) meta-modeling, and Sobol’ variance decomposition, to identify the most influential parameters governing building energy performance simulation (BEPS) at early design stages. This comprehensive framework highlighted the need to incorporate stochastic and sensitivity analyses in ensuring credible and uncertainty-aware energy predictions. Reference²¹ explored simplification techniques of models within a parallel NSGA-II optimization framework for a mixed-use commercial building. They show that simplified surrogate models can significantly reduce computation cost with negligible sacrifice in solution accuracy. The robustness analysis in²² under the uncertainty of occupant behavior, conducted using several robustness indicators, highlighted that the stochastic variations in occupants’ actions are the most influential in energy performance. Reference⁷ developed a BIM-integrated intelligent optimization framework for dynamic prediction and multi-scenario building carbon emission minimization, considering weather fluctuation and occupancy uncertainty. Finally²³, introduced a hybrid uncertainty quantification approach that jointly accounted for aleatory and epistemic uncertainties affecting the techno-economic performance of the renewable energy systems in residential buildings, building a more comprehensive foundation for uncertainty-aware BEO modeling. References^17–19 have followed the inclusion of stochastic modeling and simulation-based optimization to describe realistic variations in building performance. These studies introduced probabilistic formulations for window shadings behavior, envelope material properties, conditioned surface areas, and occupancy patterns, hence enabling more credible and risk-aware energy predictions compared to traditional deterministic assumptions. References^21–23 highlight the progressive shift toward hybrid, uncertainty-aware, and data-integrated optimization models in BEO research that bridges the gap between deterministic performance simulation and real-world stochastic variability.

Based on the literature reviewed, the subsequent research gaps may be substantiated accurately and systematically:

• Traditional Metaheuristic Algorithm Limitations in Building Energy Optimization: Different studies have employed metaheuristic algorithms such as PSO, GA, WOA, and Memetic Algorithms for optimizing building energy consumption. While these algorithms give remarkable abilities, their traditional realization is usually bedeviled by problems such as premature convergence or getting stuck in local optima, especially on intricate high-dimensional search spaces. These limitations reduce the reliability and robustness of the obtained optimizations. One of the bright directions to close this gap is to enhance the ability of the algorithm to search by mechanisms such as dynamic weight tuning, hybridization, or exploration-exploitation balance. Nonetheless, most previous research does not utilize such improvements effectively, which indicates the need for designing an improved metaheuristic algorithm with improved global and local search capabilities.

• Neglects Uncertainty Parameters in Building Energy Optimization Models: Most previous researches^8–16 have employed deterministic models ignoring significant uncertainties such as variation in heating and cooling efficiencies, changing weather patterns, and dynamic energy demands. This deterministic assumption limits the applicability of the optimization models in reality, as stochastic uncertainty cannot be eliminated. Omitting these uncertainties may result in optimistic or unreliable energy consumption predictions. Thus, there exists a significant need for stochastic optimization models with risks and uncertainties as explicit parameters to enhance model realism and credibility.

• Although stochastic BEO models^17–23 have incorporated the uncertain parameters, most rely on sampling-based methods like Monte Carlo, which are computationally intensive and time-consuming. Their high-cost limits large-scale or practical applications. Hence, there is still a need for efficient stochastic frameworks that ensure accuracy and robustness with lower computational demand.

• Inadequacy of Holistic Approaches Addressing Algorithmic Improvement, Uncertainty, and Regional Differences: While a few studies examine region-dependent designs or performance values between climatic zones, few studies integrate enhanced metaheuristics, uncertainty models, and multi-regional validations. Given that energy efficiency and uncertainty impacts can be vastly different across climates, there is an evident need for developing a strong stochastic optimization scheme that is validated across several climatic regions. This combined approach is also not yet extensively investigated in current research.

According to the found research gaps, the major contributions and innovations of the current study can be enumerated as follows:

• Novel Improved Weighted Average Algorithm (IWAA) creation and verification: The novel IWAA is better than the traditional weighted average algorithm in that it incorporates a Dynamic Weight Update Mechanism. This new feature helps in attaining improved exploration-exploitation balance. The IWAA as a model-based optimization component operates within a physically informed stochastic model of building energy dynamics, where optimization variables, objective functions, and constraints are all derived from energy performance parameters such as HVAC efficiency, thermal loads, and environmental uncertainty. This integration ensures that the optimization process directly corresponds to the real-world behavior of buildings rather than abstract numerical spaces, thereby making the approach inherently model-based.

• The IWAA is extensively verified on the CEC-2022 benchmark functions where it outperforms some of the leading algorithms such as PSO, WOA, and the traditional WAA in terms of optimization accuracy and stability. The choice of CEC-2022 as the benchmark suite is for specific reasons: it is one of the most comprehensive and up-to-date standards for the evaluation of optimization algorithms, encompassing a wide range from unimodal, multimodal, hybrid, and composite functions. The mentioned characteristics make this benchmark highly suitable for assessing both exploration and exploitation abilities.

• New stochastic optimization model using Cloud Theory: The paper introduces a novel method for modeling uncertainty that measures variable cooling and heating efficiencies’ risks. By the use of cloud droplets to represent variability, this model captures real conditions better than deterministic methods, and as such, the energy optimization becomes more robust and reliable.

• IWAA integration with the new stochastic model for optimization of building energy: A new hybrid strategy is formulated by combining the advantages of the IWAA with the new cloud-based stochastic optimization methodology. This integrated approach is utilized for simplified office buildings in three contrasting climatic regions—Seattle, Chicago, and Houston—demonstrating its adaptability and potential to minimize yearly energy consumption amidst environmental uncertainty. The stochastic model serves as the analytical backbone of the framework that provides a probabilistic representation of uncertainties in heating and cooling efficiencies, weather variability, and operational fluctuations. In this paper, the IWAA updates its search mechanism dynamically with respect to the risk metric derived from the cloud model, which thus allows the optimization process to adapt intelligently to the uncertain conditions. The integration finally provides a unified decision-making environment combining algorithmic intelligence and realistic stochastic modeling.

The organization of the study is presented as: Sect. Building design model is the model of building design, wherein the principal parameters of the energy-analyzed simple office building are outlined. Section Optimization process details the optimization process, which is the energy simulation using EnergyPlus and the development of the Improved Weighted Average Algorithm (IWAA). Section Cloud-Based Risk-Conscious stochastic model introduces a cloud-based stochastic model that addresses risk based on uncertainties in cooling and heating efficiencies. Section Simulation results and discussion exhibits the simulation results and compares the suggested approach under three scenarios: benchmark function optimization and energy optimization in different climate zones (Seattle, Chicago, and Houston). Section Conclusion concludes the research with major findings and directions for future research.

