Physics-informed neural network based system and method for analysis of dielectric response measurements

Abstract

The present invention relates to a method and system for determining equivalent physical quantities using dielectric response measurement data. The method comprises receiving dielectric response measurement data related to a sample of interest, applying the data to a machine learning model, and generating a first set of electrical characteristics of a circuit model that employs equivalent circuit components to represent the operational characteristics of the sample. A set of latent parameters associated with the circuit model is learned based on the dielectric response data and the generated electrical characteristics. This invention is particularly applicable in scenarios involving material characterization and condition monitoring of electrical systems, where the circuit model may represent complex interactions and degradation mechanisms within the sample, thereby enabling the extraction of critical physical parameters for analysis and monitoring purposes.

Description

[0001] TITLE

[0002] Physics-Informed Neural Network based system and method for analysis of dielectric response measurements

[0003] TECHNICAL FIELD

[0004] The present invention relates to the technical field of dielectric response measurement processing. In one aspect of the present invention it relates to system and method for material characterization using dielectric response measurements. In another aspect the invention relates to condition monitoring of electrical equipment using dielectric response measurements.

[0005] BACKGROUND OF THE INVENTION

[0006] Dielectric response (DR) measurements, also referred to as dielectric spectroscopy or impedance spectroscopy, are widely utilized for characterizing the dielectric properties of materials across various applications. These measurements can be performed in both the frequency domain (DFR) and time domain (DTR) and are essential tools in fields ranging from fundamental research to process and quality control in manufacturing. DR techniques are employed not only in the analysis of conventional dielectric materials but also in advanced applications such as supercapacitors, batteries, fuel cells, and electrolysers. Furthermore, DR measurements are often used for monitoring the condition of high-voltage (HV) equipment and systems, including insulation materials in High Voltage Direct Current (HVDC) transmission systems.

[0007] One of the primary concerns in HV insulation systems is the aging of dielectric materials, which can lead to system failures. DR measurements, including both frequency domain (DFR) and time domain (DTR) techniques, are extensively employed for characterizing the dielectric properties of HV components. These measurements are crucial for condition monitoring, as they provide insights into the material's status and potential degradation due to factors such as aging, contamination, and moisture ingress.

[0008] However, conventional methods have notable limitations. Specifically, they struggle to provide a comprehensive overview when dealing with large datasets and the multifaceted dependencies on various factors like material composition, temperature effects, and aging history. Additionally, ML-based methods require vast amounts of data for accurate training and often function as 'black boxes,' offering limited insight into the physical interpretation of the data.

[0009] Traditional methods for analyzing DR data, such as manual inspection by experienced personnel or fitting data to mathematical models, are limited by their inability to efficiently handle large datasets and the multifaceted nature of the data. Furthermore, while machine learning (ML) methods, particularly neural networks (NN), offer the potential for automated analysis, they require vast amounts of data for accurate model training and often function as 'black boxes,' providing little insight into the physical processes underlying the data.

[0010] Given these drawbacks, there is a substantial need for a method that leverages the strengths of modern ML tools while also providing physical insights into the data, and requiring a reduced volume of data for effective modelling.

[0011] SUMMARY OF THE INVENTION

[0012] In a first aspect of the present invention there is proposed a method for determining equivalent physical quantities using dielectric response measurement data. The method comprises the steps: receiving a plurality of dielectric response measurement data relating to a sample of interest; applying the dielectric response measurement data to a machine learning model and generating, using the machine learning model, a first set of electrical characteristics of a circuit model which uses equivalent circuit components to characterize operational characteristics of the sample of interest; and, learning a first set of latent parameters relating to the circuit model based on at least the dielectric response measurement data.

[0013] In embodiments the method further comprises the step of determining a set of equivalent physical quantities relating to the sample of interest based on the learnt set of latent parameters.

[0014] Depending on the scenario and application, the set of equivalent physical quantities may be the same as the first set of latent parameters. The sample of interest may be any sample on which a dielectric measurement is performed. In other words, the sample of interest the sample may be: a plurality of components in an electrical system and wherein the dielectric measurements relates to at least one component in the electrical system; and / or, a dielectric material or compositions thereof.

[0015] In one proposed embodiment the sample is a plurality of components in an electrical system and the method is a method for monitoring the condition in the electrical system.

[0016] In this embodiment, meaning wherein the sample is a plurality of components in an electrical system, the method for condition monitoring may more specifically be a method for detecting an anomaly in the electrical system. The method comprises analyzing the set of latent parameters and / or set of equivalent physical quantities to determine if an anomaly is detected. Meaning that, in this embodiment the set of equivalent physical quantities may be information regarding the anomaly, such as the anomaly type, a quantity relating thereto and / or a component associated with the anomaly detected.

[0017] In another embodiment the sample is a dielectric material or a composition thereof, and the method is a method for characterizing said dielectric material. The method comprising the step of determining a set of material characteristics relating to the sample of interest based on the learnt set of latent parameters. Meaning that, in one such embodiment the set of equivalent physical quantities may be the set of material characteristics relating to the sample.

[0018] As elaborated upon later, the method may be a method for condition monitoring of an electrical system and the condition monitoring might entail characterizing dielectric material for the purpose of determining condition characteristics of the electrical system.

[0019] One example of condition monitoring entailing characterizing dielectric materials is when a electrical system comprising capacitors is monitored and the condition of the capacitors is determined by reproducing the behavior of the degradation of the capacitor using equivalent circuit elements to represent the capacitor in the circuit model.

[0020] The circuit model is a theoretical representation that reproduces the interaction of equivalent circuit components under certain conditions. The circuit model may for example be designed to mimic the behavior of an electrical system for the purpose of monitoring for anomalies in the electrical system. In such a scenario, the equivalent circuit components may have corresponding, but possibly more complex, real components in the electrical system.

[0021] However, the circuit model may in another scenario be designed to emulate the behavior of a degraded film capacitor for the purpose of determining material characteristics. In this scenario the equivalent circuit components are chosen based on specific electrical properties that reflect the degradation mechanisms of the film capacitor. For instance, the equivalent circuit components might include resistive elements to represent leakage paths, capacitive elements to capture changes in capacitance due degradation such as aging of the polymeric material and / or local dielectric breakdown, and inductive elements to model parasitic effects. By adjusting these components to mirror the behavior observed in the degraded capacitor, the circuit model enables precise analysis of material degradation over time. This allows for the extraction of key parameters such as dielectric constant variation, loss tangent increase, and equivalent series resistance (ESR), which enable for assessing the health and longevity of the capacitor.

