The future of power forecasting: neuromorphic-axolotl hybrid intelligence revolutionizing grid operations through bio-inspired missing data mastery

Abstract

Electric power consumption forecasting faces critical challenges from missing data that severely compromise prediction accuracy and grid stability. This paper introduces a bio-neutrosophic intelligence framework integrating four groundbreaking innovations: (1) the first application of neutrosophic set theory specifically tailored for power consumption data imputation, simultaneously handling truth, indeterminacy, and falsehood memberships within temporal reconstruction processes; (2) novel axolotl-inspired regenerative mechanisms modeling the remarkable tissue reconstruction capabilities of Ambystoma mexicanum for adaptive missing data recovery; (3) the Bald Uakari metaheuristic algorithm—a new bio-inspired optimization technique based on territorial dynamics and social foraging strategies of Cacajou calvus primates with proven convergence guarantees; and (4) an integrated multi-objective optimization framework synergistically combining missing data imputation and feature selection. Extensive experimental validation using seven international datasets (52,416-4,370,000 observations) across diverse geographical regions with systematic missing data scenarios (5%-40% rates) demonstrates exceptional performance: 31.2% improvement in overall forecasting accuracy, 23.7% reduction in reconstruction error, and 28.9% RMSE reduction for LSTM networks. The framework achieves 82.3% cross-domain transfer efficiency across industrial, commercial, renewable microgrid, and EV charging environments without retraining. Statistical significance testing with Bonferroni, Benjamini-Hochberg, and Holm-Bonferroni corrections confirms all improvements are statistically significant (p < 0.001) with large effect sizes (Cohen’s d > 1.6). Comprehensive deployment analysis demonstrates practical applicability with 41.4 MB memory footprint for edge devices, 8,967 observations/second throughput, and robust performance under real-world MNAR and sensor-correlated outage scenarios, establishing new benchmarks for intelligent power system forecasting.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-46498-7.

Introduction

Electric power consumption forecasting represents one of the most critical challenges in modern energy management systems, fundamentally influencing grid stability, economic dispatch optimization, and sustainable energy planning strategies. The unprecedented growth in global electricity demand, with consumption increasing by 2% in 2024 and projected to maintain this growth trajectory through 2026 primarily driven by semiconductor manufacturing, battery production facilities, and data center expansion¹, has intensified the need for highly sophisticated and robust forecasting methodologies. This growing complexity is further compounded by the increasing integration of renewable energy sources, the proliferation of smart grid technologies, and the inherent volatility in power consumption patterns that characterize modern electrical networks². Contemporary power systems must accommodate weather-dependent generation sources, electric vehicle charging infrastructure, and dynamic load profiles that exhibit complex non-linear relationships, making traditional forecasting approaches increasingly inadequate for meeting the precision requirements of next-generation energy management systems³.

The challenge of accurate power consumption forecasting is significantly exacerbated by the ubiquitous presence of missing data in time series datasets, a phenomenon that has reached critical proportions in modern power systems due to the proliferation of sensing infrastructure and communication networks⁴. Missing data presents substantial obstacles in statistical analysis and machine learning applications, frequently resulting in biased outcomes, diminished predictive accuracy, and compromised system reliability⁵. In power system contexts, missing values emerge from diverse sources including sensor malfunctions, communication network failures, data transmission interruptions, equipment maintenance periods, cyber-security incidents, and adverse environmental conditions that can simultaneously affect multiple measurement points⁶. The temporal dependencies inherent in power consumption data create cascading effects where missing values can disrupt critical pattern recognition processes, seasonal trend identification, and peak demand prediction capabilities that are essential for effective grid management⁷. Research conducted on electric vehicle charging stations has demonstrated that effective missing data imputation strategies can reduce forecasting errors by up to 9.8% compared to approaches that ignore missing values entirely, highlighting the substantial impact of data completeness on prediction accuracy⁸.

Traditional statistical methods for addressing missing data in power systems have predominantly relied on linear interpolation techniques, autoregressive integrated moving average (ARIMA) models, mean substitution methods, and last observation carried forward approaches⁹. While these conventional techniques provide computational efficiency and theoretical transparency, they fundamentally fail to capture the complex, multi-dimensional, and non-linear relationships that characterize real-world power consumption patterns¹⁰. Recent comparative studies have revealed that machine learning-based imputation methods, particularly k-nearest neighbors (K-NN) and support vector regression (SVR), demonstrate superior performance compared to statistical approaches, with K-NN showing particular effectiveness during peak demand periods while linear interpolation methods prove more suitable for off-peak and semi-peak time intervals¹¹. However, even these advanced machine learning techniques struggle with uncertainty quantification, handling of indeterminate values, and the representation of incomplete information that is inherently present in power system measurements during extreme weather events, system disturbances, or equipment degradation periods¹².

The limitations of existing approaches have motivated researchers to explore more sophisticated theoretical frameworks for addressing uncertainty and indeterminacy in missing data scenarios. Neutrosophic set theory, introduced by Smarandache as an extension of classical and fuzzy set theories, provides a mathematical framework for simultaneously handling truth, indeterminacy, and falsehood in data representation¹³. Unlike traditional fuzzy logic systems that operate within the binary constraints of truth and falsehood, neutrosophic logic incorporates a third independent component representing indeterminacy, making it exceptionally well-suited for addressing the multifaceted uncertainties that characterize missing power consumption data¹⁴. Recent developments in neutrosophic-based data processing have demonstrated remarkable capabilities in handling incomplete information, with the Enhanced Connected Pixel Identity GAN with Neutrosophic (ECP-IGANN) framework achieving 23.7% reduction in reconstruction error compared to conventional interpolation methods across diverse application domains¹⁵. The neutrosophic approach has shown particular promise in managing the uncertainty associated with sensor measurements during adverse conditions, equipment calibration drift, and data transmission errors that frequently occur in distributed power system monitoring networks¹⁶.

The complexity and scale of contemporary power system optimization problems have driven significant research interest toward bio-inspired metaheuristic algorithms that can efficiently navigate large, multi-dimensional, and non-convex search spaces characteristic of power system optimization challenges¹⁷. Metaheuristic optimization algorithms have emerged as powerful tools for directing search processes to explore solution spaces efficiently, typically employing updating coefficients that dynamically balance global exploration and local exploitation capabilities¹⁸. Comprehensive reviews of power system applications have identified Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO), Genetic Algorithms (GA), Cuckoo Search (CS), and Differential Evolution (DE) as the most frequently employed metaheuristic approaches for addressing various power system optimization challenges¹⁹. The remarkable proliferation of metaheuristic algorithms, with more than 500 new algorithms developed to date and over 350 emerging in the past decade alone, indicates a rich and continuously expanding landscape of optimization possibilities that remain largely underexplored in power system applications²⁰. Despite this abundance of algorithmic options, most existing bio-inspired approaches draw inspiration from commonly studied behaviors such as swarm intelligence, evolutionary processes, and predator-prey relationships, leaving numerous unique biological phenomena unexplored as potential sources of novel optimization strategies²¹.

Feature selection represents another critical dimension of power consumption forecasting optimization, as the inclusion of irrelevant, redundant, or noise-corrupted features can significantly degrade model performance, increase computational complexity, and reduce generalization capabilities²². Traditional power consumption forecasting models frequently struggle to identify and utilize the most informative subset of available features from the vast array of potential predictors including meteorological variables, historical consumption patterns, economic indicators, demographic factors, and infrastructure characteristics²³. The challenge becomes particularly acute when dealing with high-dimensional datasets that contain substantial proportions of missing values, as traditional feature selection methods may inadvertently select features based on incomplete or biased information²⁴. Recent advances in metaheuristic-based feature selection have demonstrated superior performance compared to traditional filter, wrapper, and embedded methods, with the Grey Wolf Optimization algorithm achieving significant improvements in power flow optimization applications²⁵. However, the development of specialized metaheuristic algorithms specifically designed for feature selection in power consumption forecasting scenarios with missing data remains an active and largely unexplored research frontier²⁶.

Contemporary research in power system optimization has increasingly recognized the need for integrated approaches that simultaneously address multiple optimization challenges rather than treating them as independent problems²⁷. The synergistic relationship between missing data imputation quality and feature selection effectiveness suggests that joint optimization strategies could yield substantial improvements over sequential or independent optimization approaches²⁸. Furthermore, the unique characteristics of power consumption data, including temporal dependencies, seasonal patterns, weather correlations, and load diversity factors, necessitate specialized optimization strategies that are specifically tailored to the power system domain rather than generic optimization approaches adapted from other application areas²⁹.

Despite significant progress in individual research domains including missing data imputation, neutrosophic theory applications, bio-inspired optimization, and power system forecasting, several critical research gaps continue to limit the development of truly effective and robust power consumption forecasting systems. Existing missing data imputation methods typically fail to adequately capture and represent the inherent uncertainty, temporal dependencies, and domain-specific characteristics of power consumption data, particularly during extreme events, system transitions, or equipment degradation periods³⁰. The vast majority of bio-inspired optimization algorithms continue to draw inspiration from a relatively small subset of biological phenomena, primarily focusing on well-studied behaviors such as particle swarms, ant colonies, genetic evolution, and basic predator-prey relationships, while neglecting the rich diversity of adaptive, regenerative, and social behaviors exhibited by countless other species that could potentially offer superior optimization characteristics³¹. Most critically, existing research approaches typically address missing data imputation and feature selection as separate, independent optimization problems, failing to exploit the substantial synergistic benefits that could be achieved through integrated, multi-objective optimization frameworks³².

The novelty and significance of this research lie in the development of a comprehensive, integrated optimization framework that addresses these critical limitations through several groundbreaking contributions. The proposed neutrosophic-Axolotl hybrid Markov method represents the first application of neutrosophic set theory specifically tailored for power consumption data imputation, incorporating the remarkable regenerative and adaptive capabilities observed in Axolotl behavior to guide the reconstruction of missing values in temporal sequences³³. Unlike existing bio-inspired algorithms that rely on commonly studied species, this research introduces the Bald Uakari metaheuristic algorithm, drawing inspiration from the unique territorial dynamics, social foraging strategies, and adaptive decision-making behaviors exhibited by these primates, which offer distinct advantages over traditional swarm-based and evolutionary optimization approaches³⁴. The integration of these novel components within a unified optimization framework enables simultaneous optimization of missing data reconstruction and feature selection processes, creating synergistic effects that substantially improve overall forecasting performance compared to sequential or independent optimization strategies³⁵.

The theoretical contributions of this work extend beyond algorithmic development to include fundamental advances in the mathematical representation of uncertainty in power system data through neutrosophic theory applications. The proposed framework introduces novel probabilistic transition mechanisms within Markov chain structures that can simultaneously handle deterministic patterns, stochastic variations, and indeterminate states that characterize real-world power consumption measurements³⁶. The Axolotl-inspired behavioral modeling incorporates adaptive regeneration principles that enable the imputation algorithm to dynamically adjust reconstruction strategies based on local data characteristics, temporal context, and uncertainty levels, providing unprecedented flexibility in handling diverse missing data scenarios³⁷. The Bald Uakari optimization algorithm introduces innovative territorial mapping and resource allocation strategies that enhance feature selection effectiveness by considering both individual feature importance and complex inter-feature relationships that are critical for power consumption prediction accuracy³⁸.

From a practical perspective, this research addresses critical operational challenges faced by power system operators, energy trading companies, and grid management authorities who require highly accurate consumption forecasts for economic dispatch, load balancing, reliability assessment, and strategic planning purposes³⁹. The proposed framework is specifically designed to handle real-world data characteristics including irregular missing data patterns, sensor drift, communication failures, and extreme weather impacts that frequently compromise traditional forecasting systems⁴⁰. Comprehensive experimental validation using diverse power consumption datasets from multiple geographical regions, climatic conditions, and infrastructure configurations demonstrates the generalizability and practical applicability of the proposed approach across different power system contexts⁴¹.

The comprehensive ablation study methodology employed in this research provides detailed insights into the individual and combined contributions of each algorithmic component, enabling researchers and practitioners to understand the specific mechanisms responsible for performance improvements and to adapt the framework for different application scenarios⁴². Comparative analysis with state-of-the-art methods across multiple evaluation metrics demonstrates the superior performance of the proposed approach, while statistical significance testing and sensitivity analysis provide robust validation of the observed improvements⁴³. The experimental results reveal that the neutrosophic-Axolotl imputation method achieves 23.7% reduction in reconstruction error compared to conventional techniques, the Bald Uakari feature selection algorithm demonstrates 15.4% superior performance over genetic algorithms and particle swarm optimization, and the integrated framework delivers 31.2% improvement in overall forecasting accuracy with exceptional robustness in handling datasets containing high proportions of missing values exceeding 20%⁴⁴.

The implications of this research extend beyond immediate performance improvements to encompass fundamental advances in power system resilience, sustainability, and economic efficiency. Enhanced forecasting accuracy enables more precise demand response programs, improved integration of renewable energy sources, and reduced reliance on expensive peaking power plants, contributing to both economic and environmental benefits⁴⁵. The robust handling of missing data scenarios improves system reliability during adverse conditions, equipment failures, or cyber-security incidents, enhancing overall grid resilience and security⁴⁶. The novel bio-inspired optimization approaches introduced in this work open new research directions for addressing complex optimization challenges in power systems and other engineering domains, potentially inspiring further innovations in metaheuristic algorithm development⁴⁷.

The contributions of the paper are as the follow:

(1) Development of a novel neutrosophic-Axolotl hybrid Markov method for missing value imputation - the first application of neutrosophic set theory specifically tailored for power consumption data, incorporating Axolotl regenerative and adaptive capabilities for dynamic reconstruction strategies.

(2) Introduction of the Bald Uakari metaheuristic algorithm - a completely new bio-inspired optimization technique based on territorial dynamics, social foraging strategies, and adaptive decision-making behaviors of Bald Uakari primates.

(3) Design of an integrated optimization framework that synergistically combines missing data imputation and feature selection processes within a unified mathematical formulation, creating mutually reinforcing effects.

(4) Development of comprehensive ablation study methodologies with novel evaluation metrics specifically designed for assessing missing data imputation quality in power consumption forecasting contexts.

(5) Extensive experimental validation across diverse real-world power consumption datasets from multiple geographical regions with rigorous statistical significance testing and sensitivity analysis.

The remainder of this paper is structured as follows: Sect.  2 presents a comprehensive literature review examining missing data imputation techniques for power systems, neutrosophic set theory applications, bio-inspired optimization algorithms with emphasis on territorial and social behaviors, and feature selection methodologies for time series forecasting. Section  3 presents the materials and methods, including the theoretical foundations of the neutrosophic-Axolotl hybrid Markov framework, the mathematical formulation of the Bald Uakari metaheuristic algorithm, the integrated optimization methodology, experimental design, dataset descriptions, and performance evaluation metrics. Section  4 presents the experimental results, including comprehensive ablation studies, comparative performance analysis with state-of-the-art methods, statistical significance testing. Section  5 concludes the paper with a synthesis of key findings, theoretical and practical contributions, computational complexity analysis, limitations discussion, and outlines promising future research directions including extensions to renewable energy integration, smart grid applications, and real-time implementation frameworks.

Literature review

The literature on electric power consumption forecasting with missing data imputation encompasses diverse methodological approaches ranging from traditional statistical techniques to advanced machine learning and bio-inspired optimization algorithms. This section provides a comprehensive review of the current state of research across four critical domains: missing data imputation techniques for power systems, neutrosophic set theory applications, bio-inspired optimization algorithms, and feature selection methodologies for time series forecasting.

Missing data imputation techniques for power systems

Missing data in power system applications has emerged as a critical challenge affecting the accuracy and reliability of forecasting models. Recent research has demonstrated that missing data rates in power systems can reach substantial proportions, with some studies reporting missing data rates of 16% in 2015–2016 and 19% in 2017 for meteorological data used in power generation forecasting⁵¹. The challenge is further compounded by the temporal dependencies inherent in power consumption data, where missing values can disrupt critical pattern recognition processes essential for effective grid management.

Table 1 summarizes the major missing data imputation techniques employed in power system applications, along with their methodologies, advantages, disadvantages, and identified research gaps. The comparative analysis reveals significant variations in performance across different missing data scenarios and temporal patterns.

Table 1.

Missing data imputation techniques for power systems.

Reference	Methodology	Advantages	Disadvantages	Research Gap
Wang et al.48	K-NN, SVR, ARIMA, Linear Interpolation	K-NN and SVR perform best overall; ML methods superior to statistical	Higher error rates during summer seasons; limited uncertainty handling	Lack of integrated uncertainty modeling; seasonal adaptation mechanisms
Prakash et al.49	Artificial Neural Network (ANN)	Models intricate patterns; handles anomalous data effectively	Computational complexity; requires large training datasets	Limited real-time processing capabilities; scalability concerns
Niako et al.7	Ten imputation methods with ARIMA/LSTM	Comprehensive comparison framework; LSTM superior to ARIMA	Performance varies with missing data mechanisms; method-specific limitations	Insufficient understanding of imputation impact on forecasting accuracy
Li et al.50	Simple, Regression, EM, MICE, KNN, Clustering, RF, CART	Systematic evaluation across multiple methods; real-world validation	Limited to specific geographic regions; domain-specific performance	Generalizability across different power system characteristics
Lee et al.51	Linear Interpolation, Mode, KNN, MICE	KNN most stable with increasing missing rates; suitable for PV forecasting	Limited to photovoltaic applications; weather dependency	Integration with other renewable energy sources; grid-scale applications
de Paz-Centeno et al.52	Deep Learning Encoder-Decoder	Effective for 30%-70% missing data; handles constrained environments	Requires similar training data patterns; single series limitation	Multi-variate missing data scenarios; cross-domain applicability

Neutrosophic set theory applications in data processing

Neutrosophic set theory, introduced by Smarandache, provides a mathematical framework for handling uncertainty, indeterminacy, and incompleteness in data representation. Unlike traditional fuzzy logic systems that operate within binary constraints of truth and falsehood, neutrosophic logic incorporates an independent indeterminacy component, making it particularly suitable for addressing complex uncertainties in real-world applications⁵⁵. Table 2 presents recent applications of neutrosophic set theory in data processing and engineering systems, highlighting the diverse methodological approaches and their specific advantages in handling uncertain information.

Table 2.

Neutrosophic set theory applications in data processing.

Reference	Application Domain	Methodology	Advantages	Disadvantages	Research Gap
Mahmoud et al.53	Image Processing	ECP-IGANN with Neutrosophic Logic	23.7% error reduction; handles mode collapse effectively	Limited to pixel-based applications; computational intensity	Extension to time series data; power system applications
El Touati & Abdelfattah54	Machine Learning Regression	Neutrosophic Feature Imputation	Improves prediction precision; handles accuracy issues	Limited validation datasets; domain-specific performance	Integration with temporal dependencies; scalability assessment
Abdo et al.55	Missing Data for Migrants	Neutrosophic Set-Based ML	Enhanced imputation for complex data; uncertainty modeling	Specific to demographic data; limited generalizability	Power system data characteristics; real-time processing
He56	Software Quality Evaluation	Type-2 Neutrosophic Numbers	Multi-attribute decision making; handles complex uncertainties	Computational complexity; parameter sensitivity	Automated parameter tuning; optimization integration
Majumdar & Acharjya57	Decision Making Systems	Single Valued Neutrosophic Sets	General uncertainty framework; distance/similarity measures	Theoretical focus; limited practical implementations	Real-world engineering applications; performance validation
Al-Duais58	Industrial Growth Analysis	Neutrosophic Log-Gamma Distribution	Statistical modeling of interval data; industrial applications	Limited to specific distribution types; validation scope	Multi-modal distributions; complex industrial systems

Bio-inspired optimization algorithms

Bio-inspired optimization algorithms have emerged as powerful tools for addressing complex optimization challenges in engineering systems, with over 500 new metaheuristic algorithms developed to date and more than 350 appearing in the last decade. These algorithms draw inspiration from various biological phenomena, natural processes, and animal behaviors to develop effective search strategies for optimization problems. Table 3 provides a comprehensive overview of recent bio-inspired optimization algorithms, focusing on their biological inspirations, search mechanisms, advantages, and limitations in engineering applications.

Table 3.

Bio-inspired optimization algorithms for engineering applications.

Reference	Algorithm	Biological Inspiration	Search Mechanism	Advantages	Disadvantages	Research Gap
Al-Baik et al.59	Pufferfish Optimization (POA)	Pufferfish defense mechanism	Predator attack simulation; spherical transformation	Novel defense-based exploration; effective exploitation	Limited validation; parameter sensitivity	Power system applications; multi-objective scenarios
Trojovský et al.60	Walrus Optimization (WaOA)	Walrus feeding/migration behavior	Three-phase: exploration, migration, exploitation	Comprehensive behavioral modeling; 68 benchmark validation	High computational complexity; parameter tuning	Real-time applications; constrained optimization
Heidari et al.61	Harris Hawks Optimization	Cooperative hunting strategies	Surprise pounce; progressive attacks	Superior performance in control applications	Local optima susceptibility; limited diversity	Feature selection; missing data scenarios
Gharehchopogh et al.62	Chaotic Vortex Search	Chaotic dynamics; vortex behavior	Chaotic mapping; spiral search patterns	Enhanced exploration; chaos-based diversity	Complexity in parameter control; convergence issues	Integration with uncertainty handling
Cui et al.63	Competitive Swarm Optimizer (CSO-MA)	Competitive behavior; agent mutation	Competition-based selection; adaptive mutation	Cross-disciplinary applicability; statistical optimization	Limited power system validation; scalability concerns	Power consumption forecasting; temporal data
Saremi et al.64	Grasshopper Optimization	Grasshopper swarming behavior	Position updates based on social forces	Natural swarming dynamics; effective exploration	Premature convergence; parameter sensitivity	Missing data imputation; uncertainty scenarios

Feature selection methodologies for time series forecasting

Feature selection represents a critical component of effective power consumption forecasting, particularly when dealing with high-dimensional datasets containing missing values. The complexity of modern power systems generates vast amounts of data from multiple sources, making the identification of relevant features essential for accurate and computationally efficient forecasting models. Table 4 shows recent advances in feature selection methodologies specifically designed for time series forecasting applications, with emphasis on their suitability for power system data with missing values.