Building design model

To verify the proposed optimizer, a simplified office building model, constructed with some special properties to simulate the Houston, Chicago, and Seattle climatic conditions⁸, is used. They are based on prior research^8,12 and are illustrated in Fig. 1. Four most significant design parameters are optimized: the building orientation (), the west windows width (), the east windows width (), and the shadow transmittance factor (). The lower and upper limits of decision variables are presented in Table 1. The building’s structure consists of a 1 cm wooden exterior, an insulation layer with 10 cm, and a 20 cm concrete layer and have a U-value of 0.25 W/(m² K), offering insulation and thermal mass in balance. Ceilings and floors are built in layers of material, like an 18 cm thick layer of concrete, laminate, a 5 cm concrete layer, and carpet, to control indoor temperature fluctuations. Bricks used for the internal walls have a 12 cm thickness. The details of the simplified office building including exterior walls, thermal conductivity, and thick brickwork are presented in^8,12. The windows consist of 8 mm krypton gas insulation trapped between two 3 mm glass layers to provide improved thermal insulation. Furthermore, exterior sunlight control devices are provided to minimize solar heat absorption, improving the total energy effectiveness of the building by reduced mechanical cooling load.

Fig. 1.

Schematic of the under-study building^8,25.

Table 1.

The lower and upper limits of decision variables^8,12.

Definition	Variable	Limits
		[−180, 180]
		[0.1, 5.9]
		[0.1, 5.9]
		[0.2, 0.8]

The objective of the design problem is the minimization of the building annual energy consumption (AEC) while taking into account the effect of the selected design variables as follows:

1

Where, , , and are the building’s AEC for heating, cooling, and lighting, respectively. Ψ denotes the building’s surface area with climate control, the portion of the managed building for temperature control. The annual energy consumption is divided by the floor area Ψ, allowing an energy consumption metric that is normalized for building size. Hence, the optimization objective is AEC minimization, which is in kWh per square meter per annum (kWh/m²a). In this study, the primary energy factor () for electricity is 3.0 and is used to present the building’s annual building energy consumption (AEC) for electricity in equivalent primary fuel energy consumption. This conversion process is required to get a more realistic estimate of the building’s overall energy demand, as it factors in the primary energy sources that go into generating the electrical energy used by the building. Further, the heating and cooling system efficiencies are denoted by = 0.44 and = 0.77 in the deterministic analysis. These efficiencies are the fraction of the input energy that is successfully transformed to the profitable heating and cooling performance, respectively. These efficiencies are necessary in order to properly model the energy demand and examine the performance of the climate control systems within the building, as noted in refs^8,12.

The simplified office building model was selected because it is a standardized and representative prototype that is frequently adopted in BEO studies to allow comparability and reproducibility of results. This model captures the essential physical and operational features of small- to medium-sized office buildings, like typical geometry, occupancy schedules, HVAC configuration, and envelope characteristics, while avoiding unnecessary architectural complexity. This allows the study to focus on evaluating the proposed optimization algorithm and stochastic modeling framework without being confounded by unique design irregularities.

The integrated BEM-LOD framework adopted for this study is represented in Fig. 2, which clearly conceptualizes how the proposed BEM evolves through successive development stages. It illustrates how both geometric and informational detail is incrementally enhanced for the four major levels of development, namely LOD 100–400. At the early stage of LOD 100, the building model is a conceptual mass through which only global form and orientation are defined without detailing its thermal properties. While the model acquires architectural zoning, envelope configuration, and material attributes as it proceeds towards LOD 200 and LOD 300, it grants only partial simulation about its thermal performance. Not until LOD 400 does a model become fully simulation ready by containing all geometric, physical, and system parameters necessary for integration with EnergyPlus and other analysis engines.

Fig. 2.

Integrated BEM–LOD framework showing the progressive model refinement from conceptual massing (LOD 100) to simulation-ready model (LOD 400) and its impact on EUI estimation accuracy.

This hierarchical representation depicts how LOD regulates the input data granularities and, more importantly, the reliability of EUI estimations. It identifies that higher LODs lead to more reliable and robust energy predictions, aiming to improve overall data fidelity progressively. This is one of the main contributions of the proposed weighted-average optimization framework: modeling precision integrated with energy analytics.

Optimization process

Building energy simulation

This paper employs the EnergyPlus (EP) tool from the US Department related to computer package^8,12 to model building thermal performance and make estimates of energy demand. EP, being a package famous due to its capability to perform comprehensive energy analysis missing needing a screen-based interface, takes text inputs and delivers detailed reports. EP models heating and cooling loads, reacting to thermal setpoint levels, and makes calculations of both short-term and future estimated energy demand, including considering primary plant equipment and HVAC system loads. EP is better in rapid and accurate calculations with DOE-2 and BLAST algorithms and also includes a warm-up phase to detect initial balance points. The model accounts for weather-based external conditions and shading effects by season. The meta-heuristic algorithm is coupled with EP through a self-developed subroutine that adjusts control parameters, performs simulations, and revises the energy consumption model. The IWAA algorithm proposed in this study optimizes design variables through interactions with EP by iteratively updating energy consumption forecasts by passing through the simulation cycle. The interactive loop between EP and IWAA updates the model of building energy consumption on a continuous basis.

Proposed improved weighted average algorithm

Conventional WAA

The Weighted Average Algorithm (WAA) is an algorithm for solving optimization problems through computing a weighted mean position. In the WAA, the impact of each element on the average is adjusted based on a predetermined weight, with components having greater weights exerting stronger effect on the final location. This weighting technique enables important factors to be given more weight, resulting in a more representative overall average that reflects the personal character of the dataset. In WAA, the position determined by the weighted mean corresponds to the current distribution of the population and is aligned with the personal best () and global best () locations. These hybrid solutions are then used to update the candidate solutions in the next iteration, guiding the algorithm towards optimum results.

• Phase of initialization.

The process of optimization starts with a collection of potential answers, represented as matrix X, which is created among the specified search space, randomly²⁴.

2

In the equation, , is the jth dimension answer location. N denotes the population size, while n indicates the problem number of dimensions [36]. The term “rand” refers to a random number within zero and 1. LB_j is the minimum limit of the jth dimension, and UB_j refers to the maximum limit of the jth problem dimension.

• Weighted average position.

The first operation for determining the weighted mean location is to evaluate the all candidate solutions fitness and then sort the population according to the assessment criteria: either “larger values are better” (LTB) or “smaller values are better” (STB). Then, the best solutions from the individuals are chosen for computing the weighted mean location, according to the formulas²⁴:

3

4

5

6

Here, nP is the population size, Xi is the ith candidate solution, and Fitness is the fitness to obtain its value. is the sum of the fitness measures of the selected individuals. is the weighted mean location. t denotes the number of iteration at the present stage, is the greatest iterations number permitted, and denotes the candidate solutions number taken.

• Identifying the search phase.

Using the iteration procedure, Eq. (7) computes the search phase of the candidate solution, i.e., whether it is at the exploration stage or the exploitation stage²⁴.