[0022] In applications involving complex integrated circuits (ICs), the circuit model may be further designed to represent the hierarchical structure of the IC design. Just as digital ICs are constructed from high-level logic elements, which are themselves built from lower-level components such as transistors and diodes, the circuit model can be designed to reflect this multi-layered architecture. For instance, at a higher abstraction level, the circuit model might simulate the behavior of entire logic units like arithmetic / logic units (ALUs) or memory cells, while at a more granular level, it could focus on the performance of individual transistors or interconnects under specific operating conditions. This hierarchical approach allows design engineers to use Electronic Design Automation (EDA) tools not only to create and analyze integrated circuit layouts but also to simulate and predict how these layouts will perform under real-world conditions. By incorporating detailed geometric and electrical descriptions into the circuit model, engineers can more accurately anticipate potential issues such as signal integrity problems or thermal effects, thereby improving the reliability and efficiency of the final IC design.

[0023] The circuit model is designed based on the specific application of use. For example, a lumped element model may be suitable for low-frequency circuits, while a distributed element model may be suitable in certain high-frequency circuit. Other examples of possible circuit model designs include: hybrid-pi model, t-model (two-port network), smallsignal model, large-signal model, phasor model, state-space model.

[0024] The circuit model is representable as an equation, and depending on the complexity of the model the equation might be a differentiation equation, nonlinear equation or a combination thereof. The equation represented by the circuit model includes terms which are referred to as latent parameters, i.e. , variables within the equation that are not necessarily directly measurable from observations in a specific scenario. These latent parameters could represent underlying characteristics or properties of the sample that influence its operational characteristics but may not be explicitly manifested in the measurable quantities, or at least explicitly manifested in quantities measured in a specific scenario. For example, latent parameters could represent internal states, such as charge distribution, material properties, or other variables that contribute to the overall behavior of the circuit.

[0025] The machine learning model is employed to generate a first set of electrical characteristics pertaining to the theoretical circuit model using dielectric response measurement data of the sample of interest. The dielectric response measurement data may be experimental data that reflects how the sample responds to an applied electric field over time. In another embodiment the dielectric response measurement data may be synthetic data i.e., data from a simulation, or the dielectric response measurement data may be a combination of observed data and synthetic data.

[0026] The equation corresponding to the circuit model (circuit model equation) may be incorporated into the machine learning model, for example in at least one of the following ways:

[0027] - the circuit model equation is directly embedded as a constraint within the optimization process of the machine learning model, thereby enforcing compliance with the theoretical circuit model during training; and / or,

[0028] - the circuit model equation is incorporated as a regularization term within the loss function of the machine learning model, penalizing deviations from the behavior predicted by the theoretical circuit model; and / or, - the machine learning model is parameterized in accordance with the circuit model equation, such that the model architecture inherently reflects the theoretical relationships described by the circuit model; and / or,

[0029] - the machine learning model comprises a hybrid architecture, with a first portion of the machine learning model based on the circuit model equation and a second portion designed to learn from the dielectric response measurement data, allowing for both theoretical and data-driven learning; and / or,

[0030] - the learning process of the machine learning model is dynamically adjusted based on the circuit model equation, including modifications to learning rates, using various optimization methods in sequence, or parameter update rules to maintain alignment with the circuit model equation; and / or,

[0031] - the dielectric response measurement data is pre-processed using the circuit model equation before being input to the machine learning model, embedding theoretical constraints into the data prior to analysis; and / or,

[0032] - the outputs of the machine learning model are post- processed using the circuit model equation to adjust the predictions, ensuring alignment with the theoretical circuit model after initial computation.

[0033] Preferably, the equation corresponding to the circuit model is incorporated as a term in the loss function of the machine learning model. The machine learning model may then be optimized, i.e. , trained, to learn the first set of latent parameters.

[0034] In preferred embodiments the machine learning model is a neural network. Most preferably a Physics-Informed Neural Network (PI NN).

[0035] In one preferred embodiment the learning of the first set of latent parameters of the circuit model is based on at least first set of electrical characteristics and the dielectric response measurement data.

[0036] The first set of latent parameters are used to determine the set of equivalent psychical parameters. “Equivalent physical quantities” encompasses any measurable or inferred metrics that can be used to describe the behavior, properties, or performance of a system, component, or material under analysis. In certain embodiments the first set of latent parameters and the equivalent physical quantities may be the same, i.e., the step of “determining” does not necessarily entail further computation. However, in other embodiments, the determining might involve associating one or several parameters in the first set of latent parameters with: one or several components in a electrical system; or one or several materials in a composite material. The proposed method offers flexibility in that there are several alternatives for determining equivalent physical quantities based on learnt latent parameters, enabling for the skilled person to adapt the method to scenario specific design requirements.

[0037] In embodiments determining the set of equivalent physical quantities relating to the sample of interest comprises the step of: associating respective learnt parameter in the first set of latent parameters with a respective section and / or aspect of the sample of interest.

[0038] In one proposed embodiment the machine learning model comprises a loss function in which at least one component is derived from a time, space and / or frequency dependent differential equation that reproduces behavior of the circuit model. Learning the first set of latent parameters of the circuit model comprises the step:

[0039] - training the machine learning model using the loss function, the dielectric response measurement data and the generated first set of electrical characteristics.

[0040] The term "training" in this context refers to the process of adjusting the parameters of the machine learning model to minimize the loss function. Specifically, during training (i.e. , optimization), the model learns to map the dielectric response measurement data to the first set of electrical characteristics of the circuit model by optimizing its parameters. This optimization may be driven by the loss function, which incorporates time, space and / or frequency-dependent differential equations that reproduce the behavior of the circuit model. The training process iteratively refines the model parameters to better align the generated electrical characteristics with the expected behavior as described by the circuit model.

