Description:FIELD OF THE INVENTION

The present invention relates to the interdisciplinary field of fractional-order control systems, stochastic processes, and smart grid engineering, with a specific focus on enhancing the stability and resilience of smart grids operating under uncertain and fluctuating load conditions. More particularly, it concerns the application of advanced mathematical tools—specifically Hilfer fractional differential operators—to model, analyze, and control the dynamic behavior of modern electrical power systems in the presence of stochastic disturbances such as random load variations, renewable energy intermittency, and communication delays. The invention lies at the intersection of applied mathematics, systems engineering, and energy informatics, providing a novel control architecture that incorporates the memory-preserving and non-local characteristics of Hilfer fractional derivatives. This approach allows for more accurate modeling of physical processes in power networks that exhibit both long-term memory and probabilistic uncertainties. By embedding Hilfer-type fractional operators into stochastic differential equations and control laws, the invention enables the design of robust and adaptive controllers capable of ensuring grid stability, efficient energy dispatch, and fault-tolerant operation in complex, decentralized, and data-driven smart grid environments. It further supports real-time decision-making under uncertainty through fractional stochastic dynamic programming frameworks. This invention addresses a significant gap in current smart grid control strategies, which often overlook fractional dynamics and suffer from limited robustness against non-deterministic influences. The proposed framework is applicable to wide-ranging smart grid infrastructures, including microgrids, distributed energy systems, and cyber-physical grid interfaces, positioning it as a key technological advancement for future energy systems seeking resilience, flexibility, and sustainability.

Background of the proposed invention:

The modern electrical grid is undergoing a profound transformation, evolving from a traditionally centralized structure to a highly decentralized and dynamic network known as the smart grid. This evolution is driven by the growing penetration of renewable energy sources, such as solar and wind power, increased demand-side participation, the integration of energy storage systems, and the advancement of information and communication technologies. While these changes promise greater efficiency, environmental sustainability, and economic benefits, they also introduce unprecedented levels of uncertainty and complexity into grid operations. The inherent variability and intermittency of renewable energy sources, coupled with unpredictable load demands and stochastic disturbances, pose significant challenges to the stability and reliability of power systems. Conventional control strategies, typically designed under deterministic assumptions and based on integer-order models, are increasingly inadequate for managing such systems. These methods often fail to account for the memory-dependent, hereditary, and non-local characteristics of real-world physical and cyber-physical processes, particularly under conditions of noise, delay, and randomness. As a result, there is a pressing need for novel modeling and control frameworks that can capture the intricacies of smart grid behavior under stochastic influence. In response to this challenge, researchers have turned to fractional calculus—a mathematical framework that generalizes classical differential equations to non-integer orders—as a powerful tool for modeling complex dynamical systems. Fractional-order models have been shown to more accurately represent processes with memory and hereditary properties, as commonly observed in power system components such as batteries, supercapacitors, transformers, and electrical loads. Among the various fractional derivatives developed, the Hilfer fractional derivative stands out due to its ability to interpolate between the Caputo and Riemann–Liouville formulations through a tunable parameter, thus offering a greater degree of flexibility and model fidelity. The Hilfer operator has gained attention for its capacity to represent hybrid memory behavior, which is particularly relevant in the smart grid context where short-term dynamics (such as voltage or frequency oscillations) and long-term trends (such as cumulative energy consumption) coexist. At the same time, stochastic calculus has emerged as an essential mathematical apparatus to address randomness and uncertainty in engineering systems. Stochastic differential equations (SDEs), particularly those driven by Wiener processes (Brownian motion), are commonly used to model systems subject to continuous-time random fluctuations. However, integrating stochastic calculus with fractional-order dynamics—especially using generalized operators like the Hilfer derivative—remains a relatively nascent research domain with vast untapped potential for practical applications. The proposed invention builds upon these foundational developments by introducing a stochastic control framework for smart grid stability that is governed by Hilfer-type fractional differential equations. The key innovation lies in combining the non-local memory features of fractional calculus with the probabilistic treatment of uncertainties in a unified model that can be leveraged for optimal control design. In this framework, the dynamic equations of key grid components, such as voltage regulators, inverters, distributed generators, and energy storage systems, are reformulated using Hilfer fractional derivatives to more accurately reflect their complex dynamic behavior. Load variations, renewable generation fluctuations, and communication delays are modeled as stochastic processes, with the assumption that they follow Wiener or Poisson-type randomness, depending on the application. This approach provides a more realistic representation of grid dynamics under operational uncertainties. Moreover, the control strategy is designed using a stochastic optimal control approach based on dynamic programming principles extended to fractional-order systems. The Hamilton–Jacobi–Bellman (HJB) equation, a central tool in stochastic control theory, is reformulated to include Hilfer fractional terms, allowing for the derivation of control laws that are both optimal and robust under uncertainty. The novelty of the proposed invention also extends to stability analysis. Traditional notions of stability, such as Lyapunov stability, are revisited in the context of fractional and stochastic systems. Mean-square stability and asymptotic convergence are analyzed under the influence of both memory effects and noise, providing rigorous mathematical guarantees for the system’s behavior over time. The model is implemented and validated using high-fidelity simulations on benchmark smart grid test systems, such as the IEEE 14-bus and 33-bus microgrid configurations. These simulations demonstrate the superior performance of the Hilfer-based stochastic control system in terms of voltage and frequency regulation, load-following capabilities, and disturbance rejection when compared to classical PID and linear-quadratic regulator (LQR) controllers. Furthermore, the proposed framework supports real-time adaptability, as the tunable parameter in the Hilfer operator can be adjusted based on real-time grid conditions or historical data, providing an additional layer of control customization. In terms of practical implications, this invention addresses several critical challenges faced by modern power systems, including blackouts due to frequency instability, poor voltage profiles under fluctuating demand, and inefficient utilization of distributed energy resources. By enabling a more accurate and robust control methodology, it contributes directly to enhancing grid reliability, improving power quality, and facilitating the seamless integration of renewable energy sources. Additionally, the mathematical generality of the proposed framework allows for its extension beyond smart grids to other domains where stochasticity and memory effects play a critical role, such as robotics, biological systems, financial engineering, and aerospace control systems. The integration of Hilfer fractional calculus into stochastic control also paves the way for new theoretical developments in fractional dynamic systems, including the formulation of new theorems for existence, uniqueness, and stability of solutions under mixed stochastic-fractional regimes. Despite the growing interest in fractional calculus applications, the specific utilization of Hilfer operators in stochastic smart grid control remains largely unexplored, giving this invention a pioneering character. This work is the first of its kind to propose a structured, implementable methodology for real-time control of smart grids using Hilfer fractional stochastic models, positioning it as a landmark advancement in the field of intelligent energy systems. Consequently, the background of this proposed invention resides in the confluence of three transformative ideas—fractional-order modeling, stochastic control, and smart grid evolution—culminating in a novel control paradigm that addresses current limitations and anticipates future needs of resilient, adaptive, and intelligent power infrastructures.

Summary of the proposed invention:

The proposed invention introduces a pioneering stochastic control system that integrates Hilfer fractional operators into the dynamic modeling and regulation of smart grids operating under uncertain and highly variable load conditions. As smart grids continue to evolve into complex, decentralized, and cyber-physical infrastructures powered by renewable energy sources, energy storage systems, and digital communication technologies, their inherent susceptibility to random disturbances, fluctuating demand, and delayed information exchange becomes more pronounced. These challenges necessitate the development of advanced mathematical models and control strategies capable of handling such uncertainties while ensuring stability, robustness, and performance reliability. Traditional integer-order control methods, although effective in deterministic settings, fall short in capturing the non-local, memory-dependent behavior of real-world systems, especially when subjected to noise, delay, and probabilistic fluctuations. In contrast, fractional-order calculus provides a more general and accurate framework for modeling such systems, allowing for the inclusion of hereditary properties and memory effects that are characteristic of many components in the electrical power network. The Hilfer fractional operator, in particular, extends the modeling flexibility by bridging the Caputo and Riemann–Liouville definitions through a tunable parameter, thereby enabling a spectrum of dynamic behaviors to be modeled depending on the system’s memory profile. By embedding this operator within a stochastic differential equation framework, the invention creates a hybrid mathematical model that effectively describes both the fractional memory dynamics and stochastic uncertainties of the grid. The load uncertainties, intermittent power generation from renewables, and stochastic user demand are modeled using Wiener processes, providing a realistic representation of real-time smart grid behavior. The invention further contributes to control system design by proposing a stochastic optimal control law derived through a modified dynamic programming approach that accounts for the fractional-order nature of the system. Specifically, the fractional Hamilton–Jacobi–Bellman (HJB) equation is formulated using Hilfer derivatives, leading to optimal control strategies that minimize a cost functional representing grid stability objectives, such as voltage regulation, frequency stability, and minimization of control effort. This results in a feedback control law that adapts to system changes in real-time and maintains operational stability despite uncertain and rapidly changing load profiles. Additionally, the invention provides a robust stability analysis framework based on mean-square stability theory and Lyapunov functions adapted to fractional stochastic systems. This theoretical foundation ensures that the control system will perform reliably under both expected and extreme conditions, preventing instabilities that could result in blackouts, equipment damage, or loss of power quality. The invention is validated through extensive numerical simulations on benchmark test systems such as the IEEE 14-bus and 33-bus configurations. The results demonstrate superior performance in maintaining system voltage and frequency profiles, as well as improved resilience against load perturbations and communication delays compared to conventional control schemes, including PID and LQR controllers. Moreover, the Hilfer operator’s tunable parameter introduces a new degree of freedom in control design, allowing operators to dynamically adjust the memory weighting in the control response, which can be particularly advantageous in systems with evolving operational conditions or component degradation over time. Beyond theoretical contributions, the proposed invention offers significant practical benefits. In smart grid applications, the ability to model and control systems with high fidelity to real-world behavior is essential for reducing operational costs, increasing energy efficiency, and enhancing reliability. The proposed system can be implemented in control centers, microgrid management systems, distributed energy resource controllers, and inverter-based power converters, offering a unified and scalable solution to grid stability challenges. Furthermore, the model supports the integration of real-time data analytics, machine learning, and adaptive parameter estimation, opening pathways to further enhance performance through data-driven tuning and predictive diagnostics. The invention is also highly relevant to distributed generation environments, where local variability and autonomy require advanced local control capabilities that can coordinate within a broader stochastic environment. From a broader perspective, this invention stands at the confluence of three transformative areas in modern engineering—fractional calculus, stochastic systems, and smart energy networks. It provides a foundational framework for future developments in energy system control, enabling smarter and more adaptive control architectures that can operate effectively under uncertainty and complexity. In addition, the mathematical constructs developed here have potential applications in other engineering fields, such as biomedical systems, robotics, aerospace engineering, and financial modeling, wherever systems exhibit both stochastic behavior and memory-dependent dynamics. The novelty of the invention lies not only in the application of Hilfer fractional operators to smart grid control but also in the comprehensive integration of modeling, control design, stability analysis, and practical implementation into a cohesive framework. No existing control architecture provides this level of flexibility, fidelity, and robustness in the context of fractional stochastic systems. As a result, this invention is poised to set a new standard for control design in smart grid and cyber-physical energy system domains. It addresses critical needs in energy reliability, grid flexibility, and adaptive infrastructure by introducing tools and methods that are both mathematically rigorous and practically implementable. Moreover, the invention is highly scalable and can be extended to support larger, more complex grids with multi-agent coordination, hierarchical control, and real-time data integration. It can also be tailored to support demand response programs, electric vehicle integration, and renewable energy forecasting, further enhancing its relevance to next-generation smart grid applications. In conclusion, this invention represents a significant advancement in the field of intelligent energy system control by combining the generalized fractional dynamics of Hilfer operators with robust stochastic control theory. Through a unified mathematical and engineering approach, it provides a new paradigm for modeling and controlling smart grids under uncertainty, ensuring both theoretical soundness and practical viability for future power systems that are expected to be increasingly dynamic, distributed, and data-driven.
Brief description of the proposed invention:

The proposed invention presents a comprehensive and pioneering approach to improving the stability, adaptability, and robustness of smart grids under load uncertainties by developing a stochastic control system embedded with Hilfer fractional operators, a novel mathematical construct that generalizes classical fractional derivatives. As smart grids evolve into highly decentralized, cyber-physical infrastructures with deep integration of renewable energy sources, distributed generation units, electric vehicles, and energy storage systems, they are increasingly subject to unpredictable load variations, intermittent renewable outputs, and communication lags—all of which introduce stochastic behavior into the system. Traditional control strategies such as PID and linear-quadratic regulators (LQR), which rely on integer-order models and deterministic assumptions, often fall short in handling these complexities due to their inability to capture long-memory effects, dynamic uncertainties, and the stochastic nature of real-world smart grid operations. The invention addresses these challenges by formulating the system dynamics through stochastic differential equations (SDEs) governed by Hilfer fractional derivatives, which are capable of modeling both short-term and long-range memory processes through an adjustable interpolation parameter. This flexibility allows for more accurate representation of physical components such as batteries, transformers, and supercapacitors, which exhibit memory-dependent behavior, and provides the mathematical machinery to describe a broader class of dynamic behaviors. The stochasticity arising from load fluctuations, communication noise, renewable generation variability, and sensor inaccuracies is modeled using Wiener processes (Brownian motion), while the Hilfer operator captures the hereditary and non-local effects, creating a dual-layered model that is both temporally adaptive and probabilistically robust. This hybrid modeling framework enables a deeper understanding of how smart grid subsystems interact under random disturbances and varying temporal influences. Central to the invention is the development of an optimal control law using a stochastic dynamic programming framework that incorporates the fractional nature of the system through a modified version of the Hamilton–Jacobi–Bellman (HJB) equation extended to Hilfer-type derivatives. The control objective function is defined to minimize a composite cost involving state deviations (voltage, frequency), control energy, and robustness metrics. The resulting control policy is both state-feedback-based and memory-aware, dynamically adjusting the control inputs in real-time to reflect changes in both the system state and the stochastic environment. Additionally, the proposed invention includes a comprehensive stability analysis using fractional Lyapunov theory and mean-square stability concepts, providing guarantees that the controlled system remains bounded and convergent even under persistent stochastic excitation. This contributes to ensuring reliable performance of the grid, avoiding frequency drifts, voltage sags, or systemic failures in response to sudden load surges or drops. Another significant innovation is the tunability offered by the Hilfer fractional operator, which introduces a memory weighting parameter and a fractional order that can be calibrated based on real-time performance metrics or historical data using adaptive algorithms. This makes the controller not only robust but also self-optimizing, able to adjust its temporal response characteristics depending on the evolving grid conditions, such as the aging of components, seasonal demand changes, or shifts in generation patterns. The invention is validated through detailed simulation studies on standardized smart grid test cases, including the IEEE 14-bus and 33-bus microgrid models. These simulations involve various realistic scenarios such as renewable intermittency, load demand spikes, and communication delays, demonstrating superior performance in terms of frequency regulation, voltage stability, reduced settling time, and disturbance rejection when compared to conventional controllers. Furthermore, the proposed control system is compatible with existing smart grid communication protocols and can be implemented within modern supervisory control and data acquisition (SCADA) systems or energy management systems (EMS), making it practically viable for deployment in both centralized and distributed control architectures. The invention’s design also allows for modularity and scalability, supporting its extension to larger networks or integration with multi-agent systems that manage multiple distributed energy resources in real-time. Beyond smart grid applications, the proposed stochastic control system with Hilfer fractional modeling can be extended to other engineering domains where systems experience memory effects and are subject to uncertainty, such as thermal process control, biomedical systems, structural health monitoring, and financial system modeling. The flexibility of the Hilfer operator enables domain-specific tuning and generalization, making it a powerful tool for hybrid dynamical systems. Additionally, the mathematical framework introduced by this invention opens new research avenues in fractional stochastic systems, including new existence and uniqueness theorems, spectral analysis of operators, and novel optimization strategies tailored for Hilfer-type dynamics. Importantly, this invention fills a critical research and application gap, as prior work in the smart grid domain has largely focused on either deterministic fractional control or stochastic modeling using only integer-order dynamics, without addressing the complex interplay between memory-dependent dynamics and probabilistic uncertainties in a unified framework. By bridging this gap, the proposed invention offers a transformative contribution to intelligent energy system design, enabling smart grids to maintain resilience, flexibility, and high performance even as they become increasingly complex and data-driven. Moreover, the invention supports future enhancements through the integration of data-driven learning, machine intelligence, and real-time adaptive control, enabling the control system to evolve with grid modernization efforts. This aligns with global initiatives for decarbonization, energy security, and climate resilience by ensuring that the underlying control infrastructure can support high-penetration renewables and variable demand patterns. In summary, this invention offers a novel, rigorous, and implementable solution for smart grid stability under uncertainty, uniting Hilfer fractional calculus and stochastic control theory in a practical engineering context. It redefines how control systems can be designed for energy networks that are not only adaptive and resilient but also intelligent and self-regulating, thereby setting a new benchmark
in the control of complex, uncertain, and memory-embedded systems.
, Claims:We Claim:

1. A stochastic control system for smart grid stability under load uncertainty, comprising a controller configured to utilize Hilfer fractional differential equations to model and regulate grid dynamics, wherein said equations integrate both Riemann–Liouville and Caputo derivatives via a tunable interpolation parameter.

2. The system of claim 1, wherein the controller models stochastic disturbances in grid operations—such as load fluctuations and renewable energy intermittency—using Wiener processes embedded within Hilfer fractional stochastic differential equations.

3. The system of claim 1, wherein the Hilfer fractional operator enables dynamic adjustment of the memory characteristics of the control model, allowing fine-tuning between short-term and long-term response behaviors.

4. A method for enhancing smart grid stability, comprising the steps of: modeling power system components with Hilfer-type fractional differential equations; incorporating stochastic inputs representing load uncertainties; and generating optimal control actions using a modified Hamilton–Jacobi–Bellman (HJB) framework adapted to fractional-order stochastic systems.

5. The system of claim 4, wherein the optimal control actions are derived based on a minimization of a cost functional that includes voltage deviation, frequency deviation, and control energy.

6. The system of claim 1, further comprising a stability assurance module that applies mean-square Lyapunov criteria tailored to Hilfer fractional stochastic systems to ensure convergence and boundedness of the control response under uncertainty.

7. A real-time adaptive tuning mechanism for the system of claim 1, wherein the Hilfer operator’s interpolation parameter and fractional order are dynamically adjusted based on real-time grid performance metrics or historical data trends.

8. The system of claim 1, implemented within a smart grid supervisory control and data acquisition (SCADA) or energy management system (EMS), wherein the control system interfaces with distributed energy resources, storage systems, and communication networks.

9. A multi-scale control strategy for microgrids and distributed energy systems based on claim 1, wherein different Hilfer fractional parameters are assigned to subsystems with varying memory characteristics, achieving coordinated stability under heterogeneous uncertainty profiles.

10. A computer-implemented method for smart grid control comprising: receiving real-time sensor data from grid nodes; solving Hilfer-type fractional stochastic differential equations; computing optimal control actions using a fractional HJB solver; and issuing control signals to actuators to stabilize voltage and frequency in the presence of stochastic load variations