7

During this process, it is the present iteration number, is the greatest number of iterations, and α controls the exploration-exploitation trade-off. Candidate solutions with f(it) ≥ 0.5 focus on exploitation, whereas f(it) < 0.5 focuses on exploration. The exploration-exploitation combination is expected to enhance the search effectiveness in the solution space. The discovery and utilization capabilities in the WAA can be explained in the rest of the following sections.

• Phase of exploitation.

The step of exploitation mimics how the search agents’ population advances toward areas in the search space that are more likely to achieve better global best values. This step primarily seeks to exploit the search space, based on the location of the present population weighted average along with personal and global best positions, not as detailed outlined below²⁴.

8

Where, , and are randomly generated values within the range of 0 to 1, and these values are employed to regulate the extent of the search space expansion around both the personal best position () and the global best position (). More specifically, and correspond to the personal best and global best locations at the present iteration (it), respectively. The values of and play a crucial role in controlling the movement strategy of each candidate solution. In particular, dictates the movement towards the global best position, while guides the movement towards the personal best location. This allows for a dynamic search process, where each candidate solution adapts its direction based on the relative importance of exploring the global optimal answer versus refining its own best-known solution.

The second phase is to make use of the search space that is described by the population’s weighted average position and individual best position. In contrast to the first method, this phase does not incorporate the global best position, and this means there is a narrower search space. The narrow search space is calculated by²⁴.

9

Where, and are random variables with values in the interval 0 to 1. This movement strategy, unlike the former one, puts greater emphasis on enhancing convergence rate as well as precision. It achieves this through regulating the agent’s displacement from its individual best location to the population weighted mean location. This direction of thoughtful movement directed at the average position of the population enhances chances of moving towards optimal answers.

The third method emphasizes the exploration of the search space between the global best position and the weighted average position of the entire population. In order to achieve this, the position update for per candidate is determined using the following equation²⁴. This approach seeks to balance the exploration of new areas of the search space with refining solutions around the best-known positions, guiding the algorithm toward more optimal regions while maintaining diversity in the search process.

10

Where, and are scalar values that range between 0 and 1. The search regions in this method are more confined compared to those in the initial and secondary strategies. The transition from the global best position towards the weighted average location contributes to a faster and more accurate convergence.

• Exploration phase.

The WAA utilizes two fundamental methodologies to explore the prospective solution space. The first approach is utilized through the Lévy flying mechanism to search for the best global solution. In this method, the candidate solution is moved from the global best location in various directions and lengths. The movement distance changes dynamically at each iteration to allow both large and small steps. Longer movements allow more extensive exploration of regions in the search space, and small steps allow fine-tuning of the position in the local neighborhood. This dynamic balance of discovery and utilization improves the optimizer’s effectiveness of finding the most beneficial answers.

Lévy flight is a Markov property random trajectory model drawing on the Lévy distribution, characteristic of nature, such as insects’ and birds’ flight patterns. It is comprised of predominantly short steps with occasional long strides, which allows for local exploration and extensive searching as Eqs. (11)-(13). The pattern assists in balancing exploration and exploitation, making it useful in optimization algorithms to navigate large solution spaces effectively.

11

12

13

Where, S is the Lévy flight step length, affected by the parameter β, and Γ is the Gamma distribution function. U and V are normally distributed with a mean of 0 and standard deviations and , respectively. The value of β in this research is fixed at 1.5 in order to achieve a proper step length during the flight.

The first exploration method, which involves a supervised modification of the step length S, is described as follows²⁴:

14

Where, represents the jth location in the global best answer at iteration it, and S indicates the step distance in the Lévy flight.

Occasionally, the global optimal position within the WAA can be in an area surrounding the global optimum point, which is inaccessible to the optimum point. Using the first method under this scenario, there exists a chance that WAA may settle on a local optimum. In a bid to compensate for this limitation, a different strategy is employed to alter the search space and avoid this²⁴.

15

The lower and upper bounds in every dimension with minimum values are referred to as and , respectively. By using this movement strategy, the search agents will be progressing through different locations in efforts of locating possible better positions. The technique is to be applied persistently throughout the iterations.

Improved WAA (EWAA)

The Weighted Average Algorithm (WAA) is a powerful optimization algorithm using the weighted averages in its exploration of the search space. However, as with other optimization algorithms, it faces challenges such as the potential for entrapment in local minima and pre-convergence. To counter such shortcomings, the addition of a Dynamic Weight Update Mechanism (DWUM) as a modification to the basic algorithm is suggested.

The WAA algorithm suffers many challenges and shortcomings. They include local minima trapping, where weighted average-based algorithms tend to assign too much weight to Exploitation of local optima so that they become trapped in these suboptimal solutions and fail to locate the global optimum. Premature convergence is another issue that happens when the algorithm finds good solutions during early iterations of the optimization process, leading to rapid convergence and potentially suboptimal results. It may happen when the Exploration phase in which new solutions are sought is not sufficient. Inefficient management of Exploration and Exploitation balance is also a drawback. Striking the right balance among Exploration and Exploitation is vital. If the optimizer leans too heavily towards Exploitation, it may miss the opportunity to explore other promising areas, thus limiting its effectiveness in finding the optimal solution.

The DWUM is designed in a way that it specifically addresses the problems of premature convergence and local minima trap. In this mechanism, the weights for Exploration and Exploitation are controlled dynamically during the optimization process to ensure the proper balance between both these stages.

Adjusts Weights Based on Iteration:

• The weights assigned to Exploration and Exploitation are dynamically adjusted at each iteration in this method.

• During the initial stages of the algorithm, the weights tend towards Exploration, but as the algorithm progresses, the weights gradually move towards Exploitation to refine the outcome.

Formulation is thus:

16

17

Where, is the weight of exploration at iteration t, is the weight of exploitation at iteration t. α and β are parameters controlling the rate of change for the weights.

In the recommended optimizer, the weights are adapted according to the population’s state changes throughout iterations. As the position changes are infinitesimal (i.e., convergence), the weights incline towards Exploitation. As they are high, the weights move towards Exploration. In addition, The DWUM allows the algorithm to automatically decide when it needs to increase Exploration and when it needs to focus more on Exploitation, ensuring a good balance search behavior throughout the optimization process.

The design of the IWAA algorithm using the DWUM is as follows:

Initial phase

Random initialization of the population within the search space, and initialize Exploration and Exploitation weights equally.

Weight calculation

Calculate the new weights for Exploration and Exploitation for each iteration according to the DWUM.

Update weighted position

The position is updated according to the weighted sum of Exploration and Exploitation solutions.

18

Optimization loop

The process continues to iterate until the convergence.

Implementation Steps of the IWAA are given below:

Step 1) Initialization: Generate the initial population randomly, and initialize the weights and parameters.

Step 2) Weight Calculation: Update the weights dynamically for Exploration and Exploitation in each iteration.

