<p>  </p> <p>A <em>digital twin</em> (DT), which keeps track of nuclear reactor history to provide real-time predictions, has been recently proposed for nuclear reactor monitoring. A digital twin can be implemented using either a differential equations-based physics model, or a data-driven machine learning model<strong>. </strong>The principal challenge in physics model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, while the main challenge in data-driven DT appears in the extensive training requirements and potential lack of predictive ability. </p> <p>In this thesis, we investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. In this way, PINNs establish theoretical constraints and biases to supplement measurement data and provide solution to several limitations of purely data-driven machine learning (ML) models. We develop a PINN model to solve the point kinetic equations (PKEs), which are time dependent stiff nonlinear ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring the spatial dependence of the neutron flux. PKEs portray the kinetic behavior of the system, and this kind of approach is the basis for most analyses of reactor systems, except in cases where flux shapes are known to vary with time. This system describes the nuclear parameters such as neutron density concentration, the delayed neutron precursor density concentration and reactivity. Both neutron density and delayed neutron precursor density concentrations are the vital parameters for safety and the transient behavior of the reactor power. </p> <p>The PINN model solution of PKEs is developed to monitor a start-up transient of the Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The facility under modeling, PUR-1, is a pool type small research reactor located in West Lafayette Indiana. It is an all-digital light water reactor (LWR) submerged into a deep-water pool and has a power output of 10kW. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. </p> <p>The findings of this thesis research indicate that the PINN model achieved highest performance and lowest errors in data interpolation. In the case of extrapolation data, three different test cases were considered, the first where the extrapolation is performed in a five-seconds interval, the second where the extrapolation is performed in a 10-seconds interval, and the third where the extrapolation is performed in a 15-seconds interval. The extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.</p>

A power system consists of several critical components necessary for providing electricity from the producers to the consumers. Monitoring the lifetime of power system components becomes vital since they are subjected to electrical currents and high temperatures, which affect their ageing. Estimating the component's ageing rate close to the end of its lifetime is the motivation behind our project. Knowing the ageing rate and life expectancy, we can possibly better utilize and re-utilize existing power components and their parts. In return, we could achieve better material utilization, reduce costs, and improve sustainability designs, contributing to the circular industry development of power system components. Monitoring the thermal distribution and the degradation of the insulation materials informs the estimation of the components' health state. Moreover, further study of the employed paper material of their insulation system can lead to a deeper understanding of its thermal characterization and a possible consequent improvement. Our study aims to create a model that couples the physical equations that govern the deterioration of the insulation systems of power components with modern machine learning algorithms.  As the data is limited and complex in the field of components' ageing, Physics-Informed Neural Networks (PINNs) can help to overcome the problem. PINNs exploit the prior knowledge stored in partial differential equations (PDEs) or ordinary differential equations (ODEs) modelling the involved systems. This prior knowledge becomes a regularization agent, constraining the space of available solutions and consequently reducing the training data needed.  This thesis is divided into two parts: the first focuses on the insulation system of power transformers, and the second is an exploration of the paper material concentrating on cellulose nanofibrils (CNFs) classification. The first part includes modelling the thermal distribution and the degradation of the cellulose inside the power transformer. The deterioration of one of the two systems can lead to severe consequences for the other. Both abilities of PINNs to approximate the solution of the equations and to find the parameters that best describe the data are explored. The second part could be conceived as a standalone; however, it leads to a further understanding of the paper material. Several CNFs materials and concentrations are presented, and this thesis proposes a basic unsupervised learning using clustering algorithms like k-means and Gaussian Mixture Models (GMMs) for their classification. / Ett kraftsystem består av många kritiska komponenter som är nödvändiga för att leverera el från producenter till konsumenter. Att övervaka livslängden på kraftsystemets komponenter är avgörande eftersom de utsätts för elektriska strömmar och höga temperaturer som påverkar deras åldrande. Att uppskatta komponentens åldringshastighet nära slutet av dess livslängd är motivationen bakom vårt projekt. Genom att känna till åldringshastigheten och den förväntade livslängden kan vi eventuellt utnyttja och återanvända befintliga kraftkomponenter och deras delar   bättre. I gengäld kan vi uppnå bättre materialutnyttjande, minska kostnaderna och förbättra hållbarhetsdesignen vilket bidrar till den cirkulära industriutvecklingen av kraftsystemskomponenter. Övervakning av värmefördelningen och nedbrytningen av isoleringsmaterialen indikerar komponenternas hälsotillstånd. Dessutom kan ytterligare studier av pappersmaterial i kraftkomponenternas isoleringssystem leda till en djupare förståelse av dess termiska karaktärisering och en möjlig förbättring.  Vår studie syftar till att skapa en modell som kombinerar de fysiska ekvationer som styr försämringen av isoleringssystemen i kraftkomponenter med moderna algoritmer för maskininlärning. Eftersom datan är begränsad och komplex när det gäller komponenters åldrande kan  fysikinformerade neurala nätverk (PINNs) hjälpa till att lösa problemet. PINNs utnyttjar den förkunskap som finns lagrad i partiella differentialekvationer (PDE) eller ordinära differentialekvationer (ODE) för att modellera system och använder dessa ekvationer för att begränsa antalet tillgängliga lösningar och därmed minska den mängd träningsdata som behövs.  Denna avhandling är uppdelad i två delar: den första fokuserar på krafttransformatorers isoleringssystem, och den andra är en undersökning av pappersmaterialet som används med fokus på klassificering av cellulosananofibriller (CNF). Den första delen omfattar modellering av värmefördelningen och nedbrytningen av cellulosan inuti krafttransformatorn. En försämring av ett av de två systemen kan leda till allvarliga konsekvenser för det andra. Både PINNs förmåga att approximera lösningen av ekvationerna och att hitta de parametrar som bäst beskriver datan undersöks. Den andra delen skulle kunna ses som en fristående del, men den leder till en utökad förståelse av själva pappersmaterialet. Flera CNF-material och koncentrationer presenteras och denna avhandling föreslår en simpel oövervakad inlärning med klusteralgoritmer som k-means och Gaussian Mixture Models (GMMs) för deras klassificering. / <p>QC 20231010</p>