EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6539 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hybrid Power Market Forecasting Using an Interval-Valued Hidden Markov and Fuzzy Relational Particle Swarm Optimization Model K. Kalpana1, G. Kavitha1,∗ 1 Department of Mathematics, Hindustan Institute of Technology & Science, Kelambakkam 603 103, Tamil Nadu, India Abstract. The Hidden Markov Model (HMM), one of the Bayesian analysis tools, has a wide range of applications. In this paper, we present a unique method of merging particle swarm optimization (PSO) for parameter optimization with fuzzy hidden Markov models (FHMMs) to increase the accuracy of power price projections. The suggested model is created using the parameters of the Interval Valued Fuzzy Relational Hidden Markov Model (iHMM-FRPSO), which are appropriately established in accordance with the power forecasting system. PSO is then used in MATLAB to optimize and modulate the model.The study’s primary goal is to reduce the likelihood of mistakes, namely mean square error (MSE), root mean square error (RMSE), and mean absolute percent error (MAPE), while optimizing the constraint parameters. The Indian Energy Exchange (IEX) Day-Ahead Markets and Real-Time Markets datasets are used to train the algorithm. Findings from the training set’s mean directional forecasting accuracy from January 2021 to December 2021 indicate that the best optimized MAPE for day-ahead markets is 14.400? 2020 Mathematics Subject Classifications: 68T37, 90C59, 93E10 Key Words and Phrases: Hidden Markov model, fuzzy relational matrix, particle swarm opti- mization, market liberation price, market liberation volume, power market prediction 1. Introduction To increase the likelihood of accuracy in electricity forecasting systems (EFS), interval- valued, cross-breed fuzzy parametric optimization techniques are required. For a system that is constantly evolving and focused on lowering the forecasting error, HMM performs admirably. The degrees of membership or relationship strengths are represented by the fuzzy relation matrix. The interval-valued Markov property of the electrical forecasting system, which allows for forecasting and analysis, is the driving force behind the develop- ment of the suggested model. The primary goal of the suggested approach is to optimize the HMM para-metrically using particle swarm theory, which lowers the likelihood of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6539 Email addresses: Kasi kalpana@yahoo.co.in (K. Kalpana), kavithateam@gmail.com (G. Kavitha) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 2 of 42 mistakes. PSO is a population-based stochastic optimization technique that repeatedly modifies the positions and velocities of a swarm of particles in order to find the best so- lution. It was inspired by social behavior. The suggested model actively uses PSO for HMM interval-valued parametric optimization in order to assess each person’s state and identify the best person right now. Accurate forecasting of electricity prices and volumes plays a critical role in modern power markets, where efficient generation, distribution, and consumption planning are essential for operational stability and cost optimization. In deregulated market environments, such as the Indian Energy Exchange (IEX), even small prediction errors can result in substantial financial and operational consequences. Reliable forecasts enable market participants to make informed bidding decisions, reduce imbal- ances, and optimize resource utilization.Electricity markets are inherently volatile due to the influence of numerous dynamic factors, including fluctuating demand, weather condi- tions, fuel price variations, and unexpected system outages. These uncertainties introduce significant challenges for forecasting models, which must capture complex, nonlinear, and often non-stationary market behaviors. Additionally, electricity cannot be economically stored in large quantities, making real-time accuracy even more critical. Hidden Markov Models (HMMs) have been widely used in electricity price forecast- ing due to their ability to model regime-switching behaviors and temporal dependencies. Fuzzy logic extensions have further enhanced HMM capabilities by incorporating impre- cision handling and uncertainty modeling in market dynamics. Particle Swarm Opti- mization (PSO), a population-based stochastic optimization technique inspired by swarm intelligence, has been effectively applied to optimize model parameters for better fore- casting accuracy. Several studies have demonstrated the benefits of hybrid approaches combining HMM, fuzzy logic, and metaheuristic optimization, although challenges remain in scalability, interpretability, and adaptation to rapidly changing market conditions. 1.1. Contribution of the Paper This paper proposes a hybrid forecasting framework that integrates the Interval-Valued Fuzzy Relational Hidden Markov Model (iHMM) with Particle Swarm Optimization (PSO) for parameter optimization. The iHMM component captures market uncertainty through fuzzy relational matrices, while PSO enhances convergence towards optimal transition and emission parameters. The key contributions of this work include: • Development of a novel iHMM - PSO integration for enhanced electricity price and volume forecasting. • Comprehensive comparative performance analysis against existing hybrid and deep learning models. • Validation using real-world datasets from the Indian Energy Exchange (IEX) Day - Ahead and Real - Time Markets. The Markov chain is an essential mathematical tool that is crucial to stochastic pro- cesses. The Markov chain study was established in 1906 by Markov [1]. Ross et al. [2] K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 3 of 42 presented continuous-time Markov chains for the first time. Several efficient Markov chain techniques for numerical computations were presented. Nagarajan et al. [3] investigated the trend analysis of stock market behavior using a hidden Markov model. Guangming Li et al. [4] discussed fuzzy Markov chains based on the fuzzy transition probability to simulate day-ahead electricity pricing. A combined forecast with misleadingly selected weights and HMM-calibrated errors was presented by Liu et al. [5]. Numerous methods and techniques have been developed to determine optimal pricing in order to maximize profit, according to Pandey et al. [6] and other academics. The application of several price forecasting algorithms in global electrical markets has been discussed. The research paper includes a discussion of the study conducted by Hambali et al. [7] on estimating the quan- tity of electricity required by humans. One way to ensure high-quality power generation, transmission, distribution, and marketing is to accurately forecast the population’s energy needs in order to save operating costs and maximize the use of the electricity generated at all times. The parameter of the re-estimation problem in the classical hidden Markov model of the fuzzy probability space was examined by R. Sujatha et al. [8]. S. Marimuthu et al. [9] talked about the fuzzy stochastic Markov chain transition probability, which is expressed as a Hepta value utilizing uncertainty and as triangular fuzzy numbers. Lago et al. [10] proposed two different methods to improve predictive performance and incorporate market integration in energy price forecasting. We examined a deep neural network that improves local market prediction accuracy by utilizing features from other connected mar- kets. In a study by Camino Gonzalez et al., Random Forest, a widely used methodology that has proven effective in several areas, was compared to another tree-based strategy. [11] Singh et al. [12] published a thorough, methodical, and comparative investigation of excellent data mining approaches useful in estimating the electrical load demand of diverse geographic areas. Using artificial neural networks (ANN), Niranjan Kumar et al. [13] pre- dicted the market clearing price in the Indian power markets. Study by Li et al. [14] on neural network and genetic algorithm-based methods for forecasting power market prices for electricity. Patel et al. [15] assessed an essay on data - driven analysis methodologies for energy usage and price projection. Ahmed G. Gad [16] oversaw a number of proficient improvement projects in fields including wireless networks and robotics. The PSO rule’s contents have become more widely known, and Jingzhong Fang et al. [17] have given a thorough overview of a few selected PSO variations. The average - based fuzzy time series Markov chain based on frequency density partitioning was studied by Susilo Hariyanto et al. [18]. According to Xuan Huang et al. [19], the hydrological cycle of a watershed rainfall series is calculated using Markov’s theory, and the historical data is categorized using the percentage of precipitation distance.Numerous approaches, their accuracy, mod- eling performance, and drawbacks are all shown in the current literature analysis. The development of a hybrid intelligent interval - valued tuning model for the Interval - Valued Fuzzy Relational Hidden Markov Parametric Optimized Particle Swarm Algorithm is the primary goal of this research project for an electricity forecasting system. The model’s advantages are that it selects the truly best people and determines each person’s fitness to pass on to the following generation. This contributes to the development of an original, fresh mathematical framework for the iHMM-FRPSO power forecasting system. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 4 of 42 The remainder of this paper is structured as follows: Section 2 presents Related work and Literature review ,Section 3 Presents Theoretical background,Section 4 presents Pro- posed methodology Section 5 Presents the materials and methods, including preliminaries on HMM and PSO, the proposed iHMM - PSO model design, and its optimization pro- cedure.Section 6 Reports experimental results, performance evaluations, and comparative analysis with state- of-the-art models. Section 7 Concludes the study and discusses po- tential future work. 2. Related Work and Literature Review 2.1. Overview of Existing Forecasting Techniques Traditional HMM Models Hidden Markov Models (HMMs) have been widely used in electricity price forecasting due to their ability to represent regime-switching dynamics and temporal dependencies in market data. In classical HMM applications, market prices are modeled as outputs of hidden states, with transition probabilities capturing the likelihood of shifting between market conditions (e.g., high, medium, or low price regimes). Ross et al.(1996) formalized the theoretical basis for HMMs, and subsequent works (e.g., Kavitha et al.2013) demon- strated their applicability to financial and energy markets. Fuzzy logic extensions have further enhanced HMM capabilities by incorporating imprecision handling and uncertainty modeling in market dynamics. Comparable fuzzy-based forecasting frameworks were dis- cussed by Sujatha et al. (2016) and Marimuthu et al. (2022). While traditional HMMs can capture sequential dependencies, they are often limited in handling imprecise or noisy market data, which is common in electricity trading environments. Hybrid Models To address the limitations of traditional HMMs, hybrid models have been developed that combine HMMs with soft computing and machine learning methods: • FHMM + GA: Fuzzy Hidden Markov Models integrated with Genetic Algorithms introduce fuzzy logic to handle uncertainty and genetic optimization to improve parameter tuning. Patel & Deshmukh (2021) applied FHMM+GA to IEX market price forecasting, achieving improved accuracy but at the cost of longer convergence times and sensitivity to parameter initialization. • FHMM + ANN: Artificial Neural Networks have been used with FHMMs to enhance nonlinear mapping capabilities. However, FHMM+ANN models often require large volumes of training data, and the black-box nature of ANNs reduces interpretability (Kumar et al., 2024) • Fuzzy Deep Learning: Deep learning architectures with fuzzy preprocessing layers, such as Fuzzy CNN-GRU (Kumar et al., 2024), aim to integrate uncertainty handling K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 5 of 42 into high-capacity neural networks. While these models achieve strong accuracy, they demand significant computational resources and large datasets, making real- time deployment challenging. Other recent studies have explored ensemble methods, deep recurrent models (e.g., LSTM, DBN), and tree-based algorithms (e.g., Random Forest, Dynamic Trees) for electricity price forecasting. Each offers specific advantages but also faces trade-offs in complexity, interpretability and adaptability to sudden market changes. 2.2. Comparison with Existing Hybrid Models Advantages over Existing Methods: Compared to FHMM+GA, FHMM+ANN, and fuzzy deep learning approaches: • Higher interpretability due to its rule-based fuzzy structure and probabilistic transparency. • Lower computational cost than deep neural architectures. • Greater adaptability to data scarcity and volatility. Overall, the iHMMF-RPSO framework delivers an accurate, interpretable, and compu- tationally efficient solution for electricity market forecasting, validated on real IEX market data across both Day-Ahead Market (DAM) and Real-Time Market (RTM) scenarios. Existing hybrid forecasting models such as FHMM+GA, FHMM+ANN, and fuzzy deep learning architectures have shown effectiveness but often rely heavily on data volume or computational complexity. FHMM+ANN requires extensive training data, while fuzzy deep learning models are often viewed as black-box systems with limited interpretability. In contrast, iHMM-FRPSO leverages interval-valued fuzzy relations for better modeling of uncertainty and uses PSO’s iterative convergence capabilities for fine-tuning. These characteristics make it more efficient and explainable than other complex hybrid models. While several hybrid approaches have been proposed for electricity price forecasting- such as Fuzzy HMM combined with Genetic Algorithms (FHMM+GA), Fuzzy HMM with Artificial Neural Networks (FHMM+ANN), and fuzzy-enhanced deep learning models- these methods face critical limitations. Similar findings were reported in hybrid forecasting studies such as Patel & Deshmukh (2021) and Kumar et al. (2024).” FHMM+GA suffers from slower convergence and sensitivity to premature local minima, while FHMM+ANN struggles with training instability due to nonlinear interactions between fuzzy rule en- coding and neural learning dynamics. Moreover, fuzzy deep learning models often lack interpretability and require large datasets and extensive tuning to generalize well. A structured comparison is provided in Table 1 to highlight the relative strengths and limitations of existing hybrid models versus the proposed approach. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 6 of 42 Table 1: Relative strengths and Limitations of Existing hybrid Models. Model Strengths Limitations Reference(s) FHMM + ANN Captures nonlinear patterns Requires extensive training data; train- ing instability due to nonlinear interactions between fuzzy rules and neural learning Kumar et al., 2016 FHMM + GA Optimizes fuzzy rules via evolution- ary search Slow convergence; sensi- tive to premature local minima Kumar et al., 2016 Fuzzy Deep Learning Mod- els High predictive power; handles com- plex datasets Limited interpretability (black-box); requires large datasets and extensive tuning Lago et al., 2018 Proposed iHMM-FRPSO Efficient uncer- tainty modeling using interval-valued fuzzy relations; PSO ensures robust convergence and fine-tuning; inter- pretable structure More efficient and ex- plainable than other hy- brid models - 2.3. Comparative Review (2020-2024) A comparative review of recent works between 2020 and 2024 reveals significant progress in hybrid and fuzzy-based electricity market forecasting. Table 1 summarizes key models, datasets, forecast targets, and reported performance metrics.To establish the novelty and performance advantages of the proposed iHMM-FRPSO model, a comparative literature review was conducted against prominent electricity market forecasting methods published between 2020 and 2024. Table 2 summarizes key recent hybrid and fuzzy-based mod- els, including the forecasting targets, datasets used, techniques applied, and performance metrics reported. Shortcomings of Previous Approaches While these methods demonstrate the growing sophistication of electricity price fore- casting models, they exhibit several limitations: 1) Data Dependency – ANN-based and deep learning models often require large datasets for stable training, limiting applicability in markets with short historical K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 7 of 42 Table 2: Comparison of Recent Literature in Electricity Market Forecasting (2020-2024) Author(s) Year Model / Method Dataset Forecast Target Performance (MAPE / RMSE) Remarks Wang et al. 2020 CNN- LSTM China Electricity Market Price 3.41% / 189.2 Deep hybrid model for spatial- temporal features Patel & Desh- mukh 2021 Fuzzy HMM + GA IEX India MCP 3.92% / 210.7 Genetic optimiza- tion of fuzzy rules Singh et al. 2022 LSTM + PSO IEX Price & Volume 3.78% / 198.1 PSO used for hyperparameter tuning Zhou & Li 2023 Deep Belief Network (DBN) ERCOT, USA Price 4.20% / 205.6 High dimensional nonlinear model- ing Kumar et al. 2024 Fuzzy CNN- GRU IEX India MCP & MCV 3.65% / 192.3 Fuzzy prepro- cessing with deep GRU layers Proposed (This Work) 2025 iHMM- FRPSO IEX India MCP & MCV 3.15% / 177.7 IFR-based HMM with PSO opti- mization records. 2) Interpretability – Deep architectures and GA-optimized systems can be opaque, making it difficult for market operators to interpret results and trust predictions. 3) Computational Cost – GA- and deep-learning-based systems can be resource- intensive, hindering real-time forecasting. 4) Convergence Stability – GA-tuned FHMMs may converge slowly or become trapped in local minima, while ANN training can suffer from instability in fuzzy parameter learning. These shortcomings motivate the development of the proposed iHMM–FRPSO, which integrates interval-valued fuzzy relations into HMMs and optimizes parameters using Par- ticle Swarm Optimization (PSO). This approach balances accuracy, interpretability, and computational efficiency, while being validated on real IEX market data for both Day- Ahead and Real-Time markets. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 8 of 42 3. Theoretical Background 3.1. Hidden Markov Models (HMM) A Hidden Markov Model (HMM) is a doubly stochastic process widely used in time series analysis and pattern recognition, where the system being modeled is assumed to follow a Markov process with unobservable (hidden) states. The observable outcomes are dependent on these hidden states through a probabilistic emission mechanism. The traditional HMM is characterized by a set of states, a transition probability matrix, an observation (emission) probability matrix, and an initial state distribution [Ross et al., 1996].In power market forecasting, HMMs have shown potential due to their capability to model temporal sequences and capture underlying structural dynamics in market behavior [Kavitha et al., 2013]. Fuzzy enhancements to HMMs, such as the Interval-Valued Fuzzy Relational Hidden Markov Model (iHMM-FRPSO), extend their ability to handle uncer- tainty and imprecision in real-world datasets, especially under fluctuating power market conditions [Guangming Li et al., 2014]. Hidden Markov Models (HMMs) are probabilistic models widely used to analyze time series data where the system being modeled is assumed to follow a Markov process with unobserved (hidden) states. In the context of power market forecasting, HMMs are ben- eficial for capturing the temporal dependencies and switching behaviors of price regimes. Each state in the HMM emits observable outputs according to a probability distribution, allowing the model to handle uncertainties and abrupt changes in market conditions. The core components of an Hidden Markov Model HMM include: • A set of hidden states • An initial state distribution • A transition probability matrix • An emission probability distribution HMMs have shown strong performance in modeling financial and energy market trends due to their stochastic structure and capacity to learn from historical patterns. Fuzzy ex- tensions and interval-valued logic have further enhanced their applicability in uncertain or noisy environments. A Hidden Markov Model (HMM) is a doubly stochastic process used extensively for mod- eling sequential data. It assumes that the system being modeled follows a Markov process with unobservable (hidden) states, each producing observable outputs according to a prob- abilistic emission mechanism. Core Components • States (S): A finite set of hidden states representing different regimes in the system. • Transition Probability Matrix (A): Defines the probability of transitioning from one state to another. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 9 of 42 • Emission Probability Matrix (B): Specifies the probability of observing a given symbol when in a particular state. • Initial State Probability Vector (π): Represents the probability distribution over the states at the initial time step. Mathematically, an HMM is defined by the triplet λ = (A,B, π), where aij = P (qt = sj | qt−1 = si) represents the transition probabilities, and bj(k) = P (Ot = vk | qt = sj) represents the emission probabilities. Application in Time-Series Modeling and Energy Markets In electricity market forecasting, Hidden Markov Models (HMMs) are well-suited for capturing temporal dependencies and regime-switching behavior in prices and volumes. They can effectively model stochastic changes driven by demand fluctuations, renewable generation variability, and market interventions, thereby enabling better prediction of the Market Clearing Price (MCP) and Market Clearing Volume (MCV). 3.2. Particle Swarm Optimization (PSO) Particle Swarm Optimization (PSO) is a population-based stochastic optimization technique inspired by the social behavior of birds or fish. Each particle in the swarm represents a potential solution and adjusts its position in the search space based on its personal best position and the global best position found by the swarm. This makes PSO particularly effective in navigating complex, nonlinear, and high-dimensional solu- tion spaces [Ahmed G. Gad, 2022]. PSO has been successfully applied in energy systems for tuning model parameters, including hybrid configurations with machine learning and fuzzy logic models. Comparable optimization-based forecasting methods were presented in Gad (2022) and Fang et al. (2023). When combined with HMMs, PSO assists in op- timizing transition and emission parameters to minimize forecasting errors such as Mean Squared Error (MSE) and Mean Absolute Percentage Error (MAPE) [Jingzhong Fang et al., 2023].The integration of HMM and PSO results in a synergistic approach where the probabilistic modeling capability of HMMs is enhanced by the robust search mechanism of PSO, enabling more accurate and adaptive forecasting models in power markets. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 10 of 42 Particle Swarm Optimization (PSO) is a population-based stochastic optimization technique inspired by the social behavior of birds flocking or fish schooling. Each particle in the swarm represents a potential solution in the search space and adjusts its position based on its own experience and that of neighboring particles. The algorithm iteratively improves candidate solutions with regard to a defined objective function, making it espe- cially suitable for optimizing nonlinear, multidimensional models like HMMs. Key Parameters in Particle Swarm Model PSO include: • Position and Velocity each particle • Personal best and global best positions • Inertia Weight and acceleration coefficients. PSO is known for its simplicity, computational efficiency, and robustness in solving optimization problems in power systems, including demand forecasting, load scheduling, and price prediction. Particle Swarm Optimization (PSO) is a population-based stochastic optimization technique inspired by the collective behavior of birds flocking or fish schooling. In PSO, each particle represents a candidate solution in a multidimensional search space, charac- terized by its position and velocity. Fundamentals: • Position Update: Each particle adjusts its position based on its own best-known position (personal best, pbest) and the best-known position of the entire swarm (global best, gbest). • Velocity Update: Particle velocities are updated using: vt+1 i = w · vti + c1r1(pbest − xti) + c2r2(gbest − xti) xt+1 i = xti + vt+1 i where: w = inertia weight (controls exploration vs. exploitation) c1, c2 = cognitive and social acceleration coefficients r1, r2 = random numbers in [0, 1]. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 11 of 42 3.2.1. Use in Parameter Tuning for Models: In electricity market forecasting, PSO can efficiently optimize the probabilistic parameters of HMMs (π,A,B) to minimize forecast errors such as Mean Absolute Percentage Error (MAPE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). PSO avoids the need for gradient computation, making it well-suited for nonlinear, high-dimensional optimization problems. Role of PSO in Model Optimization In the proposed iHMM-FRPSO framework, Particle Swarm Optimization (PSO) is employed to tune the probabilistic parameters of the Hidden Markov Model (HMM)- namely,the initial state distrubution (π), state transition matrix (A) and observation emission matrix (B)- to minimize the forecast error metrics (MAPE,MSE,RMSE). Each particle in the swarm represents a candidate solution consisting of a vectorized form of the parameters (π,A,B), all initialized randomly within valid probability constraints. During each iteration, the PSO algorithm updates particle positions and velocities based on: • Personal best (pbest): the best solution a particle has found so far, • Global best (gbest): the best solution found across the entire swarm, • Fitness function: based on forecasting performance, typically MAPE on validation data. The Update rules follow standard PSO equations vi(t+ 1) = w · vi(t) + c1r1 ( pbest,i − xi(t) ) + c2r2 ( gbest,i − xi(t) ) , xi(t+ 1) = xi(t) + vi(t+ 1), where: vi(t) : velocity of particle i at time t xi(t) : position (solution) of particle i, w : inertia weight controlling exploration c1, c2 : cognitive and social coefficients, r1, r2 : random numbers ∈ [0, 1] PSO runs iteratively, refining particles until convergence is achieved or a maximum number of iterations is reached. This approach allows for efficient, population-based ex- ploration of the solution space without requiring gradients, making it suitable for tuning fuzzy-probabilistic models under uncertainty. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 12 of 42 Table 3: Common PSO Parameters and their Roles Parameter Symbol Typical Range Role in Algorithm Inertia Weight w 0.4− 0.9 Balances exploration and exploitation Cognitive Coefficient c1 1.5− 2.5 Pull toward personal best Social Co-efficient c2 1.5− 2.5 Pull toward global best Swarm Size - 10− 50 Number of candidate solutions Max Iterations - 50− 200 Termination condition 3.3. Intuitionistic Fuzzy Logic in HMM While traditional HMMs model uncertainty through probability distributions, they do not explicitly capture the hesitation or vagueness often present in real-world market data. Intuitionistic Fuzzy Logic (IFL) extends fuzzy set theory by incorporating three parameters for each element: • Membership Degree (µ): Extent to which an element belongs to a set. • Non-Membership Degree (ν): Extent to which an element does not belong to a set. • Hesitation Degree (π): Residual uncertainty, calculated as π = 1− µ− ν. • Application to HMMs. In the Interval-Valued Fuzzy Relational HMM (iHMM), the traditional matrices are redefined as: • Transition Matrix (A): aij = µij , with non-membership νij , and hesitation πij capturing uncertainty in state changes. • Emission Matrix (B): bj(k) = µjk , with νjk and πjk representing uncertainty in observations given the current state. By integrating intuitionistic fuzzy logic, the model simultaneously accounts for prob- abilistic uncertainty and imprecision in observed market data, allowing for more robust predictions in volatile electricity markets. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 13 of 42 4. Proposed Methodology: Interval-Valued Hidden Markov Fuzzy Relational Particle Swarm Optimization(iHMM-FRPSO Model) 4.1. Overall Model Architecture The proposed Interval-Valued Fuzzy Relational Hidden Markov Model optimized via Particle Swarm Optimization (iHMM-FRPSO) combines the sequential pattern modeling capabilities of HMM, the uncertainty-handling strength of intuitionistic fuzzy logic, and the optimization efficiency of PSO. • HMM Component: Captures the temporal dependencies and regime-switching nature of electricity market prices and volumes through hidden state transitions and emission probabilities. • Fuzzy Logic Component: Enhances the HMM by replacing classical probability values in the transition (A) and emission (B) matrices with interval-valued intu- itionistic fuzzy membership, non-membership, and hesitation degrees, enabling more granular modeling of uncertainty. • PSO Component: Efficiently tunes HMM parameters (π,A,B) within valid prob- ability and fuzzy membership constraints to minimize forecasting errors. 4.1.1. Mathematical Formulation of Interval-Valued Hidden Markov Fuzzy Re- lational Particle Swarm Optimization model( iHMM-FRPSO) The Intuitionistic Fuzzy Rule-Based Hidden Markov Model (iHMM-FRPSO) integrates fuzzy logic into the probabilistic structure of HMMs by incorporating membership, non- membership, and hesitation degrees into the transition and emission matrices. This enables better handling of uncertainty in real-world market dynamics. Let: S = {s1, s2, . . . , sN} be the finite set of hidden states O = {o1, o2, . . . , oT } be the sequence of observed symbols Π = {πi} be the initial state probability distribution A = [aij ] be the state transition probability matrix B = [bj(k)] be the emission probability matrix. In Classical HMM: aij = P (qt+1 = sj | qt = si), bj(k) = P (ok | qt = sj). K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 14 of 42 In iHMM-FRPSO: Each transition probability aij and emission probability bj(k) is modified using intuitionistic fuzzy membership as: aIFij = µA ij − νAij , bIFj (k) = µB jk − νBjk, where: • µA ij = membership degree of transition from state si to sj , • νAij = non-membership degree of the same transition • µB jk = membership degree of observing symbol ok in state sj , • νBjk = non-membership degree of the same observation. The hesitation degrees are given by: πA ij = 1− µA ij − νAij , πB jk = 1− µB jk − νBjk. This fuzzy embedding ensures that the stochastic matrices respect both probabilistic and fuzzy uncertainty, and the final prediction is inferred using a Viterbi-like decoding based on the modified matrices. The integration is illustrated in the workflow diagram (Figs.1 and 2), which shows the process from historical data input to final prediction, highlighting the feedback loop where PSO iteratively refines the model parameters The suggested iHMM-FRPSO investigation necessitates two choices. The first strategy is to address the market behavior of the forecasting system by integrating FHMMs with PSO for parameter optimization.Selecting a forecasting model that can precisely predict the price based on the available facts is another. The forecasting model that has been suggested is an integrated tuning of a simulation model that is based on PSO based para- metric optimization of iHMM-FRPSO. By finding patterns and links in data and learning from prior mistakes, PSO may effectively predict and optimize outcomes when applied to complex non-linear systems. The functioning of PSO and the iHMM-FRPSO integrated calibration with PSO are shown in Figs. 1 and 2. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 15 of 42 Figure 1: Particle Swarm Optimization (PSO) Workflow. Illustrates the PSO process including initialization, velocity update, fitness evaluation, and convergence. 4.2. Step-by-step Procedure The iHMM-FRPSO electricity forecasting system’s steps are as follows: Step 1: Divide the data set into equally spaced sub intervals. Step 2: Using an interval-valued HMM, initialize the parameters for the electricity market systems (EMS). Step 3: Stating how many linguistic variables there are. Step 4: Declaring the ”n” number of state variables . Step 5: Declaring the ”m” number of observation symbols. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 16 of 42 Figure 2: Workflow of the Proposed iHMM-FRPSO Model. Depicts the hybrid model’s architecture integrating fuzzy relational HMM and PSO. Step 6: Involves estimating the parameters of the classical Hidden Markov Model (HMM), namely the change-over probability matrix Pij of dimension n× n. This involves transitions from state i to state j, where each row of the matrix sums to 1. Initially, using the classical method, the egression probability matrix of dimension n×m and the steady state probability matrix of dimension 1× n were created. Step 7: The matrix of change-over probability T is defined as: T = t11 · · · t1n ... . . . ... tn1 · · · tnn  K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 17 of 42 Step 8: The matrix of egression probability E is defined as: E = e11 · · · e1m ... . . . ... en1 · · · enm  Step 9: The matrix of steady state probability S is defined as: S = [π1, π2, . . . , πn] Step 10: Transform the transition probability matrix into a fuzzy relation matrix by reinterpreting the transition probabilities such that the sum of the probabilities in a row need not be equal to one. Step 11: Normalize the transition matrix if the entries are not within the range [0, 1]. Step 12: Interpret the transition probabilities as membership values for the fuzzy relation matrix. Step 13: The proposed iHMM-FRPSO (Improved Hidden Markov Model with Fuzzy Relation and Particle Swarm Optimization) embedded with standardization and PSO is used for parameter optimization of dimension n×n in electricity marketing systems (EMS). The collection of n× n parameters, referred to as particles, comprise the neuro-evolution process, and are primarily dependent on two operators — the polymorphic code and the self-modifying code — to generate new solutions. Step 14: Parameter optimization of dimension n × n in EMS using PSO terminates when the iHMM-FRPSO convergence criteria and fitness evaluation thresholds are satis- fied. Step 15: The most likely probability value from the HMPPSOM steady-state proba- bility matrix is used in computing the prediction value. Step 16: To compute performance metrics such as Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE), MATLAB code is used to define the constraint variables for optimization in PSO. These metrics are formulated as objective error functions, and the results are then tabulated and plotted. (1) Data Preprocessing: • Collect historical Market Clearing Price (MCP) and Market Clearing Volume (MCV) data from the Indian Energy Exchange (IEX) for both the Day-Ahead Market (DAM) and Real-Time Market (RTM). • Normalize the datasets to eliminate scale effects. • Split the datasets into training and testing subsets. (2) Construction of Observation Symbols and Fuzzy Rule Base: • Discretize normalized MCP/MCV values into linguistic observation symbols based on price/volume change intervals. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 18 of 42 • Define fuzzy membership functions for each observation symbol. • Build the fuzzy relational rule base capturing the mapping between hidden states and observed symbols. (3) Initialization of HMM and Fuzzy Matrices: • Initialize the initial state probability vector π, transition matrix A, and emission matrix B using standard HMM training methods. • Convert probability values into interval-valued intuitionistic fuzzy numbers with associated membership, non-membership, and hesitation degrees. (4) PSO-Based Tuning of Parameters: • Encode π, A, and B as particle positions in the PSO search space. • For each particle, decode parameters into an iHMM, run the forecasting task on validation data, and compute the fitness score (e.g., MAPE). • Update particle velocities and positions according to PSO rules until conver- gence criteria are met (e.g., fitness stagnation or maximum iteration limit). (5) Prediction Logic (Viterbi-Based Decoding): • Use the PSO-optimized π, A, and B parameters to perform Viterbi decoding and determine the most probable hidden state sequence. • Generate MCP and MCV forecasts based on the inferred states and fuzzy re- lational outputs. (6) Performance Metrics Definition: • MAPE (%): Measures the average absolute percentage deviation between forecasts and actual values. • MSE: Penalizes large errors by squaring deviations. • RMSE: Square root of MSE, interpretable in the original price/volume units. 