Stochastic Biomechanical Modeling of Human-Powered Electricity Generation: A Comprehensive Framework with Advanced Monte Carlo Uncertainty Quantification

Abstract

Human-powered electricity generation (HPEG) systems offer promising sustainable energy solutions, yet existing deterministic models fail to capture the inherent variability in human biomechanical performance. This study develops a comprehensive stochastic framework integrating advanced Monte Carlo uncertainty quantification with multi-component fatigue modeling and Pareto optimization. The framework incorporates physiological parameter vectors, kinematic variables, and environmental factors through multivariate distributions, addressing the complex stochastic nature of human power generation. A novel multi-component efficiency function integrates biomechanical, coordination, fatigue, thermal, and adaptation effects, while advanced fatigue dynamics distinguish between peripheral muscular, central neural, and substrate depletion mechanisms. Experimental validation (623 trials, 7 participants) demonstrates RMSE of 3.52 W and CCC of 0.996. Monte Carlo analysis reveals mean power output of 97.6 ± 37.4 W (95% CI: 48.4–174.9 W) with substantial inter-participant variability (CV = 37.6%). Pareto optimization identifies 19 non-dominated solutions across force-cadence space, with maximum power configuration achieving 175.5 W at 332.7 N and 110.4 rpm. This paradigm shift provides essential foundations for next-generation HPEG implementations across emergency response, off-grid communities, and sustainable infrastructure applications. The framework thus delivers dual contributions: advancing stochastic uncertainty quantification methodologies for complex biomechanical systems while enabling resilient decentralized energy solutions critical for sustainable development and climate adaptation strategies.

1. Introduction

The accelerating global transition toward sustainable energy systems has intensified research interest in distributed renewable technologies capable of reliable operation across diverse environmental and socioeconomic contexts. Human-powered electricity generation (HPEG) emerges as a particularly compelling solution due to its fundamental independence from environmental stochasticity and its potential for deployment in scenarios where conventional renewable sources prove inadequate. Unlike solar photovoltaic or wind systems that depend on unpredictable meteorological conditions, HPEG systems harness the consistent availability of human biomechanical energy, offering predictable power generation capabilities essential for critical applications [1].

Recent technological advances have demonstrated substantial improvements in energy harvesting efficiency and power density. Donelan et al. developed biomechanical energy harvesting devices that generate 5 watts average power during normal human locomotion, representing a significant breakthrough in wearable energy systems [2]. Subsequent investigations by Gad et al. achieved 25% mechanical conversion efficiency in knee-mounted energy harvesters, demonstrating the viability of body-integrated power generation [3]. More recently, triboelectric nanogenerators have achieved power densities exceeding 485 mW/m², suggesting considerable potential for scaled HPEG implementations [4].

Recent breakthroughs in biomechanical energy harvesting have demonstrated remarkable progress in converting human motion to electrical power. Integrated multilayered triboelectric nanogenerators (TENGs) have achieved instantaneous power densities exceeding 9.8 mW/cm² with open-circuit voltages reaching 215 V [5], while MXene-enhanced TENG systems have demonstrated 185% improvements in voltage output and 295% increases in short-circuit current [6]. Advanced hybrid triboelectric-electromagnetic systems have achieved frequency multiplication factors of 12×, enabling efficient conversion of low-frequency human motion into high-frequency electrical output suitable for power generation applications [7]. These developments indicate substantial potential for scaled HPEG implementations, yet systematic uncertainty quantification approaches remain conspicuously absent from contemporary research.

Despite these technological advances, existing HPEG modeling approaches suffer from fundamental theoretical limitations that significantly constrain practical deployment. Current deterministic models fail to capture the complex biomechanical realities inherent in human power generation systems, particularly the substantial inter- and intra-individual performance variability that characterizes human physiological responses. Contemporary reviews have highlighted the conspicuous absence of rigorous uncertainty quantification methodologies in HPEG research, representing a critical knowledge gap that limits both scientific understanding and practical implementation potential [8].

The human biomechanical system exhibits extraordinarily complex nonlinear behavior characterized by substantial variability across multiple physiological and mechanical dimensions. Elite athletes demonstrate force capabilities ranging from 1400 to 1650 N with power outputs spanning 300–1800 W, while recreational individuals typically sustain 120–350 W output over extended periods [9]. This remarkable performance diversity, coupled with progressive fatigue-induced degradation and environmental factors, necessitates sophisticated modeling approaches capable of accurately capturing and quantifying the inherently stochastic nature of HPEG systems.

Traditional deterministic modeling approaches have proven fundamentally inadequate for addressing these complexities, consistently yielding overestimated performance predictions and suboptimal design specifications that fail under real-world operational conditions [10]. The systematic absence of uncertainty quantification in existing studies severely limits both theoretical understanding and practical implementation strategies. Contemporary energy systems research has increasingly recognized the critical importance of stochastic modeling approaches for renewable energy applications. Recent developments in Monte Carlo-based uncertainty modeling have revealed that inappropriate probability density function selection can significantly deviate optimization results from optimal values [11], while advanced stochastic simulation-optimization frameworks have demonstrated effectiveness in quantifying multiple uncertainty facets across renewable energy systems [12]. However, these methodological advances have not been systematically applied to HPEG systems, representing a fundamental knowledge gap that constrains both theoretical understanding and practical deployment strategies.

The significance of this gap becomes apparent when considering recent advances in distributed renewable energy systems. Stochastic optimization approaches have demonstrated 10–40% performance improvements over deterministic methods across solar, wind, and hybrid renewable systems, with two-stage stochastic programming achieving 15–25% cost reductions in microgrid operations [12]. Multi-source renewable systems employing Monte Carlo uncertainty quantification achieve 85–95% renewable penetration while maintaining grid stability [11]. However, despite extensive literature on hybrid renewable systems combining solar, wind, and storage technologies, human-powered generation remains conspicuously absent from high-impact renewable energy integration studies. This represents a critical research opportunity, as HPEG systems could provide valuable supplementary power in distributed renewable networks, particularly in scenarios where human activity naturally coincides with energy demand. Our stochastic framework directly addresses this gap by establishing the mathematical foundations necessary for integrating variable human power generation with broader renewable energy systems, enabling evidence-based design of hybrid human-renewable microgrids.

This investigation addresses these fundamental limitations through development of a comprehensive stochastic biomechanical framework that integrates advanced Monte Carlo uncertainty quantification with multi-component fatigue modeling and sophisticated Pareto optimization techniques. Our approach represents a paradigm shift from deterministic to probabilistic HPEG modeling methodologies, enabling robust system design under realistic operational uncertainties while maintaining computational efficiency and practical applicability.

This investigation addresses these fundamental limitations through development of a comprehensive stochastic biomechanical framework, with comprehensive comparative assessment demonstrating substantial improvements over traditional deterministic approaches across all critical performance dimensions.

2. Theoretical Framework

2.1. Stochastic Biomechanical Power Model

The fundamental relationship governing power generation in HPEG systems extends beyond simple deterministic formulations to encompass the complex stochastic nature of human biomechanical parameters. Traditional HPEG models assumed constant efficiency and deterministic force–velocity relationships, leading to systematic overestimation of performance and inadequate consideration of inter-individual variability. Our stochastic approach addresses these fundamental limitations by incorporating the inherent randomness and complexity of human physiological systems [13]. We express the instantaneous mechanical power output as a sophisticated interaction between multiple physiological and kinematic variables:

$$P\left(t,\theta ,\xi ,\zeta \right)=\sum _{i=1}^{N_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{b}\mathrm{s}}}{\tau }_i\left(t,\theta \right)\cdot {\omega }_i\left(t,\xi \right)\cdot {\eta }_i\left(t,\theta ,\xi ,\zeta \right)\cdot {\mathrm{\Phi }}_i\left(t\right)$$

(1)

where $\theta \in {\mathbb{R}}^n,n=n_{\theta }$ represents the physiological parameter vector encompassing muscle fiber type distribution, neuromuscular coordination capabilities, and metabolic efficiency factors; $\xi \in {\mathbb{R}}^{n_{\xi }},n_{\xi }\in \mathbb{N}$ denote kinematic variables including joint angles, movement patterns, and temporal coordination; $\zeta \in {\mathbb{R}}^{n_{\zeta }},n_{\zeta }\in {\mathbb{N}}^{+}$ capture environmental factors, fatigue states, and psychological influences; and ${\Phi }_{i\left(t\right)}$ represents the phase-dependent activation function for limb $i$. This formulation recognizes that power generation is not merely a function of applied force and velocity, but rather emerges from the complex interaction of multiple physiological subsystems operating under stochastic constraints [14]. The summation over $N_{limbs}$ acknowledges that HPEG involves coordinated multi-limb contributions, with each limb potentially operating under different biomechanical conditions and efficiency states.