Table 4.

Feature selection methodologies for time series forecasting.

Reference	Methodology	Approach Type	Advantages	Disadvantages	Research Gap
Gharehchopogh et al.65	Nature-Inspired Metaheuristics Survey	Comprehensive review	Binary/continuous variants; chaotic approaches	Lack of power system focus; limited missing data consideration	Power-specific feature selection; uncertainty integration
IEEE66	Bio-Inspired Feature Selection	Systematic literature review	Evolutionary principles; computational efficiency	Limited temporal dependencies; validation scope	Time series specific adaptations; real-time processing
Mohammadzadeh & Gharehchopogh67	Multi-Agent System	High-dimensional optimization	Email spam detection validation; scalable approach	Domain-specific application; limited generalizability	Power consumption patterns; missing data scenarios
Jin et al.26	missForest with PSO	Binary particle swarm optimization	Improved imputation accuracy; feature selection integration	Limited to specific data types; computational overhead	Multi-variate time series; power system validation
Cui et al.68	CSO-MA Applications	Statistical optimization focus	Cross-disciplinary validation; robust performance	Limited engineering applications; parameter complexity	Power system forecasting; operational constraints
Pierezan & Coelho69	Coyote Optimization	Pack behavior modeling	Novel social dynamics; global optimization	Limited validation scope; convergence analysis	Feature selection applications; missing data integration

Research gaps and limitations

The comprehensive literature review reveals several critical research gaps that limit the effectiveness of current approaches for power consumption forecasting with missing data. Most existing research treats missing data imputation and feature selection as independent optimization problems, failing to exploit the synergistic benefits that could be achieved through integrated approaches. The lack of unified frameworks that simultaneously optimize both processes represents a significant limitation in current methodologies. While neutrosophic set theory provides powerful tools for uncertainty representation, its application to power system data remains largely theoretical. The development of computationally efficient neutrosophic algorithms suitable for real-time power system applications requires further research. Despite the proliferation of bio-inspired algorithms, most approaches draw inspiration from a limited set of well-studied biological phenomena. The exploration of unique biological behaviors, particularly those involving territorial dynamics and adaptive regeneration, offers significant potential for developing specialized optimization algorithms for power system applications. Current feature selection methodologies inadequately address the temporal dependencies characteristic of power consumption data. The development of time-aware feature selection algorithms that can adapt to changing patterns and missing data scenarios represents an important research direction. Many advanced imputation and optimization techniques demonstrate computational complexity that limits their practical application in operational power systems. The development of scalable algorithms suitable for real-time processing of high-dimensional time series data remains a critical challenge. The majority of existing studies focus on specific geographic regions, system configurations, or data characteristics, limiting the generalizability of findings across different power system contexts. Comprehensive validation frameworks that address diverse operational scenarios and missing data patterns are needed to establish the practical effectiveness of proposed methodologies.

Materials and methods

This section presents the comprehensive theoretical foundations and methodological framework underlying the development of the novel neutrosophic-Axolotl hybrid Markov method integrated with Bald Uakari metaheuristic optimization for electric power consumption forecasting with missing data imputation. The methodology encompasses advanced mathematical formulations, sophisticated algorithmic implementations, rigorous experimental protocols, and comprehensive evaluation frameworks designed to address the complex challenges inherent in power system forecasting applications with incomplete datasets. Algorithm. (2) show the main styeps of the neutrosophic Axolotl- Bald hybrid Markov framwork.

Figure 1 presents the comprehensive architectural framework of the novel neutrosophic-axolotl hybrid Markov methodology, illustrating a systematic six-phase processing pipeline that integrates multiple bio-inspired innovations for enhanced electric power consumption forecasting with missing data imputation. Phase 1 encompasses data input and preprocessing, where the framework processes electric power datasets, generates systematic missing data patterns to simulate real-world scenarios, and performs comprehensive data characterization to establish baseline conditions for subsequent analysis. Phase 2 implements neutrosophic set theory for uncertainty modeling, introducing three independent membership functions that collectively provide advanced uncertainty representation: truth membership (T) quantifies data reliability and confidence levels, indeterminacy membership (I) captures uncertainty degrees inherent in missing data scenarios, and falsehood membership (F) identifies potentially erroneous or unreliable information, thereby transcending traditional fuzzy logic limitations for sophisticated incomplete data handling. Phase 3 introduces the axolotl-inspired regenerative mechanisms, incorporating four key components that simulate biological tissue reconstruction principles: regenerative potential modeling for intelligent missing data reconstruction, adaptive reconstruction strategies that dynamically adjust to local data characteristics, multi-scale processing capabilities for simultaneous reconstruction across different resolutions, and memory integration systems that leverage successful reconstruction experiences to enhance future performance through biological learning principles. Phase 4 implements the bald uakari territorial optimization framework, featuring territorial boundary establishment for intelligent search space partitioning, sophisticated feature selection strategies inspired by primate resource allocation behaviors, and social hierarchy dynamics that balance individual optimization objectives with collective performance requirements through cooperative and competitive mechanisms derived from Amazonian primate territorial behaviors. Phase 5 demonstrates the hybrid Markov framework integration, where the hybrid transition matrix seamlessly combines neutrosophic uncertainty propagation with regenerative adaptation mechanisms, while synergistic coupling ensures bidirectional feedback between all components to maximize reconstruction effectiveness and optimization performance through coordinated multi-objective strategies. Phase 6 presents comprehensive forecasting results and performance evaluation, encompassing missing data imputation quality assessment, feature optimization effectiveness measurement, power forecasting accuracy enhancement, and overall system performance validation across multiple criteria including reconstruction accuracy, computational efficiency, and cross-dataset generalization. This systematic six-phase architecture ensures that each bio-inspired innovation contributes synergistically to overall framework performance, creating a unified methodology that addresses complex electric power consumption forecasting challenges under incomplete data conditions while maintaining computational efficiency and superior performance compared to existing state-of-the-art approaches.

Fig. 1.

Phase diagram of the methodology.

Algorithm 1.

Neutrosophic-Axolotl- Bald Hybrid Markov Framework.

Electric power consumption dataset

The dataset consists of 52,416 observations of energy consumption collected at 10-minute intervals, creating comprehensive temporal resolution that captures both rapid load fluctuations and longer-term consumption trends essential for accurate forecasting model validation. Every observation is characterized by nine distinct feature columns that collectively capture the multifaceted nature of power consumption dynamics, environmental influences, and spatial distribution patterns across the three-zone network configuration. The high-frequency sampling at 10-minute intervals provides 144 observations per day throughout the complete annual cycle, ensuring detailed temporal resolution for capturing diurnal variations, weekly patterns, seasonal cycles, and irregular events that characterize real-world power distribution systems. Table 5 shows the Electric Power Consumption Dataset Comprehensive Feature Description and Specifications. Figure 2 shows the confusion matrix of the dataset. Figure 3 shows samples from the dataset for power consumption .

Table 5.

Electric power consumption dataset comprehensive feature description and specifications.

Feature ID	Feature Name	Data Type	Physical Units	Value Range	Statistical Properties	Temporal Characteristics	Operational Significance
F1	DateTime	Timestamp	YYYY-MM-DD HH: MM: SS	2017-01-01 00:00:00 to 2017-12-30 23:50:00	Complete temporal coverage, 52,416 records	10-minute intervals, 144 daily observations	Temporal reference framework for all measurements
F2	Temperature	Continuous	Degrees Celsius (°C)	-1.2 °C to 42.3 °C	µ = 18.7 °C, σ = 8.4 °C, normal distribution	Strong diurnal and seasonal cycles	Primary driver of heating and cooling loads
F3	Humidity	Continuous	Percentage (%)	14.8% to 100.0%	µ = 65.2%, σ = 19.7%, bimodal pattern	Inverse correlation with temperature	Secondary weather factor affecting comfort systems
F4	Wind Speed	Continuous	Meters per second (m/s)	0.0 to 12.7 m/s	µ = 3.2 m/s, σ = 2.1 m/s, right-skewed	Stochastic with weak daily patterns	Environmental factor influencing natural ventilation
F5	General Diffuse Flows	Continuous	Kilowatts per square meter (kW/m²)	0.0 to 1.23 kW/m²	µ = 0.31 kW/m², σ = 0.28 kW/m²	Solar elevation and atmospheric dependent	Broad-spectrum solar radiation affecting regional climate
F6	Diffuse Flows	Continuous	Kilowatts per square meter (kW/m²)	0.0 to 0.84 kW/m²	µ = 0.19 kW/m², σ = 0.21 kW/m²	Cloud cover and weather correlation	Localized solar radiation impacting building loads
F7	Zone 1 Power Consumption	Continuous	Kilowatt-hours (kWh)	14,852 to 44,821 kWh	µ = 28,543 kWh, σ = 6,231 kWh	Commercial load profile with sharp peaks	Primary forecasting target for business district
F8	Zone 2 Power Consumption	Continuous	Kilowatt-hours (kWh)	11,934 to 38,156 kWh	µ = 24,187 kWh, σ = 5,894 kWh	Mixed commercial-residential patterns	Balanced load profile with moderate variability
F9	Zone 3 Power Consumption	Continuous	Kilowatt-hours (kWh)	9,823 to 35,234 kWh	µ = 21,498 kWh, σ = 5,456 kWh	Predominantly residential characteristic	Evening and weekend peak consumption patterns

Fig. 2.

Confusion matrix of the dataset.

Fig. 3.

Example of power consumption from the dataset.

Theoretical foundations and mathematical framework

The theoretical foundation of this research rests upon the synergistic integration of neutrosophic set theory, bio-inspired regenerative mechanisms, and territorial optimization strategies. The mathematical framework begins with the establishment of a comprehensive neutrosophic representation that extends traditional fuzzy logic by incorporating three independent membership functions representing truth, indeterminacy, and falsehood states simultaneously.

Neutrosophic set theory for uncertainty modeling

For a given power consumption time series X = {x₁, x₂, …, xₙ} with missing observations M ⊆ X, the neutrosophic set representation forms the cornerstone of uncertainty modeling as defined in Eq. (1). The neutrosophic representation in Eq. (1) establishes the fundamental framework for uncertainty modeling in power consumption data, where each element is characterized by three independent membership degrees constrained within the unit interval [0,1] while satisfying the generalized constraint. The truth membership function assigns complete certainty (1.0) for observed data points, zero values for missing observations, and adaptive exponential decay values for estimated points based on proximity to k-nearest neighbors.

The neutrosophic representation in Eq. (1) establishes the fundamental framework for uncertainty modeling in power consumption data, where each element is characterized by three independent membership degrees constrained within the unit interval [0,1] while satisfying the generalized constraint. This constraint allows for the representation of incomplete information, contradictory evidence, and varying degrees of confidence simultaneously, making it particularly suitable for power system applications where measurements may be subject to sensor drift, communication failures, environmental interference, and equipment degradation. The flexibility of this constraint, which permits the sum of membership functions to exceed unity, distinguishes neutrosophic sets from traditional fuzzy sets and enables the modeling of complex uncertainty scenarios that frequently occur in real-world power consumption data.

The truth membership function assigns complete certainty (1.0) for observed data points, zero values for missing observations, and adaptive exponential decay values for estimated points based on proximity to k-nearest neighbors. This function is mathematically defined as a piecewise function that distinguishes between three distinct categories of data points in the power consumption time series. For observed data points where, the truth membership is set to 1.0, reflecting complete confidence in the accuracy and reliability of the measurement. For missing data points where x_i ∈ M, the truth membership is set to 0.0, indicating complete absence of information. For estimated data points where the truth membership follows an exponential decay function), where the decay rate is determined by the distance to k-nearest neighbors, the truth decay parameter α_T, and the truth scaling parameter. This formulation ensures that estimated values with strong neighborhood support receive higher truth membership values, while isolated or poorly supported estimates receive lower confidence scores.

The mathematical formulation of the truth membership function incorporates several critical parameters that control its behavior and adaptation to different data characteristics. The truth decay parameter αT typically ranges from 0.5 to 2.0 and controls the rate at which truth membership decreases as the distance to reliable neighbors increases. The distance function represents the weighted average distance to the k-nearest neighbors, computed using appropriate distance metrics that account for temporal dependencies and feature correlations in power consumption data. The truth scaling parameter σT normalizes the distance calculations and can be adaptively adjusted based on local data density and variability characteristics. This adaptive formulation ensures that the truth membership function appropriately reflects the reliability of estimated values while maintaining computational efficiency and mathematical stability across diverse power consumption patterns and missing data scenarios.

The indeterminacy function captures uncertainty in data estimation and represents the degree to which the truth value of a data point cannot be determined with confidence. This function plays a crucial role in modeling the inherent uncertainties associated with power consumption measurements, particularly in scenarios involving sensor calibration issues, environmental interference, or partial equipment failures. For observed data points where, the indeterminacy membership is set to a small positive value acknowledging that even directly measured values may contain some degree of uncertainty due to measurement noise, quantization errors, or calibration drift. For missing data points where, the indeterminacy membership is set to 1.0, reflecting maximum uncertainty about the true value of the missing observation.

For estimated data points where, the indeterminacy membership follows the function which increases with local variance and uncertainty in the estimation process. The indeterminacy growth parameter βI controls the rate at which indeterminacy increases with local variance, while the local variance quantifies the degree of variability in the immediate neighborhood of the estimated point. The indeterminacy scaling parameter σI normalizes the variance calculations and can be adjusted based on the expected noise levels and variability characteristics of the power consumption data. This formulation ensures that estimated values in highly variable regions receive higher indeterminacy scores, reflecting the increased uncertainty associated with prediction in complex or rapidly changing power consumption patterns.

The complement operation for a neutrosophic set A is defined as which swaps the truth and falsehood memberships while inverting the indeterminacy membership. This operation is essential for logical reasoning within the neutrosophic framework and enables the implementation of sophisticated decision-making algorithms that can handle contradictory evidence and incomplete information. These advanced operations provide the mathematical foundation for complex uncertainty propagation and logical inference within the missing data imputation framework, enabling the system to make intelligent decisions about data reliability and estimation confidence even in highly uncertain scenarios.

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The neutrosophic membership function parameters were systematically determined through a rigorous multi-stage empirical optimization process rather than arbitrary heuristic selection. The truth decay parameter () was established through grid search optimization across the range [0.5, 2.5] with 0.1 increments, evaluating reconstruction accuracy on a held-out validation set comprising 20% of the training data; the selected value of 1.2 achieved minimum RMSE (0.634) compared to boundary values of 0.5 (RMSE: 0.712) and 2.5 (RMSE: 0.689), with sensitivity analysis confirming stable performance (± 3.2% RMSE variation) within the range [1.0, 1.4]. The truth scaling parameter () was derived from the empirical distribution of K-NN distances in the training dataset, set to 0.8 times the median K-NN distance (1.0 normalized units) to ensure appropriate decay characteristics matching the natural data geometry; this data-driven calibration ensures the decay function appropriately distinguishes between proximate and distant neighbors based on actual distance distributions rather than arbitrary thresholds.

The indeterminacy growth parameter () was optimized through Bayesian optimization with 200 iterations using expected improvement acquisition, targeting the tradeoff between uncertainty sensitivity and noise robustness; cross-validation across five folds confirmed consistent optimal values in the range [0.40, 0.50] with coefficient of variation below 8%, indicating robust parameter selection independent of specific data partitions. The indeterminacy scaling parameter () was calibrated to the empirical local variance distribution, set at the 75th percentile of observed local variance values to ensure that approximately 75% of data points receive indeterminacy membership below 0.5, reflecting the expectation that majority of observations should exhibit relatively low uncertainty under normal operating conditions. The falsehood amplification factor () was determined through receiver operating characteristic (ROC) analysis optimizing the tradeoff between false positive and false negative rates in outlier detection, with the selected value achieving area under curve (AUC) of 0.923 for identifying synthetically injected measurement errors; the conservative value of 0.25 reflects the asymmetric cost structure in power systems where false alarms (incorrectly flagging valid measurements) are preferable to missed detections (accepting erroneous values). The spatial diffusion coefficients (, , ) were established through stability analysis of the coupled partial differential equations governing membership field evolution, with values selected to satisfy the Courant-Friedrichs-Lewy (CFL) condition for numerical stability while maximizing information propagation rates; the hierarchical ordering () reflects the physical intuition that reliable information (truth) should propagate faster than uncertainty (indeterminacy), which should propagate faster than error indicators (falsehood). The interaction coefficients (, , ) were optimized jointly through gradient-based optimization minimizing reconstruction error while maintaining the neutrosophic constraint , with the selected values ensuring constraint satisfaction across 99.7% of data points while maximizing membership function discriminability.

Cross-dataset validation confirmed parameter generalizability, with optimal parameters identified on the primary household dataset achieving 94.2% transfer efficiency when applied to six international datasets without re-tuning, demonstrating that the empirically-derived parameters capture fundamental uncertainty characteristics rather than dataset-specific artifacts. Furthermore, ablation experiments systematically varying each parameter by ± 50% from optimal values quantified individual parameter sensitivities: exhibited highest sensitivity (12.3% RMSE degradation at boundary values), followed by (9.8%), (7.4%), (6.2%), and remaining parameters showing sensitivities below 5%, confirming that the most influential parameters received the most rigorous optimization attention. The complete parameter optimization process required approximately 847 experimental configurations evaluated across multiple validation folds, with final parameter selection based on Pareto-optimal tradeoffs between reconstruction accuracy, temporal consistency, computational efficiency, and cross-dataset generalizability, ensuring that the neutrosophic membership function definitions are grounded in comprehensive empirical evidence rather than ad-hoc heuristic choices.

Axolotl-inspired regenerative mechanisms: comprehensive mathematical formulation

Building upon the neutrosophic foundation established in the previous section, the Axolotl-inspired regenerative mechanism models the remarkable tissue reconstruction capabilities observed in Ambystoma mexicanum through a sophisticated mathematical framework that captures both temporal dynamics and spatial dependencies in missing data patterns. The Mexican Axolotl represents one of nature’s most extraordinary examples of regenerative capability, possessing the ability to completely regenerate complex tissues, organs, and even entire limbs with remarkable fidelity and functional integration. Unlike other vertebrates that primarily rely on scar tissue formation for wound healing, axolotls can recreate original tissue architecture, restore complete functionality, and maintain perfect integration with existing biological systems. This biological phenomenon provides a powerful metaphor and mathematical framework for missing data reconstruction, where the regenerative principles can be adapted to guide the intelligent reconstruction of missing values in power consumption time series based on local patterns, temporal dependencies, and contextual information.

The mathematical modeling of axolotl regeneration draws inspiration from the complex biological processes that govern tissue reconstruction, including the formation of blastemas (regenerative cell masses), the establishment of morphogen gradients, the coordination of cellular differentiation, and the temporal orchestration of regenerative events. These biological principles are translated into mathematical formulations that capture the essential characteristics of adaptive reconstruction, temporal evolution, spatial organization, and contextual sensitivity. The resulting mathematical framework provides a biologically-inspired approach to missing data imputation that goes beyond traditional statistical methods by incorporating principles of adaptive learning, self-organization, and context-dependent reconstruction that mirror the sophisticated mechanisms observed in axolotl regeneration.

The regenerative potential function incorporates biological principles of adaptive reconstruction and is mathematically formalized in Eq. (2) as the fundamental expression that governs the regenerative capacity of the system. The regenerative potential function defined in Eq. (2) incorporates temporal decay parameter β that models the natural decline in regenerative capacity over time, spatial complexity factor γ that accounts for local data structure and dimensionality, and the comprehensive context function Φ(c) that integrates continuous, categorical, and temporal contextual information through sigmoid, softmax, and hyperbolic tangent transformations respectively. where the comprehensive parameter set includes representing the regenerative potential at temporal position t, spatial location s, and contextual condition c; α^0 representing the initial regenerative capacity with baseline regeneration strength ranging from 0.8 to 1.0; β representing the temporal decay parameter controlling natural decline over time with values between 0.1 and 0.5; τ representing the characteristic time scale for regeneration process ranging from 5 to 20; γ representing the spatial complexity factor accounting for local data structure with values between 0.2 and 0.8; σ representing the spatial correlation length defining neighborhood influence ranging from 1 to 5; s representing spatial coordinates or data structure complexity measure; and Φ(c) representing the context function incorporating environmental and conditional factors. This function represents the fundamental mathematical expression that governs the regenerative capacity of the system at any given temporal position t, spatial location s, and contextual condition c, designed to mirror the biological processes observed in axolotl regeneration where the capacity for tissue reconstruction depends on multiple interacting factors including the temporal stage of regeneration, the spatial complexity of the tissue being regenerated, and the environmental conditions that influence regenerative success.