[0041] In one preferred embodiment, it is proposed that one or more of the equivalent circuit components in the circuit model is replaced by a sub-circuit model which describes operational characteristics of a section of the sample of interest and / or one or several components of a equivalent circuit describing the samples. Using this design, the circuit model achieves a multi-layered architecture enabling the circuit model to reproduce more complex behavior. The sub-circuit model preferably uses equivalent circuit components to characterize operational characteristics of said section in the sample of interest. The equivalent circuit components of the circuit model and / or the sub-circuit model may in one embodiment reproduce behavior of at least one of the following circuit components: resistance, capacitance, inductance, and / or impedance.

[0042] The sub-circuit model may for example be used to replace an equivalent capacitance component to reproduce the degradation of a physical capacitance component in a electrical system.

[0043] It is also preferable that each equivalent circuit component is represented by a term in the time and / or frequency dependent differential equation that reproduces behavior of the circuit model and / or sub-circuit model. Each term is then constant or have a functional value dependent on one or more variables, including but not limited to temperature, electric field strength, voltage, aging history, material composition, and / or morphology of the sample of interest.

[0044] In one proposed embodiment determining the set of equivalent physical quantities of the sample of interest comprises the step of: applying a set of inputs to a sub-ML model and generating, from the sub-ML model, a second set of electrical characteristics relating to the sub-circuit model; and, learning a second set of latent parameters relating to the sub-circuit model based on the loss function of the sub-circuit model, the generated set second of electrical characteristics of the sub-circuit model and the set of inputs.

[0045] The sub-ML model may be utilized to learn the second set of latent parameters relating to the sub-circuit model. In the proposed embodiment the sub-ML model comprises a loss function in which at least one component is derived from a time, space and / or frequency dependent differential equation that reproduces behavior of the sub-circuit model.

[0046] In preferred embodiments the sub-ML model is a neural network. Most preferably a Physics-Informed Neural Network (PINN).

[0047] The sub-ML model may be incorporated in a number of different ways. In certain scenarios it is preferable that there are several sub-ML models which are incorporated as recited in different embodiments. In one embodiment the second set of latent parameters is used as input into the circuit model for generating the first set of electrical characteristics. In this embodiment generating the first set of electrical characteristics of the circuit model based further includes: applying the second set of latent parameters of the sub-circuit model and the dielectric response measurement data to the machine learning model and generating, using the machine learning model, the first set of electrical characteristics of the circuit model.

[0048] The set of inputs may in this embodiment preferably comprise dielectric response measurement data.

[0049] In another embodiment the second set of latent parameters is used as input into the circuit model for learning the first set of latent parameters. In this embodiment training of the machine learning model further includes:

[0050] - training the machine learning model using the loss function of the circuit model, the second set of latent parameters of the sub-circuit model, the dielectric response measurement data and the generated first set of electrical characteristics of the circuit model.

[0051] In this embodiment it is proposed that at least one parameter in the second set of latent parameters of the sub-circuit model are applied as frozen parameters and / or initial parameters allowed to change, and that the set of inputs preferably comprises at least one of: the generated first set of electrical characteristics of the circuit model and / or the acquired dielectric response measurement data.

[0052] In yet another embodiment, the machine learning model and sub-ML model are integrated into one. In this embodiment it is proposed that the equations reproducing the operational characteristics of the circuit model and the sub-circuit model are integrated into one or several time and / or frequency dependent differential equation and that the machine learning model and sub-ML model may be the same machine learning model. The determined set of equivalent physical quantities then includes the at least one equivalent physical quantities relating to a sub section of the sample of interest.

[0053] In one variation of this embodiment, the machine learning model is a Physics-Informed Neural Network (PINN), and each coupled differential equation contributes an additional component to the total loss function of the PINN model.

[0054] For example, the sample of interest may be modelled by a system of differential equations that represent different physical processes or sections within the sample. These equations, which reproduce the operational characteristics of both the overall circuit model and its sub-models, are incorporated into the same PINN. In this implementation, the PI NN's loss function may be augmented by the inclusion of each differential equation, ensuring that the PINN learns the underlying physics governing the system.

[0055] The set of inputs applied to the sub-ML model may comprise synthetic data, calculated / learnt data or data gathered from sensors. In other words, the set of inputs is not limited data relating to electrical characteristics but could relate other types of data such as temperature.

[0056] In line with this, the sub-ML model may also incorporate advanced functional values that represent equivalent circuit components of the sub-circuit model. These functional values may integrate various domain-specific models, each describing distinct physical phenomena relevant to the sample of interest. For example, an equivalent circuit component may be characterized by a thermodynamic model that accounts for energy and entropy, or by an electromagnetic model that captures field interactions within the system. By embedding these complex models, the sub-ML model may not only process data that is synthetically generated, calculated, or sensor-acquired but also utilize these models to enhance the fidelity of the circuit model’s predictions.

[0057] Complementing the dielectric response measurement data with a diverse set of input data sources may significantly enhance the accuracy and robustness of the sub-ML model and / or machine learning model in determining equivalent physical quantities. By integrating data from various domains — ranging from digital imaging and morphological analysis to chemical composition and environmental stress factors — the model can develop a more comprehensive understanding of the underlying physical phenomena. This multi-faceted approach allows for the identification of subtle correlations and interactions that may not be apparent when relying solely on dielectric response data. Additionally, the incorporation of physical, chemical, and environmental property data further enables the sub-circuit models to account for complex dependencies and variations in material behavior under different conditions. The synergistic use of these diverse inputs contributes to the generation of more reliable and interpretable model outputs, thereby facilitating more informed decision-making in applications such as material characterization and condition monitoring of electrical systems.

[0058] In one embodiment, the method the set of input data may include not only dielectric response measurement data but also additional data relevant to the electro-mechanical- thermal aging of the sample of interest. Such additional data may include measurements from electro-mechanical aging tests, or combined electro-mechanical-thermal aging tests, conducted with or without other aging influence conditions like chemical aging factors.

[0059] The additional input data may include various electrical, thermal, mechanical, and chemical properties of the sample, which are monitored during the aging tests. For instance, the data may encompass voltage, current, temperature, capacitance, resistivity (including both series and parallel resistance), and equivalent series resistance (ESR) losses, all as functions of time and applied load. Furthermore, the method may incorporate input data relating to mechanical vibrations, acoustic emissions, or other signals that reflect complex dependencies between the operational characteristics of the system and the construction and materials of its components.