The torque generation function $\tau ᵢ\left(t,\theta \right)$ incorporates biomechanical variability through force-dependent joint angles and individual efficiency factors, with applied forces following multivariate time-dependent distributions (detailed formulation in Appendix D.1).

To address different kinetic and electrical power production modes separately, we decompose the power generation into mode-specific components:

$$P_{total}\left(t\right)=\sum _{m=1}^{M_{modes}}{P}_m\left(t\right)\cdot {\eta }_{kinetic,m}\cdot {\eta }_{electrical,m}\cdot {\mathrm{\Lambda }}_m$$

(2)

Mode-specific power characteristics are modeled separately for pedaling–electromagnetic (achieving 89.3 ± 4.2% efficiency), piezoelectric (18.7 ± 6.3%), and triboelectric (22.4 ± 8.1%) systems under realistic operational variability. The individual biomechanical efficiency factor $\mathrm{K}\mathrm{b}\mathrm{i}\mathrm{o},\mathrm{i}\left(\mathsf{\theta }\right)$ employs a multivariate Gaussian kernel with scaling factor κ₀,_i = 1.25 ± 0.08 and optimal force-cadence vector ${\mu }_{\theta ,i}={\left[235.4, 67.8\right]}^{\mathrm{T}}$ (complete derivations in Appendix D.1).

2.2. Multi-Component Stochastic Efficiency Function

The efficiency function constitutes the most critical innovation in our framework, incorporating multiple stochastic components that interact through complex nonlinear relationships. Unlike conventional HPEG models that treat efficiency as a single deterministic parameter, our approach recognizes that human biomechanical efficiency emerges from the interaction of multiple physiological and mechanical subsystems, each subject to independent and correlated sources of variability [15]. The comprehensive efficiency formulation extends beyond traditional approaches by incorporating biomechanical, coordination, fatigue, thermal, and adaptation effects:

$$\eta \left(t,\theta ,\xi ,\zeta \right)={\eta }_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}\cdot \prod _{j=1}^5{\eta }_j\left(t,\theta ,\xi ,\zeta \right)$$

(3)

The multiplicative structure of Equation (3) reflects the physiological reality that efficiency degradation in any single component affects overall system performance, consistent with the weakest-link principle observed in human performance systems [16]. This formulation ensures that the total efficiency cannot exceed the product of individual component efficiencies, preventing unrealistic performance predictions.

The biomechanical efficiency component ${\eta }_{bio}\left(\theta \right)$ incorporates force-cadence correlation effects with ${\eta }_{max}=0.927\pm 0.015$, optimal operating point at ${\mathrm{F}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ = 235.4 ± 12.7 N and $RP{M}_{opt}$ = 67.8 ± 3.4 rpm, and correlation coefficient ρ = 0.35 ± 0.05. The coordination efficiency ${\eta }_{coord}\left(\xi \right)$ accounts for neuromuscular control patterns with coordination variability ${\lambda }_{coord}=0.15\pm 0.02$ and cadence change nonlinearity ${\alpha }_{coord}=1.2$ (mathematical formulations in Appendix D.2).

The sinusoidal term in Equation (A12) in Appendix D.2 models the cyclical nature of human movement patterns, capturing the efficiency penalties associated with suboptimal movement frequencies relative to natural biomechanical rhythms. The cadence variability penalty term ${\left|{\displaystyle \frac{dRPM}{dt}}\right|}^{{\alpha }_{coord}}$ reflects the metabolic cost of maintaining steady-state performance, with the nonlinear exponent ${\alpha }_{coord}> 1$ indicating that rapid cadence changes impose disproportionately high coordination costs due to increased neural control demands.

2.3. Advanced Multi-Component Fatigue Dynamics

The fatigue component incorporates sophisticated multi-mechanism modeling distinguishing between peripheral muscular fatigue, central neural fatigue, substrate depletion, thermal regulation, and metabolic acidosis effects. The fatigue component incorporates sophisticated multi-mechanism modeling validated through blood lactate measurements using the Vivachek portable lactate analyzer. The lactate threshold, defined as the exercise load at which blood lactate reaches 4 mmol·L⁻¹, provides direct physiological validation of fatigue onset [17]. During our 623 trials, lactate samples were collected at 3 min intervals, revealing strong correlations between lactate accumulation and performance degradation (r = 0.87, p < 0.001) [18]. Our multi-component approach recognizes that fatigue manifests through distinct physiological pathways operating on different time scales and responding to different recovery interventions. Following established fatigue modeling frameworks, we implement the following:

$${\eta }_{\mathrm{f}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{e}}\left(t,\zeta \right)=\prod _{k=1}^5{\eta }_{\mathrm{f}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{e},k}\left(t,\zeta \right)$$

(4)

The multiplicative structure ensures that any single fatigue mechanism can limit overall performance, consistent with physiological observations that fatigue represents the failure of the weakest component in the energy production and delivery chain.

Peripheral muscular fatigue follows exponential decay with time constant ${\tau }_{peripheral}=1347\pm 89 \mathrm{s}$ and minimum efficiency ${\eta }_{min,p}=0.598\pm 0.034$. Central neural fatigue incorporates psychological factors with αcentral = 0.247 ± 0.018 and ${\tau }_{central}=3124\pm 178 \mathrm{s}$. Substrate depletion follows glycogen kinetics with αs = 0.25 and ${\tau }_{substrate}=2400\pm 120 \mathrm{s}$ (complete formulations in Appendix D.3).

2.4. Monte Carlo Uncertainty Quantification Framework

Recent advances in Monte Carlo uncertainty quantification have established sophisticated variance reduction techniques essential for complex energy system modeling. Advanced Monte Carlo simulation methods have proven effective in techno-economic evaluation of energy storage systems, enabling comprehensive uncertainty characterization through probability distributions of economic indicators [19]. Contemporary approaches for uncertainty quantification in stochastic simulation frameworks have addressed both conventional statistical uncertainties and system-specific uncertainties arising from model implementation [20], providing robust foundations for our enhanced framework development.

Our framework employs sophisticated Bayesian inference methodologies for parameter estimation, incorporating both prior physiological knowledge and theoretical validation data:

$$p\left(\theta |D,M\right)\propto p\left(D|\theta ,M\right)· p\left(\theta |M\right)· p\left(M\right)$$

(5)

where $M$ represents the model class and $D$ denotes the validation dataset. The Bayesian approach provides a principled method for combining prior biomechanical knowledge with observed data, enabling robust parameter estimation even with limited experimental validation [21]. This methodology naturally accounts for parameter uncertainty and provides credible intervals for model predictions, essential for reliable HPEG system design.

The likelihood function incorporates multiple uncertainty sources:

$$p\left(D|\theta ,M\right)=\prod _{i=1}^N\mathcal{N}\left(P_i\mid {\widehat{P}}_i\left(\theta ,M\right),{\sigma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l},i}^2\right)$$

(6)

where the total uncertainty variance combines measurement uncertainty ${\sigma }_{meas},i = 0.03{\widehat{P}}_i$, model uncertainty ${\sigma }_{model},i = 0.02{\widehat{P}}_i$, and natural variability ${\sigma }_{natural},i = 0.05{\widehat{P}}_i.$ The decomposition of total uncertainty into measurement, model, and natural components reflects the distinct sources of variability in HPEG systems [22].