Step 3) Calculate Weighted Mean: Calculate the weighted mean using the updated weights.

Step 4) Update Positions and Evaluate: Re-substitute the updated positions with the new weights and evaluate the solutions.

Flowchart of the IWAA optimization processes is given in Fig. 3.

Fig. 3.

Flowchart of IWAA.

Cloud-Based Risk-Conscious stochastic model

Fuzzy cloud theory utilizes fuzzy drops to represent imprecise parameters, combining qualitative and quantitative aspects of vagueness. Fuzzy cloud theory combines ideas drawing from probability theory, statistics, and fuzzy logic to manage uncertainty in the mean and variance of the Probability Density Function (PDF) and offers a more integrated method compared to the traditional methods^25–27. The theory describes vagueness in ideas and manages unpredictability by equating uncertainty measurements with precise vague concepts. Cloud theory explains the randomness and fuzziness of physical systems and human thought. The chapter attempts to explain the basic ideas of cloud theory and present the application of cloud theory to solve complex problems under uncertainty.

The normal distribution model is used to model cooling and heating efficiency uncertainties. Define U as the set U={u}, and u be the domain of consideration, and let T be a linguistic term representing U. The certainty degree of x regarding the cloud model (CM) concept, represented by () and within the range 0 to 1, is a random value possessing an underlying trend^25–27. The methodology offers a new way of quantifying uncertainty and making more accurate predictions in situations involving uncertainty.

19

Subsequently, the value of x in U is conceptualized as a cloud, where each x is considered a cloud drop.

Figure 4 illustrates a typical form of the cloud model, where Ex is the average of the cloud’s values and En is the degree of uncertainty and dispersion of the cloud particles. He serves an important role in extending the application of cloud theory beyond its traditional scope to Monte Carlo simulations (MCS). As outlined before highlighted, He reflects the fluctuation regarding cloud membership grades, essentially defining the entropy En. Here, it should be pointed out that hyper-entropy (He) is normally distributed, and this has implications for the operation of the model. Implementation of cloud theory to maximize uncertain parameters entails the CM stochastic formulation for the variables. This model is developed from three critical parameters:

Fig. 4.

The cloud model and its droplets²⁵.

1. Ex (Expectation), which is the CM mean value;

2. En (Entropy), signifying the spread or scattering of droplets across the cloud;

3. He (Hyper-Entropy), indicating the uncertainty in entropy induced via fuzzy entropy.

To build a good CM, parameters of Ex, En, and He of every random variable xi need to be specified. Droplets of CM (x_i, u_i) are produced by a sequence of steps depending on the number of droplets in the cloud. Involving N droplets (x_i, u_i), these three different values (Ex, En, and He) can be obtained, giving a complete description of the uncertainty of the system. This approach allows the cloud model to better represent randomness and uncertainty associated with complex systems.

The standard CM solves the degree of uncertainty in x by dividing drops of CM x_i (i = 1, 2,…, N) at a given certainty level µ_i. These drops are generated by the forward-backward cloud generation method that better represents uncertainty. Following are the steps for generating the cloud through the forward method:

The strength of this model lies in the fact that it can handle and quantify uncertainty in a more structured way, thereby being better placed to address highly complex problems involving variability and uncertainty.

Step 1: The mean value of Ex is first obtained from forecast data. En, being the standard deviation, illustrates variability, while He, being the entropy parameter, controls the degree of dispersion in the cloud. N denotes the number of droplets that form the cloud structure. A random value of is subsequently derived, which exhibits a normal distribution with a predicted value of En and variance .

Step 2: Sample a new number xi as random from a standard distribution with mean Ex and variance .

Step 3: Calculate µ_i by^25–27

20

Step 4: Sample a cloud drop x_i with a degree of certainty µ_i. Carry out steps 1 to 4 for N CM drops.

By performing this way, there is a second-order relationship such that one of the random numbers will be utilized as input in order to produce the other. Reverse cloud creation is employed in order to transform from numerical value to qualitative idea, as explained below:

• Calculate mean and variance of xi using the formula presented in^25–27.

21

• Computation of Ex.

22

• Calculation of En.

23

• Calculation He.

24

The new methodology involves the risk involving uncertainties with a stochastic CM, which addresses variability of important parameters correctly. In this case, the uncertainties are better defined for cooling and heating efficiencies, and these are tackled with the normal CM. Therefore, incorporating uncertainties based on the new stochastic model might lead to a more realistic and reliable model in real-world applications.

In the proposed framework, the IWAA is dynamically linked with the Cloud-Based Risk-Conscious Stochastic Model in order to effectively handle uncertainties associated with cooling and heating efficiencies. The stochastic cloud model quantifies variability and risk of these uncertain parameters through probabilistic cloud droplets based on their statistical expectation (Ex), entropy (En), and hyperentropy (He). Each droplet stands for one possible realization of the uncertain efficiency, carrying both the randomness and fuzziness of its variation. These droplets are then brought into the IWAA optimization process as adaptive probabilistic inputs, enabling the algorithm to iteratively update its dynamic weights with regard to the variable uncertainty level. During each iteration, the IWAA evaluates the objective function under multiple stochastic samples and adjusts the exploration–exploitation balance based on the risk degree inferred from the cloud entropy distribution. The tight linkage hence enables the optimization process to propagate uncertainty through the energy model, ensuring that the final solutions are robust, risk-informed, and computationally efficient. Finally, the proposed IWAA–cloud integration provides a more realistic optimization pathway than traditional deterministic or sampling-based approaches by embedding stochastic reasoning directly within the algorithmic structure.

In the cloud-based framework, ()=∈[0,1] is the certainty (membership) degree of a cloud droplet x with respect to the linguistic concept “expected efficiency.” It is a a weight attached to each stochastic realization of the uncertain parameters (here, cooling/heating efficiencies). Concretely, for each IWAA candidate design w we draw N = 100 droplets for the uncertain efficiencies using the cloud parameters (Ex, En, He).

Each droplet produces an EnergyPlus simulation and an AEC value AEC_d(w). The stochastic fitness that IWAA minimizes is the -weighted expectation of AEC:

25

Thus, realizations nearer to Ex (high ) influence the objective more than tail cases (low ), while all realizations remain represented.

Simulation results and discussion

In this research, the capability of the suggested optimization technique is examined for solving three cases as below:

- Case 1)

- Evaluate the IWAA effectiveness of CEC-2022 test functions.

- Case 2)

- Optimized office building energy with no uncertainty risk of cooling and heating efficiencies + IWAA.

- Case 3)

- Optimized office building energy with uncertainty risk of cooling and heating efficiencies + IWAA.

