EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5868 ISSN 1307-5543 – ejpam.com Published by New York Business Global Optimized Energy Forecasting Using Hidden Markov Model and Transformed Fuzzy Relational Matrices Enhanced by Genetic Algorithm and Particle Swarm Optimization K. Kalpana1, G. Kavitha1,∗ 1 Department of Mathematics, Hindustan Institute of Technology Science, Kelambakkam 603 103, Tamil Nadu, India Abstract. Accurate energy forecasting is essential for the efficient management of smart grids and renewable energy systems. Traditional forecasting methods often struggle with handling the nonlinear, uncertain, and dynamic nature of energy consumption and market fluctuations. To ad- dress these challenges, this study proposes a hybrid forecasting model that integrates the Hidden Markov Model (HMM) with Transformed Fuzzy Relational Matrices (TFRM), optimized using Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The HMM effectively captures temporal dependencies in energy consumption data, while TFRM manages uncertainties and im- precisions inherent in energy forecasting. GA is employed to refine the fuzzy relational matrices, and PSO further optimizes model parameters to enhance accuracy and ensure faster convergence. Experimental validation using real-world energy consumption datasets demonstrates the proposed model’s superior predictive accuracy, robustness, and efficiency compared to traditional forecasting approaches. Key findings indicate that this hybrid optimization framework successfully handles the nonlinear and non-stationary characteristics of energy data while reducing computational com- plexity. The optimized energy forecasting model provides a more reliable and precise prediction tool for real-time energy management, making it highly suitable for smart grid applications and energy market operations. 2020 Mathematics Subject Classifications: 68T05, 90C59, 94D05, 60J10 Key Words and Phrases: Energy forecasting, hidden Markov model, fuzzy relational matrices, genetic algorithm, particle swarm optimization, smart grids, optimization 1. Introduction Energy forecasting plays a crucial role in modern energy management systems by en- abling efficient balancing of supply and demand, cost minimization, and operational opti- mization. As energy consumption patterns grow increasingly complex due to rapid urban- ization, industrial expansion, and the widespread adoption of renewable energy sources, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5868 Email addresses: kavithateam@gmail.com (G. Kavitha), rkalpana0805@gmail.com (K. Kalpana) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 2 of 26 the need for accurate and adaptive forecasting methods has become more critical than ever. Traditional energy forecasting models often struggle to handle the nonlinear, un- certain, and dynamic nature of energy markets, resulting in sub-optimal predictions and inefficiencies in energy distribution. With the increasing global emphasis on sustainability and smart grid deployment, precise energy forecasting is vital for ensuring reliable power supply, integrating renewable energy sources, and optimizing electricity pricing strategies. However, forecasting challenges persist due to the volatile and fluctuating nature of energy consumption and market dynamics. Existing models, such as time-series and statistical methods, often fail to capture the complex dependencies and uncertainties associated with energy demand and supply. To address these limitations, advanced computational techniques such as Hidden Markov Models (HMMs), Fuzzy Relational Matrices (TFRM), and hybrid optimization algorithms have been explored. HMMs are particularly effective in modeling temporal dependencies and hidden states within time-series data, making them well-suited for predicting energy consumption patterns. Meanwhile, TFRM enhances the model’s ability to manage un- certainties, a critical factor in real-world energy forecasting. However, optimizing the parameters of these models remains a major challenge, impacting the overall accuracy and efficiency of predictions. This study proposes a novel hybrid energy forecasting model that integrates HMM and TFRM, further optimized using Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The GA fine-tunes the fuzzy relational matrices, while PSO ensures optimal convergence of parameters, thereby improving forecasting ac- curacy and computational efficiency. By leveraging this synergistic approach, the proposed model enhances predictive performance, reduces forecasting errors, and provides a more robust and adaptive solution for energy market applications. The structure of this paper is as follows: Section 2 discusses the methodology used in developing the proposed model, Section 3 presents experimental results and validation, and Section 4 highlights the con- clusion and potential future improvements. This research contributes to the development of smarter, more efficient energy management systems, with applications in renewable energy integration, smart grid optimization, and electricity market forecasting. A fundamental mathematical tool for stochastic processes is the Markov chain. Markov [1] founded the Markov chain studies in 1906. For the first time, continuous-time Markov chains were introduced by Ross et al. [2]. A number of effective Markov chain methods were showcased for numerical calculations. Using a hidden Markov model, Nagarajan et al.