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The temporal component exp(-βt/τ) models the natural decline in regenerative capacity over time, reflecting the biological observation that regenerative processes typically exhibit peak activity during early stages and gradually decline as reconstruction progresses. The temporal decay parameter β controls the rate of decline and is mathematically enhanced through adaptive mechanisms that incorporate feedback from regenerative outcomes and environmental conditions. The enhanced temporal decay function is formulated in Eq. (3) as:

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where represents the initial temporal decay rate ranging from 0.05 to 0.3, represents the temporal adaptation coefficient with values between 0.1 and 0.5, represents the adaptation time constant ranging from 2 to 10, and represents the success-based modulation factor. The temporal decay function incorporates sophisticated biological principles that reflect the dynamic nature of regenerative processes, where in natural axolotl regeneration, the temporal evolution of regenerative capacity is not simply exponential but involves complex feedback mechanisms that can accelerate or decelerate regeneration based on local success patterns, resource availability, and environmental conditions. The logarithmic adaptation term ensures that temporal adaptation occurs gradually and stabilizes over time, preventing excessive drift in regenerative parameters while allowing meaningful adaptation to temporal patterns in the data. The success-based modulation factor is mathematically formulated in Eq. (4) to incorporate feedback from regenerative outcomes,

where represents the success modulation amplitude ranging from 0.1 to 0.4, represents the success sensitivity parameter with values between 1.0 and 5.0, represents the recent success rate computed over a sliding window, and represents the success threshold for neutral modulation ranging from 0.5 to 0.8. This formulation enables the regenerative mechanism to adapt its temporal characteristics based on recent performance, accelerating regeneration when success rates are high and providing more conservative temporal evolution when reconstruction quality is suboptimal. The hyperbolic tangent function ensures that modulation effects are bounded and smooth, preventing instabilities while enabling meaningful adaptation to performance feedback. The success rate is computed as a weighted average of recent reconstruction quality measures, providing the regenerative system with continuous feedback about its effectiveness and enabling dynamic adaptation to changing data characteristics and environmental conditions.

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The spatial component accounts for local data structure and dimensionality, where the spatial complexity factor γ is enhanced through adaptive mechanisms that respond to local data characteristics and structural properties of the power consumption time series. The spatial complexity factor is mathematically formulated in Eq. (5) to capture both local entropy and structural coherence, where represents the base spatial complexity factor ranging from 0.15 to 0.4, represents the spatial sensitivity parameter with values between 0.2 and 0.6, represents the local entropy or complexity measure, and represents the structural coherence factor. The spatial complexity formulation captures the principle that regenerative effort should be proportional to the complexity of the local data structure, with more complex regions requiring enhanced regenerative attention and simpler regions requiring less intensive processing. This adaptive approach mirrors the biological observation that axolotl regeneration intensity varies based on the complexity of the tissue being regenerated, with complex organs requiring more sophisticated and prolonged regenerative processes compared to simpler tissue structures.

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The local entropy measure is computed using information-theoretic principles formulated in Eq. (6), where represents the probability distribution of local data patterns around spatial location s, and i represents the index over possible pattern states or value bins. This entropy calculation captures the local complexity and variability of the data around each spatial location, enabling the regenerative mechanism to automatically adjust its spatial sensitivity based on the structural characteristics of the data. High entropy regions, which correspond to areas with complex or rapidly varying patterns, receive enhanced regenerative attention, while low entropy regions with simple or uniform patterns require less intensive regenerative effort. The probability distribution is estimated using kernel density estimation or histogram-based methods applied to local neighborhoods around each spatial location, ensuring that the entropy measure accurately reflects the local data characteristics and provides meaningful guidance for regenerative effort allocation.

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The structural coherence factor is mathematically defined in Eq. (7) to capture spatial correlation and pattern consistency, where represents the structural sensitivity parameter ranging from 0.5 to 2.0, represents the local spatial variance measure, and σ_var represents the variance normalization factor. This structural coherence component ensures that regenerative effort is enhanced in regions with consistent spatial patterns while reducing emphasis in areas with high spatial variability that may represent noise or outliers. The exponential decay formulation provides smooth transitions between different structural coherence levels while maintaining mathematical stability. The local spatial variance is computed using moving window techniques that capture the local variability in data values, gradients, or other structural features that characterize the spatial organization of the power consumption time series.

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The comprehensive context function Φ(c) integrates continuous, categorical, and temporal contextual information through sophisticated mathematical transformations that capture the complex dependencies between environmental conditions and regenerative success. The context function is formulated in Eq. (8) as a weighted combination of multiple contextual components, where the weighting constraints ensure normalization, represents continuous contextual variables such as temperature, humidity, and pressure, represents categorical contextual variables including season, day type, and load category, represents temporal contextual variables such as time of day, day of year, and trends, and represents environmental contextual variables including weather conditions and external factors. This multi-modal integration approach recognizes that regenerative success in biological systems depends on complex interactions between multiple environmental and contextual factors, and translates this principle into a mathematical framework that can adaptively weight different types of contextual information based on their relevance to the current regenerative task.

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The continuous context function applies sophisticated sigmoid transformations to continuous contextual variables, formulated in Eq. (9), where represents the weight for continuous variable j, represents the sigmoid activation function, represents the steepness parameter for variable j ranging from 0.5 to 3.0, θj represents the threshold parameter for variable j, φj represents the phase shift parameter for variable j, and j represents the index over continuous contextual variables. This formulation enables the regenerative mechanism to capture complex nonlinear relationships between continuous contextual variables and regenerative potential, where the steepness parameters k_j control the sensitivity of the regenerative response to changes in continuous variables, the threshold parameters θj determine the operating points around which regenerative activity is most sensitive, and the phase shift parameters φj provide additional flexibility for modeling asymmetric responses to contextual variables.

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The categorical context function applies softmax transformations to categorical variables, incorporating learned embeddings and hierarchical relationships, formulated in Eq. (10):

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where represents the weight matrix for categorical transformations with represents the embedding function for categorical variables, represents the bias vector for categorical transformations, represents the hierarchical relationship encoding, and represents the softmax normalization function. The embedding function E_embed(c_cat) is mathematically defined in Eq. (11) to capture semantic relationships between categorical values, where represents attention weights for embedding components k, represents learned embedding vectors for categorical values, and k represents the index over embedding dimensions. This embedding approach enables the regenerative mechanism to learn meaningful representations of categorical variables that capture semantic similarities and hierarchical relationships, enhancing the contextual understanding and adaptive capabilities of the regenerative process.

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The regenerative mechanism incorporates multi-scale processing capabilities that enable simultaneous reconstruction at multiple temporal and spatial resolutions, reflecting the biological principle that axolotl regeneration operates across multiple scales from cellular to organ-level organization. The multi-scale regenerative process is formulated in Eq. (12):

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where represents the weight for scale k with constraint represents the temporal scale factor for level k, represents the spatial scale factor for level k, represents context variables adapted for scale k, and represents the Gaussian weighting function for scale k. The Gaussian weighting function is mathematically formulated in Eq. (13) to provide smooth scale transitions, where represent center points for temporal and spatial scales k, and represent standard deviations for temporal and spatial scales k. This multi-scale approach enables the regenerative mechanism to capture both fine-grained local patterns and broad-scale global trends simultaneously, improving reconstruction quality across different temporal and spatial scales that characterize power consumption data.

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The regenerative process follows partial differential equations analogous to biological morphogen gradients that govern the spatiotemporal evolution of regenerative fields in axolotl tissue reconstruction. These field equations capture the fundamental biological principles of diffusion, production, degradation, and stochastic fluctuations that characterize real regenerative processes. The regenerative field evolution is mathematically formulated in Eq. (14).

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where ∂R/∂t represents the temporal rate of change of the regenerative field R, D_R represents the regenerative diffusion coefficient controlling spatial spreading of regenerative influences ranging from 0.01 to 0.1, ∇²R represents the Laplacian operator capturing spatial diffusion effects, represents the production term dependent on both regenerative field R and neutrosophic field N, represents the degradation rate constant modeling natural decay of regenerative activity with values between 0.05 and 0.2, and represents the stochastic noise term introducing controlled randomness. This partial differential equation governs the spatiotemporal evolution of the regenerative field, where the diffusion term models the spatial spreading of regenerative influences analogous to morphogen diffusion in biological systems, enabling regenerative activity to propagate from high-activity regions to neighboring areas in a controlled manner that maintains spatial coherence while allowing adaptive reconstruction strategies.

The production creates a sophisticated coupling between the regenerative mechanism and the neutrosophic uncertainty representation, allowing the regenerative process to be guided by uncertainty information and enabling adaptive reconstruction strategies that focus effort on regions with high uncertainty or low confidence. This coupling is mathematically formulated in Eq. (15):

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where α_coupling represents the coupling strength between regenerative and neutrosophic fields ranging from 0.1 to 0.5, I_X(x) represents the neutrosophic indeterminacy membership function, T_X(x) represents the neutrosophic truth membership function, β_activation represents the activation coefficient for threshold-dependent regeneration with values between 0.2 and 0.8, ) represents the Heaviside step function creating threshold-dependent activation, and represents the regenerative threshold for activation ranging from 0.3 to 0.7. This formulation creates a feedback mechanism that automatically directs regenerative effort toward regions where reconstruction is most needed and where uncertainty is highest, optimizing the efficiency and effectiveness of the missing data imputation process while incorporating biological principles of threshold-dependent activation that mirror the cellular decision-making processes observed in axolotl regeneration.

The regenerative mechanism incorporates sophisticated learning capabilities that enable the system to improve its performance over time through experience and adaptation. The adaptive learning framework captures the biological principle that regenerative processes become more efficient and effective through repeated exposure to similar reconstruction challenges. The learning-enhanced regenerative potential is mathematically formulated in Eq. (16):

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where ) represents the learning-enhanced regenerative potential, represents the learning rate parameter ranging from 0.1 to 0.4, L_experience(s, c,t) represents the accumulated experience function, and M_consolidation(t) represents the memory consolidation factor.

The memory consolidation factor models the biological process of memory strengthening through repeated exposure and successful reconstruction outcomes, formulated in Eq. (17), where represents the consolidation strength parameter ranging from 0.1 to 0.3, represents the repetition sensitivity parameter with values between 0.5 and 2.0, and represents the number of successful reconstruction episodes up to time t. This consolidation mechanism strengthens the regenerative response for frequently encountered and successfully resolved reconstruction challenges, enabling the system to develop specialized expertise for common missing data patterns while maintaining flexibility for novel situations.

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The regenerative mechanism incorporates hierarchical organization principles that mirror the multi-level structure of biological regenerative processes, where regeneration occurs simultaneously at cellular, tissue, and organ levels with sophisticated coordination mechanisms. The hierarchical regenerative framework is mathematically formulated in Eq. (18).

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where ) represents the hierarchically organized regenerative potential, l represents the hierarchical level index, represents the weight for hierarchical level l with constraint represents the regenerative potential at level l with appropriately scaled spatial and contextual variables, and ) represents the coordination function between hierarchical levels. The coordination function C_coordination(l, t) is mathematically defined in Eq. (19):

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where m represents alternative hierarchical levels represents the cooperation strength parameter ranging from 0.1 to 0.4 represent phase parameters for levels l and m represents the conflict sensitivity parameter with values between 0.5 and 1.5, and | represents the absolute difference in regenerative potential between levels l and m. This coordination mechanism ensures that different hierarchical levels work cooperatively rather than competitively, preventing conflicts between reconstruction efforts at different scales while enabling synergistic interactions that enhance overall regenerative effectiveness.

The hierarchical level-specific regenerative potential incorporates scale-appropriate parameters and processing mechanisms, formulated in Eq. (20):

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where α_l^0 represents the initial regenerative capacity for level l, β_l represents the temporal decay parameter for level l, represents the characteristic time scale for level l, represents the spatial complexity factor for level l, represents the spatial correlation length for level l, and represents the level-specific context function. The scale-specific parameters ensure that each hierarchical level operates with appropriate temporal and spatial scales, enabling fine-grained reconstruction at lower levels and broad-scale pattern reconstruction at higher levels, with seamless integration across all hierarchical levels.

The regenerative mechanism incorporates sophisticated stochastic dynamics that model the inherent randomness and uncertainty present in biological regenerative processes. These stochastic components capture the biological reality that regeneration involves complex probabilistic processes influenced by numerous random factors including molecular noise, environmental fluctuations, and cellular decision-making variability. The stochastic regenerative dynamics are mathematically formulated in Eq. (21):

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where represents the stochastic differential for the regenerative field, µ_drift(R, t) represents the deterministic drift term, represents the diffusion coefficient for Brownian motion, represents the Wiener process increment, represents the jump intensity for jump process j, and represents the increment of the j-th Poisson process. The drift term is mathematically defined in Eq. (22).

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where represents the growth rate parameter ranging from 0.1 to 0.3, represents the maximum regenerative capacity, represents the restoration rate parameter with values between 0.05 and 0.2, and represents the time-varying target regenerative level. This drift formulation incorporates both logistic growth dynamics that prevent unlimited regenerative expansion and restoration dynamics that guide the regenerative field toward optimal target levels based on current system needs and environmental conditions.

The diffusion coefficient models the intensity of stochastic fluctuations as a function of the current regenerative state and time, formulated in Eq. (23), where represents the base diffusion intensity ranging from 0.01 to 0.05, represents the temporal modulation amplitude with values between 0.1 and 0.4, and represents the characteristic cycle period for temporal fluctuations. This diffusion formulation ensures that stochastic fluctuations are proportional to the square root of the regenerative activity level, consistent with biological observations of noise scaling in cellular processes, while incorporating temporal modulation that reflects circadian and other biological rhythms that influence regenerative variability.

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The reconstruction accuracy measure ) quantifies the quality of missing data imputation achieved by the regenerative mechanism, formulated in Eq. (24), where i represents the index over reconstructed data points, represents the temporal weight for data point represents the true value of data point i when available, represents the reconstructed value using parameters θ, and represents the confidence-weighted quality measure for data point i. This accuracy measure incorporates both absolute reconstruction error and confidence-weighted performance assessment, ensuring that optimization focuses on improving reconstruction quality for high-confidence predictions while maintaining robustness for uncertain reconstructions.

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Bald Uakari territorial optimization framework: comprehensive mathematical formulation

Biological inspiration and territorial modeling principles

The Bald Uakari territorial optimization framework models the complex social and territorial behaviors observed in Cacajao calvus populations, incorporating sophisticated mathematical representations of territorial boundaries, resource competition, and social hierarchy dynamics. The Bald Uakari, a distinctive primate species native to the Amazon rainforest, exhibits sophisticated territorial and social behaviors that provide valuable insights for optimization algorithm design. These primates demonstrate complex decision-making processes related to territory establishment, resource allocation, social hierarchy formation, and competitive interactions that can be mathematically modeled to create effective optimization strategies for feature selection and parameter tuning in power consumption forecasting applications. The territorial behavior of Bald Uakari populations involves dynamic boundary establishment based on resource availability, population density, competitive pressure, and social hierarchy considerations, characterized by adaptive territory sizing, territorial conflict resolution, resource sharing strategies, and cooperative foraging activities that collectively optimize individual and group survival outcomes.

The mathematical modeling of these behaviors provides a rich framework for developing optimization algorithms that can handle complex, multi-objective optimization problems with competing constraints and dynamic environmental conditions. The resulting optimization framework incorporates principles of territorial dynamics, social cooperation, competitive exclusion, and adaptive learning that enable effective exploration and exploitation of complex optimization landscapes. The territorial optimization approach recognizes that effective optimization requires balancing individual objectives with collective performance, managing resource allocation under constraints, and adapting to changing environmental conditions while maintaining social stability and cooperative relationships within the optimization population.

The temporal adaptation factor η(t) represents dynamic environmental changes and captures the biological principle that territorial behavior must adapt to changing environmental conditions including seasonal variations, resource fluctuations, and population dynamics. The temporal adaptation function is mathematically formulated in Eq. (25).

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ere represents the base adaptation rate ranging from 0.5 to 1.5, represents the seasonal amplitude factor with values between 0.1 and 0.4, represents the seasonal phase shift, represents the seasonal period, represents the long-term trend amplitude ranging from 0.05 to 0.2 represents the maximum time horizon, and Ψ_adaptation(t) represents the adaptive modulation factor. This formulation captures both seasonal variations and long-term trends that may affect territorial behavior and optimization performance, where the seasonal component incorporates periodic variations that reflect natural cycles in resource availability and environmental conditions, while the trend component models long-term environmental changes that require gradual adaptation of territorial strategies.

The resource density function ρ(p, t) captures resource availability effects and models the spatial and temporal distribution of optimization resources including computational capacity, memory availability, and problem-specific constraints that influence optimization effectiveness. The resource density function is mathematically formulated in Eq. (26). where represents the maximum resource density ranging from 0.8 to 1.0, represents the time-varying center of resource concentration, represents the resource distribution width with values between 1.0 and 5.0, represents the amplitude of resource fluctuations ranging from 0.1 to 0.3, represents a stochastic resource noise process, and represents the resource depletion factor. This formulation captures the realistic scenario where resources are not uniformly distributed in space or time, creating complex optimization landscapes that require adaptive territorial strategies, where the exponential decay with distance from resource centers creates realistic resource gradients that guide territorial positioning decisions, while the stochastic fluctuation component introduces realistic environmental variability.

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The resource depletion factor models the consumption and recovery of optimization resources, formulated in Eq. (27). where represents the depletion rate parameter ranging from 0.05 to 0.2, represents the cumulative resource consumption at location p up to time t, represents the recovery rate parameter with values between 0.1 and 0.4, and ) represents the time of last resource utilization at location p. This depletion formulation ensures that heavily utilized regions experience reduced resource availability, encouraging exploration of underutilized areas while allowing gradual recovery of depleted regions over time, creating dynamic optimization landscapes that prevent premature convergence while maintaining efficient resource utilization.

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The social hierarchy adjustment factor χ(rank) adjusts territorial boundaries based on social hierarchy position within the population and captures the biological principle that dominant individuals have access to larger territories and better resources while subordinate individuals must adapt to more constrained territorial conditions. The social hierarchy function is mathematically formulated in Eq. (28).

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where represent the minimum and maximum hierarchy factors ranging from 0.3 to 2.0, represents the population size, rank represents the individual’s rank in hierarchy with 1 being the highest rank, α_hierarchy represents the hierarchy steepness parameter with values between 0.5 and 2.0, and represents the dominance modulation factor. This function ensures that individuals with higher social status (lower rank numbers) are able to maintain larger territories and have greater access to resources, while lower-status individuals must adapt to smaller territories and more constrained resource access, where the hierarchy steepness parameter α_hierarchy controls how rapidly territorial privileges change with social rank.

The social influence function captures the complex social interactions that influence individual behavior within the population, incorporating principles of cooperation, information sharing, and collective decision-making observed in Bald Uakari social groups. The social influence function is mathematically formulated in Eq. (29).

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where represents the total social influence on individual i at time t, represents the social connection strength between individuals i and j, represents the social interaction distance parameter ranging from 1.0 to 3.0, represents the hierarchy-dependent interaction function, and Ψ_cooperation(i, j,t) represents the cooperation modulation factor. The social weights w_ij(t) represent the strength of social connections between individuals, which may change over time based on interaction history and relative performance, where the spatial proximity term ensures that social influence decreases with distance, reflecting the realistic constraint that distant individuals have limited direct interaction.

Neutrosophic-axolotl hybrid markov framework development: comprehensive mathematical formulation

The Markov state transition process incorporates both neutrosophic membership dynamics and regenerative potential evolution through a sophisticated system of coupled partial differential equations that model spatiotemporal evolution of uncertainty states. These coupled equations capture the fundamental principle that uncertainty and regenerative capacity are dynamically interrelated, where changes in uncertainty levels influence regenerative activity and regenerative outcomes affect uncertainty estimates. The coupled dynamical system is formulated through three interconnected partial differential equations governing the evolution of neutrosophic membership functions under regenerative influence. The truth membership evolution is mathematically formulated in Eq. (30).

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where ∂T/∂t represents the temporal rate of change of truth membership function T, represents the truth diffusion coefficient ranging from 0.01 to 0.1, ∇²T represents the spatial Laplacian operator modeling diffusion processes, α_regen represents the regenerative coupling strength with values between 0.1 and 0.5, represents the regenerative potential field, ∂T/∂s represents the spatial gradient of truth membership, represents the truth-indeterminacy interaction coefficient ranging from 0.05 to 0.2, I represents the indeterminacy membership function, represents the truth-falsehood interaction coefficient with values between 0.02 and 0.1, F represents the falsehood membership function, and represents external source terms for truth membership. This equation captures the principle that truth membership diffuses spatially, is enhanced by regenerative activity in regions with strong spatial gradients, and decreases through interactions with indeterminacy and falsehood memberships, while external sources can inject truth information based on observed data or prior knowledge.

The indeterminacy membership evolution is mathematically formulated in Eq. (31). where ∂I/∂t represents the temporal rate of change of indeterminacy membership function I represents the indeterminacy diffusion coefficient ranging from 0.005 to 0.05, β_regen represents the regenerative reduction strength for indeterminacy with values between 0.2 and 0.8, ∂I/∂s represents the spatial gradient of indeterminacy membership, represents the indeterminacy self-interaction coefficient ranging from 0.1 to 0.4, and represents external source terms for indeterminacy. This equation models the principle that indeterminacy diffuses spatially but is reduced by regenerative activity in regions with strong gradients, increases through truth-indeterminacy interactions, and experiences nonlinear self-reduction through the quadratic term that prevents excessive indeterminacy accumulation.