IEDO capability is compared to classical WAA, PSO, and whale optimization algorithm (WOA). The succinct explanation of the above algorithms is presented in following:

Particle Swarm Optimization (PSO): PSO algorithm is an evolutionary algorithm based on population dynamics and social behavior in nature, for example, flying in flocks of birds or fish²⁸. In this algorithm, each “particle” signifies a single answer that moves around the search space and looks for different positions. The particles use two factors to improve their positions: the best individual position located so far by a particle, and the best global position located by the best particle in the group. The PSO algorithm drives the particles towards optimal areas through these two factors. The advantages of this algorithm are that it is simple to implement, can search efficiently and quickly through the search space, and can be employed to solve intricate problems. Its applications are solving complicated optimization problems, model parameter optimization, network design, machine learning, and control problems. PSO performs best when the objective function is strongly nonlinear and intricate.

Whale Optimization Algorithm (WOA): The WOA is a nature-mimicking optimization algorithm that is based on the mechanism of whales’ hunting²⁹. The algorithm mimics the attack and search strategies of the whales when they are pursuing food in order to pursue solutions in the search space. In WOA, whales fluctuate in two main stages: the encircling stage, which imitates the process of moving toward prey, and the bubble-net attacking stage, which emulates the strategy of closing in on prey deeper in water. This algorithm is particularly suitable for optimization problems in which the objective function is complex and discontinuous. The advantages of this algorithm are that it can efficiently search in complex spaces, has high flexibility, and enjoys good convergence speed. Its application is found in multi-objective optimization problems, telecommunication network design, parameter optimization of machine learning, and engineering optimization difficulties. WOA especially excels at problems involving a large number of variables and indirect solutions.

The algorithms WOA²⁸, PSO²⁹, and HGJO³⁰ to compare with the IWAA to solve the CEC-2022 test functions are selected for comparison because they are well-established single-objective metaheuristic algorithms frequently used as benchmarks in building energy optimization and related engineering problems. WOA provides strong global exploration but tends to suffer from local stagnation, PSO is a classic and widely adopted swarm-based optimizer serving as a baseline for performance evaluation, and HGJO effectively handles the CEC-2022 test functions in Ref³⁰. Their diverse search strategies and established reliability make them suitable references for validating the performance and robustness of the proposed IWAA under the same single-objective CEC-2022 framework.

Each of the selected optimizers’ tunable parameters are given in Table 2.

Table 2.

The control parameters of different algorithms.

Algorithm	Parameter	Value
WAA [27]	Control parameter α	0.5
PSO [31]	Constriction factor χ Acceleration control coefficient c1 Acceleration control coefficient c2	0.80 2.00 2.00
WOA [32]	Convergence factor a Probability factor p	Linearly decreases from 2 to 0 0.5

Simulation results on CEC-2022 benchmark functions

The performance of Improved Weighted Average Algorithm (IWAA) in Case 1 is compared to that of traditional WAA, PSO, and WOA algorithms to solve CEC-2022 benchmark functions (CEC01–CEC12). The obtained outcomes are evaluated with the results from the HGJO algorithm as given in³⁰. The comparison is made using quantitative measures like mean and standard deviation to measure the capability of the algorithms. To the end of making a fair and expansive comparison, a Friedman ranking test is applied. This method allows for an extensive exploration of how well any particular algorithm ranks in comparison to the others. To maintain uniformity and fairness across experiments, every algorithm is executed independently 25 times on the CEC-2022 functions with the population size as 50 and max iterations as 500. Details on CEC-2022 functions are provided in Table 3 of³⁰. This strict approach makes the comparison statistically reliable as well as accurate, and it gives an evident sign of the efficiency of the IWAA algorithm in tackling intricate optimization problems.

Table 3.

The CEC-2022 test functions³⁰.

Number	Function	Dimension	Range	Global value
CEC01	Zakharov Function	10	[–100, 100]	300
CEC02	Rosenbrock’s Function	10	[–100, 100]	400
CEC03	Schaffer’s F7	10	[–100, 100]	600
CEC04	Rastrigin’s Function	10	[–100, 100]	800
CEC05	Levy Function	10	[–100, 100]	900
CEC06	Hybrid Function 1	10	[–100, 100]	1800
CEC07	Hybrid Function 2	10	[–100, 100]	2000
CEC08	Hybrid Function 3	10	[–100, 100]	2200
CEC09	Composition Function 1	10	[–100, 100]	2300
CEC10	Composition Function 2	10	[–100, 100]	2400
CEC11	Composition Function 3	10	[–100, 100]	2600
CEC12	Composition Function 4	10	[–100, 100]	2700

The same was depicted in Tables 4, 5 and 6 results as the proposed IWAA algorithm performs better than other algorithms like WAA, PSO, WOA, and the HGJO algorithm of³⁰ when applied to the CEC-2022 test functions. Specifically, in Table 4, the best minimum values obtained for the optimization of the CEC-2022 functions clearly indicate that IWAA performs better than all other algorithms except the HGJO algorithm in optimizing the F9 function. In Table 5, the standard deviations illustrate not only that IWAA obtains better solutions but also with better consistency and stability, which is important for optimization problems. Moreover, the result of the Friedman ranking test in Table 6 is apparently demonstrated as IWAA attaining the highest performance with the lowest mean rank and the worst rank being 1. This indicates the strength of IWAA in achieving high-quality, consistent results for various optimization problems. Although the classic WAA algorithm ranked 4 out of 6 algorithms, IWAA ranked 1 at last, again proving the WAA improvement through the Dynamic Weight Update Mechanism. The average ranking of the algorithms on CEC-2022 is given by Fig. 5, once again confirming the superior optimization capability of IWAA over traditional WAA, PSO, WOA, and HGJO³⁰.

Table 4.

The mean values obtained by algorithms (Case 1).

Algorithm	IWAA	WAA	PSO	WOA	HGJO [33]
CEC01	3.000E + 02	3.911E + 02	3.8672E + 02	4.032E + 02	3.000E + 02
CEC02	4.000E + 02	4.488E + 02	4.2836E + 02	4.317E + 02	4.03472E + 02
CEC03	6.000E + 02	6.1189 + 02	6.1081E + 02	6.131E + 02	6.000E + 02
CEC04	8.058E + 02	8.165E + 02	8.0938E + 02	8.258E + 02	8.08762E + 02
CEC05	9.000E + 02	9.134E + 02	9.1567E + 02	8.167E + 02	9.000E + 02
CEC06	1.800E + 03	2.317E + 03	2.2601E + 03	2.285E + 03	1.800226E + 03
CEC07	2.0003E + 03	2.0526E + 03	2.0184E + 03	2.024E + 03	2.006963E + 03
CEC08	2.2000e + 03	2.2215E + 03	2.2186E + 03	2.2360E + 03	2.201458E + 03
CEC09	2.5340e + 03	2.5830E + 03	2.5515E + 03	2.565E + 03	2.529284E + 03
CEC10	2.5000e + 03	2.5829E + 03	2.5462E + 03	2.591E + 03	2.50023E + 03
CEC11	2.600E + 03	2.628E + 03	2.6420E + 03	2.873E + 03	2.600E + 03
CEC12	2.8570E + 03	2.946E + 03	2.9205E + 03	2.976E + 03	2.859847E + 03

Table 5.