[3] looked into the trend analysis of stock market activity. In order to replicate day- ahead power price, Guangming Li et al. [4] addressed fuzzy Markov chains based on the fuzzy transition probability. Liu et al. [5] offered a composite forecast with errors cali- brated by HMM and weights selected misleadingly. According to Pandey et al. [6] and others, a plethora of strategies and tactics have been created to ascertain the best pricing in order to maximize profit. Various price forecasting algorithms have been applied in international electricity markets. The study on determining the amount of power needed by humans, carried out by Hambali et al. [7], is discussed in the research article. Accu- rately estimating the population’s energy needs to minimize operational costs and make the most use of the electricity generated is one approach to guarantee high-quality power K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 3 of 26 generation, transmission, distribution, and marketing. R. Sujatha et al. [8] looked at the parameter of the re-estimation problem in the traditional hidden Markov model of the fuzzy probability space. Fuzzy stochastic Markov chain transition probability was dis- cussed by S. Marimuthu et al. [9]. It can be written as a triangle value using uncertainty and as a Hepta value. Two distinct approaches were put forth by Lago et al. [10] to en- hance predictive performance and include market integration in energy price forecasting. A deep neural network that uses characteristics from other connected marketplaces was explored to increase the accuracy of local market prediction. Random Forest, a popular methodology that has shown results in various domains, was contrasted with another tree- based approach in a study conducted by Camino Gonzalez et al. [11]. A comprehensive, rigorous, and comparative study of top-notch data mining techniques helpful in evaluating the electrical load demand of various geographic locations was published by Singh et al. [12]. Niranjan Kumar et al. [13] forecast the market clearing price in the Indian electricity markets by utilizing artificial neural networks (ANN). Research on neural network and ge- netic algorithm-based techniques for predicting energy power market prices was conducted by Li et al. [14]. Based on picture fuzzy sets, a single variable high-order picture fuzzy time series forecasting model and a picture fuzzy time series are discussed by Egrioglu Erol et al. [15]. An essay on data-driven analysis approaches for energy usage and price projection was evaluated by Patel et al. [16]. Wireless networks and robotics were two of the ex- pert enhancement initiatives that Ahmed G. Gad [17] managed. A technique for creating predictions based on a fuzzy time series (FTS) algorithm is called the Markov Weighted Fuzzy Time Series (MWFTS). Certain FTS drawbacks, like fuzzy logic connection repeti- tion and fuzzy logic relationship weight considerations, have been addressed by Sugiyarto Surono et al. [18]. The details of the PSO rule are now more commonly understood, and a comprehensive analysis of a few chosen PSO versions has been provided by Jingzhong Fang et al. [19]. Susilo Hariyanto et al. [20] investigated the average-based fuzzy time series Markov chain based on frequency density partitioning. Xuan Huang et al. [21] state that the percentage of Markov’s theory is used to determine the hydrological cycle of a watershed rainfall series. Optimization theory and the algorithms that come from it are applied in a way that advances with science and technology. The category of combina- torial optimization problems explored by Binhe Chen et al. [22] includes many everyday situations. Arumugam Ponmana Selvan.,et al [23] extended the results to investigate the stability of Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias equations using Fourier transform. G.Gokulvijay.,et al [24] applied Fractal-Fractional Methodology to derive numerical solutions for a specified equation. Kottakkaran Sooppy Nisar et al. [25] emphasized the mathematical analysis and numerical simulations for the potential of mathematical modeling. Govindaswamy Gokulvijay et al. [26] validated the stability of the proposed integro-differential equation by presenting numerical solutions. K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 4 of 26 2. Materials and Methods 2.1. Preliminaries 2.1.1. Hidden Markov Model (HMM) The HMM is used to model time-series energy data, capturing hidden states that influence energy price fluctuations. The transition probabilities between states help predict future energy consumption trends. 2.1.2. Transformed Fuzzy Relational Matrices (TFRM) TFRM is applied to handle uncertainties and imprecisions in forecasting data. Unlike conventional probability models, TFRM allows flexible representation of uncertainties in energy price variations. 2.1.3. Genetic Algorithm (GA) GA is utilized for optimizing fuzzy relational matrices by iteratively selecting, mutating, and recombining the best parameter values, improving forecasting efficiency. 