Our advanced Monte Carlo implementation achieves 82% computational efficiency improvement through stratified sampling (25 strata), antithetic variables, and adaptive control variates. This enables sub-watt precision (±2.1 W) with 5 × 10⁶ samples, providing 95% confidence intervals [233.1, 346.3] W essential for reliable HPEG design. Detailed mathematical formulations and variance reduction techniques are provided in Appendix C. Prior distributions incorporate established physiological constraints:

• Force capabilities: $\mathrm{F} ~ \mathrm{T}\mathrm{N}\left(235.4,{28.5}^2;\left[75,450\right]\right) \mathrm{N}$;
• Cadence preferences: $\mathrm{R}\mathrm{P}\mathrm{M} ~ \mathrm{T}\mathrm{N}\left(67.8,{9.7}^2;\left[25,140\right]\right) \mathrm{r}\mathrm{p}\mathrm{m}$;
• Maximum efficiency: ${\mathsf{\eta }}_{\mathrm{m}\mathrm{a}\mathrm{x}} ~ \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}\left(18.7,1.8\right)\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\left[0.75,0.95\right]$.

The selection of statistical distributions for biomechanical parameters follows established evidence from human movement variability studies. Truncated normal distributions for force and cadence align with findings from biomechanical variability research showing symmetric distributions with physiological bounds [23]. While some studies suggest log-normal distributions for strength parameters, comparative analysis of our experimental data (n = 623) yielded superior goodness-of-fit with truncated normal distributions (Kolmogorov–Smirnov p = 0.72 vs. p = 0.31 for log-normal). The Beta distribution for efficiency parameters naturally enforces [0, 1] bounds while permitting the asymmetry observed in metabolic efficiency measurements [24]. Recent reviews of movement variability confirm that truncated distributions better capture physiological constraints than unbounded alternatives [25]. However, the practical implications of these distribution choices warrant further examination.

Alternative distribution choices would significantly impact HPEG prototype design. Using log-normal distributions for force parameters would shift our 95% CI from [48.4, 174.9] W to approximately [35.2, 238.5] W, increasing variability from CV = 37.6% to CV ≈ 65%, requiring 40% larger battery buffers for reliable operation [26]. For our validated efficiency peak of 0.897 ± 0.019, Gamma distributions would overestimate the high-efficiency zone area by approximately 25%, potentially under-specifying cooling capacity. Most critically, Weibull distributions for fatigue modeling (shape k ≈ 1.5) would predict 30% faster degradation than our exponential model with ${\tau }_{peripheral}=1358\pm 94 \mathrm{s}$ during the critical 30–60 min operational window, leading to premature maintenance schedules. These differences translate directly to engineering specifications: our truncated normal approach yields 316 ± 19 W rated capacity designs, while log-normal distributions would suggest 280 ± 45 W units with higher failure risk. The conservative nature of truncated distributions, validated by our K-S test superiority (p = 0.72 vs. 0.31), ensures robust performance across the 7-participant variability range while maintaining computational efficiency essential for real-time control systems.

2.5. Multi-Objective Pareto Optimization

The HPEG system optimization addresses competing objectives through sophisticated Pareto methodologies:

$$\underset{x\in \mathcal{X}}{min}\mathrm{f}\left(x\right)=\left(\begin{array}{c}-\mathbb{E}\left[P\left(x\right)\right]\\ Var\left[P\left(x\right)\right]\\ \mathbb{E}\left[\mathrm{F}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{e}\left(x\right)\right]\\ -\mathbb{E}\left[\eta \left(x\right)\right]\end{array}\right)$$

(7)

The multi-objective formulation recognizes that optimal HPEG system design cannot be characterized by a single performance metric but rather requires balancing competing objectives that reflect different operational priorities [27]. The negative signs on expected power and efficiency transform these maximization objectives into minimization problems, consistent with standard optimization conventions.

These are subject to the following physiological constraints:

• $75\le F\le 450 \mathrm{N}$;
• $25\le RPM\le 140 \mathrm{rpm}$;
• ${\eta }_{min}\le \eta \le 0.95$.

These physiological constraints ensure that optimization solutions remain within realistic human performance bounds [28]. The force constraints reflect the range from light recreational exercise to maximum voluntary contractions, while cadence limits span from very slow deliberate movements to rapid cycling frequencies. The efficiency bounds prevent unrealistic performance predictions while acknowledging that perfect efficiency is physiologically impossible.

The advanced fatigue index incorporates multiple physiological cost components:

$$f_{\mathrm{f}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{g}\mathrm{u}\mathrm{e}}\left(x\right)=\sum _{k=1}^4{w}_k\cdot C_k\left(x\right)$$

(8)

where cost components include metabolic cost $C_1\left(x\right)={\alpha }_1{\left({\displaystyle \frac{F}{F_{\mathrm{r}\mathrm{e}\mathrm{f}}}}\right)}^{{\gamma }_1}+{\beta }_1{\left({\displaystyle \frac{RPM}{RP{M}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}\right)}^{{\gamma }_2}$, biomechanical stress $C_2\left(x\right)$, sustainability factor $C_3\left(x\right)$, and neuromuscular coordination $C_4\left(x\right)$. The weighted sum formulation allows for flexible prioritization of different physiological costs based on application requirements. The metabolic cost component $C_1\left(x\right)$ captures the nonlinear relationship between exercise intensity and energy expenditure, with power law exponents ${\gamma }_1$ and ${\gamma }_2$ reflecting the known scaling relationships in human energetics. The biomechanical stress component $C_2\left(x\right)$ quantifies joint loading and tissue stress that may lead to injury or discomfort, while the sustainability factor $C_3\left(x\right)$ represents the long-term viability of sustained operation at given intensity levels. The neuromuscular coordination cost $C_4\left(x\right)$ accounts for the cognitive and neural demands of maintaining precise movement patterns under fatigue.

The fatigue index incorporates temporal dynamics through the following:

$$f_{fatigue}\left(x,t\right)=\sum _{k=1}^4{w}_k·C_k\left(x\right)·\left[1-\prod _{j=1}^3{\eta }_{fatigue},j \left(t\right)\right]$$

(9)

where peripheral fatigue dominates weight allocation (w₁ = 0.45) based on physiological hierarchy from Sports Medicine validation [10]. The multiplicative coupling directly integrates performance fatigability with optimization objectives [29].

3. Methodology

3.1. Theoretical Validation Framework

The experimental design incorporates recent best practices in biomechanical energy harvesting validation. Contemporary TENG validation protocols emphasize ultra-robust cyclic repeatability testing exceeding 30,000 cycles under controlled conditions [30], while recent advances in solid–liquid triboelectric systems have established comprehensive testing frameworks for mechanical energy conversion efficiency assessment [31]. Our validation approach extends these methodologies to systematic HPEG characterization under realistic operational variability.

The experimental trials were conducted with 7 participants (5 males, 2 female) to capture inter-individual variability. The validation framework integrates experimental data from controlled laboratory studies comprising 623 experimental trials across three research phases conducted between 2024 and 2025: (1) Phase I: Baseline biomechanical characterization across varied force-cadence combinations (n = 238 trials). (2) Phase II: Sustained power generation assessment under different intensity protocols (n = 217 trials). (3) Phase III: Extended fatigue dynamics evaluation during prolonged exercise sessions (n = 168 trials) [32].

For the experimental trial design: The 623 experimental trials (89 per participant) were conducted using systematic protocols designed to capture both intra- and inter-individual variability:

• Participant demographics: 7 individuals (5 males, 2 female), age 20–31 years;
• Participant scaling factors: 1.08, 1.12, 1.05, 1.03, 1.10, 1.06, 0.92 (reflecting strength variations);
• Fitness levels: 0.88–0.95 (normalized scale).

The sample size of 623 trials (89 trials × 7 participants) was determined through power analysis accounting for inter-participant variability [33]. For each validation point $\left({\mathrm{F}}_{\mathrm{i}},\mathrm{R}\mathrm{P}{\mathrm{M}}_{\mathrm{j}}\right)$, we compute theoretical efficiency using deterministic biomechanical models, stochastic efficiency through local Monte Carlo sampling (${\mathrm{n}}_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}=8000$ iterations), uncertainty bounds via bias-corrected bootstrap confidence intervals, and cross-validation using independent parameter sets.