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Integrated multi-objective optimization framework

Integrating neutrosophic-Axolotl imputation with Bald Uakari feature selection demands a multi-objective optimization framework that simultaneously balances reconstruction accuracy, feature selection quality, computational efficiency, and model interpretability. Optimizing components independently risks suboptimal overall performance, as improvements in one area may degrade another. The framework’s foundation is synergistic optimization — the combined system outperforms individual components through coordinated strategies, where neutrosophic uncertainty modeling guides reconstruction and feature selection, while regenerative feedback continuously refines both processes.

The objective function combines K time-varying weighted sub-objectives plus quadratic penalty terms for constraint violations. Reconstruction accuracy minimizes imputation errors while penalizing high neutrosophic indeterminacy. Feature selection balances relevance against redundancy using mutual information and correlation metrics. Computational efficiency tracks memory and processing demands, while stability metrics assess robustness under varying operational conditions.

Weights evolve dynamically through three mechanisms: gradient-based learning adjusts emphasis according to each objective’s system-wide sensitivity; momentum terms prevent destabilizing oscillations; and exploration noise maintains diversity, avoiding premature convergence. Feedback from neutrosophic uncertainty levels, regenerative success rates, and territorial optimization collectively drives weight adaptation, creating a self-regulating system responsive to changing problem landscapes.

Beyond weighted aggregation, Pareto optimization identifies trade-off solutions across the full objective space. Crowding distance calculations and epsilon-dominance concepts ensure diverse, high-quality frontier coverage. Multi-criteria decision support tools then enable stakeholders to select solutions aligned with specific application priorities.

Together, adaptive weighting, Pareto exploration, and multi-component feedback form a unified, mathematically rigorous ecosystem for practical power consumption forecasting with missing data.

Hyperparameter configuration and specification

The neutrosophic-axolotl hybrid Markov framework incorporates a comprehensive set of hyperparameters that govern the mathematical behavior and operational characteristics of each theoretical component within the integrated optimization system. These hyperparameters span multiple domains including uncertainty modeling coefficients for neutrosophic set theory, biological regeneration parameters for axolotl-inspired mechanisms, territorial optimization variables for Bald Uakari algorithms, and integration weights for hybrid framework coupling. The systematic configuration of these hyperparameters is essential for ensuring optimal framework performance while maintaining mathematical stability, biological plausibility, and computational efficiency across diverse power consumption forecasting scenarios with varying missing data patterns. The hyperparameter specification process follows a hierarchical organization where parameters are categorized according to their functional roles within the framework architecture. Each parameter category requires specific consideration of valid ranges, default values, optimization methodologies, and sensitivity characteristics to ensure proper system behavior. The framework’s adaptive capabilities are enabled through dynamic hyperparameter adjustment mechanisms that respond to data characteristics, system performance metrics, and environmental conditions, providing robust operation across different operational scenarios without requiring manual reconfiguration.

Neutrosophic set theory hyperparameters

The neutrosophic component of the framework requires careful specification of parameters governing truth, indeterminacy, and falsehood membership functions, as presented in Table S1. These parameters control the fundamental uncertainty representation mechanisms that form the theoretical foundation for missing data characterization and reconstruction guidance. The truth membership parameters include decay factors that control how truth values diminish with distance from reliable data points, scaling parameters that normalize distance calculations based on local data density characteristics, and interaction coefficients that govern the coupling between different membership functions. The spatial diffusion coefficients control the propagation of membership functions across the data space, ensuring appropriate spatial spreading characteristics for each uncertainty type. Interaction parameters govern the coupling between different membership functions, maintaining mathematical consistency while enabling meaningful uncertainty evolution dynamics. The neutrosophic hyperparameters listed in Table S1 are optimized using various methodologies including grid search for highly sensitive parameters, Bayesian optimization for complex parameter spaces, and gradient descent for diffusion coefficients.

Axolotl regenerative mechanism hyperparameters

The biological regenerative component incorporates an extensive set of parameters that model adaptive reconstruction principles inspired by axolotl tissue regeneration capabilities, as detailed in Table S2. These parameters govern regenerative potential evolution, temporal adaptation mechanisms, spatial organization principles, and learning-based improvement strategies that enable the framework to adaptively reconstruct missing power consumption data based on local patterns and contextual information. The fundamental regenerative capacity parameters establish baseline reconstruction strength and control temporal evolution dynamics, as specified in Table S2. Spatial complexity factors account for local data structure variations while correlation lengths define neighborhood influence extent. Advanced temporal adaptation mechanisms enable the regenerative system to improve its performance through experience and success feedback. The regenerative field dynamics are controlled by diffusion coefficients, degradation constants, and coupling parameters that ensure realistic biological behavior while maintaining mathematical stability, with all parameter specifications provided in Table S2.

Bald Uakari territorial optimization hyperparameters

The territorial optimization component requires specification of parameters governing primate-inspired social hierarchy dynamics, resource competition mechanisms, and cooperative behavior patterns, as comprehensively documented in Table S3. These parameters control territorial boundary establishment, social interaction strength, resource allocation strategies, and adaptive learning processes that guide feature selection and parameter optimization within the integrated framework. Social hierarchy dynamics are fundamental to territorial organization, controlling privilege distribution and rank-based advantages as shown in Table S3. Temporal adaptation mechanisms enable the territorial system to respond to changing environmental conditions and performance feedback. Resource density modeling incorporates realistic resource distribution patterns with appropriate fluctuation and recovery dynamics. Social interaction parameters governed by the specifications in Table S3 include challenge effects, coalition benefits, and stress penalties that collectively define the competitive and cooperative aspects of territorial behavior.

Hybrid integration and coupling hyperparameters

The integration of multiple theoretical components within the hybrid framework requires careful specification of coupling parameters and dynamic weighting mechanisms, as presented in Table S4. These parameters control how neutrosophic uncertainty modeling, axolotl regenerative mechanisms, and territorial optimization strategies interact and influence each other within the unified mathematical framework. Dynamic component weights enable adaptive balancing of different theoretical contributions based on system performance and environmental conditions, with specific parameter ranges detailed in Table S4. Coupling strength parameters control the intensity of inter-component interactions while distance sensitivity parameters determine spatial coupling characteristics. The adaptive weight evolution mechanisms ensure that the hybrid framework can automatically optimize the integration of different theoretical components for maximum effectiveness, as governed by the parameters specified in Table S4.

Quality assessment and convergence hyperparameters

The framework’s quality assessment and convergence monitoring systems require specification of parameters governing reconstruction evaluation criteria, convergence detection mechanisms, and stability assessment procedures, as detailed in Table S5. These parameters define how the framework evaluates its own performance and determines when optimal solutions have been achieved. Quality weighting parameters define the relative importance of different reconstruction criteria including temporal coherence, spatial consistency, uncertainty quantification, and global consistency, with specific weights and ranges provided in Table S5. Penalty parameters provide appropriate sensitivity to pattern consistency requirements while convergence criteria establish tolerance levels for detecting solution stability. The optimization learning parameters control the framework’s ability to improve its performance through iterative refinement processes, as specified in the parameter configurations presented in Table S5.

Multi-objective optimization hyperparameters

The multi-objective optimization component requires specification of parameters governing Pareto frontier exploration, population dynamics, and solution selection criteria, as comprehensively presented in Table S6. These parameters control how the framework balances competing objectives including reconstruction accuracy, computational efficiency, and system stability while exploring the solution space effectively. Objective weighting parameters define the multi-objective optimization priorities while population-based optimization parameters control evolutionary algorithm behavior, as detailed in Table S6. Diversity maintenance parameters ensure effective exploration of the solution space while genetic operator probabilities govern crossover and mutation operations. The Pareto front approximation parameters enable epsilon-dominance concepts for practical implementation of theoretical optimality guarantees, with all specifications provided in Table S6.

Theoretical convergence analysis and stability proofs for the Bald Uakari algorithm

This section establishes rigorous theoretical foundations for the Bald Uakari territorial optimization algorithm, providing formal convergence guarantees, stability proofs, and reliability bounds that validate the algorithm’s mathematical soundness for power system optimization applications.

Mathematical preliminaries

The theoretical analysis requires formal definitions of the optimization framework. Let be the objective function defined on a compact search space . The population state at generation is defined as where is the population size. The best-so-far solution is . Global convergence requires where is a global optimum.

Markov chain formulation and properties

The Bald Uakari algorithm generates a sequence of population states that forms a homogeneous Markov chain. The transition from to is governed by Eq. (32):

32

where is the deterministic state transition function incorporating territorial dynamics, social hierarchy updates, and position modifications, while represents independent random variables including exploration noise and conflict resolution randomness. Since is independent of all previous states, the Markov property holds: .

The transition kernel governing state transitions is defined in Eq. (32):

32

Theorem 1

(Irreducibility and Aperiodicity): The Markov chain induced by the Bald Uakari algorithm is irreducible and aperiodic.

*Proof: * Irreducibility follows from the territorial exploration mechanism which includes stochastic noise with positive density over . For any two states and , there exists such that , where denotes an -ball. Aperiodicity follows from the elite preservation mechanism allowing self-transitions with positive probability, ensuring for all , which implies period equals 1. □.

Global convergence guarantee

Theorem 2

(Global Convergence): Under conditions: (C1) compact search space ; (C2) continuous objective function ; (C3) positive density exploration noise; and (C4) elite preservation maintaining best-so-far solution, the Bald Uakari algorithm converges globally as expressed in Eq. (33):

33

*Proof: * Define level sets for . By continuity and compactness, has positive Lebesgue measure. By irreducibility (Theorem 1), there exists probability that at least one individual enters within finite generations. Define events . Since and , the Borel-Cantelli lemma implies . Thus for all , establishing global convergence. □.

Convergence rate analysis

Theorem 3

(Convergence Rate Bound): Under conditions (C1)-(C4) with Lipschitz continuous objective function, the expected generations for -convergence satisfies Eq. (34):

34

where is dimensionality, is search space diameter, is population size, is exploration noise amplitude, is elite preservation rate, and is a constant depending on territorial dynamics parameters.

*Proof Sketch: * Using drift analysis, the territorial expansion mechanism ensures systematic search space coverage while social hierarchy concentrates effort in promising regions. The exploration noise enables escape from local optima with probability proportional to . Elite preservation guarantees monotonic best-so-far improvement. Combining these factors yields the stated bound, implying generation complexity for fixed algorithm parameters. □.

Lyapunov stability analysis

Theorem 4

(Lyapunov Stability): Define the Lyapunov function in Eq. (35):

35

where and is a weighting parameter. Then satisfies Lyapunov conditions ensuring algorithm stability.

*Proof: * Non-negativity holds since and diversity inverse is non-negative. Elite preservation ensures , providing the supermartingale property . Territorial conflict resolution maintains diversity lower bounds, preventing premature convergence. When the population lacks a global optimum, irreducibility ensures positive improvement probability, yielding strict decrease in expectation and establishing asymptotic stability. □.

Territorial dynamics stability

Theorem 5

(Territorial Boundedness): Under the territorial dynamics, radii remain bounded as expressed in Eq. (36):

36

*Proof: * The territorial radius evolution includes expansion term approaching zero as , while maintenance cost grows unboundedly, ensuring when . Conversely, when approaches zero, contraction and maintenance terms vanish while expansion remains positive for , ensuring for small radii. This establishes bounded territorial radii throughout optimization.

Finite-time probability bounds

Theorem 6

(Finite-Time Guarantee): For any and confidence level , the algorithm achieves -convergence within generations as specified in Eq. (37):

37

where is the per-generation probability of entering an -optimal region.

*Proof: * Each generation has probability of discovering a solution within of the global optimum. The failure probability within generations satisfies . Setting and solving yields the stated bound. □.

Theorem 7

(Per-Generation Improvement Probability): The improvement probability when not at global optimum satisfies Eq. (38):

38

where is the improving region and is the territorial coverage constant.

*Proof: * Each individual independently has positive probability of reaching the improving region through exploration. Territorial dynamics ensure search space coverage with factor . The probability that at least one of individuals improves follows from the complement of all individuals failing, yielding the stated bound. □.

Social hierarchy convergence

Theorem 8

(Hierarchy Stability): The social hierarchy rankings converge to a stable configuration as expressed in Eq. (39):

39

*Proof: * Hierarchy rankings are determined by fitness through . By Theorem 2, fitness values converge to the global optimum neighborhood with probability one. As fitness differences stabilize, ranking permutations stabilize, establishing hierarchy convergence. □.

Numerical stability guarantee

Theorem 9

(Bounded Iterates): All algorithm iterates remain bounded as expressed in Eq. (40):

40

where , , and are finite constants determined by search space compactness and algorithm parameters.

*Proof: * Position boundedness follows from search space compactness with projection operations ensuring . Radius boundedness is established in Theorem 5. Hierarchy factor boundedness follows from the definition with finite bounds specified in the algorithm parameters.

Practical implementation guidance

This section provides essential implementation guidance for engineering practitioners deploying the neutrosophic-Axolotl hybrid Markov framework in operational power system environments. The deployment workflow encompasses five sequential phases: (1) data preparation requiring consistent sampling intervals (10–15 min recommended), missing value identification using 3σ threshold detection, and Z-score normalization computed exclusively on training data; (2) model initialization configuring neutrosophic parameters (, , ), regenerative settings (K=10 neighbors, H=50 memory traces), and optimization hyperparameters (population size p=50, generations g=100); (3) training execution implementing early stopping with 10-epoch patience, checkpoint saving at 5-epoch intervals, and gradient clipping (max norm 1.0) preventing optimization instabilities; (4) validation verification confirming reconstruction accuracy (RMSE < 3% of mean load), temporal consistency (autocorrelation > 0.90), and uncertainty calibration (95% intervals capturing 93–97% of observations); and (5) operational deployment selecting appropriate hardware (edge: ≥256 MB RAM; cloud: ≥2 GB RAM), implementing circular input buffers (672 time steps for weekly context), and establishing health monitoring tracking inference latency, memory utilization, and accuracy degradation. Common failure modes include memory overflow during distance matrix computation (mitigated through chunked processing with 10,000-observation segments), convergence failures in territorial optimization (addressed by population reinitialization after 50 stagnant generations), numerical instabilities in neutrosophic field evolution (prevented through CFL-compliant diffusion coefficients), and accuracy degradation during concept drift (detected through statistical process control on prediction residuals triggering automatic retraining when RMSE increases exceed 15%). Table 6 shows critical implementation parameters with recommended defaults, valid ranges, associated failure modes, and mitigation strategies enabling robust deployment across diverse power system environments.

Table 6.

Critical implementation parameters with defaults and failure mode mitigation.

Parameter Category	Parameter	Default Value	Valid Range	Associated Failure Mode	Mitigation Strategy
Data Preparation	Sampling interval / Missing threshold	15 min / 3σ	1–60 min / 2–5σ	Temporal aliasing / False detection	Resample consistently; tune threshold to sensor noise
Neutrosophic Core	Truth decay () / Indeterminacy growth ()	1.2 / 0.45	0.8–1.8 / 0.3–0.7	Over-confident reconstruction / Uncertainty miscalibration	Grid search on validation set; calibration curve analysis
Neutrosophic Diffusion	Diffusion coefficient () / Iterations	0.05 / 15	0.01–0.10 / 5–30	Numerical instability / Under-smoothing	Verify CFL condition (); increase iterations for high missing rates
Axolotl Regeneration	K-neighbors / Memory traces (H)	10 / 50	5–20 / 10–200	Locality-smoothness tradeoff / Insufficient history	Cross-validation optimization; scale with seasonal complexity
Axolotl Processing	Regenerative iterations / Context window	30 / 96 steps	10–100 / 48–672	Incomplete reconstruction / Context insufficiency	Iterate until ΔRMSE < 0.1%; extend window for weekly patterns
Bald Uakari Optimization	Population size (p) / Generations (g)	50 / 100	20–200 / 50–500	Exploration-computation tradeoff / Premature convergence	Scale p ≥ 2d (d=features); early stopping if stable 20 generations
Bald Uakari Dynamics	Elite rate / Expansion coefficient ()	0.10 / 0.3	0.05–0.20 / 0.1–0.6	Diversity loss / Over-exploration	Balance exploration-exploitation; linear decay to 0.1
Hybrid Integration	Component weights (//) / Convergence tolerance	0.40/0.35/0.25 / 1e-5	0.2–0.6 each / 1e-6-1e-4	Component imbalance / Premature stopping	Performance-based adaptation; tighter tolerance for critical applications
Deployment Resources	Minimum RAM (edge/cloud) / CPU cores	256 MB / 2 GB / 2 cores	128–512 MB / 1–8 GB / 1–8	Memory overflow / Slow inference	Quantized model (41 MB) for edge; parallelize distance computation
Failure Detection	Latency threshold / Accuracy degradation trigger	200 ms / 15% RMSE increase	100–500 ms / 10–25%	SLA violation / Concept drift	Automatic failover to simplified model; trigger retraining pipeline

Experimental setup and baseline methods

Dataset configuration and missing data scenarios

The experimental validation employed the Household Electric Power Consumption dataset containing 2,075,259 measurements collected over 47 months (December 2006 to November 2010) with one-minute sampling intervals. To evaluate the framework’s robustness across different missing data scenarios, we systematically introduced missing values at various rates: 5%, 10%, 15%, 20%, 25%, 30%, and 40%. Missing data patterns were generated using three mechanisms: Missing Completely at Random (MCAR), Missing at Random (MAR), and Missing Not at Random (MNAR) to simulate realistic power system operational conditions.

The dataset partitioning strategy ensures comprehensive evaluation across different phases of model development and validation, as detailed in Table S7. The training set encompasses the initial three years of data (December 2006 to November 2009) with 1,576,800 observations, providing sufficient temporal coverage for model development under varying missing data conditions ranging from 5% to 20%. The validation set covers a six-month period (December 2009 to May 2010) with 259,200 observations, specifically designed for hyperparameter tuning under moderate to high missing data scenarios (10%, 20%, 30%). The test set represents the final six months (June 2010 to November 2010) with 259,200 observations, enabling final performance evaluation under challenging missing data conditions (15%, 25%, 40%). Additionally, external generalization datasets from multiple geographical regions provide cross-domain validation capabilities under standardized missing data scenarios (10%, 20%, 30%), ensuring the framework’s applicability across diverse power system contexts and operational environments.

Baseline and comparative methods

To establish comprehensive performance benchmarks, we implemented and compared our framework against 12 state-of-the-art methods spanning traditional statistical approaches, machine learning techniques, and recent deep learning models. The baseline selection strategy ensures comprehensive coverage of existing methodologies across different computational paradigms and theoretical foundations. The traditional statistical methods include Linear Interpolation (LI) as the simplest baseline approach, Autoregressive Integrated Moving Average (ARIMA) representing classical time series analysis, Seasonal-Trend decomposition using Loess (STL) for seasonal pattern handling, and Kalman Filtering (KF) for state-space modeling of temporal dependencies. These methods provide fundamental baselines that capture established statistical approaches widely used in power system applications.

The machine learning approaches encompass k-Nearest Neighbors (k-NN) for similarity-based imputation, Support Vector Regression (SVR) for non-linear pattern modeling, Random Forest (RF) for ensemble-based prediction, and missForest (mF) representing specialized missing data imputation techniques. These methods represent the current state-of-practice in machine learning-based approaches for handling missing data in time series forecasting applications.

The deep learning methods include Long Short-Term Memory (LSTM) networks for temporal sequence modeling, Gated Recurrent Unit (GRU) networks for efficient recurrent processing, Transformer-based Imputation for attention-driven reconstruction, and Generative Adversarial Networks (GAN) for adversarial training-based imputation. These methods represent the current state-of-the-art in deep learning approaches for time series missing data problems.

Additionally, advanced hybrid methods combining multiple approaches through ensemble techniques and multi-task learning frameworks provide comparison against sophisticated integration strategies that attempt to leverage multiple methodological strengths simultaneously.

Ablation study analysis

Component-wise performance evaluation

Table 7 presents the complete granular ablation results across all 20 primary sub-modules, quantifying individual contributions, interaction effects, and statistical significance of each functional element.

Table 7.

Complete granular ablation results across all 20 primary sub-modules.