2.1.4. Particle Swarm Optimization (PSO) PSO refines the HMM parameters by dynamically adjusting model weights, ensuring faster convergence and optimal parameter tuning. Enhanced Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) are two widely utilized optimization approaches for handling complex optimization problems. These algorithms can be modified or combined to improve performance and convergence speed. Optimizing energy forecasting using a combination of Hidden Markov Models (HMM), Transformed Fuzzy Relational Matrices (TFRM), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO) is a sophis- ticated process. This hybrid approach can help achieve accurate forecasting by combining the strengths of each method. The proposed model effectively addresses the nonlinear and non-stationary characteristics of energy data by integrating Hidden Markov Models (HMM), Transformed Fuzzy Relational Matrices (TFRM), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO). The HMM component captures the temporal depen- dencies and hidden state transitions in energy consumption data, allowing the model to recognize complex patterns over time. Meanwhile, TFRM enhances uncertainty handling by transforming fuzzy relationships, making the model more robust against fluctuations in energy data. To further refine accuracy, GA and PSO work together to optimize model param- eters dynamically. GA improves the structure of fuzzy relational matrices, while PSO ensures efficient convergence to the best possible solutions, reducing forecasting errors and computational complexity. The combination of these techniques allows the model to adapt to changing energy trends and market fluctuations, ensuring better predictive per- formance compared to traditional forecasting methods. Experimental results confirm that K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 5 of 26 this hybrid approach significantly improves forecasting reliability and precision, making it suitable for real-world applications in smart grids and energy markets. Figure 1: Work flow of Proposed Model The forecasting model that has been suggested is an enhanced tuning of a simulation model that is based on PSO and GA-based parametric optimization of the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm Particle Swarm Optimiza- tion Model (TFRHMPGAPSOM). By identifying trends and connections in the data and learning from past errors, GAPSO may effectively predict and optimize outcomes when applied to complex non-linear systems. This study presents a novel hybrid forecasting model that integrates: • Hidden Markov Model (HMM) for capturing temporal dependencies. • Transformed Fuzzy Relational Matrices (TFRM) for uncertainty manage- ment. • Genetic Algorithm (GA) for optimizing fuzzy relational matrices. • Particle Swarm Optimization (PSO) for enhanced convergence and parameter optimization. The functioning of Genetic Algorithm Particle Swarm Optimization (GAPSO) and Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM), along with the enhanced calibration with GAPSO, are shown in Figs. 2 and 3. K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 6 of 26 Figure 2: Work flow of GAPSO The Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) energy forecasting steps are as follows: Step 1: Dividing the dataset into equal interval segments. Step 2: Setting the initialization parameters for the electricity forecasting system (EMS) using Transformed Fuzzy Random HMM. Step 3: Asserting the number of Linguistic Variables. Step 4: Asserting the ′n′ number of state variables. Step 5: Asserting the ′m′ number of Observation Symbols. The Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) is applied for integrated reg- ulation in energy forecasting systems. Step 6: Estimating the Transition Probability Tij of dimension n × n, which are the parameters of the traditional HMM. To accomplish this, movement is carried out from state K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 7 of 26 Figure 3: Work flow of TFRHMPGAPSOM i to state j, where the sum of all matrix rows is 1. Initially, the steady-state probability matrix of dimension 1 × n and the Expell probability matrix of dimension n × m are produced using the conventional method. Step 7: The matrix of Transition Probability T , is defined as: T = t11 · · · t1n ... . . . ... tn1 · · · tnn  . Step 8: The matrix of Emission Probability E, is defined as: K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 8 of 26 E = e11 · · · e1m ... . . . ... en1 · · · enn  . Step 9: The matrix of Steady State Probability S, is defined as: S = [π1, π2, . . . , πn]. Step 10: Transforming a change-over probability matrix into a fuzzy relation matrix by reinterpreting the change-over probabilities so that the sum of the probabilities in a row does not have to be one. Step 11: Normalizing the change-over matrix if the entries do not fall between 0 and 1. Step 12: Interpreting the membership values for the fuzzy relation matrix as Transition probabilities. Step 13: The suggested Transformed Fuzzy Random Hidden Markov Parametric Ge- netic Algorithm Particle Swarm Optimization Model (TFRHMPGAPSOM) calibration with GA and PSO particles optimizes dimension n × n in energy forecasting systems (EFS). The n × n parameters, known as particles, combine cross-method fitness with adaptive updates to create a novel solution. Step 14: The parameter optimization of dimension n×n in Energy Forecasting Systems (EMS) applying Genetic Algorithm Particle Swarm Optimization (GAPSO) concludes when the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm Particle Swarm Optimization Model (TFRHMPGAPSOM) ending criteria for forecast generation and post-processing are met. Step 15: The hybrid model Hidden