[0060] By integrating additional input data into the sub-ML model, the method enhances the capacity to analyze and characterize the sample's aging and degradation processes more accurately. The sub-ML model may use this enriched dataset to generate a more comprehensive set of electrical characteristics of the sub-circuit model, thereby improving the learning of latent parameters and the subsequent determination of equivalent physical quantities.

[0061] In one embodiment the set of inputs comprises at least one of: the generated first set of electrical characteristics of the circuit model; the learnt latent parameter of the circuit model; and / or the acquired dielectric response measurement data.

[0062] In another embodiment, the set of inputs comprise any of: digital imaging data, such as photographic data and / or microscope data; morphological analysis, such as surface morphology, bulk morphology or crystallinity analysis; chemical composition and material analysis data, such as data relating to additives and impurities, aging and degradation, surface analysis, bulk analysis or molecular weight distribution; surface and chemical treatment data, such as chemical treatments, corona treatments, or plasma treatment data; electrical properties, such as insulation breakdown strength or aging-related electrical data; process-related data, such as manufacturing processes or electrochemical processes; environmental stress data, such as stress conditions or historical data.

[0063] In yet another embodiment, the set of inputs comprises data from at least one of: physical property data such as thermocouples, strain gauges, LVDTs (Linear Variable Differential Transformers), ultrasonic sensors, optical micrometers, hardness testers (e.g., Rockwell, Vickers), or microscopes (e.g., SEM, TEM); chemical property data, such as information from XRF (X-ray Fluorescence), FTIR (Fourier Transform Infrared Spectroscopy), mass spectrometers, pH sensors, and EDS (Energy Dispersive Spectroscopy); electrical property data, resistivity meters, Hall Effect sensors, and impedance analyzers; optical property data, such as measurements from spectrophotometers, refractometers, ellipsometers, colorimeters, or Raman spectrometers.

[0064] In further embodiments, the set of inputs comprise data form at least one of: temperature monitoring data from sensors such as RTDs (Resistance Temperature Detectors), thermistors, infrared sensors, and fiber optic temperature sensors; vibration monitoring data from sensors such as accelerometers, velocity sensors, and proximity probes; electrical parameter monitoring data, collect by for example current transformers (CTs), voltage transformers (VTs), power meters, Rogowski coils, insulation resistance testers, or partial discharge sensors; environmental monitoring data, such as readings from humidity sensors, gas sensors (e.g., SF6, CO2), dust sensors, or acoustic emission sensors.

[0065] In this embodiment the set of inputs comprises at least one of: the generated first set of electrical characteristics of the circuit model; the learnt latent parameter of the circuit model; the acquired dielectric response measurement data; digital imaging data, such as photographic data and / or microscope data; morphological analysis, such as surface morphology, bulk morphology or crystallinity analysis; chemical composition and material analysis data, such as data relating to additives and impurities, aging and degradation, surface analysis, bulk analysis or molecular weight distribution; surface and chemical treatment data, such as chemical treatments, corona treatments, or plasma treatment data; electrical properties, such as insulation breakdown strength or aging-related electrical data; process-related data, such as manufacturing processes or electrochemical processes; environmental stress data, such as stress conditions or historical data; physical property data such as thermocouples, strain gauges, LVDTs (Linear Variable Differential Transformers), ultrasonic sensors, optical micrometers, hardness testers (e.g., Rockwell, Vickers), or microscopes (e.g., SEM, TEM); chemical property data, such as information from XRF (X-ray Fluorescence), FTIR (Fourier Transform Infrared Spectroscopy), mass spectrometers, pH sensors, and EDS (Energy Dispersive Spectroscopy); electrical property data, resistivity meters, Hall Effect sensors, and impedance analyzers; optical property data, such as measurements from spectrophotometers, refractometers, ellipsometers, colorimeters, or Raman spectrometers; and / or wherein the set of inputs comprise data form at least one of: temperature monitoring data from sensors such as RTDs (Resistance Temperature Detectors), thermistors, infrared sensors, and fiber optic temperature sensors; vibration monitoring data from sensors such as accelerometers, velocity sensors, and proximity probes; electrical parameter monitoring data, collect by for example current transformers (CTs), voltage transformers (VTs), power meters, Rogowski coils, insulation resistance testers, or partial discharge sensors; environmental monitoring data, such as readings from humidity sensors, gas sensors (e.g., SF6, CO2), dust sensors, or acoustic emission sensors.

[0066] In one embodiment, at least one term representing an equivalent circuit component of the circuit model and / or sub-circuit model may a functional value incorporating one of the following:

[0067] Electromagnetic models describing field interactions, such as a model based Maxwell’s Equations for modelling the behavior of electric and magnetic fields across various media; or, wave propagation models that describe the transmission, reflection, refraction, and absorption of electromagnetic waves through different materials; or, antenna models that characterize the radiation patterns and efficiency of antennas in both transmitting and receiving electromagnetic signals.

[0068] - Thermodynamic models describing energy and entropy, such as equation of state models that relate pressure, volume, and temperature in various substances; or, phase change models used to predict the behavior of materials during transitions like melting, evaporation, or solidification; or, entropy and enthalpy models that describe energy transformations, particularly in systems undergoing heat exchange or chemical reactions.

[0069] Structural models describing mechanical integrity, including finite element models (FEM) used to analyze stresses, strains, and deformations in structures under various loads, ensuring their safety and durability; or, fracture mechanics models that predict the initiation and propagation of cracks within materials and structures, crucial for assessing the risk of failure; or, buckling analysis models that determine the stability of structures when subjected to compressive loads. Fluid dynamics models describing fluid behavior, such as the Navier-Stokes Equations that govern the motion of viscous fluid substances, forming the foundation of computational fluid dynamics (CFD) simulations; or, turbulence models that describe chaotic, non-linear fluid motion, often requiring specialized approaches like k-e or Reynolds-Averaged Navier-Stokes (RANS) models to accurately capture these complex phenomena; or, multiphase flow models that describe the behavior of mixtures of different phases, such as gas-liquid or liquidsolid flows.

