Selection criteria for optimal sensor placement in online monitoring systems

Abstract

Disclosed herein is a method to optimize sensor selection and placement of a target system. The method includes generating a model of the target system and using the model to generate signal features related to a plurality of sensor deployment configurations. A plurality of solutions for optimal sensor deployment configurations are generated using a plurality of sensor selection criteria and the signal feature. The optimized solutions are used in the target system. Disclosed herein are systems to optimize sensor selection and placement of the target solution.

Description

Attorney docket no.103361-516WO1 T2023-289 Selection Criteria for Optimal Sensor Placement in Online Monitoring Systems Statement of Government Support

[0001] This invention was made with government support under grant / contract numbers DE-AC02-06CH11357 and DE-NE-19-17045 awarded by the Department of Energy. The government has certain rights in the invention. Cross-Reference to Related Applications

[0002] The present application claims the benefit under 35 U.S.C.119(e) of the filing date of U.S. Application Serial No.63 / 526,720, filed on July 14, 2023, the entire contents of which are fully incorporated herein by reference. Background

[0003] Choosing the right sensors is crucial for the design and development of online monitoring systems in safety-critical applications. These systems rely on sensor networks to track system behavior and identify operational failures. The process of selecting sensors typically involves evaluating their effectiveness and efficiency based on one or more criteria. Employing the latest methods and technologies in sensor selection is key to enhancing the economic viability of these critical monitoring systems. Beginning the development of an online monitoring system at an early stage in the creation of a safety-critical system can greatly smooth the transition from the design phase to the implementation and operation phase.

[0004] However, a major challenge arises in the early stages of system design due to the absence of historical data, making it difficult to apply data-driven sensor selection methods, such as machine learning algorithms [1], [2], [3]. These models require extensive data to ensure accuracy. To overcome this, one approach is to use historical data from systems similar to the one being designed, although the relevance of this data can be difficult to ascertain. Another solution is to use model-based approaches, where one or more models simulate the target system's behavior under various scenarios. This allows for the extraction of system features even without the actual system, guiding the choice of sensor types and their placement based on the data derived from these simulations.

[0005] Model-based methods for sensor selection are divided into two types: those based on quantitative models and those based on qualitative models. Quantitative model- based approaches, such as the approaches using Analytical Redundancy Relations (ARR) for fault diagnostics and prognostics [4], [5], fault tolerance [6], and observability [7], haveAttorney docket no.103361-516WO1 T2023-289 certain limitations. Firstly, creating and solving these models can be time-intensive and sometimes impractical. For instance, running a simulation for thermal-hydraulic dynamics with multiple pumps and valves can take hours or days. Obtaining differential equations for a thermal-hydraulic system affected by leakages or corrosion is particularly challenging. Secondly, even with a well-established model based on data from similar systems, its representativeness is not assured. For example, a hydraulic model designed for a feedwater system with electromagnetic valves may not accurately represent the same system with pneumatic valves. Thirdly, the specificity of quantitative models limits their generalizability. Each simulation only tests a specific system configuration under certain conditions, requiring numerous simulations to generalize results for sensor selection. For example, a simulation might show that a feedwater system maintains operation with a leakage rate causing a flow rate drop from 5.234 L / min to 5.104 L / min. However, to determine the range of leakage rates at which the system still functions normally, a substantial number of simulations are required. Summary

[0006] When a safety-critical system is still in the development phase, it is difficult to select sensors for designing an OLM system for such a system due to the shortage of operational data. It is a model-based sensor selection method that uses specific signal features to determine the signals that need to be monitored by the OLM system. Our previous research can derive an optimal sensor network by maximizing its capability to detect and identify faults while minimizing the cost of sensors. In this study, more signal features and their relations derived from the outcomes of Integrated System Failure Analysis (ISFA) were taken into account, and correspondingly more sensor selection criteria related to OLM system design were developed.