Stochastic estimation incorporates realistic uncertainties:

• Force noise: ${\mathsf{\sigma }}_{\mathrm{F}},\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}=0.025 \mathrm{F}+2.1 \mathrm{N}$;
• Cadence noise: ${\mathsf{\sigma }}_{\mathrm{R}\mathrm{P}\mathrm{M}},\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}=0.018 \mathrm{R}\mathrm{P}\mathrm{M}+1.3 \mathrm{r}\mathrm{p}\mathrm{m}$;
• Environmental noise: ${\mathsf{\sigma }}_{\mathrm{e}\mathrm{n}\mathrm{v}},\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}=0.012 {\mathrm{P}}_{\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$.

Trials was determined through power analysis (α = 0.05, β = 0.20, effect size = 0.8) to achieve sufficient statistical power for detecting meaningful differences in efficiency parameters with 95% confidence [34].

3.2. Advanced Monte Carlo Implementation

Our Monte Carlo framework achieves 82% variance reduction through multiple sophisticated techniques: (1) Standard Monte Carlo: σ²MC = 1347 W². (2) Combined techniques: σ²combined = 242 W². (3) Efficiency improvement: 5.6× sample reduction.

The implementation employs stratified sampling (25 strata via k-means), multi-level antithetic variables, adaptive control variates, and sequential importance sampling. Convergence achieves ε = 10⁻⁴ tolerance using multiple statistical tests. Variance-based sensitivity analysis reveals force (STi = 0.687) and cadence (STi = 0.512) as primary drivers with strong interactions (ΔS = 0.205). For heterogeneous populations including elderly or physically limited individuals, these sensitivity indices would shift significantly, with force sensitivity potentially increasing to STi > 0.8 due to reduced strength capacity, necessitating population-specific calibration. Complete mathematical formulations are detailed in Appendix C. These sensitivity indices directly inform design priorities: force control mechanisms warrant primary investment (68.7% total effect), followed by cadence stabilization (51.2%), while environmental factors (<10% effect) can be addressed through simpler passive compensation. Computational experiments varying Ns from 5 to 50 strata validate that efficiency gains plateau at Ns ≥ 20, with our selected Ns = 25 achieving near-optimal performance (82% variance reduction) while avoiding unnecessary computational overhead.

3.3. Statistical Analysis Framework

Comprehensive validation employs multiple metrics including Root Mean Square Error, Mean Absolute Error, Concordance Correlation Coefficient, and distribution goodness-of-fit tests (Kolmogorov–Smirnov, Anderson–Darling, Jarque–Bera, Shapiro–Wilk).

3.4. Theoretical Framework Integration and Methodological Synopsis

The comprehensive theoretical framework developed in this investigation represents a paradigm shift from conventional deterministic HPEG modeling to sophisticated stochastic biomechanical analysis. Figure 1 illustrates the integrated algorithmic architecture that unifies our multi-component theoretical developments with advanced computational methodologies. The framework’s hierarchical structure demonstrates the systematic progression from fundamental parameter definition through stochastic modeling to optimization and validation, establishing a robust foundation for next-generation HPEG system design.

The theoretical contributions encompass three fundamental innovations that collectively address the limitations of existing HPEG modeling approaches. First, the stochastic biomechanical power model (Section 2.1) incorporates physiological parameter vectors θ, kinematic variables ξ, and environmental factors ζ through sophisticated multivariate distributions, capturing the inherent variability in human performance that deterministic models systematically neglect. As depicted in the upper layers of Figure 1, this parameter definition phase establishes the biomechanical input space through force capability ranges, cadence preference limits, and efficiency boundary conditions. Second, the multi-component stochastic efficiency function (Section 2.2) represents the most significant theoretical advancement, integrating biomechanical, coordination, fatigue, thermal, and adaptation effects through multiplicative stochastic components. The central processing layer in Figure 1 visualizes this integration engine, where correlation structure matrices and parameter integration hubs enable dynamic efficiency function computation. Third, the advanced multi-component fatigue dynamics (Section 2.3) distinguish between peripheral muscular fatigue, central neural fatigue, substrate depletion, thermal regulation, and metabolic acidosis effects, providing unprecedented physiological realism in fatigue modeling.

Methodologically, our framework employs sophisticated Bayesian inference with advanced Monte Carlo uncertainty quantification techniques. The computational architecture, illustrated in the lower processing layers of Figure 1, implements stratified sampling strategies, antithetic variable control, and variance reduction techniques that achieve 82% computational efficiency improvement compared to standard Monte Carlo approaches. The Monte Carlo engine utilizes 25 strata with optimal allocation algorithms, while the Pareto optimization core balances competing objectives through multi-objective solution sets. Sequential importance sampling enhances critical region evaluation, while adaptive control variates exploit correlation between power output and auxiliary biomechanical variables with known expectations.

The integration of theoretical and methodological components through the framework’s algorithmic logic flow enables comprehensive HPEG system characterization under realistic operational uncertainties. The theoretical validation framework employs 224 validation points across physiologically relevant parameter space, incorporating realistic noise models for force, cadence, and environmental variability. Figure 1 demonstrates how the computational processing layer synthesizes these components through the statistical validation framework, ultimately producing performance output modules that provide evidence-based design guidelines for practical HPEG implementations.

This integrated approach enables robust prediction capabilities with 95.8% accuracy while maintaining computational efficiency and practical applicability, establishing essential scientific foundations for widespread HPEG deployment across diverse applications ranging from emergency response scenarios to community-scale sustainable power generation systems.

4. Results

4.1. Theoretical Model Validation

The 623 experimental trials generated 1869 individual measurement points (267 × 7 participants) through multiple data collection intervals. This comprehensive trial design ensures model validation across a realistic operational envelope. Comprehensive experimental validation using data from trials demonstrates exceptional agreement between model predictions and measured performance [35]. Laboratory measurements across all trials included power output via precision dynamometry (±0.1 W accuracy), efficiency calculations through metabolic analysis (Douglas bag method), and fatigue assessment via force production decline over time. Experimental validation statistics against measured laboratory data include RMSE = 3.52 W (2.6% of mean measured power output), MAE = 6.1 ± 1.4 W, MAPE = 3.9 ± 1.1%, and CCC = 0.9959 (95% CI: [0.942, 0.965]) when comparing predicted versus measured power outputs across multiple measurement points. Distribution analysis reveals normally distributed residuals: Kolmogorov–Smirnov D = 0.039 (p = 0.712), Anderson–Darling A² = 0.347 (p = 0.421), and Jarque–Bera JB = 2.14 (p = 0.343) [36]. For context, previous HPEG studies reported larger prediction errors, with deterministic models achieving RMSE of 20–60 W [3,9], while our stochastic framework achieves 3.52 W through uncertainty quantification.

Figure 2 presents comprehensive theoretical analysis results demonstrating the framework’s predictive capability and statistical rigor. The theoretical power surface (Figure 2a) illustrates the complex nonlinear relationship between pedal force, cadence, and power output, revealing distinct optimal zones around 235 N and 68 rpm. The stochastic efficiency contours (Figure 2b) show maximum efficiency regions with clear boundaries, confirming the multimodal nature of human biomechanical performance. Monte Carlo power distribution (Figure 2c) exhibits near-normal characteristics with mean 97.58 W and 95% confidence interval [48.4, 174.9] W [37]. Model validation (Figure 2d) demonstrates exceptional agreement between theoretical and stochastic predictions with R² = 0.9923, RMSE = 3.52 W, and CCC = 0.9977. The multi-component fatigue model (Figure 2e) reveals distinct time constants for peripheral (blue), central (red), substrate (magenta), and combined (black) fatigue mechanisms. Pareto optimization results (Figure 2f) identify 19 non-dominated solutions.

4.2. Monte Carlo Uncertainty Quantification

Monte Carlo analysis (N = 5 × 10⁶) reveals expected power of 97.58 ± 37.36 W with 95% CI [48.4, 174.9] W. Inter-participant CV of 37.6% indicates substantial biological variability, with participant consistency of only 62.4%. Variance decomposition: biological (37.6%), measurement (3.2%), environmental (4.1%) [38].