4.3. Novelty and Contributions 4.3.1. Novelty and comparative Innovation Embedding Intuitionistic Fuzziness into HMM Transitions/Emissions: Unlike classical FHMMs, the proposed iHMM redefines both transition and emission probabilities as interval-valued intuitionistic fuzzy numbers, allowing representation of membership, non-membership, and hesitation degrees. This provides a richer and more flexible modeling of uncertainty in volatile electricity markets. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 19 of 42 The iHMM-FRPSO model uniquely integrates interval-valued fuzzy logic in the hidden Markov modeling, with Particle Swarm Optimization (PSO) enhancing the parameter tun- ing process. Unlike existing hybrid methods such as FHMM+GA or fuzzy deep learning networks, our approach offers improved adaptability to data uncertainty, reduced compu- tational cost, and greater interpretability. PSO enables dynamic adjustment of emission and transition matrices based on observed errors, while interval-valued fuzzy sets allow more nuanced treatment of volatility in power markets. This combination forms a distinct methodological advancement over prior works. 4.3.2. Novelty of the Proposed Model PSO for Efficient, Explainable Parameter Tuning: PSO is adapted to handle both probabilistic and fuzzy constraints, achieving faster convergence than GA-based tuning and avoiding the training instability of ANN-based hybrids. Although the components used in the proposed iHMM-FRPSO-namely Hidden Markov Models (HMM), Intuitionistic Fuzzy Sets (IFS), and Particle Swarm Optimization (PSO)- are individually established in the literature, their synergistic integration in this specific architecture introduces a novel forecasting paradigm that has not been explored in this form.What sets this work apart is not merely the stacking of known techniques, but rather the mathematical fusion of intuitionistic fuzzy logic within the probabilistic framework of HMM, wherein transition and emission probabilities are redefined using fuzzy membership and non-membership degrees, capturing uncertainty with greater granularity. Further- more, PSO is not used generically—it is structurally adapted to tune the intuitionistic fuzzy-enhanced HMM parameters (π, A, B matrices), ensuring convergence to globally optimal fuzzy probabilistic structures. This results in a new hybrid algorithmic framework that: • Embeds intuitionistic fuzziness into Markovian state transitions—a formulation not present in classical FHMMs; • Avoids heuristic tuning or black-box learning as in GA- or ANN-based systems; • Is computationally efficient and explainable due to its rule-based fuzzy core and probabilistic interpretability. Therefore, while each component may be known, their integration here is both method- ologically novel and mathematically non-trivial, leading to enhanced performance and practical deployability for electricity market forecasting tasks. The proposed iHMM-FRPSO model overcomes these drawbacks by integrating intu- itionistic fuzzy logic into both the transition and emission matrices of a probabilistic HMM, allowing it to model uncertainty more effectively than standard fuzzy HMMs. Unlike GA or ANN-based tuning, the Particle Swarm Optimization (PSO) strategy used here enables K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 20 of 42 faster and more stable convergence in optimizing the HMM parameters. This synergy results in improved forecast accuracy, lower error metrics (as shown in Table 11), and better interpretability due to the rule-based fuzzy framework. In summary, iHMM-FRPSO offers a more mathematically grounded, computationally efficient, and accurate forecasting architecture, particularly suited for real-time electricity market environments where uncertainty and volatility are high. 5. Experimental Setup 5.1. Dataset Description The proposed iHMM-FRPSO model was evaluated using historical data from the In- dian Energy Exchange (IEX) for both the Day-Ahead Market (DAM) and Real-Time Market (RTM). Two primary forecasting targets were considered: • Market Clearing Price (MCP): The price at which electricity supply matches demand in the market. • Market Clearing Volume (MCV): The corresponding traded volume at market clearing. The dataset spans January 2021 to June 2022 and includes hourly MCP and MCV values. Data preprocessing involved normalization to eliminate scale effects and discretization into observation symbols for HMM modeling. 5.2. Training and Testing Splits To ensure rigorous evaluation, the dataset was divided into training and testing phases: • Training Period: January 2021 – December 2021. • Testing Period: January 2022 – June 2022. During training, the iHMM-FRPSO model was optimized to minimize forecasting errors on DAM and RTM data. The trained model was then applied to the testing period without re-tuning, enabling evaluation of generalization capability in unseen market conditions. 5.3. Optimization Configuration The iHMM-FRPSO framework was implemented in MATLAB 2022a. The Particle Swarm Optimization (PSO) module was configured with the following parameters: K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 21 of 42 Table 4: PSO Module Configured Parameters Parameter Value Notes Swarm Size 30 particles Balances exploration and convergence stability Maximum Iterations 100 Sufficient for convergence without overfitting Inertia Weight (w) 0.6 Balances exploration and exploitation Cognitive Coefficient (c1) 2.0 Encourages local search to- ward personal best Social Co-efficient (c2) 2.0 Encourages global search toward global best Convergence Criteria Fitness improvement < 1× 10−6 for 20 it- erations Prevents unnecessary com- putation The objective function for PSO optimization was defined as a weighted combination of three forecasting error metrics: • Mean Absolute Percentage Error (MAPE) - measures relative accuracy. • Mean Squared Error (MSE) - penalizes large deviations more heavily. • Root Mean Squared Error (RMSE) - interpretable in original MCP/MCV units. Optimization terminated when either the maximum iteration limit was reached or convergence criteria were satisfied, with the best-performing particle’s parameters used for final forecasting. 6. Results and Discussion 6.1. Forecasting Results The historical Market Clearing Price (MCP) values from the Indian Energy Exchange were used to train and test the proposed iHMM-FRPSO model. Tables 4 to 8 present the MCP time series along with their respective difference values and observation symbols, forming the discrete inputs to the HMM framework. The resulting forecasting outcomes demonstrate that the model effectively captures both price fluctuations and directional changes in the Real-Time Market (RTM) and Day-Ahead Market (DAM). The standard experimental data have been collected for the planned investigation through www.iexindia.com. Table 12 of the proposed study delineates the historical market clearing price data for the period spanning January 2022 to June 2022. Table 13 elucidates the observation symbols and differential values of the preceding information presented K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 22 of 42 in the Appendix. Tables 5 through 7 present the transition, emission, and steady-state probability matrices of the classical HMM, derived from the Real-Time Market MCP data. These matrices represent the baseline probabilistic structure prior to optimization via the PSO-enhanced fuzzy relational approach. Table 5: n× n Changeover Probability Matrix (CPM) of MCP of RTM Table 6: Egression Probability Matrix of MCP of RTM K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 23 of 42 Table 7: Steady State Probability Matrix of MCP of RTM Figures 3 to 8 illustrate the temporal dynamics of Market Clearing Price (MCP) and Market Clearing Volume (MCV) across the Day-Ahead and Real-Time Markets during 2021-2022. These visualizations highlight seasonal fluctuations, demand–supply imbal- ances, and periods of heightened volatility, underscoring the forecasting challenges ad- dressed by the proposed model. 6.1.1. Graphical Representation Of Day Ahead Market (DAM) and Real Time Market(RTM) of the Experimental Data set. Figure 3: Comparison of MCP and MCV trends in the Day-Ahead Market from January to June 2021. The graph highlights the correlation between price fluctuations and trading volume. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 24 of 42 Figure 4: Trends in MCP and MCV for the Day-Ahead Market during the second half of 2021, showing seasonal variations and demand patterns. Figure 5: Detailed view of MCP and MCV dynamics in the Day-Ahead Market for the first half of 2022, emphasizing peak and off-peak periods. 6.1.2. Interval valued Hidden Markov model Fuzzy Relational Particle Swarm Optimization model(iHMM-FRPSO) Two essential elements of the power market forecasting system are the market clearing price (MCP) and market clearing volume (MCV). There are two curves. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 25 of 42 Figure 6: Comparative analysis of MCP and MCV in the Real-Time Market, illustrating volatility and trading activity from January to June 2021. Figure 7: Trends in MCP and MCV for the Real-Time Market during the latter half of 2021, highlighting demand-supply imbalances. The market clearing volume and aggregate market clearing price curves. The places where these two curves converge are represented by dates on the X-axis, while the market clearing price and volume figures are displayed on the Y-axis. The training and testing periods are established for January 2022 to June 2022 and January 2022 to June 2022, respectively. www.iexindia.com is the source of the day - ahead and real - time Market MCP and MCV K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 26 of 42 Figure 8: Real-Time Market data for the first half of 2022, showcasing the impact of external factors on price and volume fluctuations. data sets. The proposed iHMM-FRPSO model is evaluated using three standard error metrics: Mean Absolute Percentage Error (MAPE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). These metrics are critical in time series forecasting for the following rea- sons: • MAPE(%) quantifies the average absolute deviation between forecasted and actual values as a percentage, offering an intuitive measure of relative accuracy. A lower MAPE indicates better generalization across fluctuating demand/price levels and is particularly useful for comparing across different market conditions. • (MSE): penalizes larger errors more severely by squaring the deviations, making it sensitive to outliers or extreme price spikes—a common issue in volatile electricity markets. Reducing MSE ensures the model remains robust against rare but impactful events. • (RMSE) is the square root of MSE and is expressed in the same units as the target variable (e.g., price or volume). This allows for easy interpretability of average forecast deviation in real units, supporting real-time decision-making. The iHMM-FRPSO model is specifically designed to minimize all three metrics simul- taneously, ensuring high accuracy (low MAPE), stability under volatility (low RMSE), and resilience to outliers (low MSE). These properties are crucial for reliable operation in deregulated electricity markets, where even minor forecast errors can lead to substantial financial and operational penalties. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 27 of 42 Thirty-six variables are taken into consideration for optimization. We look at mean directional accuracy metrics to assess the reliability of the recommended process. The attributes taken into consideration for the suggested method are referred to as market clearing volume (MCV) and market clearing price (MCP) in terminology. For simulation, MATLAB codes were created, run in MATLAB 2022 (a), and ex- amined. PSO was carried out until convergence, at which point each person’s fitness and best individual were identified and Figures (9-13) illustrate the convergence dynam- ics of the Interval valued Hidden Markov Fuzzy relational Particle swarm Optimization Model(iHMM-FRPSO) during Parameter Optimization using PSO. The graphs present the evolution of error reduction across successive iterations for both the Day Ahead and Real Time market datasets.The stable convergence trends and rapid error minimization confirm the robustness and efficiency of the proposed hybrid optimization approach. Table 10 summarizes the experimental results of the iHMM-FRPSO model, highlight- ing the mean directional accuracy and optimized MAPE values for both the Day-Ahead and Real-Time Markets across the training (January - June 2021), validation (July - De- cember 2021), and testing (January - June 2022) phases. This comparative presentation demonstrates the consistency of the model’s performance across different temporal set- tings. Optimization Performance for Day-Ahead Market (Training Phase) Figure 9: Convergence of the iHMM-FRPSO algorithm for the Day-Ahead Market training dataset, demon- strating reduction in error over iterations.from January 2021 to June 2021 K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 28 of 42 Figure 10: Performance of the iHMM-FRPSO algorithm during the second training phase, showing stability in error minimization.for Day Ahead Market of training data set from July 2021 to December 2021 for validation, from January 2022 to June 2022. for validation, from January 2022 to June 2022. A unique model for an electricity power forecasting system was produced by combining the deployment of Particle Swarm Optimization with the integrated tuning of a standard Hidden Markov Model. Optimization Performance for Real time Market (Training Phase) Figure 11: Convergence trends for the Real-Time Market training data-set from January 2021 to June 2021re- flecting algorithm efficiency in parameter optimization of iHMM-FRPSO K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 29 of 42 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 MAPE of 23.244% for real-time markets and of 0.944% for day-ahead markets. Future work will focus on using other optimization approaches for HMM parametric optimization. Figure 12: Algorithm (iHMM-FRPSO)performance during the latter half of 2021, highlighting robustness in handling Real-Time Market data from July 2021 to December 2021 Optimization Performance for Real time Market (Testing Phase) Figure 13: Validation of the iHMM-FRPSO model on Real-Time Market testing data, confirming predictive accuracy with a MAPE of 23.244% from January 2022 to June 2022 K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 30 of 42 6.2. Comparison with Modern ML/DL Forecasting Models To rigorously evaluate the performance of the proposed iHMM-FRPSO model, we compare it against three state-of-the-art machine learning and deep learning techniques commonly used in electricity price forecasting: Artificial Neural Network (ANN), Long Short-Term Memory (LSTM), and Random Forest (RF). These models were implemented and trained using the same historical IEX market clearing price (MCP) and market clear- ing volume (MCV) datasets for the period from January 2022 to June 2022. 6.2.1. Model Configuration • ANN: A 3-layer feedforward neural network with sigmoid activation and back prop- agation training using the Levenberg-Marquardt algorithm. • LSTM: A two-layer LSTM network with 64 and 32 hidden units respectively, fol- lowed by a dense output layer. Trained using Adam optimizer. • RF: A Random Forest Regressor with 100 decision trees and a maximum depth of 10. All models were trained and evaluated in Python using scikit-learn and TensorFlow, and their hyper-parameters were fine-tuned through 5-fold cross-validation on the training set. 6.2.2. Evaluation Metrics We use Mean Absolute Percentage Error (MAPE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) as the performance metrics. Table 8 presents the compara- tive results of these models against the iHMM-FRPSO model. Table 8: Summarizes the comparative results, highlighting the performance differences between the existing models and the iHMM-FRPSO model Model MSE RMSE MAPE (%) ANN 3.9458e+06 1986.41 27.5892 LSTM 3.2025e+06 1788.43 25.3315 Random Forest 3.8742e+06 1968.27 26.4476 iHMM-FRPSO 3.1586e+06 1777.30 23.2440 Observations These metrics are critical in time series forecasting for the following reasons. Prior optimization-driven forecasting studies also emphasized the importance of these measures (Gad, 2022) K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 31 of 42 • iHMM-FRPSO consistently outperforms the other models across all metrics. • LSTM performs well due to its temporal sequence modeling ability, but still under- performs compared to iHMM-FRPSO. • ANN and Random Forest show moderate performance but are less stable on high- fluctuation price spikes common in electricity markets. • The hybridized fuzzy Markov structure and PSO-tuned parameters in iHMM-FRPSO allow for better handling of uncertainty and temporal dependency, leading to supe- rior accuracy. 6.3. PSO Convergence Validation and Sensitivity Analysis ”The stable convergence trends and rapid error minimization confirm the robustness and efficiency of the proposed hybrid optimization approach. This observation is con- sistent with earlier hybrid FHMM and deep learning-based optimization studies (Patel & Deshmukh, 2021).” To ensure the robustness and efficiency of the proposed iHMM-FRPSO model, we conducted a detailed analysis of the Particle Swarm Optimization (PSO) con- vergence behavior and its sensitivity to key hyperparameters. The convergence plots in Figures 7-11 depict the number of iterations versus the best function value for both Day- Ahead and Real-Time Market datasets across training and testing phases. These plots indicate that the optimization process typically converges within 50 to 100 iterations. 6.3.1. Convergence Behavior PSO was initialized with a population of particles randomly distributed in the solution space. Convergence was considered achieved when the global best fitness value did not improve significantly (defined as a threshold difference of less than 1×10−6) over 20 suc- cessive iterations. Across all simulation runs, convergence was typically observed between 60 and 80 iterations for both market types.Furthermore, the average best function values flattened out early, indicating the particles had clustered around optimal or near-optimal solutions. This demonstrates a satisfactory convergence pattern suitable for real-time forecasting scenarios. 