[0070] - Acoustic models describing sound propagation, including wave equation models that describe how sound waves travel through different media, vital in the design of audio devices and architectural acoustics; or, acoustic impedance models that characterize how sound is transmitted and reflected at material interfaces, important for noise control and soundproofing; or, noise prediction models used in environmental engineering to estimate the impact of noise sources and to design effective mitigation measures.

[0071] Optical models describing light behavior, such as ray tracing models that simulate the path of light through optical systems, accounting for phenomena like reflection, refraction, and absorption, widely used in imaging and lighting design; or, wave optics models that describe wave phenomena such as diffraction and interference, essential in understanding the behavior of light in scenarios where wave effects are significant; or, photonic bandgap models used in the design of photonic crystals and devices that control the flow of light.

[0072] Energy models describing power systems, such as load flow models used to calculate the flow of electrical power in a network, optimizing the generation and distribution of electricity; or, energy storage models that describe the behavior and efficiency of energy storage systems like batteries, supercapacitors, and flywheels, critical for the integration of renewable energy sources; or, renewable energy integration models that analyze the impact and efficiency of integrating renewable energy sources into existing power grids.

[0073] In embodiments is it proposed that there are multiple sub-circuit model; preferably wherein the plurality of sub-circuit model are describing operational characteristics of plurality of sections in the sample of interest. For example, each sub-circuit model may represent one section or the sub-circuit models may at times overlap. The invention allows for flexibility in this aspect so that the specific architecture can be designed for varying applications areas.

[0074] In other words, the system may incorporate a plurality of sub-models, each tailored to describe specific operational characteristics of different sections of the sample of interest. The term "sections" can refer to distinct spatial regions within an electrical system, to various layers or materials within a dielectric composite, or even different section across a frequency spectrum. For instance, in an electrical system, different sub-models could correspond to different circuit components or sections, such as capacitive, resistive, or inductive elements, each influencing the overall behavior of the system. Similarly, in a dielectric composite, sub-models may be designated to analyze properties of individual layers or materials, such as variations in permittivity, conductivity, or response to an electric field.

[0075] In certain embodiments, it may be advantageous to deploy distinct sub-models to address varying frequency domains within the dielectric response measurement data. This approach allows for a more precise analysis of the material's behavior across a wide frequency spectrum. For instance, one sub-model could be specialized for low-frequency analysis, capturing long-term polarization effects, while another could focus on high- frequency behaviors, which are more sensitive to changes in material composition or structural integrity.

[0076] The integration of multiple sub-models may also enhance the overall accuracy of the analysis by allowing each sub-model to concentrate on a particular aspect or characteristic of the sample. This modular approach can lead to more refined parameter estimation and improved diagnostics, particularly in complex systems where different sections exhibit distinct electrical behaviors. By employing sub-models tuned to specific sections and frequency ranges, the system can more accurately characterize the sample’s operational characteristics and provide a comprehensive understanding of its dielectric properties.

[0077] In one proposed embodiment, the training of the machine learning model comprises preprocessing of data used in the training process. The preprocessing comprises at least one of the steps: applying a linear scaling operation (i.e. , a linear normalization operation) to the dielectric response measurement data and / or the generated first set of electrical characteristics, such that the scaled data values are close to unity (i.e., scaled such that the absolute data values fall within a predefined range around 1, e.g., range between 0 and 10, preferably 0 to 1.1. ); applying a logarithmic transformation to the dielectric response measurement data and / or the generated first set of electrical characteristics, wherein the logarithm is selected from the group consisting of natural logarithm, base-10 logarithm, or any other logarithm (i.e., another logarithmic transformation appropriate for the data range), to bring the data onto the same order of magnitude (i.e., to compress the data values into a similar order of magnitude);

[0078] - transforming the dielectric response measurement data into a feature vector using random Fourier features (i.e., applying random Fourier features to project the dielectric response measurement data into a higher-dimensional space), wherein the feature vector may defined as y(X)=(cos(BX),sin(BX)) with B being a predetermined matrix or vector (e.g., With B being a predefined or selected matrix or vector based on the problem's characteristics).

[0079] In embodiments, the machine learning model is a neural network and the training of the neural network further comprises the at least one of the steps:

[0080] - factorizing the weights of the neural network using a random initialization process and applying the factorized weights during the training process, wherein the factorization is designed to promote the network's ability to capture high-frequency behavior in the input data (i.e., promote the detection and learning of high- frequency patterns or variations within the data.); applying a loss balancing mechanism (e.g., a adaptive scaling mechanism for the individual loss terms) during the training process, wherein the balancing mechanism dynamically adjusts the contribution of each loss component to promote acceptable model performance across predetermined metrics; defining a hard boundary constraint for the output of the neural network, wherein the constraint enforces that certain outputs must exactly match predefined values; and incorporating the hard boundary constraint into the training process, such that the network learns to adhere to the predefined values under the specified conditions. In a second aspect of the present invention there is proposed a system for determining equivalent physical quantities using dielectric response measurement data, the system comprising:

[0081] - one or more processors; and,

[0082] - a non-transitory memory coupled to the one or more processors, wherein the memory comprises instructions configured to cause the processors to perform operations for: receiving a plurality of dielectric response measurement data relating to a sample of interest; applying the dielectric response measurement data to a machine learning model stored in the non-transitory memory and executable by the one or more processors, wherein the machine learning model is configure to generate a first set of electrical characteristics of a circuit model, wherein the circuit model uses equivalent circuit components to characterize operational characteristics of the sample of interest; learning a first set of latent parameters relating to the circuit model based on the dielectric response measurement data; and, determining a set of equivalent physical quantities relating to the sample of interest based on the learnt set of latent parameters.

[0083] In an embodiment according to the second aspect the sample is an electrical system, and the system is a system for condition monitoring of the electrical system. The electrical system comprises at least one dielectric material or a composition thereof, and the machine learning model is configured to determine a set of material characteristics relating to the sample of interest based on the learnt set of latent parameters.

[0084] In another embodiment according to the second aspect the dielectric material or composition thereof is a capacitor, and one or several material degradation characteristics of the capacitor is monitored. The material degradation characteristics is indicative of the present and / or future condition of the capacitor.