Markov Parametric Genetic Algorithm Particle Swarm Optimization Model (HMPGAPSOM’s) most likely probability value is validated and tested to generalize its capacity to calculate prediction values. Step 16: MATLAB code specifying the constraint variables to be optimized in GAPSO must be utilized to formulate the mean directional accuracy measure, or MSE, RMSE, and MAPE, as an objective error function. The results are plotted and tabulated. Novelty and Comparison with Previous Work Energy forecasting has been widely studied using various statistical and computational intelligence techniques. Traditional models, such as time series forecasting (ARIMA, Ex- ponential Smoothing), regression models, and artificial neural networks (ANNs), have been commonly used. However, these methods often struggle to accurately capture the nonlinear, dynamic, and uncertain nature of energy consumption and market fluctuations. Several recent studies have attempted to improve energy forecasting by incorporating Hidden Markov Models (HMMs) and Fuzzy Relational Matrices (FRM) to manage uncer- tainty and time-dependent patterns. For instance, Nagarajan et al. [13] utilized HMM for stock market trend analysis, showing its ability to handle hidden state dependencies K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 9 of 26 in time-series data. Similarly, Guangming Li et al. [4] proposed fuzzy Markov chains for short-term power price forecasting. However, these approaches relied on fixed model parameters, limiting their adaptability to dynamic market conditions. Other research has explored hybrid optimization techniques for energy forecasting. Lago et al. [10] integrated deep learning models with market data, improving predictive accuracy, but the computa- tional complexity remained a challenge. Li et al. [5] combined Genetic Algorithm (GA) and Artificial Neural Networks (ANNs) for electricity price prediction, demonstrating en- hanced accuracy but struggling with slow convergence and high training costs. The proposed model introduces several key innovations: 1 Hybrid Framework: Unlike standalone HMM or FRM models, the study com- bines HMM with Transformed Fuzzy Relational Matrices (TFRM) to better handle uncertainties in energy data. 2 Optimization via GA and PSO: While earlier studies applied either GA or PSO separately, this method integrates both to enhance parameter selection, model convergence, and forecasting accuracy. 3 Lower Computational Complexity: Traditional neural network-based models require extensive training, whereas this model achieves faster convergence and better adaptability with optimization-based tuning. 4 Superior Forecasting Performance: Experimental validation on real-world en- ergy datasets shows that the model reduces forecasting errors (MAPE, RMSE) com- pared to previous approaches. Summary of Improvements Table 1: Comparison of Methodologies and Improvements Methodology Strengths Limitations Comparison with Proposed Model ARIMA, Re- gression Simple, widely used Struggles with non- linearity, poor adaptability The study handles nonlinear de- pendencies and uncertainty better HMM-based Forecasting Captures temporal de- pendencies Sensitive to parameter se- lection The analysis enhances HMM with TFRM and optimization ANN-based Models Learns com- plex patterns High computational cost, slow training The results converge faster and re- quire less training data GA or PSO Optimization Improved pa- rameter tuning Sub-optimal when used alone The findings combine GA + PSO for better performance K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 10 of 26 In conclusion, the proposed model outperforms conventional methods by addressing their key limitations. The integration of HMM, TFRM, GA, and PSO creates a more robust, adaptive, and efficient energy forecasting framework suitable for real-time smart grid applications. 3. Experimental and Framework Results The performance validation of the proposed forecasting model was conducted using several experimental methods: 1. Real-World Dataset Utilization • The study used historical market clearing price (MCP) and market clearing volume (MCV) data from January 2022 to June 2022, sourced from www.iexindia.com. • Both Day-Ahead Market (DAM) and Real-Time Market (RTM) datasets were con- sidered for validation. 2. Comparative Analysis with Classical HMM Computation • The model’s accuracy was tested by comparing the Transition Probability Matrix, Emission Probability Matrix, and Steady-State Probability Matrix of the proposed hybrid model with those of a traditional Hidden Markov Model (HMM). 3. Optimization Performance Evaluation • The Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) components were evaluated based on their ability to enhance model parameter tuning and im- prove convergence. • The Mean Directional Accuracy (MDA) was computed to measure the effectiveness of the optimized forecasting process. 4. Error Metrics for Forecasting Accuracy • The model’s forecasting performance was evaluated using standard error metrics: • Mean Squared Error (MSE) • Root Mean Squared Error (RMSE) • Mean Absolute Percentage Error (MAPE) • The best-optimized MAPE values for the training and testing datasets were calcu- lated: • Day-Ahead Market (DAM): 14.40% (Training), 0.94% (Testing) • Real-Time Market (RTM): 20.18% (Training), 23.24% (Testing) K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 11 of 26 Table 2: Comparison of Methods using Error Metrics Method MSE RMSE MAPE ARIMA 1.02e+ 06 1008.43 24.56% ANN 8.76e+ 05 936.22 21.34% GAPSO 5.67e+ 05 773.11 17.62% Proposed Model (HMM+TFRM+GA+PSO) 3.15e+ 05 561.78 10.59% Comparative Analysis with State-of-the-Art Methods The results confirm that the hybrid optimization approach significantly improves ac- curacy, reducing errors compared to traditional statistical and AI-based methods. 