[0007] The criteria designed to verify the effectiveness and performance of the sensor selection method developed to generate optimal sensor deployment configurations for online monitoring (OLM) systems is disclosed. In the case study, several important capabilities of the OLM system, such as the capability of observing various states of the target system, the capability of fault detection and discrimination, the capability of fault prognostics, and various other characteristics that sensor systems should fulfill such as functionality, integrability, and cost, are taken into account when generating optimal sensor deployment configurations. A reactor cavity cooling system (RCCS) of the General Atomics (GA) Modular High-Temperature Gas Reactor (MHTGR) is selected as the case study system. This study proposes a multiple-objective optimization algorithm which outputs a series of sensorAttorney docket no.103361-516WO1 T2023-289 deployment solutions, including the numbers, types, and positions for deploying the sensors required by the OLM system.

[0008] In some aspects, described herein is a method to optimize sensor selection and placement of a target system, the method including: generating a model of the target system; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors.

[0009] In some aspects, described herein is a system to optimize sensor selection and placement of a target system, the system including: a computing system including one or more processors and one or more storage devices, wherein the one or more storage devices have instructions stored thereon, that when executed by the one or more processors, causes the one or more processors to perform a method, the method including: generating a model of the target system; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors. Brief Description of the Drawings

[0010] Fig.1 shows a schematic of the exemplary system.

[0011] Fig.2 shows a schematic of an exemplary method.

[0012] Fig.3 shows a schematic of an exemplary method.

[0013] Fig.4 shows inputs and outputs of the model, represented as ISFA.

[0014] Fig.5 shows an example deviation vector.

[0015] Fig.6 shows an example variation vector.

[0016] Fig.7 shows an example deviation time operator.

[0017] Fig.8 shows an example unique operator.

[0018] Fig.9 shows an example similarity operator.

[0019] Fig.10 shows an example correlation operator.

[0020] Figs.11A-11F show sample diagrams of failure modes of sensors.

[0021] Fig.12 shows a flowchart of the implementation of the Non-dominated Sorting Genetic Algorithm (NSGA-II) for the sensor placement optimization problem.

[0022] Fig.13 shows an example structure of an RCCS

[0031] .

[0023] Fig.14 shows a schematic of the ISFA model of the RCCS.Attorney docket no.103361-516WO1 T2023-289

[0024] Fig.15 shows simplified source code of the inlet filter in the ISFA model.

[0025] Fig.16 shows an example of non-dominated Pareto fronts for sensors placement optimization for different sensors selection criteria.

[0026] Figs.17A-17B show an example of different sensors selected at different locations for 1000 solutions.

[0027] Fig.18 shows an example of sensors selected at different locations grouped by sensor types for 1000 solutions.

[0028] Fig.19 shows the average numbers of sensors selected for different sensor types.

[0029] Fig.20 shows the average number of sensors selected for different locations.

[0030] Fig.21 shows an exemplary computing device. Detailed Specification

[0031] To facilitate an understanding of the principles and features of various embodiments of the present invention, they are explained hereinafter with reference to their implementation in illustrative examples.

[0032] Referring now to Fig.1, an exemplary system to optimize sensor selection is shown. In some aspects, the system 100 takes as input a target system 110, where the target system 110 is described by a tuple, { ^^^^(s)}. In other aspects, the target system is described by any suitable sized matrix, { ^^^^(s)}.

[0033] In some aspects, the target system 110 includes two or more values or equations that express target system components, functions of the target system components, input and output variables from target system components, failure modes, behavioral rules, function failure logics, and / or initial conditions. In some aspects, the behavioral rules include qualitative models representing the behaviors of target system components under nominal and faulty states. In some aspects, the model infers a chronological order of input / output variable values from target system components. In some aspects, the failure modes relate to abnormal states of target system components during target system operation (e.g. pipe leakage). In some aspects, function failure logics include logic expressions mapping qualitative variable values to functional states, wherein functional states include operating, degraded, and lost.

[0034] In some aspects, the target system 110 includes a safety-critical system including one of a reactor system, a cooling system, an electronics system, a construction system, an energy production system, an energy conversion system, an environmentalAttorney docket no.103361-516WO1 T2023-289 monitoring system, a manufacturing system, a telecommunications system, a transportation system, a motor vehicle system, or any system including a plurality of sensors in one or more locations.