[0085] In yet another embodiment according to the second aspect the sample is a electrical system, and the system is a system for detecting an anomaly in the electrical system. The anomaly in the electrical system is detected based on deviations in the equivalent physical quantities from expected values. In embodiments according to the second aspect wherein the sample is a dielectric material or a composition thereof, and the system is a system for characterizing said dielectric material and the machine learning model is configured to determine a set of material characteristics relating to the sample of interest based on the learnt set of latent parameters.

[0086] In a embodiment according to the second aspect the system comprises one or more sensors configured to generate dielectric response measurement data relating to the sample of interest.

[0087] In embodiments according to the second aspect the proposed system is configured to perform the method according to any of the embodiments according to the first aspect of the present invention.

[0088] In further aspects there is proposed a computer-implemented method for determining equivalent physical quantities using dielectric response measurement data, the method comprises: receiving, by one or more processors, a plurality of dielectric response measurement data relating to a sample of interest; applying, by one or more processors, the dielectric response measurement data to a machine learning model and generating, using the machine learning model, a first set of electrical characteristics of a circuit model, wherein the circuit model uses equivalent circuit components to characterize operational characteristics of the sample of interest; learning, by one or more processors, a first set of latent parameters relating to the circuit model based on the dielectric response measurement data; and, determining, by one or more processors, a set of equivalent physical quantities relating to the sample of interest based on the learnt set of latent parameters.

[0089] DEFINITIONS

[0090] The term “circuit model” refers to a theoretical representation of an sample, (e.g., a electrical system, component, dielectric material or a composition thereof). The circuit model is built by components such as resistors, capacitors, inductors, voltage sources, and current sources. The circuit model reproduces behavior of the sample using these components. The design of the circuit model decides what behavior of the sample is reproduced.

[0091] “Electrical quantities” refers to quantities such as voltage, current, resistance, capacitance, and inductance are foundational measurements in electrical engineering. These are the basic metrics used to describe and quantify electrical properties.

[0092] “Electrical Characteristics” comprises electrical quantities but also parameters or information derived from electrical quantities. In general, electrical characteristics indicate more about the behavior and performance of electrical systems and components.

[0093] “Operational characteristics” refers to the specific behaviors, properties, or performance parameters of the sample of interest. These characteristics could include electrical, thermal, mechanical, or other relevant quantities that the equivalent circuit components are designed to simulate or represent. The purpose is to accurately reflect how the sample behaves under certain conditions when subjected to a specific analysis or measurement technique.

[0094] “Equivalent physical quantities” encompasses any measurable or inferred metrics that can be used to describe the behavior, properties, or performance of a system, component, or material under analysis. These quantities are not limited to established physical metrics like voltage, current, resistance, capacitance, or inductance, but also include any derived or inferred parameters that may provide insight into the system's characteristics.

[0095] Equivalent physical quantities might include, but are not limited to:

[0096] Impedance-related metrics (e.g., complex impedance, reactance, or admittance) Frequency-dependent properties (e.g., resonance frequencies, phase angles, or spectral densities)

[0097] Material-specific metrics (e.g., dielectric constant, permittivity, permeability, conductivity resistivity, surface resistivity or surface conductivity)

[0098] - Transient response parameters (e.g., rise time, decay rate, or overshoot) Energy-related measures (e.g., energy dissipation, power loss, dielectric loss, tangent delta, or storage capacity)

[0099] Nonlinear or emergent properties (e.g., hysteresis, non-linear capacitance or inductance, or any behavior resulting from complex interactions within the system) Environmental response characteristics (e.g., temperature coefficients, thermal expansion, or humidity sensitivity)

[0100] Statistical or probabilistic parameters (e.g., variance, noise factors, or reliability indices)

[0101] These equivalent physical quantities may represent real, complex, or abstracted metrics that, together, provide a comprehensive understanding of the system's and / or material’s performance, behavior, or underlying physical phenomena.

[0102] The term "sub-circuit model" is intentionally broad, encompassing various types of models such as electromagnetic models, thermodynamic models, fluid dynamic models, acoustic models, and others that describe different physical aspects of the sample.

[0103] DETAILED DESCRIPTION OF EXEMPLIFYING EMBODIMENTS

[0104] In figure 1, a process 100 for determining equivalent physical quantities using dielectric response measurement data is shown. The process 101 involves receiving a plurality of dielectric response measurement data related to a sample of interest. This data may be preprocessed for subsequent analysis and serves as the input to process 102. The dielectric response data may include both time-domain and frequency-domain measurements.

[0105] In process 102, the dielectric response measurement data is applied to a machine learning model. This model is specifically designed to interpret and analyze the data, generating a first set of electrical characteristics that correspond to a theoretical circuit model. This circuit model uses equivalent circuit components, such as resistors, capacitors, and inductors, to characterize the operational behavior of the sample of interest. The machine learning model’s output may provide a detailed representation of how the sample behaves under different conditions.

[0106] Next, in process 103, the machine learning model learns a first set of latent parameters associated with the circuit model. These latent parameters represent underlying physical properties of the sample that may not directly measurable but which may significantly influence its behavior. In embodiments the learning process involves optimizing the machine learning model using a loss function that incorporates the circuit model equation, thereby ensuring that the machine learning model's predictions align with established physical principles.

[0107] Finally, in process 104, a set of equivalent physical quantities relating to the sample of interest is determined based on the learned latent parameters. These quantities may include metrics such as dielectric constant, impedance, and other material-specific parameters that provide insights into the sample's condition and performance.

[0108] In one embodiment, the process may further comprise integrating a sub-model to describe a specific section of the sample. This sub-model is applied within process 103, where it generates a second set of latent parameters that reflect the operational characteristics of that section. The machine learning model may then combine these second set of latent parameters with the first set of latent parameters, allowing for a detailed and sectionspecific analysis. This approach may enhance the accuracy of the determined equivalent physical quantities by incorporating localized variations within the sample.

[0109] Figure 2 illustrates a system 200 for determining equivalent physical quantities 205 using dielectric response measurement data 204. The system 200 comprises a processing unit 201 , which includes:

[0110] One or more processors 202, and

[0111] - A non-transitory memory 203 coupled to the one or more processors 202.