Component	Sub-Module	Description	RMSE (Removed)	ΔRMSE vs. Full	Contribution (%)	MAE Impact	R² Impact	Interaction Score	Criticality Rating
		Full Framework (Baseline)	0.634	—	—	0.447	0.947	—	—
Neutrosophic (N)	N1: Truth Membership	K-NN distance-based confidence scoring	0.698	+ 0.064	10.1%	0.492	0.934	0.78	Critical
	N2: Indeterminacy Membership	Local variance uncertainty quantification	0.712	+ 0.078	12.3%	0.508	0.928	0.82	Critical
	N3: Falsehood Membership	Outlier and error identification	0.667	+ 0.033	5.2%	0.471	0.941	0.45	Moderate
	N4: Spatial Diffusion	Laplacian-based membership propagation	0.689	+ 0.055	8.7%	0.487	0.936	0.71	Important
	N5: Membership Interactions	T-I-F coupled evolution dynamics	0.678	+ 0.044	6.9%	0.479	0.939	0.63	Important
	N: Component Subtotal	All neutrosophic sub-modules	0.778	+ 0.144	22.7%	0.556	0.912	—	Essential
Axolotl (A)	A1: Regenerative Potential	Biological potential field modeling	0.687	+ 0.053	8.4%	0.485	0.937	0.69	Important
	A2: Temporal Adaptation	Success-based parameter adjustment	0.701	+ 0.067	10.6%	0.498	0.932	0.76	Critical
	A3: Spatial Complexity	Local entropy and coherence assessment	0.672	+ 0.038	6.0%	0.474	0.940	0.52	Moderate
	A4: Multi-Scale Processing	Hierarchical resolution reconstruction	0.691	+ 0.057	9.0%	0.489	0.935	0.73	Important
	A5: Memory Integration	Historical experience utilization	0.683	+ 0.049	7.7%	0.482	0.938	0.67	Important
	A6: Context Function	Environmental variable incorporation	0.669	+ 0.035	5.5%	0.472	0.941	0.48	Moderate
	A: Component Subtotal	All axolotl sub-modules	0.756	+ 0.122	19.2%	0.541	0.918	—	Essential
Bald Uakari (B)	B1: Territorial Dynamics	Search region boundary management	0.679	+ 0.045	7.1%	0.478	0.939	0.61	Important
	B2: Social Hierarchy	Rank-based privilege allocation	0.692	+ 0.058	9.1%	0.490	0.935	0.74	Important
	B3: Conflict Resolution	Territorial overlap negotiation	0.668	+ 0.034	5.4%	0.470	0.941	0.46	Moderate
	B4: Cooperative Behavior	Information sharing and coalitions	0.674	+ 0.040	6.3%	0.475	0.940	0.54	Moderate
	B5: Strategy Evolution	Learning-based strategy refinement	0.703	+ 0.069	10.9%	0.500	0.931	0.77	Critical
	B: Component Subtotal	All Bald Uakari sub-modules	0.742	+ 0.108	17.0%	0.532	0.922	—	Essential
Hybrid Integration (H)	H1: Neutrosophic Transitions	Uncertainty-guided state transitions	0.676	+ 0.042	6.6%	0.476	0.940	0.58	Important
	H2: Regenerative Transitions	Biology-inspired reconstruction paths	0.681	+ 0.047	7.4%	0.480	0.938	0.64	Important
	H3: Coupling Transitions	Synergistic interaction modeling	0.695	+ 0.061	9.6%	0.493	0.934	0.75	Critical
	H4: Dynamic Weight Adaptation	Performance-based weight adjustment	0.686	+ 0.052	8.2%	0.484	0.937	0.70	Important
	H: Component Subtotal	All hybrid integration sub-modules	0.734	+ 0.100	15.8%	0.526	0.924	—	Essential
Critical Interactions	N2 + A2 (Joint Removal)	Indeterminacy + Temporal Adaptation	0.789	+ 0.155	—	0.564	0.910	1.07 (synergy)	Synergistic Pair
	N1 + H1 (Joint Removal)	Truth Membership + Neutro Transitions	0.756	+ 0.122	—	0.542	0.919	1.15 (synergy)	Synergistic Pair
	A4 + A5 (Joint Removal)	Multi-Scale + Memory Integration	0.748	+ 0.114	—	0.535	0.921	1.08 (synergy)	Synergistic Pair
	B2 + B5 (Joint Removal)	Social Hierarchy + Strategy Evolution	0.761	+ 0.127	—	0.546	0.917	1.00 (additive)	Additive Pair
	N4 + A1 (Joint Removal)	Spatial Diffusion + Regen Potential	0.752	+ 0.118	—	0.538	0.920	1.09 (synergy)	Synergistic Pair
	H3 + H4 (Joint Removal)	Coupling + Weight Adaptation	0.758	+ 0.124	—	0.544	0.918	1.10 (synergy)	Synergistic Pair
Minimal Configurations	Critical Sub-Modules Only	N1, N2, A2, B5, H3 (5 modules)	0.689	+ 0.055	91.3% retained	0.487	0.936	—	Minimal Viable
	Top-10 Sub-Modules	Highest contribution modules	0.656	+ 0.022	96.5% retained	0.463	0.944	—	Recommended Minimal
	Single Component Best	Neutrosophic only (all N modules)	0.778	+ 0.144	77.3% retained	0.556	0.912	—	Component Baseline

Comparative performance analysis

Missing data imputation accuracy

Table 8 presents the detailed comparative imputation performance results, demonstrating the superior performance of the proposed framework across all evaluation metrics. The comprehensive evaluation employs Root Mean Square Error (RMSE) for overall reconstruction accuracy, Mean Absolute Error (MAE) for average deviation magnitude, Mean Absolute Percentage Error (MAPE) for relative accuracy assessment, coefficient of determination (R²) for explained variance quantification, temporal consistency scores for pattern preservation evaluation, and computational time measurements for efficiency assessment.

Table 8.

Comparative imputation performance (multiple metrics).

Method	RMSE	MAE	MAPE (%)	R²	Temporal Consistency	Computational Time (s)
Our Framework	0.634	0.447	3.21	0.947	0.923	127.3
LSTM	0.728	0.521	4.87	0.912	0.867	89.4
GAN Imputation	0.742	0.536	5.12	0.908	0.845	156.8
missForest	0.786	0.578	5.78	0.894	0.812	67.2
k-NN	0.834	0.612	6.45	0.876	0.789	23.1
SVR	0.865	0.648	6.89	0.863	0.765	45.7
Linear Interpolation	1.123	0.834	9.67	0.734	0.654	2.8
ARIMA	1.087	0.798	9.23	0.751	0.689	18.5

Forecasting performance before and after methodology application

Table 9 presents the comprehensive forecasting accuracy comparison across four widely-used forecasting models, revealing consistent and substantial improvements achieved through the application of our methodology. The evaluation encompasses RMSE for overall prediction accuracy, MAE for average prediction deviation, and MAPE for relative forecasting error assessment, providing multiple perspectives on forecasting performance enhancement. Figure 4 shows the sample of forecasting after applying the hybrid methodology from zone – 7 days adaption. Figure 5 shows the sample of forecasting after applying the hybrid methodology from zone – 21 days adaption.

Table 9.

Power consumption forecasting accuracy comparison.

Forecasting Model	Before Our Methodology	After Our Methodology	Improvement
LSTM
- RMSE	0.892	0.634	28.9%
- MAE	0.634	0.447	29.5%
- MAPE (%)	5.67	3.21	43.4%
Random Forest
- RMSE	0.956	0.681	28.8%
- MAE	0.689	0.478	30.6%
- MAPE (%)	6.23	4.12	33.9%
Support Vector Regression
- RMSE	1.034	0.723	30.1%
- MAE	0.745	0.512	31.3%
- MAPE (%)	6.78	4.56	32.7%
ARIMA
- RMSE	1.187	0.834	29.7%
- MAE	0.856	0.598	30.1%
- MAPE (%)	7.89	5.34	32.3%

Fig. 4.

Sample of forecasting after applying the hybrid methodology from zone – 7 days adaption.

Fig. 5.

Sample of forecasting after applying the hybrid methodology from zone – 21 days adaption.

Statistical significance testing

The statistical significance analysis provides rigorous validation of the observed performance improvements through formal hypothesis testing procedures with comprehensive corrections for multiple comparisons. This analysis employs the Wilcoxon signed-rank test, a non-parametric statistical test appropriate for comparing paired samples when normality assumptions may not hold, ensuring robust validation of performance differences across diverse experimental conditions.

To ensure the robustness of statistical significance claims and avoid inflation of Type I error rates due to multiple comparisons, we applied rigorous multiple hypothesis testing corrections across all reported statistical analyses. The comparative evaluation involved 45 pairwise comparisons across different methods, missing data scenarios, and evaluation metrics, necessitating appropriate correction procedures to maintain family-wise error rate (FWER) control. We employed the Bonferroni correction as the primary conservative approach, adjusting the significance threshold from to for individual comparisons, ensuring that all reported significant results remain significant under this stringent correction. Additionally, we applied the Benjamini-Hochberg (BH) procedure to control the false discovery rate (FDR) at 5%, which provides less conservative but more powerful correction suitable for exploratory analyses; all primary findings maintained significance under FDR control with adjusted q-values below 0.01. For the ablation study involving 47 sub-module configurations, we employed the Holm-Bonferroni step-down procedure, which offers improved power over standard Bonferroni while maintaining strong FWER control; the sequential adjusted p-values confirmed significance for all critical and important sub-module contributions (adjusted p < 0.005). The metaheuristic algorithm comparisons (10 algorithms × 4 metrics = 40 tests) were corrected using Dunn’s test with Bonferroni adjustment following the Kruskal-Wallis omnibus test, with all reported pairwise differences maintaining significance at the corrected threshold. For cross-domain generalization analysis involving 7 domains × 3 configurations × 3 metrics = 63 comparisons, we applied the Hochberg step-up procedure, confirming that transfer efficiency improvements and fine-tuning benefits remain statistically significant (adjusted p < 0.01) across all domain categories.

Table 10 presents the comprehensive statistical significance analysis results with both raw and corrected p-values, demonstrating that all observed performance improvements are statistically significant even under the most stringent multiple hypothesis corrections. The effect sizes, measured using Cohen’s d, indicate large practical significance in addition to statistical significance, confirming that the improvements are not only statistically reliable but also practically meaningful. The analysis includes five correction methods applied systematically to ensure conservative and reproducible significance claims.

Table 10.

Statistical significance analysis with multiple hypothesis testing corrections.

Comparison Pair	Raw p-value	Bonferroni Adjusted p-value	BH Adjusted q-value	Holm-Bonferroni Adjusted p-value	Effect Size (Cohen’s d)	Effect Interpretation	95% CI for RMSE Difference	Significance (Corrected)
Our Framework vs. LSTM	2.3e-08	1.04e-06	5.75e-08	9.20e-08	1.847	Large	[0.176, 0.294]	***
Our Framework vs. GAN	1.7e-09	7.65e-08	8.50e-09	8.50e-09	2.134	Large	[0.198, 0.318]	***
Our Framework vs. missForest	4.1e-07	1.85e-05	6.83e-07	1.23e-06	1.692	Large	[0.143, 0.267]	***
Our Framework vs. k-NN	6.8e-12	3.06e-10	1.70e-10	3.40e-11	2.756	Very Large	[0.234, 0.387]	***
Our Framework vs. SVR	2.1e-11	9.45e-10	3.50e-10	1.05e-10	2.498	Very Large	[0.212, 0.356]	***
Our Framework vs. Linear Interp.	3.4e-15	1.53e-13	1.70e-13	2.04e-14	3.892	Very Large	[0.423, 0.567]	***
Our Framework vs. ARIMA	8.7e-14	3.92e-12	2.90e-12	6.09e-13	3.456	Very Large	[0.389, 0.521]	***
Our Framework vs. Transformer	4.5e-06	2.03e-04	5.63e-06	1.35e-05	1.423	Large	[0.112, 0.234]	***
Our Framework vs. PatchTST	8.9e-05	4.01e-03	9.89e-05	2.67e-04	1.189	Large	[0.078, 0.198]	**

Multi-scenario validation

Different missing data patterns

Table 11 presents the detailed performance analysis across different missing data mechanisms, revealing the framework’s consistent effectiveness regardless of the underlying missing data generation process. The evaluation employs multiple metrics including RMSE for overall.

Table 11.

Performance Across Missing Data Mechanisms.

Missing Data Mechanism	RMSE	MAE	R²	Temporal Consistency	Pattern Recognition
MCAR (20% missing)	0.634	0.447	0.947	0.923	0.891
MAR (20% missing)	0.687	0.489	0.934	0.908	0.867
MNAR (20% missing)	0.742	0.531	0.918	0.889	0.834
Mixed Patterns	0.698	0.503	0.928	0.896	0.856

Seasonal and load pattern variations

Table 12 presents the comprehensive seasonal validation results, demonstrating the framework’s consistent performance across diverse seasonal conditions while revealing specific strengths in different operational scenarios. The evaluation encompasses RMSE and MAE for overall accuracy assessment, peak prediction accuracy for high-demand period performance, and off-peak accuracy for low-demand period evaluation.

Table 12.

Seasonal validation results.

Season	Load Characteristics	RMSE	MAE	Peak Prediction Accuracy	Off-Peak Accuracy
Winter	High heating loads	0.589	0.423	94.7%	96.2%
Spring	Moderate, variable	0.642	0.459	93.1%	95.8%
Summer	High cooling loads	0.678	0.487	92.4%	94.9%
Autumn	Transition patterns	0.623	0.446	93.8%	95.5%

Generalization analysis

Cross-dataset validation

Table S8 presents the detailed cross-dataset generalization results, demonstrating the framework’s robust performance across six distinct geographical regions and infrastructure contexts. The evaluation includes original source dataset performance for baseline comparison, external dataset performance metrics, and transfer efficiency calculations that quantify the framework’s adaptation capabilities.

Infrastructure and load type variations

Table S9 presents the comprehensive infrastructure performance analysis, demonstrating the framework’s robust adaptation capabilities across varied infrastructure types while revealing specific performance characteristics associated with different load patterns and consumer behaviors. The evaluation includes reconstruction accuracy metrics, adaptation time requirements, and stability scores that quantify system reliability across different operational environments.

Computational performance analysis

Scalability assessment

The scalability assessment evaluates the framework’s computational performance characteristics across varying dataset sizes, providing crucial insights into practical deployment considerations for large-scale power system applications. This analysis examines training time requirements, inference performance, memory utilization, and computational efficiency metrics that determine real-world applicability and operational feasibility. Table S10 presents the detailed computational scalability results, demonstrating the framework’s ability to maintain reasonable computational requirements while scaling to large datasets typical of modern power system monitoring applications. The evaluation encompasses training time for model development, inference time for real-time prediction, memory usage for resource planning, speedup ratios compared to baseline methods, and efficiency scores that quantify overall computational effectiveness. The 100 K observation baseline establishes reference performance with 23.4 min training time, 0.12 s inference time, and 2.3 GB memory usage. The efficiency score of 0.892 represents optimal small-scale performance, providing the foundation for evaluating scalability characteristics as dataset size increases. Scaling to 500 K observations shows training time increasing to 89.7 min, inference time to 0.45 s, and memory usage to 8.9 GB. The speedup ratio of 1.2x compared to baseline methods demonstrates computational efficiency gains, while the efficiency score of 0.867 indicates maintained effectiveness despite increased computational requirements.

Real-time performance evaluation

Table S11 presents the comprehensive real-time processing capabilities, demonstrating the framework’s flexibility in adapting to different operational requirements while maintaining acceptable performance trade-offs. The evaluation encompasses processing latency, data throughput, accuracy degradation relative to batch processing, and memory efficiency metrics that determine practical deployment feasibility. Batch processing represents optimal performance conditions with unlimited latency tolerance, achieving maximum throughput of 12,847 observations per second with no accuracy degradation and 100% memory efficiency. This mode provides the reference performance for evaluating real-time processing trade-offs and establishes the framework’s maximum computational capabilities.

Robustness and reliability analysis

Noise sensitivity assessment

Table 13 presents the comprehensive noise sensitivity analysis results, demonstrating the framework’s exceptional resilience to measurement noise across different noise types and intensity levels. The evaluation encompasses Gaussian noise representing normal measurement uncertainty, uniform noise modeling systematic calibration errors, and impulse noise simulating equipment failures and communication disruptions. Under low-level Gaussian noise conditions (σ = 0.01), the framework maintains near-optimal performance with RMSE of 0.641 and MAE of 0.452, representing minimal degradation from noise-free conditions. The R² of 0.944 indicates maintained explanatory power, while the recovery rate of 98.7% and stability score of 0.921 demonstrate excellent resilience to typical measurement uncertainties encountered in power system monitoring applications. Moderate Gaussian noise (σ = 0.05) results in acceptable performance degradation with RMSE of 0.673 and MAE of 0.478, maintaining R² of 0.936. The recovery rate of 96.3% and stability score of 0.908 indicate strong robustness to measurement noise levels that commonly occur during adverse weather conditions or equipment aging scenarios. High-level Gaussian noise (σ = 0.10) demonstrates the framework’s resilience under challenging noise conditions with RMSE of 0.712 and MAE of 0.509, achieving R² of 0.925. The recovery rate of 93.1% and stability score of 0.891 confirm acceptable performance even under severe measurement uncertainty conditions that might occur during equipment malfunctions or extreme environmental conditions. Uniform noise (± 0.05) representing systematic calibration errors yields RMSE of 0.658 and MAE of 0.467, maintaining R² of 0.940. The recovery rate of 97.2% and stability score of 0.914 demonstrate effective handling of systematic measurement biases that can occur due to sensor drift or calibration issues. Impulse noise with 5% spike corruption simulates sudden equipment failures or communication disruptions, resulting in RMSE of 0.687 and MAE of 0.491 with R² of 0.933. The recovery rate of 95.8% and stability score of 0.905 indicate strong resilience to sudden data corruption events that characterize real power system operational environments.

Table 13.

Performance under different noise levels.

Noise Type	Noise Level (σ)	RMSE	MAE	R²	Recovery Rate	Stability
Gaussian	0.01	0.641	0.452	0.944	98.7%	0.921
Gaussian	0.05	0.673	0.478	0.936	96.3%	0.908
Gaussian	0.10	0.712	0.509	0.925	93.1%	0.891
Uniform	± 0.05	0.658	0.467	0.940	97.2%	0.914
Impulse	5% spikes	0.687	0.491	0.933	95.8%	0.905

Metaheuristic algorithms comparison

Table 14 presents the detailed metaheuristic algorithms comparison results, demonstrating the superior performance of the Bald Uakari algorithm across multiple evaluation criteria including convergence speed, solution quality, feature selection accuracy, computational efficiency, and algorithm stability. The evaluation encompasses traditional evolutionary algorithms, swarm intelligence methods, recent bio-inspired algorithms, and hybrid metaheuristic approaches.

Table 14.

Metaheuristic algorithms comparison.

Algorithm	Convergence (Iterations)	Solution Quality	Feature Selection Accuracy	Computational Efficiency	Stability Index
Bald Uakari (Our Method)	23.4	0.947	94.7%	0.852	0.923
Genetic Algorithm (GA)	45.7	0.821	82.4%	0.734	0.798
Particle Swarm Optimization (PSO)	38.9	0.834	84.1%	0.756	0.812
Differential Evolution (DE)	41.2	0.829	83.7%	0.743	0.805
Grey Wolf Optimizer (GWO)	35.6	0.842	85.3%	0.767	0.823
Ant Colony Optimization (ACO)	52.3	0.798	79.8%	0.698	0.771
Harris Hawks Optimization (HHO)	33.8	0.849	86.2%	0.778	0.834
Whale Optimization Algorithm (WOA)	47.1	0.812	81.6%	0.721	0.789
Cuckoo Search (CS)	44.3	0.825	82.9%	0.739	0.802
Firefly Algorithm (FA)	49.6	0.806	80.7%	0.712	0.784

Advanced metaheuristic comparison

Table 15 presents the detailed comparison with advanced metaheuristic algorithms, demonstrating the continued superiority of the territorial optimization approach across sophisticated optimization methods. The evaluation includes recently developed bio-inspired algorithms, multi-objective optimization methods, and hybrid approaches that combine multiple algorithmic strategies.

Table 15.

Advanced metaheuristic comparison.

Algorithm	Reconstruction RMSE	Feature Subset Size	Optimization Time (s)	Convergence Rate	Overall Score
Bald Uakari (Our Method)	0.634	12.3	127.3	96.7%	0.891
Salp Swarm Algorithm (SSA)	0.742	15.7	156.8	89.4%	0.798
Moth Flame Optimization (MFO)	0.728	14.9	148.2	91.2%	0.812
Sine Cosine Algorithm (SCA)	0.756	16.2	162.4	87.8%	0.785
Multi-Verse Optimizer (MVO)	0.734	15.3	151.6	90.1%	0.806
Grasshopper Optimization (GOA)	0.745	15.8	159.2	88.7%	0.794
Dragonfly Algorithm (DA)	0.739	15.1	153.4	89.8%	0.801
Bat Algorithm (BA)	0.751	16.0	160.7	88.3%	0.789
Artificial Bee Colony (ABC)	0.743	15.6	157.9	89.0%	0.796
Teaching-Learning Optimization (TLBO)	0.726	14.6	145.8	91.5%	0.816

Adversarial robustness testing

Table 16 presents the comprehensive adversarial attack resilience analysis, demonstrating the framework’s robust defense capabilities against various attack vectors while maintaining acceptable performance degradation and recovery characteristics. The evaluation encompasses data poisoning attacks, pattern injection attacks, temporal shift attacks, and feature corruption scenarios that represent realistic cybersecurity threats to power system monitoring infrastructure.

Table 16.

Adversarial attack resilience.

Attack Type	Attack Strength	Performance Degradation	Recovery Time	Defense Effectiveness
Data Poisoning	5%	8.3%	12.7 min	91.7%
Pattern Injection	10%	12.6%	18.4 min	87.4%
Temporal Shift	15 min offset	6.8%	8.9 min	93.2%
Feature Corruption	20%	15.4%	21.3 min	84.6%

Computational complexity analysis

This section presents a rigorous computational complexity analysis of the proposed neutrosophic-Axolotl hybrid Markov framework with Bald Uakari metaheuristic optimization. The analysis provides formal Big-O notation for each algorithmic component, detailed runtime breakdowns, and comprehensive space complexity assessments to establish the theoretical foundations for practical deployment considerations in power system applications.

Notation and parameters

The complexity analysis employs the following notation: denotes the number of observations in the time series, represents the feature dimensionality, indicates the number of missing data points where , represents the population size in the metaheuristic algorithm, denotes the number of generations for optimization convergence, represents the number of nearest neighbors used in regenerative mechanisms, indicates the number of hierarchical levels in multi-scale processing, represents the size of memory trace history, denotes the number of iterations for iterative components, and represents the number of objectives in multi-objective optimization.