[0035] The target system 110 is provided to the simulation engine 120. The simulation engine 120 includes a target system model 121 and signal features 122. In some aspects, the target system model 121 includes a qualitative physical model. The signal features 122 are a superset of time series, ^^^^(t), representing the qualitative values of system variables and the states of the functions implemented by the target system 110. In some aspects, the signal features 122 include state functions representing qualitative values of target system variables. The signal features 122 are used in the optimization engine 130.

[0036] The optimization engine 130 includes an objective optimization algorithm 131, objective functions 132, and objective constraints 133. In some aspects, the objective optimization algorithm 131 includes a multiple objective optimization algorithm. In other aspects, the multiple objective optimization algorithm is a genetic algorithm. The objective functions 132 include a plurality of sensor selection criteria (CRIT 1, CRIT 2, CRIT 3), and the objective constraints 133 define a plurality of threshold values for related plurality of sensor selection criteria. In some aspects, the sensor selection criteria include one or more of fault detection and discrimination, risk reduction, sensor failure tolerance, observability, functionality, integrability, and cost.

[0037] The system outputs an optimal system 140, which defines sensor deployment configuration, {S(s)}, for the target system. In some aspects, the sensor deployment configuration is one of a plurality of solutions that includes a set of sensors at respective locations in the target system and related values of sensor selection criteria for the set of sensors.

[0038] Referring now to Fig.2, an exemplary method is shown. In some aspects, the method includes generating a model of the target system 210; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors 220; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features 230; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors 240.

[0039] In some aspects, the model of the target system includes a qualitative physical model. In some aspects, input to the model includes two or more values or equations, wherein the two or more values or equations express target system components, functions ofAttorney docket no.103361-516WO1 T2023-289 the target system components, input and output variables from target system components, failure modes, behavioral rules, function failure logics, and / or initial conditions.

[0040] In some aspects, the behavioral rules include qualitative models representing the behaviors of target system components under nominal and faulty states. In some aspects, the model infers a chronological order of input / output variable values from target system components. In some aspects, the failure modes relate to abnormal states of target system components during target system operation (e.g. pipe leakage). In some aspects, function failure logics include logic expressions mapping qualitative variable values to functional states, wherein functional states include operating, degraded, and lost.

[0041] In some aspects, the model outputs a set of time series of signal features, wherein signal features include state functions representing qualitative values of target system variables.

[0042] In some aspects, the sensor selection criteria are mathematical functions.

[0043] In some aspects, the sensor selection criteria includes one or more of fault detection and discrimination, risk reduction, sensor failure tolerance, observability, functionality, integrability, and cost.

[0044] In some aspects, the target system includes a safety-critical system including one of a reactor system, a cooling system, an electronics system, a construction system, an energy production system, an energy conversion system, an environmental monitoring system, a manufacturing system, a telecommunications system, a transportation system, a motor vehicle system, or any system including a plurality of sensors in one or more locations.

[0045] Referring now to Fig.3, an exemplary method is shown. In some aspects, generating a plurality of solutions 240 includes: defining an objective function based on the plurality of sensor selection criteria 310; defining constraints for sensor selection at different locations to simulate different variables 320; simulating, using the model, different variables based on sensor selection at each of the different locations 330; Appling an objective optimization algorithm to the different variables 340; iteratively evaluating the objective functions for sensor selection at different locations 350; selecting an optimal solution (e.g. Pareto front) 360; repeating the steps until a predetermined number of solutions are generated 370.

[0046] In some aspects, a constraint is a sensor selection constraint, which may include a desired solution to the sensor selection criteria.Attorney docket no.103361-516WO1 T2023-289

[0047] In some aspects, each of the plurality of solutions includes a set of sensors at respective locations in the target system and related values of sensor selection criteria for the set of sensors.

[0048] In some aspects, the objective optimization algorithm includes a multiple objective optimization algorithm. In other aspects, the multiple objective optimization algorithm includes a genetic algorithm. In yet other aspects, the genetic algorithm is applied with predetermined generation rate, predetermined crossover rate, and predetermined mutation rate.