[0112] The processing unit 201 (comprising both the processors 202 and the memory 203) is configured to execute instructions stored in the memory 203. These instructions enable the processors 202 to perform operations that include:

[0113] Receiving a plurality of dielectric response measurement data 204 relating to a sample of interest;

[0114] - Applying the dielectric response measurement data 204 to a machine learning model stored in the non-transitory memory 203 and executable by the one or more processors 202. The machine learning model is configured to generate a first set of electrical characteristics of a circuit model, where the circuit model uses equivalent circuit components to characterize operational characteristics of the sample of interest;

[0115] Learning a first set of latent parameters related to the circuit model based on the dielectric response measurement data 204; and, Determining a set of equivalent physical quantities 205 related to the sample of interest based on the learned set of latent parameters.

[0116] Figure 3 illustrates a circuit model for a scenario wherein the sample of interest is a degraded film capacitor. The degradation results in disconnection or loosening of the film's contacts.

[0117] The circuit model represent the degraded film capacitor in the following way:

[0118] The circuit is divided into two branches. The first branch models the capacitor with a capacitance C, and includes both series resistance Rsand leakage resistance RP.

[0119] The second branch represents the circuit elements that account for the degraded contact conditions between the film and the rest of the circuit, which may include loosened or disconnected contacts.

[0120] The parameters within second branch include resistances and capacitances that are influenced by the air gap and the film material, denoted with suffixes d-a (for air gap-related parameters) and d-f (for film-related parameters).

[0121] In this exemplifying embodiment, the method begins by receiving a plurality of dielectric response measurement data related to a degraded film capacitor, which is the sample of interest. This data could include time-domain or frequency-domain measurements that capture how the capacitor responds to applied electric fields. For example, the data might show how the capacitance of the film capacitor changes over a range of frequencies, or how leakage current evolves over time under a constant voltage.

[0122] Next, this dielectric response data is applied to a machine learning model. The model processes the data and generates a first set of electrical characteristics of a circuit model. The circuit model in this scenario uses equivalent circuit components as illustrated in figure 3 to characterize the operational characteristics of the degraded film capacitor. For instance, the model might generate electrical characteristics that include an increased equivalent series resistance (ESR) and a reduced capacitance, reflecting the degraded condition of the capacitor.

[0123] Based on the dielectric response data and the generated electrical characteristics, the machine learning model then learns a first set of latent parameters relating to the circuit model. These latent parameters might include variables such as the extent of disconnection between the film layers, the severity of the dielectric breakdown, or the changes in contact resistance due to loosened connections. Additionally, the model may incorporate variations in data from non-defective samples, which can serve as a valuable reference point. These parameters are not directly observable from the data but are inferred by the model to explain the observed changes in the capacitor's behavior.

[0124] Finally, the method determines a set of equivalent physical quantities relating to the degraded film capacitor based on the learnt latent parameters. These physical quantities could include the capacitor's effective capacitance, which has decreased due to the disconnection of the film, or the leakage resistance, which may have changed due to the formation of air gaps. These equivalent physical quantities provide a detailed assessment of the capacitor's condition, offering valuable insights into its remaining life expectancy and the need for maintenance or replacement.

[0125] Figure 4 illustrates a simple circuit model comprising two resistor components. In one example scenario, the resistor Ro depicted in Figure 4 can be substituted with a system of partial differential equations (PDEs) that model the dynamics of charge transport and the resulting displacement current within a high-voltage (HV) insulation system. This substitution accounts for the movement of various types of charges driven by both the external applied electrical field and any internal electric fields generated by the voltage across the original Ro component. The PDE-based model provides a more detailed representation of the complex interactions and physical processes occurring within the insulation material under the influence of these fields.

[0126] In another example, the combined effects of the resistor Ri and capacitor Ci in Figure 4, can be substituted with a physical model governed by an equation that emulates the polarization mechanisms within the material under study. This model might involve equations that describe how the material's polarization responds to an electric field, taking into account factors such as dielectric relaxation, ionic movement, or dipolar alignment. By using such a physical model, the behavior of the material under varying electrical conditions can be captured more accurately, offering deeper insights into the polarization processes than what can be achieved with simple R-C circuit representations. The exemplifying embodiment illustrated in Figure 5 pertains to a case wherein the circuit model is a thermal model of a capacitor, designed to analyze and predict the thermal behavior of the capacitor under operational conditions. The capacitor comprises a capacitor element embedded within an element case, which is further surrounded by external insulation and enclosed within an external casing.

[0127] The thermal model accounts for the power loss within the capacitor, which is a critical factor affecting the thermal stability and overall performance of the device. This power loss is derived from the measured current flowing through the capacitor and the resistance associated with the capacitor element. The resistance is learnt from an electrical model, which simulates the electrical characteristics of the capacitor under different conditions.

[0128] The thermal model may be employed for condition monitoring during the operation of the capacitor. By continuously analyzing the thermal behavior under operational conditions, the thermal model can detect anomalies indicative of potential faults or degradation. This facilitates proactive maintenance and enhances the reliability and longevity of the equipment by allowing timely interventions based on the monitored thermal data.

[0129] In addition, the illustrated thermal model enables estimating the heat generated within the capacitor due to power loss, which informs the design and selection of appropriate materials for insulation and casing to ensure safe and reliable operation. In this scenario, the proposed method, or system performing the method, also serves as a tool in optimizing the thermal management of capacitors, particularly in applications where they are subject to high currents and varying environmental conditions.

[0130] In an alternative embodiment not shown, the thermal model presented in Figure 5 may be further refined to incorporate dielectric losses resulting from the polarization of the capacitor's dielectric material. Dielectric losses are a important factor contributing to the overall heat generation within capacitors, especially under conditions where high- frequency alternating currents are present or when the dielectric material exhibits significant polarization.

[0131] This alternative embodiment extends the thermal model by integrating the effects of dielectric losses into the heat generation calculation. The thermal model includes a equivalent subcomponent representing the dielectric material's polarization behavior, which is influenced by factors such as the material's permittivity, frequency of operation, and the temperature dependency of the dielectric properties. This equivalent subcomponent interacts with the existing thermal circuit model, where the power loss due to polarization is computed alongside the resistive losses within the capacitor element.