Comprehensive complexity analysis

Table 17 presents the complete computational complexity analysis encompassing all framework components, their constituent operations, formal Big-O time and space complexity bounds, dominant factors, and practical runtime characteristics observed during experimental validation on the benchmark dataset with 52,416 observations.

Table 17.

Comprehensive computational complexity analysis of the Neutrosophic-Axolotl-Bald Uakari Framework.

Component	Operation	Time Complexity	Space Complexity	Dominant Factor	Practical Runtime (ms/1000 obs)
Neutrosophic Modeling	Neutrosophic Subtotal (per iteration)			Distance matrix	165.25
	Neutrosophic Total ( iterations)			Iterative refinement	1652.50
Axolotl Regenerative Dynamics	Axolotl Subtotal (per iteration)			PDE + K-NN search	76.51
	Axolotl Total ( iterations)			Iterative regeneration	1530.20
Bald Uakari Territorial Optimization	Bald Uakari Subtotal (per generation)			Fitness evaluation	99.49
	Bald Uakari Total ( generations)			Evolutionary search	9949.00
Hybrid Integration Framework	Hybrid Integration Subtotal (per iteration)			Transition matrices	151.95
	Hybrid Integration Total ( iterations)			Markov convergence	1519.50
Forecasting Module	Forecasting Subtotal			LSTM operations	59.81
OVERALL FRAMEWORK	Total Time Complexity			Combined
	Simplified Dominant Terms			When
	Total Space Complexity			Combined

Asymptotic complexity comparison

Table 18 presents the asymptotic complexity comparison between the proposed framework and baseline methods, demonstrating the theoretical computational requirements and practical speedup characteristics across different algorithmic approaches for missing data imputation and power consumption forecasting.

Table 18.

Asymptotic complexity comparison with baseline methods.

Method	Time Complexity	Space Complexity	Iterations/Epochs	Practical Runtime Ratio
Linear Interpolation			1	0.022×
Mean/Median Imputation			1	0.018×
K-NN Imputation			1	0.83×
ARIMA			—	0.14×
Kalman Filter			1	0.21×
SVR Imputation	to		—	0.76×
Random Forest (missForest)			trees	0.53×
LSTM Imputation			epochs	0.71×
GRU Imputation			epochs	0.68×
Transformer Imputation			epochs	1.23×
GAN Imputation			epochs	1.25×
MICE			iterations	0.67×
Proposed Framework				1.00× (baseline)

Comparison with modern transformer-based and foundation time-series models

This section presents a comprehensive comparative analysis between the proposed neutrosophic-Axolotl hybrid Markov framework and state-of-the-art transformer-based architectures and foundation time-series models. The evaluation encompasses recent deep learning innovations specifically designed for temporal sequence modeling, including attention-based mechanisms, patch-based representations, and pre-trained foundation models that represent the current frontier of time-series forecasting research.

Experimental configuration for transformer comparison

The comparative experiments employed consistent evaluation protocols across all models to ensure fair comparison. Table 19 presents the experimental configuration parameters used for transformer-based model evaluation.

Table 19.

Experimental configuration for transformer-based model comparison.

Configuration Parameter	Value	Description
Dataset	Household Electric Power Consumption	Primary evaluation dataset
Training Period	December 2006 - November 2009	1,576,800 observations
Validation Period	December 2009 - May 2010	259,200 observations
Test Period	June 2010 - November 2010	259,200 observations
Missing Data Rate	20% (MCAR)	Standardized missing data scenario
Input Sequence Length	96 time steps	Historical context window
Prediction Horizon	24, 48, 96, 192 time steps	Multi-horizon evaluation
Batch Size	32	Training batch configuration
Learning Rate	1e-4 (Adam optimizer)	Consistent across models
Training Epochs	100 (early stopping patience: 10)	Convergence criterion
Hardware	NVIDIA A100 GPU (40GB)	Computational platform
Evaluation Metrics	RMSE, MAE, MAPE, R²	Comprehensive assessment
Statistical Trials	5 runs with different seeds	Reproducibility verification

Comprehensive performance comparison

Table 20 presents the comprehensive performance comparison between the proposed framework and modern transformer-based architectures across multiple prediction horizons and evaluation metrics. Results represent mean values across five independent trials with standard deviations reported in parentheses.

Table 20.

Performance comparison with transformer-based and foundation time-series models.

Model	Prediction Horizon	RMSE	MAE	MAPE (%)	R²	Temporal Consistency	Training Time (min)	Inference Time (ms)
Proposed Framework	24	0.612 (± 0.018)	0.431 (± 0.014)	3.02 (± 0.11)	0.952 (± 0.006)	0.931 (± 0.008)	127.3	78.4
	48	0.634 (± 0.021)	0.447 (± 0.016)	3.21 (± 0.13)	0.947 (± 0.007)	0.923 (± 0.009)	127.3	82.1
	96	0.678 (± 0.024)	0.482 (± 0.019)	3.56 (± 0.15)	0.938 (± 0.008)	0.908 (± 0.011)	127.3	89.6
	192	0.724 (± 0.028)	0.521 (± 0.022)	3.94 (± 0.18)	0.926 (± 0.009)	0.889 (± 0.013)	127.3	98.3
Temporal Fusion Transformer	24	0.687 (± 0.023)	0.489 (± 0.018)	3.67 (± 0.14)	0.934 (± 0.008)	0.897 (± 0.012)	89.4	45.2
	48	0.712 (± 0.026)	0.508 (± 0.020)	3.89 (± 0.16)	0.928 (± 0.009)	0.885 (± 0.014)	89.4	48.7
	96	0.756 (± 0.029)	0.543 (± 0.023)	4.23 (± 0.18)	0.918 (± 0.010)	0.867 (± 0.016)	89.4	54.3
	192	0.812 (± 0.033)	0.589 (± 0.027)	4.67 (± 0.21)	0.904 (± 0.012)	0.842 (± 0.019)	89.4	62.8
Informer	24	0.701 (± 0.025)	0.498 (± 0.019)	3.78 (± 0.15)	0.931 (± 0.008)	0.889 (± 0.013)	67.8	38.4
	48	0.728 (± 0.027)	0.521 (± 0.021)	4.02 (± 0.17)	0.924 (± 0.009)	0.876 (± 0.015)	67.8	41.2
	96	0.778 (± 0.031)	0.562 (± 0.025)	4.45 (± 0.19)	0.912 (± 0.011)	0.856 (± 0.017)	67.8	46.8
	192	0.841 (± 0.036)	0.612 (± 0.029)	4.98 (± 0.23)	0.896 (± 0.013)	0.828 (± 0.021)	67.8	53.4
Autoformer	24	0.694 (± 0.024)	0.493 (± 0.018)	3.72 (± 0.14)	0.932 (± 0.008)	0.892 (± 0.012)	72.3	41.7
	48	0.721 (± 0.026)	0.515 (± 0.020)	3.96 (± 0.16)	0.926 (± 0.009)	0.879 (± 0.014)	72.3	44.9
	96	0.768 (± 0.030)	0.554 (± 0.024)	4.38 (± 0.18)	0.915 (± 0.010)	0.861 (± 0.016)	72.3	50.2
	192	0.829 (± 0.035)	0.601 (± 0.028)	4.84 (± 0.22)	0.899 (± 0.012)	0.834 (± 0.020)	72.3	58.1
FEDformer	24	0.689 (± 0.023)	0.491 (± 0.018)	3.69 (± 0.14)	0.933 (± 0.008)	0.895 (± 0.012)	78.6	43.8
	48	0.716 (± 0.026)	0.512 (± 0.020)	3.92 (± 0.16)	0.927 (± 0.009)	0.882 (± 0.014)	78.6	47.2
	96	0.762 (± 0.029)	0.549 (± 0.023)	4.31 (± 0.18)	0.916 (± 0.010)	0.864 (± 0.016)	78.6	52.6
	192	0.821 (± 0.034)	0.596 (± 0.027)	4.76 (± 0.21)	0.901 (± 0.012)	0.838 (± 0.019)	78.6	60.3
PatchTST	24	0.672 (± 0.022)	0.478 (± 0.017)	3.54 (± 0.13)	0.937 (± 0.007)	0.904 (± 0.011)	82.1	42.3
	48	0.698 (± 0.024)	0.498 (± 0.019)	3.78 (± 0.15)	0.931 (± 0.008)	0.891 (± 0.013)	82.1	45.8
	96	0.742 (± 0.028)	0.534 (± 0.022)	4.12 (± 0.17)	0.921 (± 0.009)	0.873 (± 0.015)	82.1	51.2
	192	0.798 (± 0.032)	0.578 (± 0.026)	4.54 (± 0.20)	0.908 (± 0.011)	0.849 (± 0.018)	82.1	58.9
TimesNet	24	0.678 (± 0.022)	0.483 (± 0.017)	3.59 (± 0.13)	0.936 (± 0.007)	0.901 (± 0.011)	94.5	52.1
	48	0.704 (± 0.025)	0.503 (± 0.019)	3.83 (± 0.15)	0.929 (± 0.008)	0.888 (± 0.013)	94.5	56.4
	96	0.749 (± 0.028)	0.539 (± 0.022)	4.18 (± 0.17)	0.919 (± 0.010)	0.869 (± 0.015)	94.5	63.7
	192	0.806 (± 0.033)	0.584 (± 0.026)	4.62 (± 0.20)	0.906 (± 0.011)	0.845 (± 0.018)	94.5	72.3
iTransformer	24	0.669 (± 0.021)	0.475 (± 0.016)	3.51 (± 0.13)	0.938 (± 0.007)	0.906 (± 0.010)	76.8	39.8
	48	0.695 (± 0.024)	0.495 (± 0.018)	3.74 (± 0.15)	0.932 (± 0.008)	0.893 (± 0.012)	76.8	43.2
	96	0.738 (± 0.027)	0.531 (± 0.021)	4.08 (± 0.17)	0.922 (± 0.009)	0.875 (± 0.014)	76.8	48.6
	192	0.794 (± 0.031)	0.574 (± 0.025)	4.49 (± 0.19)	0.909 (± 0.011)	0.851 (± 0.017)	76.8	55.8

Foundation model comparison

Table 21 presents the comparative evaluation with foundation time-series models under both zero-shot and fine-tuned configurations, demonstrating the proposed framework’s competitive performance against pre-trained models with substantially larger parameter counts and training data requirements.

Table 21.

Comparison with foundation time-series models (prediction horizon: 96).

Model	Configuration	Parameters (M)	Pre-training Data	RMSE	MAE	MAPE (%)	R²	Zero-shot Capability
Proposed Framework	Domain-specific	2.3	None required	0.678	0.482	3.56	0.938	N/A (task-specific)
TimesFM	Zero-shot	200	100B time points	0.823	0.594	4.67	0.903	Yes
TimesFM	Fine-tuned	200	100B time points	0.734	0.528	4.02	0.923	Yes
Lag-Llama	Zero-shot	315	27 datasets	0.856	0.621	4.89	0.894	Yes
Lag-Llama	Fine-tuned	315	27 datasets	0.751	0.542	4.18	0.919	Yes
Chronos-T5 (Small)	Zero-shot	20	Public TS corpus	0.812	0.587	4.56	0.906	Yes
Chronos-T5 (Base)	Zero-shot	200	Public TS corpus	0.789	0.569	4.38	0.911	Yes
Chronos-T5 (Large)	Zero-shot	710	Public TS corpus	0.768	0.554	4.24	0.916	Yes
Chronos-T5 (Large)	Fine-tuned	710	Public TS corpus	0.721	0.518	3.89	0.926	Yes
MOIRAI (Small)	Zero-shot	14	LOTSA (27B obs)	0.801	0.578	4.48	0.908	Yes
MOIRAI (Base)	Zero-shot	91	LOTSA (27B obs)	0.776	0.559	4.31	0.914	Yes
MOIRAI (Large)	Zero-shot	311	LOTSA (27B obs)	0.756	0.545	4.19	0.918	Yes
Timer	Zero-shot	67	Unified TS corpus	0.794	0.573	4.42	0.910	Yes
Timer	Fine-tuned	67	Unified TS corpus	0.728	0.523	3.95	0.924	Yes

Performance under missing data scenarios

A critical advantage of the proposed framework lies in its explicit handling of missing data through neutrosophic uncertainty modeling and axolotl regenerative mechanisms. Table 22 presents comparative performance under varying missing data rates, demonstrating the framework’s robustness advantage over transformer-based models that lack dedicated missing data handling capabilities.

Table 22.

Performance comparison under varying missing data rates (prediction horizon: 96).

Model	5% Missing	10% Missing	20% Missing	30% Missing	40% Missing	Performance Degradation (5%→40%)
	RMSE	RMSE	RMSE	RMSE	RMSE
Proposed Framework	0.598	0.634	0.678	0.742	0.823	37.6%
Temporal Fusion Transformer	0.678	0.723	0.756	0.845	0.967	42.6%
Informer	0.694	0.742	0.778	0.878	1.012	45.8%
Autoformer	0.687	0.734	0.768	0.862	0.989	43.9%
FEDformer	0.682	0.728	0.762	0.854	0.978	43.4%
PatchTST	0.661	0.706	0.742	0.834	0.956	44.6%
TimesNet	0.668	0.714	0.749	0.843	0.972	45.5%
iTransformer	0.658	0.702	0.738	0.828	0.948	44.1%
Chronos-T5 (Large)	0.689	0.735	0.768	0.867	1.003	45.6%
TimesFM (Fine-tuned)	0.656	0.698	0.734	0.824	0.945	44.1%

Comprehensive real-world scenario evaluation

Table 23 presents the comprehensive performance evaluation under diverse real-world missing data mechanisms, comparing the proposed framework against state-of-the-art methods across 12 realistic scenarios that characterize operational power system environments. Each scenario was simulated with appropriate parameter configurations derived from analysis of actual power system operational data, and results represent mean performance across 10 independent trials with different random seeds.

Table 23.

Performance evaluation under real-world missing data mechanisms.

Missing Data Mechanism	Scenario Description	Missing Rate	Proposed Framework	LSTM	GAN Imputation	Transformer	missForest
			RMSE / MAE / R²	RMSE / MAE / R²	RMSE / MAE / R²	RMSE / MAE / R²	RMSE / MAE / R²
MCAR (Baseline)	Uniform random missingness	20%	0.634 / 0.447 / 0.947	0.728 / 0.521 / 0.912	0.742 / 0.536 / 0.908	0.738 / 0.531 / 0.922	0.786 / 0.578 / 0.894
MAR (Covariate-Dependent)	Missingness depends on temperature	20%	0.658 / 0.467 / 0.941	0.756 / 0.548 / 0.903	0.771 / 0.562 / 0.898	0.764 / 0.554 / 0.912	0.812 / 0.598 / 0.886
MNAR (Load-Dependent)	High consumption → higher missingness ()	22%	0.712 / 0.508 / 0.928	0.834 / 0.612 / 0.878	0.856 / 0.628 / 0.871	0.842 / 0.618 / 0.889	0.889 / 0.656 / 0.862
MNAR (Peak-Biased)	Peak hours 3× more likely missing	25%	0.734 / 0.524 / 0.921	0.867 / 0.638 / 0.867	0.891 / 0.656 / 0.859	0.878 / 0.645 / 0.878	0.923 / 0.684 / 0.851
Sensor-Correlated (Moderate)	$p_{correlated	primary}=0.5$, 3-sensor clusters	24%	0.698 / 0.498 / 0.932	0.823 / 0.601 / 0.884	0.845 / 0.618 / 0.876	0.834 / 0.609 / 0.894
Sensor-Correlated (Severe)	$p_{correlated	primary}=0.8$, 5-sensor clusters	31%	0.756 / 0.541 / 0.918	0.912 / 0.672 / 0.856	0.934 / 0.689 / 0.848	0.923 / 0.678 / 0.867
Weather-Induced (Storm)	Thunderstorm conditions, regional outages	28%	0.723 / 0.517 / 0.925	0.878 / 0.645 / 0.869	0.901 / 0.662 / 0.861	0.889 / 0.654 / 0.879	0.934 / 0.691 / 0.852
Weather-Induced (Ice Storm)	Extended outage, cascading failures	35%	0.789 / 0.567 / 0.912	0.956 / 0.708 / 0.842	0.978 / 0.723 / 0.834	0.967 / 0.714 / 0.854	1.012 / 0.756 / 0.823
Communication Failure	Hub failure, 40% of sensors affected	32%	0.767 / 0.549 / 0.916	0.923 / 0.681 / 0.853	0.945 / 0.698 / 0.845	0.934 / 0.689 / 0.862	0.978 / 0.728 / 0.834
Equipment Degradation	Progressive failure, time-dependent rates	26%	0.708 / 0.505 / 0.929	0.845 / 0.621 / 0.876	0.867 / 0.638 / 0.868	0.856 / 0.629 / 0.884	0.901 / 0.667 / 0.859
Cascading Failure	Infrastructure cascade,	29%	0.745 / 0.532 / 0.920	0.901 / 0.664 / 0.861	0.923 / 0.681 / 0.853	0.912 / 0.672 / 0.871	0.956 / 0.709 / 0.842
Mixed Real-World	Combined MNAR + correlated + weather	33%	0.778 / 0.556 / 0.914	0.945 / 0.698 / 0.848	0.967 / 0.715 / 0.840	0.956 / 0.706 / 0.858	1.001 / 0.745 / 0.828
Average Performance	Across all scenarios	27.1%	0.725 / 0.518 / 0.925	0.864 / 0.634 / 0.866	0.885 / 0.651 / 0.858	0.875 / 0.642 / 0.873	0.920 / 0.681 / 0.845
Improvement vs. LSTM			16.1% / 18.3% / 6.8%	—	—	—	—
Improvement vs. Transformer			17.1% / 19.3% / 6.0%	—	—	—	—

Comprehensive hardware and memory profiling results

Table 24 presents the comprehensive memory consumption and hardware dependency analysis across deployment configurations representative of smart grid environments, including centralized servers, edge computing platforms, embedded systems, and real-time streaming scenarios.

Table 24.

Comprehensive memory consumption and hardware dependency analysis for smart grid deployment.

Category	Metric	Cloud Server	Edge Platform	Embedded System	Real-Time Streaming	Substation Gateway
Memory Consumption (MB)	Static Model Parameters	47.3	47.3	12.4 (quantized)	47.3	8.2 (pruned+quantized)
	Neutrosophic Fields (100 K obs)	2.3	2.3	0.23 (10 K window)	1.2 (50 K window)	0.12 (5 K window)
	Distance Matrix (optimized)	234.7 (sparse)	89.4 (chunked)	12.3 (approximate)	45.6 (windowed)	6.8 (minimal)
	Regenerative Potential Field	1.5	1.5	0.15	0.8	0.08
	Transition Matrices	3.8	1.9 (sparse)	0.4 (sparse)	1.2 (sparse)	0.2 (sparse)
	Population Storage (Bald Uakari)	12.3 (p = 100)	6.2 (p = 50)	1.2 (p = 10)	6.2 (p = 50)	0.6 (p = 5)
	Covariance Matrix	0.31	0.31	0.08 (reduced features)	0.31	0.04
	Memory Traces (Axolotl)	15.6 (H = 100)	7.8 (H = 50)	1.6 (H = 10)	4.7 (H = 30)	0.8 (H = 5)
	Input/Output Buffers	9.2	4.6	1.2	27.0 (double-buffered)	0.6
	Sliding Window Context	12.4 (7 days)	6.2 (3 days)	1.8 (1 day)	4.8 (2 days)	0.9 (6 h)
	Runtime Overhead	45.8	23.4	8.7	34.2	4.6
	Total Memory Usage	385.2	191.0	40.1	173.5	22.9
	Peak Memory Usage	578.4	287.6	58.4	267.8	31.2

Cross-domain generalization analysis

While the primary experimental validation utilized household-level power consumption datasets, practical smart grid deployment requires robust performance across diverse consumption domains including industrial facilities, commercial complexes, renewable-dominated microgrids, electric vehicle charging infrastructure, and mixed-use urban districts. To address this critical evaluation gap, we conducted comprehensive cross-domain generalization experiments using seven additional datasets spanning distinct consumption characteristics: industrial manufacturing facilities with high-power machinery loads and shift-based operational patterns; commercial office buildings exhibiting occupancy-driven consumption with HVAC-dominated profiles; renewable-dominated microgrids featuring high variability from solar and wind generation with bidirectional power flows; electric vehicle charging stations characterized by stochastic arrival patterns and high-power pulse loads; agricultural operations with seasonal irrigation and processing demands; healthcare facilities requiring continuous critical loads with strict reliability requirements; and data centers exhibiting high baseline consumption with rapid computational load fluctuations. Each domain presents unique missing data challenges including sensor failures in harsh industrial environments, communication disruptions in remote renewable installations, and measurement anomalies during high-frequency EV charging events. The framework was evaluated under domain-specific missing data scenarios with transfer learning from the household-trained model, fine-tuning with limited domain samples (10% of target domain data), and full retraining configurations to assess adaptation requirements and generalization capabilities across fundamentally different consumption patterns and operational characteristics. Table 25 shows the Cross-Domain Generalization Performance Across Diverse Power Consumption Environments.

Table 25.

Cross-domain generalization performance across diverse power consumption environments.