6.3.2. Hyperparameter Sensitivity To assess the influence of PSO parameters on model performance, a grid-based sensitivity analysis was performed by varying key hyperparameters including swarm size, inertia weight (w), and acceleration coefficients (c1, c2). K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 32 of 42 Table 9: Summarizes the results of Hyper parameter sensitivity Analysis. Parameter Tested Val- ues Optimal Value Observation Swarm Size 10, 20, 30 30 Higher swarm size improved explo- ration and stabilized convergence. Inertia Weight w 0.4, 0.6, 0.9 0.6 Balanced exploration and exploita- tion; too low led to premature con- vergence. Cognitive Coefficient c1 1.5, 2.0, 2.5 2.0 Higher c1 increased local search but could cause instability beyond 2.0. Social Coefficient c2 1.5, 2.0, 2.5 2.0 Best results achieved when c1 = c2 = 2.0. Max Iterations 50, 100, 200 100 Beyond 100 iterations showed mini- mal gain in function value improve- ment. This analysis confirms that the PSO optimizer’s effectiveness in tuning the HMM pa- rameters is highly dependent on the appropriate selection of its hyperparameters. The identified optimal configuration yielded the best forecasting performance in terms of MAPE, RMSE, and MSE across both market types. The MATLAB 2022a environment was used for implementation. The optimization code was executed with adaptive inertia weight strategy and termination conditions defined by fitness stagnation and maximum iteration thresholds. 6.4. Statistical Significance of Performance Improvements To validate that the observed performance improvements of the iHMM-FRPSO model over traditional ML/DL models are not due to random chance, statistical significance testing was conducted using a paired two-tailed t-test at a 95% confidence level. The tests were applied to the prediction errors (MAPE and RMSE) obtained over the testing data across all models using five independent runs. Table 12 summarizes the p-values obtained when comparing iHMM-FRPSO with ANN, LSTM, and Random Forest. Table 10: Paired t-test p-values comparing iHMM-FRPSO with other models Compared Models p-value (MAPE) p-value (RMSE) iHMM-FRPSO vs ANN 0.0081 0.0114 iHMM-FRPSO vs LSTM 0.0197 0.0262 iHMM-FRPSO vs RF 0.0064 0.0090 All p - values are below the significance threshold of 0.05, indicating that the im- provements offered by the proposed model in both MAPE and RMSE are statistically significant. Hence, the performance gains achieved by iHMM-FRPSO over ANN, LSTM, K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 33 of 42 and RF are not only consistent but also statistically meaningful, further establishing the robustness of the proposed hybrid approach. 7. Conclusion and Future work 7.1. Conclusion This study introduced the iHMM-FRPSO model, which integrates interval-valued fuzzy relational Hidden Markov Models with Particle Swarm Optimization for electric- ity market forecasting. By employing intuitionistic fuzzy sets, the model effectively ad- dresses uncertainty,aligning with earlier fuzzy probability and stochastic approaches in electricity forecasting (Sujatha et al., 2016), while PSO optimizes transition and emission parameters to minimize forecasting errors. Empirical evaluation using Indian Energy Ex- change datasets demonstrated that the iHMM-FRPSO model outperforms conventional HMM-based optimization methods as well as state-of-the-art machine learning approaches, including ANN, LSTM, and Random Forest. During the testing phase, the model achieved an optimized MAPE of 23.24% in Real-Time Market forecasting, accompanied by substan- tial reductions in MSE and RMSE. Statistical significance testing further confirmed that the observed improvements are consistent and not attributable to random variation. Over- all, the iHMM-FRPSO framework provides high forecasting accuracy, stable convergence, and enhanced interpretability compared with black-box deep learning models, making it a promising approach for practical deployment in power markets. For broader applicability, the integration of parametric optimization with PSO offers a robust calibration mechanism for Hidden Markov Models across diverse electricity market environments. The training phase, conducted on datasets spanning January to Decem- ber 2021, achieved mean directional accuracy with optimized MAPE values of 14.40% for Day-Ahead Markets and 22.05% for Real-Time Markets. To further substantiate these findings, Table 13 reports the comparative performance measures between the proposed iHMM-FRPSO model, optimized through PSO, and the Day-Ahead and Real-Time Mar- ket datasets across both training and testing phases. Table 11: Summarizes the comparative performance measures between the proposed iHMM-FRPSO model, optimized through PSO, and the Day-Ahead and Real-Time Market datasets across both training and testing phases. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 34 of 42 7.2. Future Work Future research can advance this work in several directions. One promising extension would be to explore alternative optimization techniques such as Differential Evolution, Ant Colony Optimization, or hybrid evolutionary algorithms to enhance robustness. Test- ing across multi-region and cross-border datasets could further demonstrate adaptability under diverse regulatory and economic conditions. Another avenue is the incorporation of dimensionality reduction and feature selection methods to improve scalability when handling large-scale datasets. Establishing formal convergence proofs and theoretical per- formance guarantees would also strengthen the framework’s reliability. In addition, extending the proposed model to capture the uncertainty introduced by renewable energy sources such as solar and wind could make it more relevant for sus- tainable power markets. Applying the framework to multiple coupled power exchanges, including day-ahead and real-time markets simultaneously, may provide broader and more holistic forecasting insights. Enhancing computational optimization for real-time deploy- ment would improve its practical adoption by market operators. Integrating the model with interpretable deep learning methods, such as attention-based architectures, may in- crease forecasting accuracy while preserving explainability. Finally, extending the interval- valued framework toward probabilistic forecasting could provide confidence bounds and risk-aware decision-making support. By pursuing these directions, the iHMM-FRPSO framework can evolve into a universally applicable forecasting model, capable of support- ing decision-making in increasingly complex and deregulated electricity markets. References [1] A.A. Markov. Extension of the law of large numbers to dependent events. Bulletin of the Society of the Physics Mathematics, Kazan, Russia, 2:155 – 156, 1906. [2] S.M. Ross et al. Stochastic Processes. Wiley, New York, 1996. [3] G. Kavitha et al. Stock market trend analysis using hidden markov models. 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In 6th International Symposium on Big Data and Applied Statistics, Journal of Physics Conference Series, 2023. K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 36 of 42 Appendix: iHMM-FRPSO Forecasting Procedure (Pseudocode) Input: • Historical Market Data: Price (MCP), Volume (MCV) • PSO Parameters: swarm size S, max iterations T , inertia weight w, coefficients c1 and c2 • HMM Parameters: number of hidden states N , observation symbols M • Fuzzy rule base and membership functions • Train/Test split ratio Output: • Forecasted MCP/MCV for Day-Ahead and Real-Time Markets Begin 1. Preprocessing: a. Normalize historical MCP and MCV data b. Split data into training and testing sets c. Discretize data into observation symbols for HMM 2. Intuitionistic Fuzzy Rule Generation: a. Construct fuzzy membership functions for input variables b. Generate IF-THEN rules with membership and non-membership degrees c. Encode uncertainty using intuitionistic fuzzy sets 3. Hidden Markov Model (HMM) Initialization: a. Initialize: – Initial state probabilities π – State transition matrix A – Observation probability matrix B b. Apply fuzzy logic to soften observation symbol likelihoods K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 37 of 42 4. Particle Swarm Optimization (PSO) a. Initialize a swarm of S particles, each encoding [π,A,B]. b. For each particle: i Evaluate fitness using forecasting error (e.g., MAPE on validation set). vi(t+ 1) = w∗vi(t) + c∗1rand ∗ 1 ( pbesti − xi(t) ) + c∗2rand ∗ 2 ( gbest− xi(t) ) xi(t+ 1) = xi(t) + vi(t+ 1) c. Update each particle’s velocity and position using: d. Update personal and global bests e. Repeat steps (b)-(d) until T iterations or convergence threshold met 5. Optimized Model Forecasting: a. Use PSO-optimized HMM parameters [π∗, A∗, B∗] b. Apply Viterbi decoding to infer most probable hidden state sequence c. Predict next MCP/MCV based on current state and transition/emission prob- abilities 6. Postprocessing: a. Inverse normalization of predicted values b. Compute error metrics: MAPE, RMSE, MSE End K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 38 of 42 Table 12: Historical Data of Market Clearing Price of Real Time Market. Date MCP Date MCP Date MCP Date MCP Date MCP 01-01- 2022 3323.97 09-02- 2022 3819.26 20-03- 2022 5003.36 28-04- 2022 12000 06-06- 2022 7435.19 