[0132] The alternative thermal model may be designed to leverage dielectric response measurement data to more accurately characterize the dielectric losses. This data-driven approach allows for dynamic adjustment of the thermal model parameters to account for changes in the dielectric properties over time or due to environmental conditions, thereby improving the accuracy of thermal predictions and aiding in e.g., condition monitoring or a material selection process.

Claims

CLAIMS1. A method for determining equivalent physical quantities using dielectric response measurement data, the method comprises: receiving a plurality of dielectric response measurement data relating to a sample of interest; applying the dielectric response measurement data to a machine learning model and generating, using the machine learning model, a first set of electrical characteristics of a circuit model, wherein the circuit model uses equivalent circuit components to characterize operational characteristics of the sample of interest; learning a first set of latent parameters relating to the circuit model based on the dielectric response measurement data; and, determining a set of equivalent physical quantities relating to the sample of interest based on the learnt set of latent parameters.

2. The method according to claim 1 , wherein the machine learning model comprises a loss function in which at least one term is derived from a time, space and / or frequency dependent differential equation that reproduces behavior of the circuit model, wherein learning the first set of latent parameters of the circuit model comprises the step:- training the machine learning model using the loss function, the dielectric response measurement data and the generated first set of electrical characteristics.

3. The method according to any previous claim, wherein the sample is either plurality of components in an electrical system and wherein the dielectric measurements relates to at least one component in the electrical system; and / or, a dielectric material or compositions thereof.

4. The method according to any previous claim, wherein one or more of the equivalent circuit components in the circuit model is replaced by a sub-circuit model which describes operational characteristics of a section of the sample of interest.

5. The method according to any previous claim, wherein the circuit model and / or the subcircuit model uses equivalent circuit components to characterize operationalcharacteristics in the sample of interest, and wherein the equivalent circuit components of the circuit model and / or the sub-circuit model reproduces behavior of at least one of the following circuit components: resistance, capacitance, inductance, impedance, and / or other circuit components relevant for describing the sample of interest; wherein each equivalent circuit component is represented by a term in: the time, space and / or frequency dependent differential equation that reproduces behavior of the circuit model; and / or a time, space and / or frequency dependent differential equation that reproduces behavior of the sub-circuit model, and wherein each term is constant or have a functional value dependent on one or more variables, including but not limited to temperature, electric field strength, voltage, aging history, material composition, and / or morphology of the sample of interest.

6. The method according to claim 4, wherein determining the set of equivalent physical quantities of the sample of interest comprises the step of: applying a set of inputs to a sub machine learning model (sub-ML model) and generating, from the sub-ML model, a second set of electrical characteristics relating to the sub-circuit model, wherein the sub-ML model comprises a loss function in which at least one term is derived from a time, space and / or frequency dependent differential equation that reproduces behavior of the subcircuit model; and, learning a second set of latent parameters relating to the sub-circuit model based on the loss function of the sub-circuit model, the generated set second of electrical characteristics of the sub-circuit model and the set of inputs.

7. The method according to claim 6, wherein the machine learning model and / or sub-ML model is a Physics-Informed Neural Network.

8. The method according to claim 6, and wherein generating the first set of electrical characteristics of the circuit model based further includes:- applying the second set of latent parameters of the sub-circuit model and the dielectric response measurement data to the machine learning model and generating, using the machine learning model, the first set of electrical characteristics of the circuit model; wherein the set of inputs preferably comprises dielectric response measurement data.

9. The method according to claim 6 and 2, wherein training of the machine learning model further includes:- training the machine learning model using the loss function of the circuit model, the second set of latent parameters of the sub-circuit model, the dielectric response measurement data and the generated first set of electrical characteristics of the circuit model;, wherein at least one parameter in the second set of latent parameters of the sub-circuit model are applied as frozen parameters and / or initial parameters allowed to change, and wherein the set of inputs preferably comprises at least one of: the generated first set of electrical characteristics of the circuit model and / or the acquired dielectric response measurement data.

10. The method according to claim 6, wherein the equations reproducing the operational characteristics of the circuit model and the sub-circuit model are integrated into one time, space and / or frequency dependent differential equation and wherein the machine learning model and sub-ML model is the same machine learning model, and the determined set of equivalent physical quantities comprising the at least one equivalent physical quantity relating to a sub section of the sample of interest.

11. The method according to claim 5, wherein at least one term representing an equivalent circuit component of the circuit model and / or sub-circuit model is a functional value incorporating one of: electromagnetic model describing field interactions; or, thermodynamic model describing energy and entropy; or, structural model describing mechanical integrity; or, fluid dynamics model describing fluid behavior; or, acoustic model describing sound propagation; or, optical model describing light behavior; or, energy model describing power systems.

12. A system for determining equivalent physical quantities using dielectric response measurement data, the system comprising: one or more processors; and, a non-transitory memory coupled to the one or more processors, wherein the memory comprises instructions configured to cause the processors to perform operations for: receiving a plurality of dielectric response measurement data relating to a sample of interest;applying the dielectric response measurement data to a machine learning model stored in the non-transitory memory and executable by the one or more processors, wherein the machine learning model is configure to generate a first set of electrical characteristics of a circuit model, wherein the circuit model uses equivalent circuit components to characterize operational characteristics of the sample of interest; learning a first set of latent parameters relating to the circuit model based on the dielectric response measurement data; and, determining a set of equivalent physical quantities relating to the sample of interest based on the learnt set of latent parameters.

13. The system according to claim 12, wherein the sample is an electrical system, and the system is a system for condition monitoring of the electrical system, wherein the electrical system comprises at least one dielectric material or a composition thereof, and wherein the machine learning model is configured to determine a set of material characteristics relating to the sample of interest based on the learnt set of latent parameters.

14. The method according to claim 13, wherein the dielectric material or composition thereof is a capacitor, and wherein one or several material degradation characteristics of the capacitor is monitored, wherein said material degradation characteristics is indicative of the present and / or future condition of the capacitor.

15. The system according to any of claim 12 to 14, wherein the system comprises one or more sensors configured to generate dielectric response measurement data relating to the sample of interest.

16. The system according to any of claim 12 to 15, wherein the machine learning model is a Physics-Informed Neural Network.

17. The system according to any of claim 12 to 16, wherein the system performs the method according to any of claim 2 to 11 .

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