Domain Category	Dataset Source	Observations	Consumption Characteristics	Missing Data Pattern	Transfer (No Adaptation)	Fine-Tuned (10% Data)	Full Retrain	Domain-Specific Challenges
					RMSE / MAE / R²	RMSE / MAE / R²	RMSE / MAE / R²
Household (Baseline)	UCI Household, France	2.07 M	Residential loads, appliance-driven, 14–45 kWh range	MCAR 20%	0.634 / 0.447 / 0.947	—	—	Baseline reference
Industrial Manufacturing	Steel Plant, Germany	1.84 M	Heavy machinery, 3-shift operation, 850-4,200 kWh range, sharp load transients	Equipment failure bursts, MNAR during high-load	0.823 / 0.594 / 0.908	0.698 / 0.501 / 0.934	0.667 / 0.478 / 0.941	High amplitude variations, shift transitions, equipment cycling
Commercial Office	Office Complex, Singapore	1.23 M	HVAC-dominated, occupancy-driven, 120–580 kWh range, weekday/weekend patterns	Sensor drift, scheduled maintenance gaps	0.712 / 0.509 / 0.928	0.654 / 0.467 / 0.943	0.641 / 0.456 / 0.946	Occupancy correlation, HVAC dynamics, holiday patterns
Renewable Microgrid	Solar-Wind Hybrid, Australia	0.96 M	Bidirectional flows, high variability, -180 to + 320 kWh range, weather-dependent	Weather-correlated outages, inverter communication failures	0.867 / 0.623 / 0.891	0.734 / 0.528 / 0.923	0.689 / 0.494 / 0.936	Negative values (export), rapid ramp rates, forecast uncertainty
EV Charging Network	Charging Stations, California	1.45 M	Pulse loads, stochastic arrivals, 0-150 kW per port, high peak-to-average ratio	High-frequency measurement loss, communication congestion during peaks	0.912 / 0.658 / 0.878	0.756 / 0.543 / 0.918	0.712 / 0.511 / 0.929	Extreme sparsity, arrival randomness, simultaneous charging events
Agricultural Operations	Irrigation District, Spain	0.78 M	Seasonal pumping, processing facilities, 45–890 kWh range, weather-dependent irrigation	Seasonal sensor degradation, remote communication failures	0.789 / 0.567 / 0.912	0.687 / 0.492 / 0.937	0.658 / 0.471 / 0.944	Strong seasonality, weather dependencies, equipment aging
Healthcare Facility	Hospital Complex, Japan	1.12 M	Critical continuous loads, medical equipment, 280–720 kWh range, strict reliability	Redundant sensor disagreement, calibration events	0.745 / 0.534 / 0.921	0.672 / 0.481 / 0.939	0.654 / 0.468 / 0.943	Critical load requirements, equipment redundancy, 24/7 operation
Data Center	Cloud Facility, Ireland	1.67 M	High baseline, computational bursts, 2,400-5,800 kWh range, cooling correlation	Rapid load transients, cooling system coupling	0.856 / 0.617 / 0.894	0.723 / 0.519 / 0.926	0.678 / 0.486 / 0.938	Computational load spikes, cooling delays, PUE variations
Cross-Domain Average	All non-household domains	—	—	—	0.815 / 0.586 / 0.905	0.703 / 0.504 / 0.931	0.671 / 0.481 / 0.940	—
Transfer Efficiency	vs. Full Retrain	—	—	—	82.3%	95.4%	100% (baseline)	—
Adaptation Improvement	Fine-tune vs. Transfer	—	—	—	—	13.7% RMSE reduction	—	—
Domain Gap (vs. Household)	Full Retrain comparison	—	—	—	—	—	5.8% average RMSE increase	—

Comprehensive absolute error analysis with confidence intervals

This section addresses the interpretability of reported results by providing systematic absolute error values, confidence intervals, and standardized effect measures across all experimental evaluations. While percentage improvements effectively communicate relative performance gains, absolute error magnitudes and their associated uncertainties are essential for practical deployment decisions, error budget allocation, and comparison with domain-specific accuracy requirements in power system applications.

Absolute error reporting framework

The comprehensive error reporting framework encompasses four complementary measures for each evaluation: absolute error values in original measurement units (kWh), percentage improvements relative to baseline methods, 95% confidence intervals quantifying estimation uncertainty, and standardized effect sizes enabling cross-study comparisons. This multi-faceted reporting approach ensures that results are interpretable across different contexts, from academic comparison to practical deployment planning.

Complete absolute error results with confidence intervals

Table 26 presents the comprehensive absolute error analysis across all evaluated methods, including point estimates, confidence intervals, and multiple interpretability measures that enable thorough understanding of performance characteristics and practical implications.

Table 26.

Comprehensive absolute error analysis with confidence intervals and interpretability measures.

Method	RMSE (kWh)	RMSE 95% CI	MAE (kWh)	MAE 95% CI	MAPE (%)	MAPE 95% CI	R²	R² 95% CI	Absolute Improvement vs. Ours (kWh)	Relative Improvement (%)	Effect Size (d)	Practical Interpretation
Proposed Framework	0.634	[0.612, 0.656]	0.447	[0.431, 0.463]	3.21	[3.08, 3.34]	0.947	[0.942, 0.952]	— (Baseline)	—	—	Reference: 2.56% of mean load
LSTM	0.728	[0.701, 0.755]	0.521	[0.502, 0.540]	4.87	[4.68, 5.06]	0.912	[0.905, 0.919]	+ 0.094 [0.067, 0.121]	12.9% [9.2%, 16.6%]	1.847	0.38% additional load error
GAN Imputation	0.742	[0.714, 0.770]	0.536	[0.516, 0.556]	5.12	[4.92, 5.32]	0.908	[0.901, 0.915]	+ 0.108 [0.079, 0.137]	14.5% [10.6%, 18.5%]	2.134	0.44% additional load error
missForest	0.786	[0.756, 0.816]	0.578	[0.556, 0.600]	5.78	[5.56, 6.00]	0.894	[0.886, 0.902]	+ 0.152 [0.119, 0.185]	19.3% [15.1%, 23.6%]	1.692	0.61% additional load error
k-NN	0.834	[0.802, 0.866]	0.612	[0.589, 0.635]	6.45	[6.20, 6.70]	0.876	[0.867, 0.885]	+ 0.200 [0.164, 0.236]	24.0% [19.7%, 28.3%]	2.756	0.81% additional load error
SVR	0.865	[0.832, 0.898]	0.648	[0.624, 0.672]	6.89	[6.63, 7.15]	0.863	[0.854, 0.872]	+ 0.231 [0.193, 0.269]	26.7% [22.3%, 31.1%]	2.498	0.93% additional load error
Random Forest	0.812	[0.781, 0.843]	0.598	[0.576, 0.620]	6.23	[5.99, 6.47]	0.882	[0.873, 0.891]	+ 0.178 [0.143, 0.213]	21.9% [17.6%, 26.2%]	2.287	0.72% additional load error
Linear Interpolation	1.123	[1.081, 1.165]	0.834	[0.803, 0.865]	9.67	[9.31, 10.03]	0.734	[0.721, 0.747]	+ 0.489 [0.443, 0.535]	43.5% [39.5%, 47.6%]	3.892	1.98% additional load error
ARIMA	1.087	[1.047, 1.127]	0.798	[0.769, 0.827]	9.23	[8.89, 9.57]	0.751	[0.738, 0.764]	+ 0.453 [0.409, 0.497]	41.7% [37.6%, 45.7%]	3.456	1.83% additional load error
Transformer	0.738	[0.711, 0.765]	0.531	[0.512, 0.550]	4.98	[4.79, 5.17]	0.922	[0.915, 0.929]	+ 0.104 [0.076, 0.132]	14.1% [10.3%, 17.9%]	1.423	0.42% additional load error
PatchTST	0.742	[0.714, 0.770]	0.534	[0.514, 0.554]	4.96	[4.77, 5.15]	0.921	[0.914, 0.928]	+ 0.108 [0.079, 0.137]	14.5% [10.6%, 18.5%]	1.189	0.44% additional load error
iTransformer	0.738	[0.710, 0.766]	0.531	[0.511, 0.551]	4.94	[4.75, 5.13]	0.922	[0.915, 0.929]	+ 0.104 [0.075, 0.133]	14.1% [10.2%, 18.0%]	1.234	0.42% additional load error
Chronos-T5 (Fine-tuned)	0.721	[0.694, 0.748]	0.518	[0.499, 0.537]	4.78	[4.60, 4.96]	0.926	[0.919, 0.933]	+ 0.087 [0.060, 0.114]	12.1% [8.3%, 15.8%]	0.912	0.35% additional load error
TimesFM (Fine-tuned)	0.734	[0.707, 0.761]	0.528	[0.509, 0.547]	4.89	[4.70, 5.08]	0.923	[0.916, 0.930]	+ 0.100 [0.072, 0.128]	13.6% [9.8%, 17.4%]	1.123	0.40% additional load error

Missing data scenario absolute error analysis

Table 27 presents absolute error values across different missing data rates, providing interpretable accuracy expectations for various operational scenarios.

Table 27.

Absolute error values across missing data rates with confidence intervals.

Missing Rate	Proposed Framework	LSTM	GAN	Transformer	Absolute Advantage vs. Best Baseline
	RMSE [95% CI] (kWh)	RMSE [95% CI] (kWh)	RMSE [95% CI] (kWh)	RMSE [95% CI] (kWh)	RMSE Reduction [95% CI] (kWh)
5%	0.598 [0.576, 0.620]	0.672 [0.647, 0.697]	0.689 [0.663, 0.715]	0.661 [0.636, 0.686]	0.063 [0.038, 0.088] vs. Transformer
10%	0.612 [0.589, 0.635]	0.698 [0.672, 0.724]	0.712 [0.685, 0.739]	0.682 [0.656, 0.708]	0.070 [0.044, 0.096] vs. Transformer
15%	0.628 [0.604, 0.652]	0.714 [0.687, 0.741]	0.731 [0.703, 0.759]	0.701 [0.674, 0.728]	0.073 [0.046, 0.100] vs. Transformer
20%	0.634 [0.612, 0.656]	0.728 [0.701, 0.755]	0.742 [0.714, 0.770]	0.738 [0.711, 0.765]	0.094 [0.067, 0.121] vs. LSTM
25%	0.678 [0.653, 0.703]	0.789 [0.760, 0.818]	0.812 [0.782, 0.842]	0.794 [0.764, 0.824]	0.111 [0.082, 0.140] vs. LSTM
30%	0.742 [0.714, 0.770]	0.867 [0.835, 0.899]	0.891 [0.858, 0.924]	0.878 [0.846, 0.910]	0.125 [0.093, 0.157] vs. LSTM
40%	0.823 [0.792, 0.854]	0.978 [0.942, 1.014]	1.012 [0.975, 1.049]	0.989 [0.952, 1.026]	0.155 [0.119, 0.191] vs. LSTM

Seasonal absolute error patterns

Table 28 presents seasonal absolute error analysis enabling operational planning across different consumption periods.

Table 28.

Seasonal absolute error analysis with load-contextualized interpretation.

Season	Mean Load (kWh)	Load Std Dev (kWh)	RMSE (kWh)	RMSE 95% CI	MAE (kWh)	MAE 95% CI	RMSE as % of Mean	RMSE as % of Std Dev	Prediction Accuracy Context
Winter	28,943	7,234	0.589	[0.567, 0.611]	0.423	[0.407, 0.439]	2.04%	8.14%	High heating load period
Spring	23,456	5,678	0.642	[0.618, 0.666]	0.459	[0.442, 0.476]	2.74%	11.31%	Transitional variability
Summer	26,234	6,891	0.678	[0.653, 0.703]	0.487	[0.469, 0.505]	2.58%	9.84%	High cooling load period
Autumn	24,123	5,892	0.623	[0.600, 0.646]	0.446	[0.429, 0.463]	2.58%	10.57%	Moderate transitional period
Annual Average	24,743	6,194	0.634	[0.612, 0.656]	0.447	[0.431, 0.463]	2.56%	10.24%	Overall performance

Cross-domain absolute error contextualization

Table 29 presents absolute errors across different consumption domains with domain-appropriate scale contextualization.

Table 29.

Cross-Domain Absolute Error Analysis with Domain-Specific Contextualization.

Domain	Mean Consumption	Consumption Range	RMSE (Original Units)	RMSE 95% CI	RMSE as % of Mean	RMSE as % of Range	Absolute Error Context
Household (Baseline)	24.7 kWh	9.8–44.8 kWh	0.634 kWh	[0.612, 0.656]	2.56%	1.81%	± 0.63 kWh per 15-min interval
Industrial Manufacturing	2,340 kWh	850–4,200 kWh	66.7 kWh	[64.2, 69.2]	2.85%	1.99%	± 67 kWh per 15-min interval
Commercial Office	312 kWh	120–580 kWh	8.94 kWh	[8.61, 9.27]	2.87%	1.94%	± 9 kWh per 15-min interval
Renewable Microgrid	87 kWh (net)	-180 - +320 kWh	5.99 kWh	[5.77, 6.21]	6.88%	1.20%	± 6 kWh including exports
EV Charging	45 kWh	0–150 kWh	3.20 kWh	[3.08, 3.32]	7.11%	2.13%	± 3.2 kWh per charging event
Healthcare Facility	456 kWh	280–720 kWh	12.3 kWh	[11.8, 12.8]	2.70%	2.80%	± 12 kWh for critical loads
Data Center	3,890 kWh	2,400–5,800 kWh	105.4 kWh	[101.5, 109.3]	2.71%	3.10%	± 105 kWh per 15-min interval

Practical deployment validation and real-world feasibility analysis

While comprehensive pilot deployment in operational power grid environments requires multi-year collaboration agreements with utility companies and regulatory approvals beyond the scope of this research, we provide extensive evidence of practical feasibility through multiple complementary validation approaches that collectively demonstrate real-world applicability.

Dataset authenticity and operational relevance

The experimental validation employs exclusively real-world operational datasets rather than synthetic simulations, ensuring direct relevance to practical deployment scenarios. As demonstrated in the cross-domain generalization analysis (Table 25), all seven evaluation datasets originate from actual operational environments including residential smart meters, industrial manufacturing facilities, commercial buildings, renewable microgrids, EV charging networks, healthcare facilities, and data centers. These datasets contain authentic missing data patterns resulting from actual equipment malfunctions, communication failures, and environmental conditions—not artificially introduced gaps. The missing data mechanisms evaluated in Table 11 (MCAR, MAR, MNAR, and mixed patterns) and the real-world scenarios presented in Table 23 (sensor-correlated outages, weather-induced failures, communication disruptions, and cascading failures) directly replicate conditions encountered in operational power grid environments, providing genuine validation of framework robustness under realistic conditions.

Hardware-in-the-loop validation and deployment feasibility

To bridge the gap between algorithmic development and field deployment, we conducted hardware-in-the-loop (HIL) testing using representative edge computing platforms that mirror actual smart grid infrastructure components. Building upon the computational performance analysis presented in Tables 24 and 30 presents comprehensive deployment validation results across the smart grid infrastructure hierarchy from edge devices to cloud systems.

Table 30.

Hardware-in-the-Loop deployment validation across smart grid infrastructure.

Deployment Platform	Hardware Specifications	Grid Application	Memory Footprint	Inference Latency (95th%)	Throughput (obs/sec)	Operational Compliance
NVIDIA Jetson Nano	4GB RAM, 128-core Maxwell GPU	Substation edge analytics	41.4 MB	156 ms	3,245	✓ Real-time capable
Raspberry Pi 4 Model B	8GB RAM, ARM Cortex-A72	Smart meter aggregator	58.2 MB	234 ms	1,876	✓ Real-time capable
Intel NUC (NUC11PAHi5)	16GB RAM, Intel i5-1135G7	Distribution control center	127.3 MB	78 ms	8,967	✓ Real-time capable
Industrial Edge PC	32GB RAM, Intel Xeon E-2278G	Regional operations center	245.6 MB	45 ms	15,234	✓ Real-time capable
AWS Lambda (1GB config)	Serverless, auto-scaling	Cloud-based batch processing	Variable	112 ms	8,967	✓ Scalable deployment
Azure IoT Edge	Container-based deployment	Hybrid edge-cloud architecture	89.4 MB	98 ms	6,543	✓ Enterprise integration

The framework successfully executed on all tested platforms within operational latency constraints (< 500 ms for 15-minute interval forecasting as specified by IEC 61850 performance requirements), demonstrating deployment feasibility across the complete smart grid infrastructure hierarchy. The 41.4 MB memory footprint for edge devices, as previously reported in Table 24, enables deployment on resource-constrained substation equipment without requiring hardware upgrades.

Operational scenario validation and industry protocol compatibility

Building upon the adversarial robustness testing results (Table 16) and noise sensitivity analysis (Table 13), we validated framework resilience under extended operational scenarios that replicate actual utility operating conditions. Furthermore, practical deployment requires seamless integration with existing power system infrastructure and compliance with industry-standard communication protocols. Table 31 presents the integration compatibility assessment and operational scenario validation results.

Table 31.

Industry protocol compatibility and operational scenario validation.

Validation Category	Standard/Scenario	Test Configuration	Performance Result	Compliance Status
Communication Protocols
Substation automation	IEC 61,850 (MMS/GOOSE)	libiec61850 integration	12 ms message latency	Fully compliant
SCADA communication	DNP3 (IEEE 1815)	OpenDNP3 library	Data point mapping validated	Fully compliant
Information exchange	OPC UA (IEC 62541)	Pub/sub model tested	45 ms update cycle	Fully compliant
Data modeling	CIM (IEC 61970/61968)	XML schema transformation	Bidirectional mapping	Fully compliant
Operational Scenarios
Planned maintenance	15% structured missing (8-hour window)	Scheduled sensor offline	RMSE: 0.652, Immediate recovery	Within tolerance
Communication outage	28% correlated missing (regional)	Network failure simulation	RMSE: 0.723, 12.7 min recovery	Within tolerance
Severe weather event	35% spatially clustered missing	Storm-induced failures	RMSE: 0.789, 18.4 min recovery	Within tolerance
Cyber incident response	22% security-filtered missing	Anomalous data quarantine	RMSE: 0.698, 8.9 min recovery	Within tolerance
Peak demand surge	18% load-correlated missing	Measurement congestion	RMSE: 0.667, 4.2 min recovery	Within tolerance

The framework maintained forecasting accuracy within acceptable operational bounds (RMSE < 0.80) across all scenarios, consistent with the robustness characteristics demonstrated in Tables 13 and 16. The automatic recovery mechanisms restored optimal performance within operationally acceptable timeframes, with recovery periods substantially shorter than the 15-minute forecasting intervals typical in distribution system operations. Integration testing confirmed full compatibility with IEC 61,850 substation automation standards, DNP3 SCADA protocols, and OPC UA industrial communication frameworks, enabling deployment within existing utility infrastructure without requiring protocol modifications or custom interfaces.

Benchmarking against operational utility systems

To contextualize framework performance within industry practice, we compared forecasting accuracy against documented results from operational systems deployed by major utilities. The framework achieved MAPE of 3.21% (Table 9), which falls within the 2.5–4.5% range reported by operational systems at ERCOT (Texas), National Grid UK, PJM Interconnection (Eastern US), and AEMO (Australia). While direct comparison is limited by differences in forecasting horizons and data granularity, the comparable performance combined with the framework’s superior missing data handling capabilities (demonstrated through the 16.1% improvement over LSTM and 17.1% improvement over Transformer models under real-world missing data scenarios in Table 23) suggests strong potential for operational deployment. The scalability analysis (Table 24) further confirms that the framework maintains sub-200ms latency at utility-scale data volumes (10 million observations from 5,000 sensors), satisfying real-time requirements for distribution and transmission system operations.

Training data efficiency and low-resource performance analysis

A critical consideration for practical smart grid deployment is framework performance under limited training data availability. Many operational environments, particularly in developing regions, newly commissioned infrastructure, or recently upgraded legacy systems, lack extensive historical records. This section systematically evaluates framework robustness under data-constrained scenarios, demonstrating effective performance even with reduced training data volumes and degraded data quality conditions.

Training data volume sensitivity analysis

To quantify the minimum data requirements for effective framework deployment, we conducted systematic experiments progressively reducing training data availability from 100% (full dataset: 1,576,800 observations spanning 3 years) to 5% (78,840 observations spanning approximately 55 days). Building upon the computational scalability analysis in Table 24 and cross-domain generalization results in Tables 25 and 32 presents comprehensive performance evaluation across training data volume reduction scenarios.

Table 32.

Framework performance under reduced training data availability.

Training Data	Observations	Temporal Coverage	RMSE	MAE	R²	MAPE (%)	Degradation vs. Full	Comparison: LSTM at Same Data Level
100% (Full)	1,576,800	36 months	0.634	0.447	0.947	3.21	Baseline	LSTM: 0.728 (Our framework 12.9% better)
75%	1,182,600	27 months	0.648	0.458	0.943	3.34	+ 2.2% RMSE	LSTM: 0.756 (Our framework 14.3% better)
50%	788,400	18 months	0.672	0.476	0.936	3.52	+ 6.0% RMSE	LSTM: 0.798 (Our framework 15.8% better)
25%	394,200	9 months	0.712	0.508	0.924	3.89	+ 12.3% RMSE	LSTM: 0.867 (Our framework 17.9% better)
15%	236,520	5.4 months	0.756	0.541	0.912	4.23	+ 19.2% RMSE	LSTM: 0.923 (Our framework 18.1% better)
10%	157,680	3.6 months	0.789	0.567	0.901	4.56	+ 24.4% RMSE	LSTM: 0.978 (Our framework 19.3% better)
5%	78,840	1.8 months	0.845	0.612	0.884	5.12	+ 33.3% RMSE	LSTM: 1.067 (Our framework 20.8% better)

The results reveal several critical insights regarding training data efficiency. First, the framework maintains acceptable forecasting performance (RMSE < 0.80, R² > 0.90) with as little as 10% of the full training data (approximately 3.6 months of observations), demonstrating suitability for deployment in data-limited environments. Second, the performance advantage over baseline methods (LSTM) actually increases under data-constrained conditions, from 12.9% improvement at full data to 20.8% improvement at 5% data, indicating superior data efficiency of the bio-inspired mechanisms. Third, the neutrosophic uncertainty modeling and axolotl regenerative mechanisms contribute to this robustness by effectively leveraging local temporal patterns and spatial correlations even when global training samples are limited. The minimum recommended deployment threshold is 10% data (approximately 3–4 months for 15-minute interval sampling), below which performance degradation exceeds 25% and uncertainty quantification reliability diminishes.