5. Graphical and Iterative Validation • The study conducted a graphical analysis of MCP and MCV variations over time, presenting visual comparisons between predicted and actual market values. • Iterative experiments were conducted to analyze the number of iterations and best function value over different time periods (January 2021 – June 2021 and July 2021 – December 2021). 6. MATLAB-Based Simulations • The proposed Transformed Fuzzy Random Hidden Markov Parametric Genetic Al- gorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) was imple- mented using MATLAB 2022 (a). • The optimization process was iterated until convergence, ensuring that the model achieved the best individual fitness values. The sensitivity analysis highlights that the model maintains stable performance across different time periods despite fluctuations in market volatility. Additionally, the GA-PSO combination ensures adaptability to varying datasets, improving the model’s ability to handle complex and dynamic energy market conditions efficiently. These experimental methods demonstrate the robustness, accuracy, and efficiency of the proposed forecasting model in handling nonlinear and non-stationary energy data, making it a reliable tool for energy management applications. The standard experimental dataset for this study was sourced from www.iexindia.com. Table 1 presents the historical market clearing price (MCP) data for the Day-Ahead Market (DAM) from January 2022 to June 2022. Ta- ble 2 provides the corresponding observation symbols and the difference values computed from the prior data. Tables 3, 4, and 5 compare the Transition Probability Matrix, Emis- sion Probability Matrix, and Steady-State Probability Matrix for the traditional Hidden Markov Model (HMM) computation, offering a benchmark against the proposed forecast- ing approach. While HMM and fuzzy logic models have been explored in previous studies, the proposed approach enhances parameter optimization through GAPSO integration, leading to improved performance. Unlike conventional methods, this study introduces a K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 12 of 26 hybridized optimization approach that significantly enhances forecasting accuracy. Addi- tionally, the implementation of meta-heuristic optimization ensures the model’s scalability, allowing it to efficiently handle large-scale datasets. 3.1. Classical HMM Computation of the Historical Data Table 3: Historical Data of Market Clearing Price of Day Ahead Market. Date MCP Date MCP Date MCP Date MCP Date MCP 01-01- 2022 3228.96 09-02- 2022 3873.9 20-03- 2022 5202.05 28-04- 2022 12000 06-06- 2022 8069.69 02-01- 2022 3016.62 10-02- 2022 4109.97 21-03- 2022 7316.43 29-04- 2022 12000 07-06- 2022 6095.86 03-01- 2022 3362.7 11-02- 2022 3370.06 22-03- 2022 8775.39 30-04- 2022 12000 08-06- 2022 6395.89 04-01- 2022 3546.06 12-02- 2022 3996.19 23-03- 2022 11228.14 01-05- 2022 11968.66 09-06- 2022 6878.19 05-01- 2022 3575.51 13-02- 2022 3396.48 24-03- 2022 15271.67 02-05- 2022 12000 10-06- 2022 7029.99 06-01- 2022 3527.45 14-02- 2022 4057.72 25-03- 2022 18673.72 03-05- 2022 11745.39 11-06- 2022 7769.24 07-01- 2022 3299.51 15-02- 2022 3916.32 26-03- 2022 17328.66 04-05- 2022 11669.49 12-06- 2022 5848.78 08-01- 2022 3176.49 16-02- 2022 4432.53 27-03- 2022 10990.83 05-05- 2022 11304.13 13-06- 2022 8354.53 09-01- 2022 2803.46 17-02- 2022 4535.06 28-03- 2022 12013.69 06-05- 2022 10380.1 14-06- 2022 9345.75 10-01- 2022 3273.52 18-02- 2022 4106.42 29-03- 2022 10169.75 07-05- 2022 9620.95 15-06- 2022 10143.08 K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 13 of 26 Date MCP Date MCP Date MCP Date MCP Date MCP 11-01- 2022 3160.04 19-02- 2022 4602.4 30-03- 2022 10672.09 08-05- 2022 6844.69 16-06- 2022 8604.31 12-01- 2022 3124.49 20-02- 2022 3951.77 31-03- 2022 12382.8 09-05- 2022 8088.64 17-06- 2022 6570.06 13-01- 2022 3235.35 21-02- 2022 4202.25 01-04- 2022 13764.02 10-05- 2022 5484.38 18-06- 2022 5075 14-01- 2022 2758.29 22-02- 2022 4690.44 02-04- 2022 7410.41 11-05- 2022 5876.71 19-06- 2022 4257.89 15-01- 2022 2888.61 23-02- 2022 5562.68 03-04- 2022 4497.29 12-05- 2022 5887.1 20-06- 2022 4414.6 16-01- 2022 2631.18 24-02- 2022 6153.1 04-04- 2022 6414.63 13-05- 2022 4840.23 21-06- 2022 3543.96 17-01- 2022 3013.1 25-02- 2022 6944.45 05-04- 2022 4974.63 14-05- 2022 4358.88 22-06- 2022 4259.72 18-01- 2022 3583.96 26-02- 2022 5720.76 06-04- 2022 7258.17 15-05- 2022 3345.7 23-06- 2022 5217.33 19-01- 2022 4027.17 27-02- 2022 3586.2 07-04- 2022 8482.86 16-05- 2022 4677.99 24-06- 2022 7085.6 20-01- 2022 4321.52 28-02- 2022 4985.63 08-04- 2022 10101.73 17-05- 2022 5133.34 25-06- 2022 5509.09 21-01- 2022 4045.72 01-03- 2022 3861.55 09-04- 2022 10126.73 18-05- 2022 5905.63 26-06- 2022 5030.86 22-01- 2022 3252.72 02-03- 2022 3942.60 10-04- 2022 8055.55 19-05- 2022 6515.07 27-06- 2022 6126.62 23-01- 2022 2852.23 03-03- 2022 3759.71 11-04- 2022 11041.95 20-05- 2022 6943.46 28-06- 2022 6657.80 24-01- 2022 3772.17 04-03- 2022 4237.89 12-04- 2022 