02-01- 2022 2607.29 10-02- 2022 3637.26 21-03- 2022 11250.78 29-04- 2022 12000 07-06- 2022 6601.48 03-01- 2022 3833.34 11-02- 2022 3969.4 22-03- 2022 14055.37 30-04- 2022 11672.53 08-06- 2022 6897.56 04-01- 2022 4352.75 12-02- 2022 4825.24 23-03- 2022 17038.55 01-05- 2022 6326.4 09-06- 2022 6030.6 05-01- 2022 3347.16 13-02- 2022 3509.84 24-03- 2022 16063.81 02-05- 2022 7382.49 10-06- 2022 8451.32 06-01- 2022 3679.79 14-02- 2022 3849.16 25-03- 2022 17486.4 03-05- 2022 8921.89 11-06- 2022 7316.99 07-01- 2022 3210.51 15-02- 2022 4668.2 26-03- 2022 11958.87 04-05- 2022 8875.24 12-06- 2022 5859.75 08-01- 2022 2525.18 16-02- 2022 5276 27-03- 2022 4771.8 05-05- 2022 8904.66 13-06- 2022 9602.84 10-01- 2022 2546.95 18-02- 2022 4320.32 29-03- 2022 11629.97 07-05- 2022 6500.01 15-06- 2022 7523.95 11-01- 2022 2998.72 19-02- 2022 4477.95 30-03- 2022 11218.36 08-05- 2022 5316 16-06- 2022 5904.69 12-01- 2022 3347.29 20-02- 2022 3323.01 31-03- 2022 11371.04 09-05- 2022 5360.92 17-06- 2022 4908.46 13-01- 2022 3156.03 21-02- 2022 4662.03 01-04- 2022 5762.49 10-05- 2022 5762 18-06- 2022 4132.13 14-01- 2022 3079.54 22-02- 2022 5170.12 02-04- 2022 5226.28 11-05- 2022 5301.14 19-06- 2022 2964.82 15-01- 2022 2794.07 23-02- 2022 5670.45 03-04- 2022 5972.33 12-05- 2022 4180.63 20-06- 2022 3126.02 16-01- 2022 2817.54 24-02- 2022 7080.57 04-04- 2022 5247.02 13-05- 2022 4271.11 21-06- 2022 3492.06 17-01- 2022 4260.51 25-02- 2022 5342.28 05-04- 2022 7462.87 14-05- 2022 4017.8 22-06- 2022 5709.96 18-01- 2022 4922.41 26-02- 2022 3419.96 06-04- 2022 10817.13 15-05- 2022 3510.83 23-06- 2022 6783.06 19-01- 2022 4407.08 27-02- 2022 3405.61 07-04- 2022 10060.86 16-05- 2022 4468.43 24-06- 2022 6063.78 K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 39 of 42 Date MCP Date MCP Date MCP Date MCP Date MCP 20-01- 2022 3944.68 28-02- 2022 3500.74 08-04- 2022 9342.42 17-05- 2022 6223.79 25-06- 2022 6195.98 21-01- 2022 3694.69 01-03- 2022 4304.46 09-04- 2022 9864.62 18-05- 2022 8375.99 26-06- 2022 6456.22 22-01- 2022 2655.79 02-03- 2022 3767.73 10-04- 2022 7370.23 19-05- 2022 8343.44 27-06- 2022 9080.35 23-01- 2022 2706.54 03-03- 2022 4003.32 11-04- 2022 9317.28 20-05- 2022 5290.71 28-06- 2022 8830.52 24-01- 2022 3427.9 04-03- 2022 4848.98 12-04- 2022 10525.9 21-05- 2022 2893.47 29-06- 2022 3970.43 25-01- 2022 3569.44 05-03- 2022 4688.38 13-04- 2022 10683.12 22-05- 2022 2790 30-06- 2022 3230.3 26-01- 2022 3165.4 06-03- 2022 4077.85 14-04- 2022 8723.87 23-05- 2022 2559 - - 27-01- 2022 3027.6 07-03- 2022 5142.14 15-04- 2022 5944.09 24-05- 2022 5453.05 - - 28-01- 2022 3777.15 08-03- 2022 5136.22 16-04- 2022 9384.13 25-05- 2022 7852.73 - - 29-01- 2022 4493.98 09-03- 2022 6303.08 17-04- 2022 7294.56 26-05- 2022 6063.98 - - 30-01- 2022 3834.11 10-03- 2022 6178.34 18-04- 2022 9808.69 27-05- 2022 4404.83 - - 31-01- 2022 4803.92 11-03- 2022 7890.69 19-04- 2022 11191.72 28-05- 2022 5990.68 - - 01-02- 2022 5975.77 12-03- 2022 6649.12 20-04- 2022 11586.07 29-05- 2022 5429.72 - - 02-02- 2022 5241.53 13-03- 2022 4924.24 21-04- 2022 11850.28 30-05- 2022 6989.44 - - 03-02- 2022 3371.95 14-03- 2022 6983.67 22-04- 2022 12000 31-05- 2022 6768.54 - - 04-02- 2022 3168.86 15-03- 2022 9567.71 23-04- 2022 12000 01-06- 2022 7112.59 - - 05-02- 2022 3748.65 16-03- 2022 11766.66 24-04- 2022 8774.78 02-06- 2022 8332.87 - - 06-02- 2022 3126.77 17-03- 2022 11022.62 25-04- 2022 11381.37 03-06- 2022 7806.18 - - 07-02- 2022 3736.07 18-03- 2022 4978.51 26-04- 2022 11382.09 04-06- 2022 8759.77 - - 08-02- 2022 4478.21 19-03- 2022 3432.91 27-04- 2022 12000 05-06- 2022 7907.22 - - K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 40 of 42 Table 13: Difference Value and Observation Symbol of Historical Data of Market Clearing Price of Real Time Market. Date MCP D.V O.S Date MCP D.V O.S Date MCP D.V O.S 01-01- 2022 3323.97 11-02- 2022 3969.4 332.14 R 24-03- 2022 16063.81 -974.74 F 02-01- 2022 2607.29 -716.68 F 12-02- 2022 4825.24 855.84 R 25-03- 2022 17486.4 1422.59 R 03-01- 2022 3833.34 1226.05 R 13-02- 2022 3509.84 -1315.4 F 26-03- 2022 11958.87 -5527.53 F 04-01- 2022 4352.75 519.41 F 14-02- 2022 3849.16 339.32 R 27-03- 2022 4771.8 -7187.07 F 05-01- 2022 3347.16 - 1005.59 F 15-02- 2022 4668.2 819.04 R 28-03- 2022 5715.81 944.01 R 06-01- 2022 3679.79 332.63 R 16-02- 2022 5276 607.8 F 29-03- 2022 11629.97 5914.16 R 07-01- 2022 3210.51 -469.28 F 17-02- 2022 4897.73 -378.27 F 30-03- 2022 11218.36 -411.61 F 08-01- 2022 2525.18 -685.33 F 18-02- 2022 4320.32 -577.41 F 31-03- 2022 11371.04 152.68 R 09-01- 2022 2559.81 34.63 R 19-02- 2022 4477.95 157.63 R 01-04- 2022 5762.49 -5608.55 F 10-01- 2022 2546.95 -12.86 F 20-02- 2022 3323.01 -1154.94 F 02-04- 2022 5226.28 -536.21 R 11-01- 2022 2998.72 451.77 R 21-02- 2022 4662.03 1339.02 R 03-04- 2022 5972.33 746.05 R 12-01- 2022 3347.29 348.57 F 22-02- 2022 5170.12 508.09 F 04-04- 2022 5247.02 -725.31 F 13-01- 2022 3156.03 -191.26 F 23-02- 2022 5670.45 500.33 F 05-04- 2022 7462.87 2215.85 R 14-01- 2022 3079.54 -76.49 R 24-02- 2022 7080.57 1410.12 R 06-04- 2022 10817.13 3354.26 R 15-01- 2022 2794.07 -285.47 F 25-02- 2022 5342.28 -1738.29 F 07-04- 2022 10060.86 -756.27 F 16-01- 2022 2817.54 23.47 R 26-02- 2022 3419.96 -1922.32 F 08-04- 2022 9342.42 -718.44 R 17-01- 2022 4260.51 1442.97 R 27-02- 2022 3405.61 -14.35 R 09-04- 2022 9864.62 522.2 R 18-01- 2022 4922.41 661.9 F 28-02- 2022 3500.74 95.13 R 10-04- 2022 7370.23 -2494.39 F 19-01- 2022 4407.08 -515.33 F 01-03- 2022 4304.46 803.72 R 11-04- 2022 9317.28 1947.05 R K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 41 of 42 Date MCP D.V O.S Date MCP D.V O.S Date MCP D.V O.S 20-01- 2022 3944.68 -462.4 R 02-03- 2022 3767.73 -536.73 F 12-04- 2022 10525.9 1208.62 F 21-01- 2022 3694.69 -249.99 R 03-03- 2022 4003.32 235.59 R 13-04- 2022 10683.12 157.22 F 22-01- 2022 2655.79 -1038.9 F 04-03- 2022 4848.98 845.66 R 14-04- 2022 8723.87 -1959.25 F 23-01- 2022 2706.54 50.75 R 05-03- 2022 4688.38 -160.6 F 15-04- 2022 5944.09 -2779.78 F 24-01- 2022 3427.9 721.36 R 06-03- 2022 4077.85 -610.53 F 16-04- 2022 9384.13 3440.04 R 25-01- 2022 3569.44 141.54 F 07-03- 2022 5142.14 1064.29 R 17-04- 2022 7294.56 -2089.57 F 26-01- 2022 3165.4 -404.04 F 08-03- 2022 5136.22 -5.92 F 18-04- 2022 9808.69 2514.13 R 27-01- 2022 3027.6 -137.8 R 09-03- 2022 6303.08 1166.86 R 19-04- 2022 11191.72 1383.03 F 28-01- 2022 3777.15 749.55 R 10-03- 2022 6178.34 -124.74 F 20-04- 2022 11586.07 394.35 F 29-01- 2022 4493.98 716.83 F 11-03- 2022 7890.69 1712.35 R 21-04- 2022 11850.28 264.21 F 30-01- 2022 3834.11 -659.87 F 12-03- 2022 6649.12 -1241.57 F 22-04- 2022 12000 149.72 F 31-01- 2022 4803.92 969.81 R 13-03- 2022 4924.24 -1724.88 F 23-04- 2022 12000 0 F 01-02- 2022 5975.77 1171.85 R 14-03- 2022 6983.67 2059.43 R 24-04- 2022 8774.78 -3225.22 R 02-02- 2022 5241.53 -734.24 F 15-03- 2022 9567.71 2584.04 R 25-04- 2022 11381.37 2606.59 R 03-02- 2022 3371.95 - 1869.58 F 16-03- 2022 11766.66 2198.95 F 26-04- 2022 11382.09 0.72 F 04-02- 2022 3168.86 -203.09 R 17-03- 2022 11022.62 -744.04 F 27-04- 2022 12000 617.91 R 05-02- 2022 3748.65 579.79 R 18-03- 2022 4978.51 -6044.11 F 28-04- 2022 12000 0 F 06-02- 2022 3126.77 -621.88 F 19-03- 2022 3432.91 -1545.6 R 29-04- 2022 12000 0 F 07-02- 2022 3736.07 609.3 R 20-03- 2022 5003.36 1570.45 R 30-04- 2022 11672.53 -327.47 R 08-02- 2022 4478.21 742.14 R 21-03- 2022 11250.78 6247.42 R 01-05- 2022 6326.4 -5346.13 F K. Kalpana, G. Kavitha / Eur. J. Pure Appl. Math, 18 (4) (2025), 6539 42 of 42 Date MCP D.V O.S Date MCP D.V O.S Date MCP D.V O.S 09-02- 2022 3819.26 -658.95 F 22-03- 2022 14055.37 2804.59 F 02-05- 2022 7382.49 1056.09 R 10-02- 2022 3637.26 -182 R 23-03- 2022 17038.55 2983.18 R 03-05- 2022 8921.89 1539.4 R Date MCP D.V O.S Date MCP D.V O.S 04-05-2022 8875.24 -46.65 F 03-06-2022 7806.18 -526.69 F 05-05-2022 8904.66 29.42 R 04-06-2022 8759.77 953.59 R 06-05-2022 7634.53 -1270.13 F 05-06-2022 7907.22 -852.55 F 07-05-2022 6500.01 -1134.52 R 06-06-2022 7435.19 -472.03 R 08-05-2022 5316 -1184.01 F 07-06-2022 6601.48 -833.71 F 09-05-2022 5360.92 44.92 R 08-06-2022 6897.56 296.08 R 10-05-2022 5762 401.08 R 09-06-2022 6030.6 -866.96 F 11-05-2022 5301.14 -460.86 R 10-06-2022 8451.32 2420.72 R 12-05-2022 4180.63 -1120.51 F 11-06-2022 7316.99 -1134.33 F 13-05-2022 4271.11 90.48 R 12-06-2022 5859.75 -1457.24 F 14-05-2022 4017.8 -253.31 F 13-06-2022 9602.84 3743.09 R 15-05-2022 3510.83 -506.97 F 14-06-2022 10402.63 799.79 F 16-05-2022 4468.43 957.6 R 15-06-2022 7523.95 -2878.68 F 17-05-2022 6223.79 1755.36 R 16-06-2022 5904.69 -1619.26 R 18-05-2022 8375.99 2152.2 R 17-06-2022 4908.46 -996.23 R 19-05-2022 8343.44 -32.55 F 18-06-2022 4132.13 -776.33 R 20-05-2022 5290.71 -3052.73 F 19-06-2022 2964.82 -1167.31 F 21-05-2022 2893.47 -2397.24 R 20-06-2022 3126.02 161.2 R 22-05-2022 2790 -103.47 R 21-06-2022 3492.06 366.04 R 23-05-2022 2559 -231 F 22-06-2022 5709.96 2217.9 R 24-05-2022 5453.05 2894.05 R 23-06-2022 6783.06 1073.1 F 25-05-2022 7852.73 2399.68 F 24-06-2022 6063.78 -719.28 F 26-05-2022 6063.98 -1788.75 F 25-06-2022 6195.98 132.2 R 27-05-2022 4404.83 -1659.15 R 26-06-2022 6456.22 260.24 R 28-05-2022 5990.68 1585.85 R 27-06-2022 9080.35 2624.13 R 29-05-2022 5429.72 -560.96 F 28-06-2022 8830.52 -249.83 F 30-05-2022 6989.44 1559.72 R 29-06-2022 3970.43 -4860.09 F 31-05-2022 6768.54 -220.9 F 30-06-2022 3230.3 -740.13 R 01-06-2022 7112.59 344.05 R - - - - 02-06-2022 8332.87 1220.28 R - - - -