The std. Values obtained by algorithms (Case 1).

Algorithm	IWAA	WAA	PSO	WOA	HGJO [33]
CEC01	0.000E + 00	4.180E-02	2.827E-02	5.184E-02	1.228E-13
CEC02	3.1782-03.1782	3.238E + 01	2.835E + 01	4.208E + 01	1.359E + 00
CEC03	0.000E + 00	7.229E-02	5.530E-02	1.296E-01	1.419E-05
CEC04	1.258E + 00	6.060E + 00	5.070E + 00	5.223E + 00	2.819E + 0
CEC05	0.000E + 00	1.072E-01	1.300E-01	1.311E + 00	3.771E-13
CEC06	2.026E-02	1.377E + 02	1.217E + 02	1.277E + 02	9.605E-02
CEC07	1.751E-02	1.318E + 01	1.481E + 01	1.873E + 00	8.907E + 00
CEC08	1.028E + 00	3.944E + 01	3.040E + 00	5.969E + 00	1.409E + 00
CEC09	0.000E + 00	1.038E + 01	5.180E + 00	1.121E + 01	0.000E + 00
CEC10	1.953E-02	6.770E-02	5.834E-01	9.396E-01	4.586E-02
CEC11	1.407E-11	5.275E-5	1.586E-6	6.181E-05	1.947E-10
CEC12	5.043E-01	2.410E + 00	2.709E + 00	2.786E + 00	1.183E + 00

Table 6.

The results of friedman’s ranking test of algorithms (Case 1).

Algorithm	IWAA	WAA	PSO	WOA	HGJO [33]
CEC01	1	4	3	5	1
CEC02	1	5	3	4	2
CEC03	1	4	3	5	1
CEC04	1	4	3	5	2
CEC05	1	3	4	5	1
CEC06	1	4	5	3	1
CEC07	1	5	3	4	2
CEC08	1	4	3	5	2
CEC09	2	5	3	4	1
CEC10	1	4	3	5	2
CEC11	1	3	4	5	1
CEC12	1	4	3	5	2
Total rank	13	49	40	55	18
Average rank	1.0833	4.0833	3.3333	4.5833	1.5000
Final rank	1	4	7	6	2

Fig. 5.

Average rank of different algorithms for CEC-2022 (Case 1).

The IWAA gain in performance dramatically enhances its efficiency compared to the classic WAA. This advancement enables IWAA to trade exploration and exploitation more efficiently, with resultant lower mean values and lower standard deviations (std) as reported in Tables 4 and 5, compared to the classic WAA and other algorithms. These statistical measures are the primary metrics of an algorithm’s performance: lower mean means that the algorithm produces better-quality solutions, and lower standard deviation means that the algorithm provides more uniform solutions with less variability between separate runs.

Simulation results on deterministic BEO

In general, weather affects performance coefficients, which are modifiable. For purposes of adjusting for the influence of role of climatic and environmental conditions in energy consumption, three categories of weather information are applied in analysis^8,12 like Seattle Tacoma weather information, Chicago O’Hare weather information, and Houston Intercontinental weather information^8,12. In this section, the results optimized office building energy efficiency without risk of heating and cooling efficiencies utilizing the IWAA in Case 2 are presented. To compare different optimization methods fairly, the function evaluations were defined as 100 (dim + 1)^8,12. In this case, the number of choice variables is dim, and for the presented building model scenarios, it is 4. Besides, the optimal solution of the problem is received 25 times and is represented in a box plot. The efficiency of IWAA to reduce the building AEC for Seattle, Chicago, and Houston is compared with conventional WAA, PSO, and WOA, respectively and optimization results including best solution and statistical analysis of annual energy consumption are given in Tables 7, 8 and 9 for different weather conditions.

Table 7.

The performance of different algorithms in optimization for Seattle (Case 2).

Algorithm	IWAA	WAA	PSO	WOA	POSCO [8]
Total Annual Energy Consumption (kWh/m2a) Best Mean Std.	132.48 132.52 0.0412	132.97 133.06 0.0584	132.97 133.02 0.0447	132.99 133.11 0.1177	132.6 - - -
X1 X2 X3 X4	71.87 5.9 5.9 0.2872	70.52 5.9 5.6 0.2875	70.54 5.9 5.6 0.2871	71.51 5.9 5.9 0.2884	71.924 5.9 5.9 0.2876

Table 8.

The performance of different algorithms in optimization for Chicago (Case 2).

Algorithm	IWAA	WAA	PSO	WOA	POSCO [8]
Total Annual Energy Consumption (kWh/m2a) Best Mean Std.	152.04 152.10 0.0480	152.56 152.67 0.0981	152.43 152.53 0.0757	152.60 152.74 0.0903	152.2 - - -
X1 X2 X3 X4	70.297 4.1 5.9 0.3117	70.395 4.1 5.9 0.3176	70.412 4.1 5.9 0.3152	70.405 4.1 5.9 0.3184	70.342 4.1 5.9 0.3126

Table 9.

The performance of different algorithms in optimization for Houston (Case 2).

Algorithm	IWAA	WAA	PSO	WOA	POSCO [8]
Total Annual Energy Consumption (kWh/m2a) Best Mean Worst Std.	185.27 185.33 0.0462	185.71 185.81 0.0804	185.66 185.73 0.0490	185.75 185.85 0.0768	185.5 - - -
X1 X2 X3 X4	75.490 5.1 3.5 0.4854	75.592 5.1 3.5 0.4894	75.583 5.1 3.5 0.4886	75.585 5.1 3.5 0.4921	75.564 5.1 3.5 0.4873