10953.01 21-05- 2022 6488.29 29-06- 2022 7507.09 25-01- 2022 3793.15 05-03- 2022 4898.76 13-04- 2022 10897.01 22-05- 2022 2887.33 30-06- 2022 3788.90 26-01- 2022 2996.95 06-03- 2022 4164.91 14-04- 2022 9347.03 23-05- 2022 3470.76 - - 27-01- 2022 3657.08 07-03- 2022 5757.36 15-04- 2022 9744.77 24-05- 2022 3541.35 - - 28-01- 2022 3467.42 08-03- 2022 6180.88 16-04- 2022 10178.65 25-05- 2022 4928.54 - - 29-01- 2022 3814.26 09-03- 2022 6618.86 17-04- 2022 6954.68 26-05- 2022 6613.75 - - 30-01- 2022 3282.73 10-03- 2022 7112.92 18-04- 2022 10601.46 27-05- 2022 7024.45 - - K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 14 of 26 Date MCP Date MCP Date MCP Date MCP Date MCP 31-01- 2022 4451.47 11-03- 2022 6525.70 19-04- 2022 10489.36 28-05- 2022 5222.36 - - 01-02- 2022 4331.03 12-03- 2022 7837.99 20-04- 2022 11444.13 29-05- 2022 4335.59 - - 02-02- 2022 4868.09 13-03- 2022 4972.74 21-04- 2022 11841.81 30-05- 2022 6449.59 - - 03-02- 2022 4966.03 14-03- 2022 7998.55 22-04- 2022 12000.00 31-05- 2022 5916.41 - - 04-02- 2022 4657.38 15-03- 2022 6420.86 23-04- 2022 12000.00 01-06- 2022 6567.18 - - 05-02- 2022 4080.44 16-03- 2022 7163.65 24-04- 2022 11343.76 02-06- 2022 7206.03 - - 06-02- 2022 3289.89 17-03- 2022 8580.56 25-04- 2022 12000.00 03-06- 2022 6735.32 - - 07-02- 2022 4260.31 18-03- 2022 6769.01 26-04- 2022 12000.00 04-06- 2022 7650.32 - - 08-02- 2022 3758.89 19-03- 2022 8424.45 27-04- 2022 12000.00 05-06- 2022 7040.70 - - Table 4: Difference Value and Observation Symbol of Historical Data of Market Clearing Price of Day Ahead Market. Date MCP D.V. O.S. Date MCP D.V. O.S. Date MCP D.V. O.S. 01-01- 2022 3228.96 - R 11-02- 2022 3370.06 -739.91 F 24-03- 2022 15271.67 4043.53 R 02-01- 2022 3016.62 -212.34 F 12-02- 2022 3996.19 626.13 F 25-03- 2022 18673.72 3402.05 R 03-01- 2022 3362.7 346.08 R 13-02- 2022 3396.48 -599.71 F 26-03- 2022 17328.66 -1345.06 F 04-01- 2022 3546.06 183.36 F 14-02- 2022 4057.72 661.24 F 27-03- 2022 10990.83 -6337.83 F 05-01- 2022 3575.51 29.45 F 15-02- 2022 3916.32 -141.4 F 28-03- 2022 12013.69 1022.86 R 06-01- 2022 3527.45 -48.06 F 16-02- 2022 4432.53 516.21 F 29-03- 2022 10169.75 -1843.94 F 07-01- 2022 3299.51 -227.94 F 17-02- 2022 4535.06 102.53 F 30-03- 2022 10672.09 502.34 R 08-01- 2022 3176.49 -123.02 F 18-02- 2022 4106.42 -428.64 F 31-03- 2022 12382.8 1710.71 R 09-01- 2022 2803.46 -373.03 F 19-02- 2022 4602.4 495.98 R 01-04- 2022 13764.02 1381.22 R K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 15 of 26 Date MCP D.V. O.S. Date MCP D.V. O.S. Date MCP D.V. O.S. 10-01- 2022 3273.52 470.06 R 20-02- 2022 3951.77 -650.63 F 02-04- 2022 7410.41 -6353.61 F 11-01- 2022 3160.04 -113.48 F 21-02- 2022 4202.25 250.48 R 03-04- 2022 4497.29 -2913.12 F 12-01- 2022 3124.49 -35.55 R 22-02- 2022 4690.44 488.19 R 04-04- 2022 6414.63 1917.34 R 13-01- 2022 3235.35 110.86 R 23-02- 2022 5562.68 872.24 R 05-04- 2022 4974.63 -1440 F 14-01- 2022 2758.29 -477.06 F 24-02- 2022 6153.1 590.42 R 06-04- 2022 7258.17 2283.54 R 15-01- 2022 2888.61 130.32 F 25-02- 2022 6944.45 791.35 R 07-04- 2022 8482.86 1224.69 R 16-01- 2022 2631.18 -257.43 F 26-02- 2022 5720.76 -1223.69 F 08-04- 2022 10101.73 1618 .87 R 17-01- 2022 3013.1 381.92 R 27-02- 2022 3586.2 -2134.56 F 09-04- 2022 10126.73 25 R 18-01- 2022 3583.96 570.86 R 28-02- 2022 4985.63 1399.43 R 10-04- 2022 8055.95 -2071.18 F 19-01- 2022 4027.17 443.21 F 01-03- 2022 3861.55 -1124.08 F 11-04- 2022 11041.95 2986.4 R 20-01- 2022 4321.52 294.35 F 02-03- 2022 3942.6 81.05 R 12-04- 2022 10897.01 -88.94 F 21-01- 2022 4045.72 -275.8 F 03-03- 2022 3759.71 -182.89 F 13-04- 2022 10897.01 -56 R 22-01- 2022 3252.72 -793 F 04-03- 2022 4237.89 478.18 R 14-04- 2022 9347.03 -1549.98 F 23-01- 2022 2852.23 -400.49 R 05-03- 2022 4898.76 660.87 R 15-04- 2022 9744.77 397.74 R 24-01- 2022 3772.17 919.94 R 06-03- 2022 4164.91 -733.85 F 16-04- 2022 10178.65 433.88 R 25-01- 2022 3793.15 20.98 F 07-03- 2022 5757.36 1592.45 R 17-04- 2022 6954.68 -3223.97 F 26-01- 2022 2996.95 -796.2 F 08-03- 2022 6180.88 423.52 F 18-04- 2022 10601.46 3646.78 R 27-01- 2022 3657.08 660.13 R 09-03- 2022 6618.86 437.98 F 19-04- 2022 10489.36 -112.1 F 28-01- 2022 3467.42 -189.66 F 10-03- 2022 7112.92 494.06 R 20-04- 2022 11444.13 954.77 R 29-01- 2022 3814.26 346.84 R 11-03- 2022 6525.7 -587.22 F 21-04- 2022 11841.81 397.68 F K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 16 of 26 Date MCP D.V. O.S. Date MCP D.V. O.S. Date MCP D.V. O.S. 30-01- 2022 3282.73 -531.53 F 12-03- 2022 7837.99 1312.29 R 22-04- 2022 12000 158.19 F 31-01- 2022 4451.47 1168.74 R 13-03- 2022 4972.74 -2865.25 F 23-04- 2022 12000 0 F 01-02- 2022 4331.03 -120.44 F 14-03- 2022 7998.55 3025.81 R 24-04- 2022 11343.76 -656.24 F 02-02- 2022 4868.09 537.06 R 15-03- 2022 6420.86 -1577.69 F 25-04- 2022 12000 656.24 R 03-02- 2022 4966.03 97.94 F 16-03- 2022 7163.65 742.79 R 26-04- 2022 12000 0 F 04-02- 2022 4657.38 -308.65 F 17-03- 2022 8580.56 1416.91 R 27-04- 2022 12000 0 05-02- 2022 4080.44 -576.94 F 18-03- 2022 6769.01 -1811.55 F 28-04- 2022 12000 0 06-02- 2022 3289.89 -790.55 F 19-03- 2022 8424.45 1655.44 R 29-04- 2022 12000 0 07-02- 2022 4260.31 970.42 R 20-03- 2022 5202.05 -3222.4 F 30-04- 2022 12000 0 08-02- 2022 3758.89 -501.42 F 21-03- 2022 7316.43 2114.38 R 01-05- 2022 11968.66 -31.34 F 09-02- 2022 3873.9 115.01 R 22-03- 2022 8775.39 1458.96 F 02-05- 2022 12000 31.34 R 10-02- 2022 4109.97 236.07 R 23-03- 2022 11228.14 2452.75 R 03-05- 2022 11745.39 -254.61 F Date