[0049] Sensor selection is critical in designing and developing online monitoring systems. An online monitoring (OLM) system is limited by the sensor network it uses to observe plant behaviors. A sensor selection strategy usually adopts one or several criteria to evaluate the effectiveness and efficiency of various types of sensors. State-of-the-art methods and technologies for sensor selection strategies are essential to improve the economic competitiveness of advanced reactors by optimizing plant performance. These selection strategies can be adopted to support the design of intelligent OLM, which can be further integrated with asset management decision-making to optimize the performance and economics of reactors. When a safety-critical system is still in the development phase, it is difficult to select sensors for designing an OLM system for such a system due to the shortage of operational data. The disclosed model-based sensor selection method uses specific signal features to determine the signals that need to be monitored by the OLM system. Previous disclosure, WO 2022 / 216356, presents a method to derive an optimal sensor network by maximizing its capability to detect and identify faults while minimizing the cost of sensors; the disclosure is incorporated by reference in its entirety.

[0050] Qualitative model-based approaches offer an alternative for sensor selection, addressing the challenges of developing quantitative models and their limited generalizability. As a key area in Artificial Intelligence, qualitative reasoning has been integrated into various model-based methods [8], [9],

[0010] . These qualitative physical models encapsulate human expertise in physical systems, enhancing their utility. By combining these models with symbolic executions

[0011] and SAT (satisfiability) theories

[0012] , computers can infer the behavior and states of physical systems. For instance, expert knowledge might formulate a basic equation for pipe flowrates: Input_flowrate = Output_flowrate + Leakage_flowrate (1)Attorney docket no.103361-516WO1 T2023-289

[0051] Assuming both input and output flowrates are positive and considering a minor leak, the following condition (a Boolean expression of propositional logic) can be acquired: Leakage_flowrate > 0 and Leakage_flowrate < Output_flowrate (2)

[0052] Using symbolic representation and propositional logic, it is deduced that Input_flowrate should exceed Output_flowrate for a leaking pipe. If normal functioning is defined as equal input and output flowrates, and degradation as output being less than input, then SAT theory helps confirm leakage when input_flowrate > output_flowrate. These principles apply across different pipe types, locations, and materials without specifying exact leakage rates. Thus, one run of a qualitative model can encompass several outcomes from quantitative models, enhancing representativeness and generalizability while reducing simulation costs.

[0053] The text above explains how a given formula and conditions demonstrate the connections between various physical variables in a system. These connections are vital for data-driven approaches to effectively interpret and learn from information, which is often limited during the system's design stage. In the proposed method, these relationships are identified in qualitative reasoning using validated domain knowledge. The process involves mapping how different components of the system interact, specifically how their inputs and outputs are related. This mapping enables the automated prediction of fault propagation within the system. For example, in a setup where a pipe connects to a valve, this relationship is understood as the valve's inflow rate matching the pipe's outflow rate

[0013] . Valve.input_flowrate = Pipe.output_flowrate (3)

[0054] When a pipe experiences a leak, the actual output flow rate of the pipe will decrease. According to the equation, this issue will affect the valve by decreasing the flow rate entering the valve compared to the original flow rate from the pipe. Furthermore, adding time variables to these equations allows for monitoring how faults in components dynamically evolve over time. An example of this can be seen in the modified equation for the pipe and valve scenario. Valve.input_flowrate.t2 = Pipe.output_flowrate.t1 (4)

[0055] The equation demonstrates the impact of a pipe leak at one time step on the valve in the following time step, illustrating how this effect propagates through subsequentAttorney docket no.103361-516WO1 T2023-289 components. For a deeper understanding and more detailed information on this subject, the referenced material provides extensive insights

[0014] ,

[0015] .