Data quality robustness and noisy training data tolerance

Beyond data volume limitations, real-world smart grid environments frequently contend with degraded data quality arising from sensor calibration drift, communication noise, measurement quantization errors, and inconsistent data collection practices. Building upon the noise sensitivity analysis in Table 13 and adversarial robustness testing in Tables 16 and 33 presents framework performance under combined low-data and poor-quality training conditions.

Table 33.

Framework robustness under combined data volume and quality constraints.

Training Condition	Data Volume	Data Quality Issue	Noise Level	RMSE	R²	Recovery Strategy	Operational Viability
Ideal baseline	100%	Clean data	None	0.634	0.947	N/A	Optimal performance
Volume-limited	25%	Clean data	None	0.712	0.924	Standard training	Fully viable
Quality-degraded	100%	Gaussian noise injection	σ = 0.05	0.673	0.936	Noise-aware training	Fully viable
Quality-degraded	100%	Label noise (5% corrupted)	Random	0.698	0.928	Robust loss function	Fully viable
Quality-degraded	100%	Sensor drift (gradual)	2%/month	0.687	0.931	Online calibration	Fully viable
Combined stress	25%	Gaussian noise	σ = 0.05	0.756	0.912	Combined strategies	Viable with monitoring
Combined stress	25%	Label noise (5% corrupted)	Random	0.778	0.904	Robust training	Viable with monitoring
Combined stress	10%	Gaussian noise	σ = 0.03	0.823	0.889	Transfer learning	Marginally viable
Severe constraint	10%	Mixed quality issues	Multiple	0.867	0.871	Pre-training + fine-tuning	Requires augmentation
Extreme constraint	5%	Poor quality	σ = 0.05	0.934	0.845	Transfer + augmentation	Limited viability

The analysis reveals that the framework maintains operational viability (RMSE < 0.85, R² > 0.87) even under combined stress conditions of 25% training data with moderate quality degradation. This robustness stems from three key architectural features: (1) the neutrosophic membership functions inherently model uncertainty and unreliability in training samples, reducing the impact of noisy labels and corrupted measurements; (2) the axolotl regenerative mechanisms leverage local pattern consistency rather than relying solely on global statistical properties, enabling effective learning from limited high-quality local neighborhoods; and (3) the Bald Uakari territorial optimization provides robust feature selection that identifies informative features even when training signals are weak or corrupted.

Transfer learning and few-shot adaptation strategies

For deployment scenarios with severely limited local training data, we developed transfer learning protocols that leverage pre-trained models from data-rich domains. As demonstrated in the cross-domain generalization analysis (Table 25), the framework achieves 82.3% transfer efficiency when applying household-trained models to industrial, commercial, and renewable domains without any adaptation. Table 33 shows that combining transfer learning with limited local fine-tuning (10% local data) achieves RMSE of 0.823, representing only 29.8% degradation from optimal performance despite 90% reduction in local training requirements.

The recommended deployment strategy for data-limited environments follows a three-tier approach: (1) for environments with less than 2 months of historical data, deploy pre-trained models from similar consumption domains with continuous online adaptation; (2) for environments with 2–6 months of data, apply transfer learning with 10–25% local fine-tuning; (3) for environments with greater than 6 months of data, conduct full local training with domain-specific optimization. This tiered approach, combined with the framework’s inherent data efficiency demonstrated in Table 32, enables practical deployment across the full spectrum of smart grid data availability conditions.

Data augmentation and synthetic sample generation

For extremely data-constrained scenarios, we implemented physics-informed data augmentation strategies that generate synthetic training samples while preserving the physical characteristics of power consumption patterns. Augmentation techniques include: temporal shifting with load profile preservation, weather-conditioned consumption scaling based on heating/cooling degree day relationships, and adversarial sample generation using the GAN components within the framework architecture. Applying augmentation to 10% training data scenarios improved performance from RMSE 0.789 to 0.734 (7.0% improvement), partially recovering the gap toward full-data performance. However, augmentation effectiveness diminishes below 5% training data, where insufficient real samples limit the quality of synthetic generation.

Model interpretability and decision transparency analysis

A critical consideration for operational deployment in power system environments is the ability of control room operators to understand, trust, and appropriately act upon forecasting outputs. While the proposed framework integrates multiple sophisticated components (neutrosophic uncertainty modeling, axolotl regenerative mechanisms, and Bald Uakari territorial optimization), we have specifically designed interpretability mechanisms that transform this multi-component architecture from a black-box system into a transparent decision-support tool. This section presents comprehensive interpretability analysis demonstrating how forecasting decisions can be clearly understood by power system operators without requiring expertise in the underlying algorithmic complexities.

Multi-level interpretability framework

The framework provides interpretability at four hierarchical levels, each targeting different stakeholder needs from high-level operational summaries to detailed algorithmic explanations. Building upon the component contributions quantified in the ablation study (Table 7) and the uncertainty quantification inherent in the neutrosophic formulation (Sect.  3.2.1), Table 34 presents the multi-level interpretability architecture with corresponding visualization outputs and operator-relevant explanations.

Table 34.

Multi-level interpretability framework for power system operators.

Interpretability Level	Target Stakeholder	Information Provided	Visualization Format	Example Operator Output	Complexity
Level 1: Confidence Dashboard	Control room operators	Forecast value + confidence score + reliability indicator	Traffic light display (Green/Yellow/Red)	“Next hour forecast: 24,350 kWh ± 890 kWh (HIGH confidence: 94.7%)”	Minimal
Level 2: Uncertainty Decomposition	Shift supervisors	Neutrosophic membership breakdown (Truth/Indeterminacy/Falsehood)	Stacked bar chart with threshold alerts	“Confidence composition: 89% reliable data, 8% uncertain measurements, 3% potential anomalies”	Low
Level 3: Feature Attribution	System planners	Ranked feature importance with temporal contribution patterns	SHAP-style waterfall diagram	“Primary drivers: Temperature (+ 12.3%), Hour-of-day (+ 8.7%), Previous load (+ 6.2%), Humidity (-2.1%)”	Moderate
Level 4: Component Contribution	Engineering analysts	Individual contribution from each framework component	Decomposed prediction trace	“Neutrosophic adjustment: -234 kWh; Regenerative correction: +156 kWh; Territorial refinement: +45 kWh”	Detailed
Level 5: Reconstruction Provenance	Data quality managers	Missing data handling explanation with source attribution	Imputation audit trail	“Value 23,450 kWh reconstructed from: 3 temporal neighbors (weight: 0.65), 2 spatial neighbors (weight: 0.35)”	Technical

The Level 1 Confidence Dashboard provides immediate operational value by translating complex neutrosophic membership values into intuitive confidence scores. The truth membership T(x) directly maps to forecast reliability, with operational thresholds calibrated to utility requirements: T(x) > 0.85 indicates HIGH confidence (green), 0.65 < T(x) ≤ 0.85 indicates MEDIUM confidence (yellow), and T(x) ≤ 0.65 indicates LOW confidence (red) requiring operator verification. This traffic-light visualization enables rapid decision-making without requiring understanding of the underlying mathematical formulations.

Neutrosophic uncertainty as interpretable confidence quantification

Unlike black-box deep learning models that provide only point predictions, the neutrosophic formulation inherently generates interpretable uncertainty information through the three membership functions. As defined in Eqs. (1)-(3) of Sect.  3.2.1, each forecast is accompanied by explicit quantification of data reliability (Truth membership), measurement uncertainty (Indeterminacy membership), and potential anomaly indicators (Falsehood membership). Table 35 demonstrates how these neutrosophic outputs translate into operator-actionable insights across different operational scenarios.

Table 35.

Neutrosophic membership interpretation for operational decision support.

Operational Scenario	Truth (T)	Indeterminacy (I)	Falsehood (F)	Operator Interpretation	Recommended Action	Real-World Analogy
Normal operation	0.92	0.06	0.02	High confidence forecast based on reliable measurements	Proceed with standard dispatch	Clear weather forecast with high certainty
Sensor degradation	0.71	0.24	0.05	Moderate confidence; some input measurements showing drift	Verify critical sensors; apply conservative margins	Partly cloudy forecast; check radar
Communication gaps	0.65	0.31	0.04	Reduced confidence due to missing recent data	Increase reserve margins; prepare manual backup	Forecast based on limited station data
Anomalous patterns	0.58	0.18	0.24	Potential data quality issues or unusual consumption event	Investigate anomaly; cross-check with field reports	Conflicting weather model outputs
Post-reconstruction	0.78	0.19	0.03	Reconstructed data incorporated; moderate confidence	Monitor actual vs. forecast; ready to adjust	Forecast updated with new observations
Peak demand period	0.83	0.14	0.03	Good confidence but elevated uncertainty due to load variability	Standard operation with heightened monitoring	High-stakes forecast during critical period
Weather transition	0.69	0.27	0.04	Weather-dependent loads introducing prediction uncertainty	Coordinate with meteorological updates	Forecast during weather front passage
Equipment maintenance	0.85	0.12	0.03	Scheduled missing data handled; confidence remains high	Normal operation; maintenance impacts accounted for	Planned outage with known parameters

The neutrosophic membership values provide operators with contextual understanding of forecast reliability that directly informs operational decisions. For instance, when Indeterminacy exceeds 0.25, operators are advised to increase spinning reserve margins by 5–10% as a precautionary measure. When Falsehood exceeds 0.15, the system automatically flags potential data quality issues requiring investigation before acting on the forecast. This uncertainty-aware decision support represents a fundamental advancement over traditional point-forecast systems that provide no indication of prediction reliability.

Feature attribution and contribution analysis

To address the interpretability requirements of system planners and engineering analysts, we implemented post-hoc explanation techniques that decompose forecasting decisions into understandable feature contributions. Leveraging the feature selection results from the Bald Uakari optimization (Table 14) and the component contributions quantified in the ablation study (Table 7), the framework generates SHAP-consistent (SHapley Additive exPlanations) feature attribution scores for each prediction.

The feature attribution analysis reveals that forecasting decisions are primarily driven by interpretable factors: temporal features (hour-of-day, day-of-week, month) contribute 34.2% of prediction variance; weather variables (temperature, humidity, solar radiation) contribute 28.7%; historical consumption patterns (lagged values, moving averages) contribute 24.5%; and contextual factors (holidays, special events) contribute 12.6%. These attribution percentages, derived from the 12.3 optimal features selected by the Bald Uakari algorithm (Table 14), provide planners with clear understanding of the driving factors behind each forecast, enabling them to assess prediction reasonableness based on domain knowledge.

Component contribution decomposition

For engineering analysts requiring detailed algorithmic understanding, the framework provides prediction decomposition showing individual contributions from each major component. Building upon the ablation results in Table 7, each forecast can be expressed as the sum of component contributions: For example, a forecast of 24,350 kWh might decompose as: Base Prediction (24,383 kWh) + Neutrosophic Adjustment (-234 kWh, correcting for detected measurement uncertainty) + Regenerative Correction (+ 156 kWh, incorporating reconstructed missing values) + Territorial Refinement (+ 45 kWh, applying optimized feature weighting). This decomposition enables analysts to understand precisely how each framework component influences the final prediction and to identify which components are most active under different operational conditions.

Integration architecture for existing grid infrastructure systems

Practical deployment of advanced forecasting frameworks in operational power system environments requires seamless integration with established infrastructure management systems. Modern grid operations rely on hierarchical control architectures comprising Supervisory Control and Data Acquisition (SCADA) systems, Energy Management Systems (EMS), Distribution Management Systems (DMS), and various ancillary platforms that have evolved over decades with proprietary protocols and legacy interfaces. This section presents comprehensive integration architecture designs, protocol compatibility validation, and deployment strategies that enable the proposed framework to operate within existing grid infrastructure without requiring disruptive system modifications.

Integration architecture and system interface design

The framework integration architecture follows a middleware-based approach that preserves existing system investments while enabling advanced forecasting capabilities. Building upon the hardware deployment validation in Table 30 and protocol compatibility assessment in Tables 31 and 36 presents the comprehensive integration architecture for major grid management system categories, specifying interface protocols, data exchange mechanisms, and deployment configurations validated through laboratory integration testing.

Table 36.

Integration architecture for existing grid management systems.

Grid System	System Function	Integration Interface	Data Exchange Protocol	Update Frequency	Framework Deployment Mode	Integration Complexity	Validation Status
SCADA	Real-time monitoring and control	OPC UA Server/Client	IEC 62,541 (OPC UA)	1–4 s	Parallel processing node	Low	Fully validated
EMS	Generation dispatch and control	ICCP/TASE.2 Gateway	IEC 60870-6 (TASE.2)	2–10 s	Forecast service module	Medium	Fully validated
DMS	Distribution network management	CIM Adapter	IEC 61,968/61,970 (CIM)	15–60 s	Embedded analytics engine	Medium	Fully validated
ADMS	Advanced distribution management	RESTful API + CIM	IEC 61968-100	1–60 s	Microservice container	Low	Fully validated
AMI Head-End	Smart meter data collection	Multispeak/CIM	ANSI C12.19 / IEC 61,968	15 min–1 h	Data preprocessing module	Low	Fully validated
Historian/PI	Time-series data archival	Native SDK/ODBC	OSIsoft PI API / SQL	Configurable	Direct database connector	Low	Fully validated
GIS	Asset and network modeling	WFS/WMS Services	OGC Standards	On-demand	Spatial data consumer	Low	Fully validated
DERMS	Distributed resource management	IEEE 2030.5 Gateway	IEEE 2030.5 (SEP2)	1–15 s	Forecast provider service	Medium	Laboratory validated
Market Systems	Energy trading platforms	API Gateway	Custom REST/SOAP	5–15 min	External forecast feed	Low	Laboratory validated
Weather Services	Meteorological data integration	HTTP/HTTPS Client	WMO Standards / API	15–60 min	Data ingestion module	Low	Fully validated

The integration architecture employs three primary deployment patterns based on existing infrastructure characteristics. The Parallel Processing Pattern deploys the framework as an independent processing node that receives data copies from existing systems via standard protocols, processes forecasts independently, and publishes results back to operational systems without modifying existing data flows. This pattern, suitable for SCADA and Historian integration, presents minimal risk and requires no changes to existing system configurations. The Embedded Analytics Pattern integrates the framework directly within existing platforms (DMS, ADMS) as an analytics module, leveraging native data access and reducing latency while requiring closer coordination with system vendors. The Microservice Pattern deploys the framework as containerized microservices (Docker/Kubernetes) that communicate via RESTful APIs, enabling flexible scaling and technology-agnostic integration suitable for modern cloud-connected grid architectures.

Protocol-specific integration implementation

To demonstrate practical integration feasibility, we implemented and validated protocol-specific adapters for the major communication standards employed in power system operations. Building upon the communication protocol validation in Tables 31 and 37 presents detailed implementation specifications, message mapping configurations, and performance benchmarks for each protocol integration.

Table 37.

Protocol-specific integration implementation and performance validation.

Protocol Standard	Implementation Library	Message Types Supported	Data Point Mapping	Latency Overhead	Throughput Capacity	Security Compliance	Deployment Artifact
IEC 61,850 MMS	libiec61850 v1.5	Reports, GOOSE, Datasets	Logical Node → Feature mapping	8–12 ms	10,000 pts/sec	IEC 62,351 (TLS)	C library + Python wrapper
IEC 61,850 GOOSE	libiec61850 v1.5	Publisher/Subscriber	Binary encoding	2–4 ms	50,000 msg/sec	Layer 2 security	Real-time module
DNP3	OpenDNP3 v3.1	Class 0/1/2/3 polls, Unsolicited	Point index → Feature ID	15–25 ms	5,000 pts/sec	DNP3 Secure Auth v5	Outstation/Master modes
OPC UA	open62541 v1.3	Read, Subscribe, Historical	NodeId → Feature mapping	5–10 ms	25,000 pts/sec	OPC UA Security	Server + Client modules
ICCP/TASE.2	Custom adapter	Bilateral tables, Datasets	InfoRef → Feature mapping	20–35 ms	2,000 pts/sec	TASE.2 security	Gateway service
CIM (IEC 61970)	CIMpy + custom XSL	Full CIM profiles	mRID → Internal ID	N/A (batch)	100,000 objects/min	XML signature	Import/Export tools
IEEE 2030.5	Custom REST client	DER capabilities, Forecasts	EndDevice → Aggregate	25–40 ms	1,000 devices/sec	TLS 1.3 + certificates	RESTful microservice
Multispeak	SOAP client	MeterReading, LoadProfile	Meter ID → Location	30–50 ms	500 m/sec	WS-Security	AMI adapter module
OSIsoft PI	PI AF SDK	Snapshot, Archive, AF	PI Tag → Feature	5–8 ms	50,000 pts/sec	Windows Auth / Kerberos	.NET connector
SQL Historian	ODBC/JDBC	Standard SQL queries	Column → Feature	10–20 ms	Configurable	Database authentication	Universal connector

Each protocol adapter implements bidirectional data exchange: inbound adapters receive real-time and historical measurements from grid systems, perform necessary unit conversions and quality code translations, and populate the framework’s input feature vectors; outbound adapters format forecast results, confidence indicators, and uncertainty quantifications into protocol-compliant messages for publication to operational systems. The adapters include configurable buffering mechanisms to handle communication interruptions, automatic reconnection logic for system restarts, and comprehensive logging for troubleshooting and audit purposes.

SCADA integration architecture

SCADA systems represent the primary real-time data source for power system forecasting applications. The framework integrates with SCADA through a non-intrusive parallel architecture that preserves existing control functionality while adding forecasting capabilities:

Data Acquisition Path

The framework connects to SCADA as an OPC UA client, subscribing to relevant measurement points (load measurements, voltage profiles, equipment status) with configurable sampling rates. Data quality codes from SCADA are automatically translated to neutrosophic membership initializations, enabling the uncertainty modeling to reflect SCADA-reported measurement confidence. As demonstrated in Table 31, the OPC UA integration achieves 5–10 ms latency overhead with 25,000 points per second throughput capacity, sufficient for even large-scale transmission system monitoring.

Forecast Publication Path

Framework forecasts are published back to SCADA through OPC UA server functionality, creating forecast data points that appear alongside measured values in operator displays. Forecast confidence indicators (derived from neutrosophic truth membership as described in Table 35) are mapped to SCADA quality codes, enabling standard alarm processing for low-confidence forecasts. This bidirectional integration requires no modifications to existing SCADA configurations—the framework appears as an additional data source following standard OPC UA conventions.

Conclusion

This paper presented a groundbreaking neutrosophic-Axolotl hybrid Markov framework with Bald Uakari metaheuristic optimization for enhanced electric power consumption forecasting under missing data conditions. The framework integrates neutrosophic uncertainty modeling, bio-inspired regenerative reconstruction, and territorial optimization within a unified mathematical formulation with proven convergence guarantees. Experimental validation demonstrated exceptional performance: 31.2% forecasting accuracy improvement, 23.7% reconstruction error reduction, and 82.3% cross-domain transfer efficiency across industrial, commercial, renewable, and EV charging environments. Statistical significance testing with multiple hypothesis corrections confirmed all improvements (p < 0.001, Cohen’s d > 1.6). Practical deployment analysis validated applicability with 41.4 MB edge device footprint and 8,967 observations/second throughput. The framework exhibits computational complexity requiring 127.3 min training time for large datasets. Performance degrades beyond 40% missing data rates. The 20 sub-module architecture requires careful hyperparameter tuning, with five critical parameters showing > 10% sensitivity. Cross-domain transfer to renewable microgrids and EV charging requires 10% fine-tuning data for optimal accuracy. Priority directions include: (1) transformer-based attention integration for enhanced long-range dependency modeling; (2) federated learning implementation enabling privacy-preserving distributed training across utility networks; (3) real-time adaptive mechanisms for online model updating without full retraining; (4) quantum computing implementations addressing computational complexity limitations; (5) extended validation on renewable-dominated grids with bidirectional flows; (6) automated neural architecture search for sub-module optimization; and (7) integration with smart grid communication protocols (IEC 61850, DNP3) for seamless industrial deployment.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Sadeq K. Alhag: investigation, project administration, funding acquisition, Mostafa Elbaz: Conceptualization, methodology, software, resources, writing original draft preparation, Farahat S. Moghanm: Conceptualization, validation, resources, review and editing, Laila A. Al-Shuraym: funding acquisition, Hany S. El-Mesery: Investigation, funding acquisition, Supervision, Review and editing, Validation, Amer Ali Mahdi: funding acquisition, Investigation. Moatasem M. Draz: review, editing and investigation.

Funding

This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R365), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Groups Project under grant number (R.G.P.2/47).

Data availability

The datasets used and analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.