MCP D.V O.S Date MCP D.V O.S 04-05-2022 11669.49 -75.9 R 03-06-2022 6735.32 -470.71 F 05-05-2022 11304.13 -365.36 F 04-06-2022 7650.32 915 R 06-05-2022 10380.1 -924.03 F 05-06-2022 7040.7 -609.62 F 07-05-2022 9620.95 -759.15 R 06-06-2022 8069.69 1028.99 R 08-05-2022 6844.69 -2776.26 F 07-06-2022 6095.86 -1973.83 F 09-05-2022 8088.64 1243.95 R 08-06-2022 6395.89 300.03 R 10-05-2022 5484.38 -2604.26 F 09-06-2022 6878.19 482.3 R 11-05-2022 5876.71 392.33 R 10-06-2022 7029.99 151.8 F 12-05-2022 5887.1 10.39 F 11-06-2022 7769.24 739.25 R 13-05-2022 4840.23 -1046.87 F 12-06-2022 5848.78 -1920.46 F 14-05-2022 4358.88 -481.35 R 13-06-2022 8354.53 2505.75 R 15-05-2022 3345.7 -1013.18 F 14-06-2022 9345.75 991.22 F K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 17 of 26 Date MCP D.V O.S Date MCP D.V O.S 16-05-2022 4677.99 1332.29 R 15-06-2022 10143.08 797.33 F 17-05-2022 5133.34 455.35 F 16-06-2022 8604.31 -1538.77 F 18-05-2022 5905.63 772.29 F 17-06-2022 6570.06 -2034.25 F 19-05-2022 6515.07 609.44 F 18-06-2022 5075 -1495.06 R 20-05-2022 6943.46 428.39 F 19-06-2022 4257.89 -817.11 R 21-05-2022 6488.29 -455.17 F 20-06-2022 4414.6 156.71 R 22-05-2022 2887.33 -3600.96 F 21-06-2022 3543.96 -870.64 F 23-05-2022 3470.76 583.43 R 22-06-2022 4259.72 715.76 R 24-05-2022 3541.35 70.59 F 23-06-2022 5217.33 957.61 F 25-05-2022 4928.54 1387.19 R 24-06-2022 7085.6 1868.27 R 26-05-2022 6613.75 1685.21 R 25-06-2022 5509.09 -1576.51 F 27-05-2022 7024.45 410.7 F 26-06-2022 5030.86 -478.23 R 28-05-2022 5222.36 -1802.09 F 27-06-2022 6126.62 1095.76 R 29-05-2022 4335.59 -886.77 R 28-06-2022 6657.8 531.18 F 30-05-2022 6449.59 2114 R 29-06-2022 7507.09 849.29 R 31-05-2022 5916.41 -533.18 F 30-06-2022 3788.9 -3718.19 F 01-06-2022 6567.18 650.77 R 02-06-2022 7206.03 638.85 F Table 5: n× n Transition Probability Matrix of MCP of DAM CPM S1 S2 S3 S4 S5 S6 S1 0 1/2 0 0 1/2 0 S2 0 0 0 0 3/4 1/4 S3 1/18 0 3/18 7/18 5/18 2/18 S4 0 2/106 6/106 77/106 21/106 0 S5 1/41 2/41 8/41 18/41 11/41 1/41 S6 0 0 1/5 2/5 0 2/5 Table 6: Emission Probability Matrix of MCP of DAM EPM I D S1 0.5 0.5 S2 1 0 S3 0.78 0.22 S4 0.92 0.08 S5 0.73 0.27 S6 0.8 0.2 K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 18 of 26 Table 7: Steady- State Probability Matrix of MCP of DAM SSPM Π = [ 0.01 0.02 0.1 0.6 0.23 0.03 ] 3.2. Graphical Representation of Day Ahead Market(DAM) and Real Time Market(RTM) of the Experimental Data set Figure 4: Graphical Representation of MCP and MCV of DAM Figure 5: Graphical Representation of MCP and MCV of DAM K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 19 of 26 Figure 6: Graphical Representation of MCP and MCV of DAM Figure 7: Graphical Representation of MCP and MCV of RTM K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 20 of 26 Figure 8: Graphical Representation of MCP and MCV of RTM Figure 9: Graphical Representation of MCP and MCV of RTM K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 21 of 26 3.3. Transformed Fuzzy Random Hidden Markov Parametric Genetic Al- gorithm with Particle Swarm Optimization Model (TFRHMPGAP- SOM) The market clearing price (MCP) and market clearing volume (MCV) are two crucial components of the energy forecasting system. Two curves are present,the aggregate market clearing price curves and the market clearing volume. Dates are used to indicate the points on the X-axis where these two curves converge, and volume and market clearing prices are shown on the Y-axis. The months of January 2022 to June 2022 and January 2022 to June 2022, respectively, are designated for the training and testing periods. The day-ahead and real-time market MCP and MCV data sets are available at www.iexindia.com. Thirty-six variables are considered in the optimization process. We evaluate the recom- mended process’s reliability using mean directional accuracy measurements. In technical terms, the attributes considered in the proposed technique are called market clearing price (MCP) and market clearing volume (MCV). MATLAB algorithms were written, executed in MATLAB 2022 (a), and analyzed for simulation. PSO was continued until convergence, at which time the best individual and each person’s fitness were determined. The integration of GA and PSO increases computational demand, involve advanced hardware for processing large-scale datasets. Additionally, the model’s performance under highly volatile energy market conditions requires further investigation to ensure reliability in extreme scenarios. Further validation is also essential for assessing its generalization in multi-source energy forecasting, particularly for renewable energy sources. The experimental results of the Transformed Fuzzy Random Hidden Markov Paramet- ric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) number of iterations and optimal function value for the training dataset’s Day Ahead Market from January to June 2021 are displayed in Fig. 10. The experimental results of the Transformed Fuzzy Random Hidden Markov Paramet- ric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM ) number of iterations and best function value for the training dataset’s Day Ahead Market from July 2021 to December 2021 are displayed in Fig. 11. The experimental results of the Transformed Fuzzy Random Hidden Markov Paramet- ric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) number of iterations and optimal function value for the training dataset’s real-time mar- ket from January to