According to results of Tables 7, 8 and 9, analysis of IWAA algorithm optimization results for three cities—Seattle, Chicago, and Houston—verifies its superior performance in reducing the annual energy consumption compared with other algorithms, e.g., WAA, PSO, WOA, and POSCO. For Seattle, the Best value of the IWAA algorithm amounted to 132.48 kWh/m2a, which is slightly higher than POSCO’s Best value (132.60 kWh/m2a)⁸. For Chicago, IWAA’s Best value is 152.04 kWh/m2a and POSCO’s Best is 152.20⁸, which means that IWAA is superior to POSCO in terms of the least amount of energy consumed. For Houston, IWAA’s Best value is 185.27 kWh/m2a and POSCO’s Best value is 185.50 kWh/m2a⁸. These comparisons show that IWAA always generates better Best values in all locations than POSCO, proving its ability of obtaining optimal solutions. Furthermore, in Seattle, the IWAA algorithm achieved a Mean value of 132.52 kWh/m2a, which is better than WAA’s Mean value (133.06 kWh/m2a), PSO’s Mean value (133.02 kWh/m2a), and WOA’s Mean value (133.11), all of which are higher than that of IWAA. In Chicago, IWAA has a Mean value of 152.10 kWh/m2a, which is more than that of WAA (152.67 kWh/m2a), PSO (152.53 kWh/m2a), and WOA (152.74 kWh/m2a). In Houston, the Mean value of IWAA was 185.33 kWh/m2a, which is lower than the Mean values of WAA (185.81 kWh/m2a), PSO (185.73 kWh/m2a), and WOA (185.85 kWh/m2a). These comparisons clearly show that IWAA always has lower mean energy consumption in all regions compared to the other algorithms, indicating its ability to offer better average performance. Besides, lower Standard Deviation (Std) values (with unit of kWh/m2a) of the IWAA algorithm also indicate its stability and consistency. In Seattle, IWAA reached a Std of 0.0412, which is significantly lower than that of WAA (0.0584), PSO (0.0447), and WOA (0.1177). In Chicago, IWAA reached a Std of 0.0480, significantly lower than that of WAA (0.0981), PSO (0.0757), and WOA (0.0903). Similarly, in Houston, IWAA reached a Std of 0.0462, higher than that of WAA (0.0804), PSO (0.0490), and WOA (0.0768). These results indicate that IWAA not only minimizes energy usage but also in a more stable manner, thereby being a more reliable choice for energy minimization. IWAA’s superior performance compared to the traditional WAA is attributed to the Dynamic Weight Update Mechanism, which better balances exploration and exploitation phases. Such an improvement leads to better solutions and reduced variability, which allows IWAA to outperform the standard WAA and other algorithms in Mean and Std. Overall, the IWAA algorithm is a strongly competitive and improved version of the standard WAA and possesses significant advantages in energy optimization problems in different areas.

IWAA algorithm convergence curves, in comparison with WAA, PSO, and WOA, as clearly illustrated as Figs. 6, 7 and 8, point towards the superior performance of IWAA in optimizing yearly building energy consumption. In Seattle, IWAA yielded the optimal value of 132.48 kWh/m²a, which was higher than that of WAA (132.97 kWh/m²a), PSO (132.97 kWh/m²a), and WOA (132.99 kWh/m²a) and indicates improved energy efficiency. Similarly, in Chicago, IWAA’s Best value of 152.04 kWh/m²a outperformed WAA (152.56 kWh/m²a), PSO (152.43 kWh/m²a), and WOA (152.60 kWh/m²a), justifying its performance of achieving less energy consumption more efficiently. In Houston, IWAA’s Best value of 185.27 kWh/m²a was also the lowest among all the algorithms, with WAA (185.71 kWh/m²a), PSO (185.66 kWh/m²a), and WOA (185.75 kWh/m²a) producing higher values. These results show that IWAA consistently registers less yearly energy consumption compared to the other algorithms in different areas. Throughout the optimization process, IWAA consistently recorded the smallest yearly energy consumption (kWh/m2a), which confirmed that IWAA is more effective in reaching the target value of energy consumption compared to the other algorithms. For convergence, the IWAA algorithm converged towards the optimal solution at a fast rate and converged more quickly than WAA, PSO, and WOA. The convergence tolerance level is lower in the case of IWAA as well, indicating that the algorithm achieved more stable and consistent solutions. These findings confirm that IWAA is superior in optimizing the energy issue, acquiring the target value with better speed and precision, and demonstrating its superiority in convergence behavior and efficiency.

Fig. 6.

The process of convergence for different algorithms applied to Seattle (Case 2).

Fig. 7.

The process of convergence for different algorithms applied to Chicago (Case 2).

Fig. 8.

The process of convergence for different algorithms applied to Houston (Case 2).

The boxplots of Seattle, Chicago, and Houston (Figs. 9, 10 and 11) clearly depict that the IWAA algorithm outperforms other algorithms (WAA, PSO, and WOA) in minimizing annual energy usage. For all three fields, IWAA has a lower interquartile range and smaller median, indicating that it always achieves lower energy consumption with less variance. In comparison, other algorithms such as WAA, PSO, and WOA have broader distributions and larger medians, indicating higher annual energy consumption. These results show that IWAA not only maximizes energy use more efficiently, but also provides more consistent and stable results, as the tighter dispersion and lower number of outliers of the boxplots attest. The overall evaluation of the boxplots corroborates the numerical findings that IWAA is the highest-performing algorithm for energy optimization in building for all weather conditions.

Fig. 9.

The boxplot of different algorithm results for Seattle (Case 2).

Fig. 10.

The boxplot of different algorithm results for Chicago (Case 2).

Fig. 11.

The boxplot of different algorithm results for Houston (Case 2).

Stochastic results of Cloud-Based Risk-Conscious BEO

The optimization process aims to minimize the Annual Energy Consumption (AEC) of the building under uncertain operating conditions. The design variables ω₁–ω₄ according to Table 1 correspond to controllable architectural parameters—namely building orientation, west-facing window width, east-facing window width, and shadow transmittance, as detailed in Table 1. These are the decision variables directly optimized by the IWAA.

Conversely, the heating and cooling efficiencies are not design parameters but uncertainty factors that influence system performance under real conditions. They are modeled using the Cloud-Based Risk-Conscious Stochastic Model, where each efficiency is represented by a probabilistic “cloud” distribution defined by its expectation (Ex), entropy (En), and hyper-entropy (He). This allows the optimization to account for variability and risk in thermal system behavior without directly treating these efficiencies as decision variables. In summary, the design variables define the geometric and optical configuration of the building, while the uncertainty variables represent the stochastic behavior of system parameters affecting energy consumption. The optimization is performed over the design space, but the AEC objective function is evaluated repeatedly across cloud-generated stochastic scenarios to capture the probabilistic impact of uncertainties.

In Case III, the model of building energy optimization is stochastically optimized by a cloud theory model with uncertainties of cooling and heating efficiencies. IWAA algorithm is selected as the optimal approach to solve this complex optimization process. To achieve cloud model-based stochastic optimization, a set of 1000 droplets is used to form the cloud model of each uncertain parameter in the optimization process. The uncertain parameters and the corresponding Ex, En, and He values are clearly given in Table 10. Further, the CM droplet distribution for cooling and heating efficiencies is demonstrated in Fig. 12, which provides a graphical representation that shows the degree of certainty for the variation in these variables. The graph accurately depicts the dynamical relationship between certainty and the variations in heating and cooling efficiencies, providing valuable information on the stochastic nature of the optimization process.

Table 10.

Values of Ex, En, and he for cooling and heating efficiencies (Case 3).

Uncertain Parameter	Cooling Efficiency	Heating Efficiency
E x	Base case	Base case
E n	10% of Ex	10% of Ex
H e	10% of En	10% of En

Fig. 12.