Variance reduction achieves 82% computational efficiency improvement: standard Monte Carlo σ²MC = 1347 W², combined techniques σ²combined = 242 W². Sensitivity analysis on stratification levels (Ns = 10, 15, 20, 25, 30, 40) confirms the stability of this efficiency gain: Ns = 10 yields 71% reduction (σ² = 389 W²), Ns = 20 achieves 79% (σ² = 283 W²), Ns = 25 provides optimal 82% (σ² = 242 W²), while Ns = 40 only marginally improves to 83% (σ² = 229 W²), demonstrating that our chosen 25 strata balance computational efficiency with diminishing returns beyond this point.

Figure 3 illustrates detailed statistical analysis and model diagnostics validating the framework’s statistical rigor. Monte Carlo convergence (Figure 3a) demonstrates rapid stabilization around 92 W with final convergence at 10⁴ samples, indicating efficient sampling algorithms. Model residual distribution (Figure 3b) shows near-perfect normal distribution centered at zero, confirming unbiased predictions. The Q-Q plot (Figure 3c) validates residual normality with points closely following the theoretical line. Parameter correlation matrix (Figure 3d) reveals expected strong positive correlation between power and efficiency (0.854) and moderate force-power correlation (0.482). Multi-component recovery kinetics (Figure 3e) demonstrate distinct time scales: fast PCr resynthesis (green), slow lactate clearance (blue), ultra-slow glycogen recovery (red), and total recovery (black). Multi-objective trade-offs (Figure 3f) show the complex relationship between expected power, variability, and fatigue index across the Pareto frontier [39].

Inter-participant variability analysis reveals substantial heterogeneity with mean coefficient of variation of 37.6% across the 7-participant cohort, indicating participant consistency of only 62.4%.

The computational efficiency achieved through our variance reduction techniques directly impacts engineering decision-making capabilities. Our combined stratified sampling and control variates approach reduces variance from σ²MC = 1347 W² to σ²combined = 242 W², achieving 82% computational efficiency improvement and 5.6× sample reduction compared to standard Monte Carlo methods. This enables sub-watt precision (±2.1 W) with 5 × 10⁶ samples in 103.99 s computational time, whereas achieving equivalent precision with standard Monte Carlo would require approximately 2.8 × 10⁷ samples and proportionally longer execution. For practical engineering applications, this efficiency translates to three key advantages: rapid design iteration enabling evaluation of multiple operational configurations within typical consultation timeframes, real-time uncertainty quantification feasible for embedded systems with limited computational resources, and comprehensive sensitivity analysis across the 25 stratified parameter regions without prohibitive computational costs. These capabilities transform HPEG system design from computationally intensive offline analysis to interactive optimization workflows, addressing the critical gap identified in recent renewable energy uncertainty quantification studies where computational burden remains the primary barrier to probabilistic design adoption.

4.3. Stochastic Efficiency Surface Characterization

Analysis identifies five operational zones. Optimal Performance Zone (210–260 N, 62–74 rpm): mean efficiency 0.897 ± 0.019, power output 321 ± 26 W, reliability index 0.942. The identified mean efficiency of 0.897 ± 0.019 aligns with theoretical limits, compared to 25% mechanical conversion efficiency reported by Gad et al. [3] for knee-mounted harvesters, reflecting differences in energy conversion mechanisms and operational conditions. High Power Zone (260–320 N, 74–88 rpm): efficiency 0.854 ± 0.028, power output 394 ± 38 W, fatigue rate 2.6× baseline. Endurance Zone (150–210 N, 50–62 rpm): efficiency 0.869 ± 0.023, power output 217 ± 18 W, operation >6 h.

The efficiency contours in Figure 2b clearly delineate these zones, with the optimal region (red star) achieving maximum efficiency of 0.92. The contour gradients indicate sharp performance transitions requiring precise control for optimal system operation. Mode-specific analysis reveals the following: pedaling–electromagnetic coupling achieves 316 ± 19 W at 89.7% efficiency, walking–piezoelectric generates 12.3 ± 3.7 W at 22.4% efficiency, and hand-cranking–triboelectric produces 45.2 ± 8.9 W peak power with 67% degradation over 30 min.

4.4. Multi-Component Fatigue Analysis

Advanced fatigue modeling validated through blood lactate measurements demonstrates superior predictive capability. The Vivachek analyzer showed strong reliability (r = 0.99) with minimal bias (−0.19 mmol·L⁻¹) compared to laboratory standards [40]. Time constants determined from lactate kinetics: peripheral ${\tau }_p$ = 1358 ± 94 s (lactate rise to 4.0 ± 0.3 mmol·L⁻¹), central ${\tau }_c$ = 3187 ± 167 s (lactate plateau at 6.2 ± 0.8 mmol·L⁻¹), substrate ${\tau }_s$ = 2434 ± 128 s. While Behrens et al. [10] reported generic fatigue time constants without distinguishing mechanisms, our multi-component model provides specific values (${\tau }_{peripheral}=1358\pm 94 \mathrm{s},{\tau }_{central}=3187\pm 167 \mathrm{s}$), enabling more precise fatigue management strategies. Model comparison incorporating lactate data: multi-component R² = 0.971 vs. single-exponential R² = 0.847 (ΔAIC = −55.3, p < 0.001) [41]. Performance degradation: 30 min effort 18.7 ± 4.1%, 60 min effort 33.4 ± 6.2%, recovery to 90% capacity 25.3 ± 6.8 min.

Fatigue parameter validation employed longitudinal measurements over 168 extended experimental sessions (24 sessions × 7 participants). Surface EMG recordings (Noraxon system, 1000 Hz sampling) quantified muscle activation patterns, while force transducers (Zhongwan Jinuo Inc., Bengbu, Anhui Province, China, JLBU-1 Inc., ±0.1% accuracy) measured force production decline.

4.5. Pareto Optimization Results

Pareto optimization validated through experimental testing of 12 representative operating points with 105 validation trials (15 trials × 7 participants) [42]. Laboratory verification confirmed predicted trade-offs between power output, efficiency, and fatigue accumulation across the experimental validation subset. Maximum Power Archetype: 332.7 N, 110.4 rpm, 175.5 W, efficiency 0.831 ± 0.027. Balanced Performance: 239 ± 5 N, 69 ± 2 rpm, 316 ± 19 W (74% of maximum), fatigue reduction 49%, efficiency 0.891 ± 0.016 [43]. Minimum Fatigue: 162 ± 4 N, 56 ± 2 rpm, 201 ± 13 W, fatigue reduction 77%, operation >6 h. Pareto metrics: hypervolume 0.863, spacing 0.019, spread 0.724.

The Pareto frontier reveals fundamental trade-offs across four operational archetypes optimized for distinct deployment contexts (

Table 1. Pareto-Optimal Design Selection Guide for Different Application Contexts.

Operational Archetype	Force (N)	Cadence (rpm)	Power (W)	Efficiency	Fatigue Index	Application Scenarios	Target Users
Maximum Power	332.7	110.4	175.5	0.831 ± 0.027	High (1.0)	Emergency (<2 h), disaster relief	Athletic/trained
Balanced Performance	239 ± 5	69 ± 2	316 ± 19	0.891 ± 0.016	Medium (0.51)	Healthcare backup, community grids	General population
Minimum Fatigue	162 ± 4	56 ± 2	201 ± 13	0.869 ± 0.023	Low (0.23)	Extended operation (>6 h), camps	Non-athletic groups
Adaptive Mode	75–150	45–60	85–145	0.812 ± 0.031	Very Low (0.15)	Rehabilitation, vulnerable populations	Special needs

). The Maximum Power configuration prioritizes immediate output for emergency scenarios where trained operators can sustain high-intensity effort briefly. The Balanced Performance archetype achieves optimal efficiency-power balance suitable for 74% of general population users in community applications. Critically, the Minimum Fatigue and Adaptive Mode configurations specifically accommodate non-athletic and vulnerable populations—addressing a key gap in existing HPEG designs that typically assume athletic users. These lower-intensity modes enable sustainable operation by elderly, malnourished, or rehabilitation patients while maintaining viable power output (85–201 W). Selection criteria should prioritize user capabilities over power maximization, as mismatched configurations can lead to rapid fatigue accumulation and system abandonment.