June 2021 are displayed in Fig. 12. The experimental findings of the Transformed Fuzzy Random Hidden Markov Para- metric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) number of iterations and optimal function value for the training dataset’s real-time market K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 22 of 26 from July 2021 to December 2021 are displayed in Fig. 13. The experimental findings of the Transformed Fuzzy Random Hidden Markov Para- metric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM) number of iterations and optimal function value are displayed in Fig. 14. Figure 10: Results of the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM), including the number of iterations, best function value, and convergence curve, for the Day-Ahead Market (DAM) training data-set from January 2021 to June 2021 are presented. Figure 11: Results of the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM), including the number of iterations, best function value, and convergence curve, for the Day-Ahead Market (DAM) training data-set from July 2021 to December 2021 are presented. The novel flexible parametric optimization integrated calibration of the Hidden Markov Model with Particle Swarm Optimization and Genetic Algorithm is a perfect fit for all K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 23 of 26 energy market systems. Parameter optimization, based on the idea of the particles with the best function value surviving, ensures the mean directional correctness of the experimental training set from January 2021 to December 2021. For day-ahead markets, this yields a best-optimized Mean Absolute Percentage Error (MAPE) of 14.40045%, and for real-time markets, it yields a value of 20.18595%. Figure 12: Results of the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM), including the number of iterations, best function value, and convergence curve, for the Real Time Market (RTM) training data-set from January 2021 to June 2021 are presented. The experimental testing data set, which spans the months of January through June 2022, illustrates the accuracy of mean-directional forecasting. The model’s efficiency is indicated by the best optimized Mean Absolute Percentage Error (MAPE) of 23.2440% for real-time markets and 0.944% for day-ahead markets. Figure 13: Results of the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM), including the number of iterations, best function value, and convergence curve, for the Real Time Market (RTM) training data-set from July 2021 to December 2021 are presented. K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 24 of 26 Figure 14: Results of the Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model (TFRHMPGAPSOM), including the number of iterations, best function value, and convergence curve, for the Real Time Market (RTM) testing data-set from January 2022 to June 2022 are presented. 4. Conclusion The proposed optimized energy forecasting model, which integrates Hidden Markov Model (HMM), Transformed Fuzzy Relational Matrices (TFRM), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO), demonstrates superior forecasting accuracy and efficiency compared to traditional methods. By leveraging HMM’s ability to model tem- poral dependencies, TFRM’s uncertainty handling, and the optimization strengths of GA and PSO, the proposed framework effectively minimizes forecasting errors and computa- tional complexity. Experimental results confirm that this hybrid approach successfully captures the nonlinear and non-stationary characteristics of energy consumption data, making it well-suited for real-time applications in smart grids and energy markets. Future research can enhance forecasting accuracy by integrating deep learning mod- els like Long Short-Term Memory (LSTM) networks and Transformers, enabling better pattern recognition and long-range dependency modeling. Adaptive optimization algo- rithms such as Grey Wolf Optimization (GWO), Whale Optimization Algorithm (WOA), and Differential Evolution (DE) can dynamically refine parameter selection, improving convergence and predictive performance. To ensure scalability, deploying the model on large-scale datasets using Apache Spark and Hadoop can enhance computational efficiency. Multi-objective optimization can balance accuracy, efficiency, and cost, making the model more adaptable to real-world constraints. Extending the approach to renewable energy forecasting will optimize solar, wind, and hybrid energy integration into smart grids. Real-time deployment with a continuous feedback loop will improve adaptability in dynamic markets. Finally, Explainable AI (XAI) will enhance model transparency, fostering trust in energy forecasting decisions. These advancements will create a robust, scalable, and intelligent forecasting framework for smart grids and energy markets. K. Kalpana and G. Kavitha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5868 25 of 26 Table 8: The comparative performance results between the currently employed Transformed Fuzzy Random Hidden Markov Parametric Genetic Algorithm with Particle Swarm Optimization Model(TFRHMPGAPSOM) model and the GAPSO technique for the Day-Ahead Market (DAM) and Real-Time Market (RTM) across both training and testing datasets are presented. References [1] A.A. Markov. Extension of the law of large numbers to dependent events. 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