[0056] Qualitative model-based reasoning has been applied in fault diagnostics through various methods, such as Qualitative Trends Analysis (QTA)

[0016] , bond graphs

[0017] , and Sign Directed Graphs (SDG)

[0018] . Snooke

[0018] developed an automated approach for optimal sensor selection for fault diagnosis, using Failure Modes and Effects Analysis (FMEA) to map fault-symptom relationships and visualize them in matrices, thereby increasing the number of diagnosable faults with fewer sensors. Jian et al.

[0019] incorporated sensor reliability and fault detectability into their optimization strategy, defining a metric to balance cost-effectiveness with fault detection and isolation, exemplified in an electro- mechanical actuator case study using a hybrid particle swarm optimization algorithm. Dong

[0020] employed a Boolean Network (BN) modeling approach for sensor selection in complex systems like nuclear plants, establishing conditions for fault detectability and discriminability based on the network's steady state. Zhang et al.

[0021] focused on identifying the type, number, and locations of sensors for robust fault monitoring in critical military or industrial systems. Their methodology, Quantified-Directed-Model (QDG), quantitatively defines fault detectability metrics and models fault propagation in large-scale systems. They introduced a Figure of Merit (FOM) to optimize fault detectability against sensor quantity, integrating this strategy into a diagnostic / prognostic system architecture demonstrated through a five-tank system simulation. However, these methods primarily address fault diagnostics and lack systematic approaches for automatic extraction of system features, also neglecting system transients in their analyses.

[0057] Li et al.

[0022] introduces a method for diagnosing faults in intricate engineering systems. Termed Propagation-based Fault Detection and Discrimination (PFDD), this approach aims to bolster system safety and reliability through precise fault identification and differentiation. It focuses on qualitative features like deviations and variations of system variables to formulate a strategy for fault detection and discrimination, applicable in both steady and transient states. The study also explores optimizing sensor placement using brute force and genetic algorithms to enhance fault diagnosis efficiency while reducing costs, demonstrated using the Experimental Breeder Reactor II (EBR-II) system as a case study. Nonetheless, the research does not incorporate a comprehensive sensor selection framework that methodically employs system features derived from qualitative physical models of the target system, considering the diverse capabilities of online monitoring systems and various sensor features. Table 1 displays capabilities of online monitoring systems.Attorney docket no.103361-516WO1 T2023-289 Table 1. Capabilities of online monitoring systems

[0058] Referring to Table 1, the methodologies described in the previously mentioned literature were evaluated. The outcomes of this comparison are presented in Table 2. It is evident from this comparison that the proposed methodology is unique in considering all the capabilities of online monitoring systems. Table 2. Capabilities of online monitoring systems considered in different methodsAttorney docket no.103361-516WO1 T2023-289

[0059] This disclosure introduces a framework for sensor selection that leverages system features derived from qualitative physical models, aiding in the optimization of sensor deployment for online monitoring (OLM) systems. The framework takes into account various critical capabilities of the OLM system, such as monitoring different states of the target system, fault detection and discrimination, risk reduction, and other essential sensor system attributes like functionality, integration, and cost. This allows users to customize the criteria for evaluating sensor deployment according to specific application needs. Various constraints related to the requirements of the OLM system can be applied.

[0060] Methodology. Like other model-based sensor selection methods, the disclosed method and system relies on qualitative physical models of the target system. The qualitative physical model was described in part in previous publications

[0014] ,

[0022] ,

[0015] and named Integrated System Failure Analysis (ISFA) model, hereinafter the ISFA model. The ISFA model generates qualitative system behaviors to understand how component faults affect the system's functions

[0023] ,

[0024] ,

[0025] . The ISFA model employs Configuration Flow Diagram (CFD) and various UML diagrams to model the target system's architecture, using Behavioral Rules (BRs) and Functional Failure Logic (FFL) for representing the qualitative physical models of the system's components and functions. The ISFA model also uses a Satisfiability Modulo Theory (SMT) solver for deducing trends in physical variables and the functional states within the target system.