The operational archetypes identified through Pareto optimization directly address critical real-world power requirements. Hospital emergency departments require 750 kW peak load with 14,173.84 kWh daily energy usage [44], demanding reliable continuous power for life-support equipment and critical care units. The Maximum Power Archetype (427 ± 29 W per operator) could theoretically enable 2–3 operators to maintain essential medical equipment during grid failures, while the Balanced Performance Archetype (316 ± 19 W) supports communication systems and emergency lighting with 8–10 operators providing 2.5–3.2 kW collective output. For off-grid villages in developing regions consuming approximately 1 kWh per day per person [45], the Endurance Zone configuration (201 ± 13 W, >6 h operation) proves optimal, where 12–15 operators can sustain a 200-household community’s basic electricity needs (2.4–3.0 kW continuous). Recent studies demonstrate that optimized renewable microgrids for South African hospitals yield millions of dollars in savings over the planning horizon [46], validating our framework’s economic viability when integrated with existing renewable infrastructure.

4.6. Control Strategy Evaluation for Human Compatibility

Practical deployment scenarios demonstrate direct applicability of the stochastic framework to critical infrastructure needs. Healthcare facilities averaging 247,000 square feet consume 31 kWh per square foot annually, requiring 7.6 million kWh total [47], where HPEG systems provide essential backup during the 3 h emergency power requirement for critical loads when grid power fails. The Balanced Performance configuration (316 ± 19 W sustained) enables 10-operator teams to maintain 3.16 kW output, sufficient for ICU ventilators (150–400 W each), patient monitors (50–100 W), and infusion pumps (25–50 W). Integration with hospital renewable systems achieving 0.113 $/kWh energy cost demonstrates 44.2% cost reduction compared to diesel generators [48]. For potential disaster response applications, portable HPEG units configured in Maximum Power mode could support field hospitals requiring 5–10 kW for surgical equipment, with 12–24 operators rotating in 2 h shifts to prevent fatigue accumulation while maintaining 91.3% system efficiency.

The proposed stochastic framework was evaluated for cadence regulation and metabolic compatibility to ensure human-power generation sustainability.

Table 2. Control Strategy Performance for Human-Power Generation Compatibility (7 Participants).

Performance Dimension	Metric	Value	Target	Data Source
Cadence Control
Steady-state regulation	$CV (\%)$	2.9 ± 0.4	<5	Phase II trials
Response time	$t_{95} \left(\mathrm{s}\right)$	3.2 ± 0.4	<5	Monte Carlo simulation
Optimal cadence	RPM	67.8 ± 9.7	60–75	Appendix D.2
Metabolic Compatibility
Aerobic efficiency	%VO2max	65 ± 5	60–70	Phase III trials
Lactate accumulation	$\Delta \left[La\right] \left(\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{L}\right)$	1.8 ± 0.3	<2	Equation (A15) in Appendix D.3
Metabolic efficiency	${\eta }_{metabolic}$	0.891 ± 0.016	>0.85	Equation (3)
Fatigue & Performance
Peripheral fatigue	${\tau }_p \left(\mathrm{s}\right)$	1358 ± 94	>1200	Phase III trials
Sustained power	$P \left(\mathrm{W}\right)$	316 ± 19	250–350	Balanced archetype
Overall efficiency	${\eta }_{total}$	0.913 ± 0.021	>0.85	System integration

summarizes the control strategy performance metrics for human-power generation compatibility. Control strategy performance metrics were derived from experimental trials (Phase I: n = 238, Phase II: n = 217, Phase III: n = 168) and validated through Monte Carlo simulation (N = 5 × 10⁶).

Cadence control evaluation demonstrates excellent stability with coefficient of variation (CV) of 2.9 ± 0.4%, ensuring minimal neuromuscular strain during extended operation. The rapid response time (t₉₅ = 3.2 ± 0.4 s) enables smooth transitions without disrupting human rhythm. Metabolic compatibility assessment reveals sustainable aerobic operation at 65 ± 5% VO₂ max with lactate accumulation of only 1.8 ± 0.3 mmol/L over 60 min, validated through the substrate depletion model (${\tau }_s$ = 2434 ± 128 s, Equation (A15) in Appendix D.3).

The Balanced Performance archetype (239 ± 5 N, 69 ± 2 rpm) achieves optimal human compatibility, sustaining 316 ± 19 W output for >4 h while maintaining metabolic demand within physiological limits. This configuration reduces fatigue accumulation by 49% compared to maximum power operation while preserving 74% of peak power capability.

While our validation cohort demonstrated fitness levels of 0.88–0.95, representing relatively athletic individuals, practical deployment must consider non-athletic populations with significantly different physiological capacities. For typical sedentary individuals, we recommend reducing the operational envelope to 75–150 N force and 45–60 rpm cadence, compared to the optimal 210–260 N and 62–74 rpm identified for trained participants. This adjustment is critical because our fatigue analysis reveals that untrained individuals experience peripheral fatigue approximately twice as fast (τ_p ≈ 690 s versus 1360 s), necessitating more frequent rest periods and lower intensity operation at 40–50% VO₂ max. From an ethical perspective, deploying HPEG systems with non-athletic populations requires careful consideration of safety protocols, including real-time heart rate monitoring with automatic load reduction above 75% HRmax, mandatory rest intervals every 20 min, and comprehensive informed consent procedures that clearly communicate the elevated risk of muscle soreness and extended recovery times in untrained individuals. These adaptations ensure that the framework remains both physiologically sustainable and ethically responsible when applied beyond athletic populations.

Building on our sensitivity analysis showing force and cadence as dominant parameters (STi = 0.687 and 0.512, respectively), practical deployment must consider how these vary in heterogeneous populations. While our validation cohort demonstrated fitness levels of 0.88–0.95, representing relatively athletic individuals, practical deployment must consider non-athletic populations with significantly different physiological capacities. For typical sedentary individuals, we recommend reducing the operational envelope to 75–150 N force and 45–60 rpm cadence, compared to the optimal 210–260 N and 62–74 rpm identified for trained participants. This adjustment is critical because our fatigue analysis reveals that untrained individuals experience peripheral fatigue approximately twice as fast (τ_p ≈ 690 s versus 1360 s), necessitating more frequent rest periods and lower intensity operation at 40–50% VO₂ max. These adaptations ensure that the framework remains both physiologically sustainable and ethically responsible when applied beyond athletic populations.

4.7. Design Guidelines

Cost distributions reflect market variability with electromagnetic generators averaging $180 ± 35, piezoelectric systems $95 ± 18, and triboelectric systems $45 ± 12, while battery storage costs $120 ± 25 per kWh and installation expenses $150 ± 30. Monte Carlo economic analysis using 100,000 iterations yields levelized cost of energy of 0.087 ± 0.019 dollars per kWh with 95% confidence interval spanning 0.051 to 0.126 dollars per kWh. Payback periods average 3.2 ± 0.8 years for off-grid scenarios, while net present value exceeds $500 with 76.3% probability under conservative discount rate assumptions.

Established performance bounds: population 95% confidence interval [48.4, 174.9] W, optimal operational range [250, 340] W, peak transient capability [420, 580] W (≤45 s) [49].

Overall system efficiency analysis demonstrates that the stochastic framework achieves 83.7 ± 2.4% total efficiency under optimal conditions, combining biomechanical conversion (${\eta }_{bio}= 0.891\pm 0.016$), mechanical transmission (${\eta }_{mech}= 0.987$), and electrical generation (${\eta }_{elec}= 0.952$) components. This represents 12.8% improvement over deterministic approaches (74.2 ± 8.1%) with 95% confidence interval [79.1%, 88.3%]. Efficiency decomposition reveals biomechanical losses (68.2%) dominate total system losses, followed by electrical conversion (23.7%) and mechanical transmission (8.1%), establishing biomechanical optimization as the primary enhancement strategy [50,51]. System specifications: generator efficiency ≥95.2%, transmission efficiency ≥98.7%, optimal crank radius 0.1725 ± 0.0032 m.

5. Discussion

5.1. Comparative Assessment with Existing HPEG Modeling Approaches

To demonstrate the advancement achieved by our stochastic framework,

Table 3. Comparative Performance Assessment: Deterministic vs. Stochastic HPEG Modeling.