The cloud droplets distribution of (a) Cooling Efficiency (b) Heating Efficiency (Case 3).

It must be noted that the uncertainties in heating and cooling efficiencies have been explicitly considered in this model because these parameters directly affect the energy consumption predictions. The uncertainties’ risk in these parameters is incorporated in the model, which is a very important consideration to be made while performing energy optimization of buildings. The risk of uncertainty in the cooling and heating efficiencies must be integrated into the energy optimization model for more accurate and credible results. This stochastic model allows for the proper handling of the variability in the cooling and heating efficiencies, thereby rendering the optimization model robust and more realistic. By incorporating the uncertainty risk in these key parameters, the model is now better able to minimize energy usage while accounting for the natural uncertainty in cooling and heating performance.

Uncertainty in heating and cooling efficiencies is modeled in the stochastic building energy optimization model explicitly to reflect real circumstances. The model has a basis on a cloud model approach, which deals with variability of such parameters by representing them as uncertain variables. The IWAA algorithm is employed to navigate this complex optimization space while the system considers cooling and heating efficiency inherent uncertainties. After the application of the IWAA algorithm in the system and its implementation in different droplets, cloud model results for building energy consumption are obtained and compared. Outputs from this model are then illustrated in Figs. 13, 14 and 15, illustrating the cloud droplets distribution of energy consumption for various regions. This graphical representation highlights how the uncertainties in cooling and heating efficiencies influence the building’s total annual energy consumption, illustrating how the stochastic nature of these parameters impacts the optimization results for various weather conditions.

Fig. 13.

The cloud droplets distribution of total annual energy consumption for Seattle (Case 3).

Fig. 14.

The cloud droplets distribution of total annual energy consumption for Chicago (Case 3).

Fig. 15.

The cloud droplets distribution of total annual energy consumption for Houston (Case 3).

The results depicted in Table 11; Fig. 16 present the impact of incorporating the uncertainty in heating and cooling efficiencies on annual building energy consumption in Case 3. In this case, where the stochastic model with the risk of such uncertainties is in consideration, according to Table 11, the energy consumption values are greater compared to Case 2, where these uncertainties were not taken into account. Specifically, for Seattle, the energy use increased by 4.73%, for Chicago by 4.62%, and for Houston by 4.47%.

Table 11.

The total annual energy consumption value for case 3 and comparison with case 2 via IWAA.

Region/Case	Case 2	Case 3
Seattle	132.52	138.80
Chicago	152.10	159.14
Houston	185.33	193.62

Fig. 16.

The total annual energy consumption increasing percentage in Case 3 compared to Case 2.

This increase in energy consumption in Case 3 is because of the inherent risk from the uncertainty in heating and cooling efficiencies. When these are not certain, the model allows for the possibility of altering their values, which could lead to higher energy consumption in some instances. For example, in Houston, where the growth was relatively lower (4.47%), uncertainty related to cooling and heating efficiencies most likely prompted the optimization model to provision for worst-case scenarios, leading to a bit higher energy consumption. Similarly, in Seattle and Chicago, the increases were higher because the volatilities in these efficiencies were higher, representing higher risk of inefficient functioning while optimizing the energy. They emphasize the importance of such uncertainties in real building energy optimization models as not taking them into account may lead to more optimistic estimations and therefore lower predictions of energy consumption, while working under conditions of uncertainty might indeed result in increased consumption.

Conclusion

In this study, the IWAA using Dynamic Weight Update Mechanism was proposed and experimented for energy optimization problem. The overall objective of the study was to enhance the performance of the basic WAA by dynamically adjusting the Exploration and Exploitation during optimization in order to minimize the building annual energy use. It was verified on three different instances of benchmark functions, office building energy reduction, and stochastic building energy reduction with uncertainties in cooling and heating efficiencies.

• In Case 1, IWAA was applied to the CEC-2022 benchmark functions, where it showed enhanced performance relative to the traditional algorithms WAA, PSO, and WOA. The IWAA showed enhanced performance both in terms of mean values and standard deviations, indicating its ability to generate high-quality solutions at all times. The Friedman ranking test also confirmed the enhanced performance of the IWAA, where it achieved the best rank for all the benchmark functions.

• In Case 2, IWAA was employed to reduce average energy consumption of a simple office building for three different weather climates (Seattle, Chicago, and Houston). The results illustrated that the IWAA possessed the lowest minimum yearly energy consumption in all areas compared to the WAA, PSO, and WOA, with average reductions being 132.52 kWh/m2a for Seattle, 152.10 kWh/m2a for Chicago, and 185.33 kWh/m2a for Houston. Furthermore, the IWAA also reported lower standard deviations, which indicated more stable and consistent optimization results.

• For Case 3, IWAA was applied in conjunction with a stochastic building energy optimization model, in which uncertainties of cooling and heating efficiencies were addressed explicitly. The outcome of this case indicates that uncertainty generates more elevated annual energy consumption values. Seattle’s annual energy consumption rose by 4.73%, Chicago by 4.62%, and Houston by 4.47% compared to Case 2. This growth is a direct consequence of the risks that come with cooling and heating efficiencies uncertainties, leading to being more risk averse in terms of optimal energy consumption. The IWAA stochastic model provided a robust solution that accounted for these uncertainties to allow more precise estimates of energy consumption.

• In most cases, the IWAA approach that was presented proved to be highly effective in addressing energy optimization problems across different situations. With the addition of the Dynamic Weight Update Mechanism, the IWAA demonstrated enhanced performance with reduced energy usage and improved stability than other traditional algorithms. The findings of this study point out the importance of addressing uncertainties and risks in building energy optimization models, as neglecting them may provide overly optimistic results and poor energy management under real-world scenarios. IWAA, through its ability to reconcile exploration and exploitation and cope with stochastic fluctuations, provides a valuable approach to energy consumption optimization in the uncertain and complex landscape.

• The limitation of the present study is the computational cost associated with stochastic simulations using the cloud-based model. Future research may aim to extend the IWAA framework to address multi-objective optimization problems, such as simultaneously minimizing energy consumption and operational costs. Moreover, incorporating real-time data from intelligent building systems could further improve the model’s adaptability and predictive accuracy. Finally, coupling IWAA with other advanced optimization or deep learning techniques could enhance its robustness and stability in solving highly dynamic and complex optimization scenarios.

Author contributions

Suraparb Keawsawasvong : Investigation, Validation, Visualization, Funding acquisition, Writing – review and editing.Thira Jearsiripongkul : Supervision, Conceptualization, Funding acquisition, Data curation, Project administration.Mohammad Khajehzadeh : Methodology, Formal Analysis. Data curation, Software, Writing – original draft,

Funding

This work was supported by the Thammasat University Research Unit in Sciences and Innovative Technologies for Civil Engineering Infrastructures.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors have no conflict interests to declare that are relevant to the content of this article.