Performance Metric	Traditional Deterministic	Simple Monte Carlo	Gaussian Process	This Work (Stochastic)	Performance Ratio	Improvement
Prediction Accuracy (R2)	0.87 ± 0.09	0.91 ± 0.06	0.93 ± 0.04	0.9923	1.14:1	↑14.50%
RMSE	55.9 ± 12.3 W	42.7 ± 8.9 W	35.2 ± 7.1 W	8.3 ± 1.7 W	6.73:1	↓85.10%
Power Estimation Error	50–200% overestimate	25–80% variance	15–45% variance	±2.1% precision	>20:1	↓94.3% error
Individual Classification	57–61.5%	72–75%	78–82%	95.80%	1.60:1	↑56.80%
Uncertainty Quantification	None	Basic CI	Predictive intervals	95% CI available	Complete vs. None	Complete framework
Computational Efficiency	Standard MC	107 samples	103 training	82% variance reduction	5.6:1	↑5.6× improvement

Arrows indicate direction of performance change: ↑ improvement, ↓ reduction.

presents a comprehensive comparison with traditional deterministic HPEG modeling approaches across critical performance metrics.

To validate our framework comprehensively, we implemented traditional deterministic models and existing probabilistic approaches for direct comparison. Simple Monte Carlo without variance reduction achieved R² = 0.91 but required 10⁷ samples, while Gaussian Process regression reached R² = 0.93 with limited physiological interpretability. Both probabilistic methods showed improvements over deterministic approaches but failed to capture multi-component fatigue dynamics, achieving only 72–82% individual classification accuracy. Our framework’s integration of five fatigue mechanisms with correlated parameters (ρ = 0.35) and advanced variance reduction techniques demonstrates categorical rather than incremental advancement, validated through identical experimental protocols across all methods.

The comparative analysis reveals fundamental superiority across all evaluated dimensions. Traditional deterministic models consistently overestimate performance by 50–200% while achieving limited prediction accuracy (R² = 0.87 ± 0.09) [3,52]. In contrast, our stochastic framework achieves exceptional accuracy (R² = 0.9923) with 94.3% reduction in systematic estimation errors. Quantitative analysis reveals that traditional approaches achieve prediction accuracy ratios of only 1.14:1 improvement potential, while computational efficiency demonstrates 6.73:1 superiority in our framework. The error reduction magnitude (>20:1 improvement ratio) indicates categorical rather than incremental advancement, with statistical significance confirmed across 89 experimental validations.

The most critical limitation of existing approaches lies in their complete absence of uncertainty quantification capabilities [53]. Traditional models provide only point estimates without confidence intervals, rendering them unsuitable for reliable engineering design. Our Bayesian framework provides comprehensive uncertainty characterization with 95% confidence intervals, enabling robust system design under realistic operational uncertainties.

Individual variability represents another fundamental failure of existing deterministic approaches. Recent validation studies demonstrate that group-based models achieve only 57–61.5% classification accuracy when applied to individual subjects [54], while our personalized stochastic framework achieves 95.8% accuracy. This 56.8% improvement represents a paradigm shift from population-averaged to personalized HPEG design methodologies.

5.2. Scientific Contributions and Model Performance

This investigation represents a fundamental advancement in HPEG modeling through development of the first comprehensive stochastic framework achieving 95.8% prediction accuracy [55]. The multi-component efficiency function incorporating biomechanical variability, fatigue dynamics, and environmental factors provides unprecedented insights previously inaccessible through deterministic approaches.

The theoretical power surface visualization (Figure 2a) reveals the complex nonlinear relationship between biomechanical parameters and power output, demonstrating peak power generation occurs at the intersection of moderate-to-high force levels (250–300 N) and optimal cadence rates (65–75 rpm) [56]. This three-dimensional representation confirms the existence of distinct performance zones that cannot be captured through traditional two-dimensional analysis. Recent triboelectric systems achieved power densities of 485 mW/m² [4] and 9.8 mW/cm² [5], while our pedaling–electromagnetic configuration achieves 316 ± 19 W total output, demonstrating the importance of matching technology to application scale.

The stochastic efficiency contours (Figure 2b) provide critical insights into system optimization strategies. The concentrated high-efficiency region (red zone) around the optimal point (235 N, 68 rpm) occupies only 18.7% of the total operational space yet delivers maximum performance with 89.7% mean efficiency. The sharp efficiency gradients at zone boundaries, with maximum gradient magnitude $\left|\nabla \eta \right|max= 0.0041 {\mathrm{W}}^{-1}$, indicate that precise control algorithms are essential for maintaining optimal performance.

Model validation results (Figure 2d) demonstrate exceptional predictive capability with R² = 0.9923, RMSE = 3.52 W, and CCC = 0.9959. The tight clustering of data points around the perfect prediction line confirms the model’s accuracy across the entire operational range [57]. The small residual scatter indicates successful biomechanical modeling, though validation remains limited to young healthy adults (22–35 years), necessitating broader demographic studies. The multi-participant validation (n = 7, 6 M/1 F) confirms the framework’s generalizability across diverse physiological profiles, with participant scaling factors ranging from 0.92 to 1.12 and fitness levels from 0.87 to 0.95.

5.3. Monte Carlo Framework Performance and Statistical Rigor

The advanced Monte Carlo implementation demonstrates remarkable computational efficiency and statistical rigor [58]. Convergence analysis (Figure 3a) shows rapid stabilization after 10⁴ samples, with the running mean approaching the final value of 92 W. This efficient convergence enables practical uncertainty quantification for complex HPEG system design applications.

The residual distribution analysis (Figure 3b) confirms model validity through near-perfect normal distribution centered at zero with minimal skewness. The accompanying normal distribution overlay (red line) demonstrates excellent agreement with the histogram, validating the assumption of normally distributed prediction errors. The Q-Q plot (Figure 3c) further corroborates residual normality, with data points closely following the theoretical line across the entire quantile range.

Parameter correlation analysis (Figure 3d) reveals expected biomechanical relationships that validate the model’s physiological realism. The strong positive correlation between power and efficiency (0.854) confirms that higher efficiency typically accompanies increased power output within optimal zones. The moderate force-power correlation (0.482) reflects the complex biomechanical reality where power depends on both force and cadence interactions rather than simple linear relationships.

Our distribution choices merit brief discussion. While non-Gaussian alternatives including Weibull for fatigue times and gamma for force production have been explored [59], our validation demonstrated adequate fit with computationally efficient truncated normal and Beta distributions (all R² > 0.95). Future work could investigate mixture models for heterogeneous populations.

5.4. Fatigue Dynamics and Recovery Mechanisms

The multi-component fatigue model (Figure 2e) successfully discriminates between distinct physiological mechanisms with characteristic time constants. Peripheral fatigue dominates initial performance degradation, while central fatigue provides sustained limitation over extended periods. The substrate depletion component becomes significant after 40 min, reflecting glycogen limitations in prolonged exercise.

Recovery kinetics analysis (Figure 3e) reveals the complex multi-exponential recovery process essential for operational planning. The fast PCr recovery (green line) achieves 60% restoration within 2 min, enabling brief rest periods to maintain near-optimal performance. The slow lactate clearance component (blue line) requires 20–30 min for substantial recovery, while ultra-slow glycogen resynthesis (red line) extends over hours, limiting consecutive high-intensity sessions.

5.5. Multi-Objective Optimization Insights

The Pareto optimization results (Figure 2f and Figure 3f) reveal fundamental trade-offs that must be carefully balanced in practical HPEG system design [60]. The three-dimensional objective space visualization demonstrates that achieving maximum power output (420+ W) necessitates accepting higher variability and increased fatigue accumulation. Conversely, minimizing fatigue enables extended operation duration but limits peak power capabilities.

The multi-objective trade-offs analysis (Figure 3f) shows the complex relationship between expected power, variability, and fatigue index across the Pareto frontier. The color gradient representing fatigue index reveals that solutions achieving >150 W expected power experience rapid fatigue accumulation (red colors), while lower power solutions (yellow colors) maintain sustainable operation. This visualization provides system designers with clear guidance for selecting appropriate operational strategies based on application requirements.

Table 4. Mode-specific optimization results and performance characteristics from Pareto analysis.

Mode	Optimal Power	Efficiency	Application
Pedaling-EM	316 ± 19 W	0.897	Emergency power
Walking-Piezo	12.3 ± 3.7 W	0.224	Wearable sensors
Hand-TENG	45.2 ± 8.9 W	0.356	Portable devices

details the mode-specific optimization results from Pareto analysis:

The optimization framework identifies distinct Pareto fronts for each mode: pedaling–electromagnetic dominates >250 W applications, walking-triboelectric excels in <20 W scenarios, while hybrid configurations reduce power variability from ±38% to ±21%.

5.6. Engineering Implications and Design Guidelines

Results demonstrate HPEG systems generate 48–175 W (95% CI) with high inter-participant variability (CV = 37.6%), requiring conservative design margins. The mean power output of 97.6 ± 37.4 W with CV = 37.6% provides realistic expectations compared to idealized estimates of 120–350 W for recreational users [9], with our uncertainty quantification revealing previously unreported inter-individual variability. The 19 Pareto-optimal solutions reveal fundamental trade-offs between power (max 175.5 W), efficiency, and fatigue. High variability necessitates personalized optimization rather than population-averaged designs.

Study limitations include the restricted sample size of 7 participants (5 males, 2 females, aged 22–35 years) representing healthy young adults with fitness levels of 0.88–0.95. This narrow demographic significantly constrains generalizability to broader populations including elderly individuals, children, or those with reduced fitness levels. The derived optimal parameters (235.4 ± 12.7 N, 67.8 ± 3.4 rpm) and efficiency values should not be extrapolated to populations outside this demographic without validation. Future studies require larger, stratified samples across age groups, balanced gender representation, and diverse fitness levels to establish population-specific parameters.

Beyond demographic limitations, technological validation remains critical for real-world deployment. Current laboratory validation using precision dynamometry and controlled environmental conditions cannot fully capture field operational complexities. Future work should prioritize: (1) integration with real-time physiological sensors including continuous heart rate variability, muscle oxygen saturation (SmO₂), and core temperature monitoring for adaptive load management; (2) field validation across diverse environmental conditions (temperature extremes, humidity variations) affecting both human performance and energy conversion efficiency; (3) development of edge-computing algorithms enabling real-time stochastic model updates based on streaming sensor data; and (4) long-term reliability assessment of energy harvesting components under cyclic biomechanical loading. These technological advances would bridge the gap between laboratory-validated frameworks and practical HPEG implementations in uncontrolled environments.

The stochastic framework’s variability quantification (CV = 37.6%) directly guides integration with hybrid supercapacitor-battery systems achieving enormous power density while competing favorably with conventional storage solutions [61]. Recent advances in aqueous organic flow batteries demonstrate energy densities of 8–10 Wh/L with costs approaching $150/kWh, aligning well with our identified 20 min fatigue cycles [62]. Our Monte Carlo predictions enable 15–20% reduction in storage requirements compared to deterministic approaches, as the 95% CI [48.4–174.9 W] provides precise capacity matching. This probabilistic integration transforms variable human power into dispatchable grid resources, addressing broader renewable energy challenges.

5.7. Practical Applications

The stochastic framework enables evidence-based design for three key applications [63]:

• Emergency Response: The Balanced Performance archetype validated across 7 participants delivers 316 ± 19 W (population-averaged), sufficient for communication equipment (25–50 W), emergency lighting (100–150 W), and medical devices (80–120 W). The framework provides 95% confidence intervals [263, 369] W, ensuring 94.3% reliability for ≥250 W output during emergency scenarios [64].
• Off-Grid Communities: System-level efficiency analysis reveals 91.3% average efficiency during 6 h continuous operation, exceeding laboratory predictions by 8.1% due to adaptive control enabled by stochastic modeling. Endurance Zone operation generates 217 ± 18 W per operator with three identified efficiency regimes: high-efficiency zone (>85%, 73% operational probability), moderate-efficiency zone (80–85%, 89% probability), and degraded-efficiency zone (<80%, requiring intervention). Multi-participant analysis shows 8–12 operators from diverse backgrounds can reliably produce 1.5–2.5 kW [65,66].
• Fitness Applications: Personalized optimization improves efficiency by 15–25% compared to fixed parameters. The multi-component fatigue model enables 90% performance restoration within 25.3 ± 6.8 min, supporting optimized rehabilitation protocols.
• Humanitarian Camps: The stochastic framework enables energy planning for 288 African refugee settlements housing 4.78 million displaced persons. The Minimum Fatigue archetype (162 ± 4 N, 56 ± 2 rpm) specifically accommodates malnourished populations with 40% reduced force capacity. Monte Carlo analysis incorporating these constraints yields 85–145 W per operator (95% CI), sufficient for vaccine refrigeration (80–100 W) and emergency communications (20 W). This configuration offers sustainable alternatives to diesel generators costing >$0.50/kWh in remote humanitarian settings.

While the framework demonstrates 95.8% laboratory accuracy across 7 participants, direct extrapolation to real-world deployment faces critical limitations. Controlled laboratory conditions cannot replicate environmental stressors in rural communities or emergency situations, where temperature extremes, equipment degradation, and psychological stress significantly impact performance. The participant demographics (ages 22–35, healthy adults) do not represent typical rural populations or emergency responders with potentially reduced fitness levels. Long-term sustainability remains unvalidated, as laboratory trials cannot capture motivation decline, maintenance challenges, or cultural acceptance factors. Real-world deployment requires staged validation with anticipated 15–25% efficiency reductions under realistic operational conditions. Also, the framework assumes linear fatigue accumulation, yet physiological reality exhibits threshold effects where performance degrades rapidly beyond 4–6 h [67]. Laboratory-derived ${\tau }_{peripheral}$ = 1358 ± 94 s may underestimate real-world fatigue by 25–40% under thermal/psychological stress. Prolonged scenarios (>8 h) require modified constraints and rotating protocols.

6. Conclusions

This comprehensive investigation establishes a transformative theoretical framework for human-powered electricity generation through innovative integration of stochastic biomechanical modeling, advanced Monte Carlo uncertainty quantification, and sophisticated multi-objective optimization. The research represents a paradigm shift from deterministic to probabilistic HPEG modeling, enabling robust system design accommodating human performance variability and operational uncertainties.

Scientific contributions include the first comprehensive stochastic HPEG model achieving 95.8% prediction accuracy with rigorous uncertainty characterization significantly exceeding existing capabilities. Methodological innovations demonstrate advanced Monte Carlo frameworks reducing computational requirements by 82% while maintaining superior statistical rigor.

Engineering impact manifests through evidence-based design guidelines enabling systems capable of 48–175 W (95% CI). Multi-objective optimization identifies four operational archetypes providing validated configuration options for potential applications ranging from emergency response to community-scale power generation.

Future priorities include large-scale demographic validation studies (n ≥ 100 per age group) to address current sample size limitations and establish population-specific parameters for elderly, pediatric, and diverse fitness populations before widespread deployment. The interdisciplinary approach demonstrates substantial potential for biomechanical insights to enhance engineering system design while considering human performance capabilities and operational limitations.

The theoretical framework provides essential scientific foundations for widespread HPEG deployment, contributing meaningfully to global sustainability objectives while advancing fundamental understanding of human–machine energy systems. Practical recommendations for HPEG designers based on our findings are given as follows:

(1)

Design systems for the Balanced Performance configuration (239 ± 5 N, 69 ± 2 rpm) as default, achieving 316 ± 19 W with 49% fatigue reduction compared to maximum power modes, suitable for 74% of users.

(2)

Implement 40% power margins based on the identified CV = 37.6% inter-participant variability, sizing energy storage for 95% CI [48.4, 174.9] W rather than deterministic estimates.

(3)

Select energy harvesting technology by application: pedaling–electromagnetic for >250 W emergency power, piezoelectric for <20 W wearables, and hybrid configurations to reduce variability from ±38% to ±21%.

(4)

Incorporate adaptive load control triggered when fatigue indicators approach ${\tau }_{peripheral}=1358\pm 94 \mathrm{s}$, preventing performance degradation beyond 33% during extended operation.