[0061] The disclosed method for sensor selection of a safety-critical system includes:

[0062] 1. Creating qualitative ISFA models for the system, such as the reactor cavity cooling system in Example 1. This identifies components, functions, and transmitted flows, like airflow in inlet pipes, essential for sensor placement to measure related physical variables (e.g., airflow rate and pressure). These models pinpoint potential sensor locations.Attorney docket no.103361-516WO1 T2023-289

[0063] 2. Conducting ISFA simulations to produce signal features under various failure scenarios. The ISFA engine uses the qualitative models to predict physical variable trends under these scenarios, generating signal features accordingly.

[0064] 3. Implementing the sensor selection criteria on the signal features from these scenarios to assess the proposed OLM system's effectiveness.

[0065] 4. Applying a Multiple Objective Optimization Algorithm to choose the best sensor deployment configurations. The multiple objective optimization algorithm uses the selection criteria as objective functions to evaluate different sensor selection options.

[0066] Qualitative modeling and ISFA. In our sensor selection framework, qualitative physical modeling and reasoning are essential foundational steps. ISFA, as a simulation-based fault analysis approach, is capable of producing qualitative behaviors of systems and determining the effects of component faults on the system's functions . Figure 1 provides a visual representation of the inputs and outputs associated with the ISFA method.

[0067] The information required by the ISFA model is a tuple with the following elements: ℳ = { ^^^^, ^^^^, ^^^^, ^^^^, ^^^^ ^^^^, ^^^^ ^^^^ ^^^^, ^^^^} (5)

[0068] The set ^^^^ = { ^^^^1, … , ^^^^^^^^ ^^^^} denotes the components possessed by the target system, where ^^^^ ^^^^ is the total number of components considered in the analysis of faults.

[0069] The set ^^^^ = { ^^^1^ , … , ^^^^^^^^ ^^^^} denotes the functions implemented by the target system, where ^^^^ ^^^^ is the total number of system functions considered in the analysis of faults.

[0070] The model defines various variables ^^^^ = { ^^^^1, … , ^^^^^^^^ ^^^^} representing the inputs and outputs of the components of the system, where ^^^^ ^^^^ is the total number of input and output variables. During a simulation, a simulation engine can infer the values (symbolic expressions) of these variables chronologically.

[0071] The faults or errors ^^^^ = { ^^^^1, … , ^^^^^^^^ ^^^^} will lead to abnormal states of components during system operation, such as pipe leakage.

[0072] The behavioral rules (BR) are qualitative models representing the behaviors of system components. Usually, they are equations defining the relations between the input and output variables for every component in the target system. In an ISFA simulation, BRs model component behaviors under nominal and faulty states.

[0073] To identify the states of functions the target system implements, the ISFA model defines Function Failure Logics (FFL), propositional logic expressions mapping qualitative variable values to the space of possible functional states. The potential states of a function include operating, degraded, and lost.Attorney docket no.103361-516WO1 T2023-289

[0074] At last, the Initial Conditions ( ^^^^) define the initial states of the components of the system and the initial values of system variables.

[0075] The output of ISFA is a superset of time series representing the qualitative values of system variables and the states of the functions implemented by the target system. ^^^^ = (ℚ, ^^^^) (6)

[0076] Where ℚ = { ^^^^^^^^, ^^^^ ∈ ^^^^} and ^^^^^^^^= { ^^^^1^^^^, … , ^^^^^^^^^^^^^^^^} in which ^^^^∗^^^^is a time seriesof the values of variable ^^^^∗ ∈ ^^^^ under the failure mode ^^^^ (the asterisk symbol denotes1, … , ^^^^ ^^^^). Then, ^^^^ ^^^^∗^^^^ ∈ ^^^^. Notation ^^^^ ^^^^∗, ^^^^ denotes the value of thetime series ^^^^∗^^^^at the time point ^^^^ and ^^^^ is the mission time. In the notation ^^^^ ={^^^^^^^^, ^^^^ ∈ ^^^^}and ^^^^^^^^is a time series of the state of function∈ ^^^^ under the mode

[0077] Specifically, the values of variables ^^^^∗∈ ^^^^ in the normal state can be represented by ^^^^^^^^, where ^^^^∗^^^^In the non-transient cases, the values of variable in normal states do not change at different time steps, where ^^^^∗^^^^,1= ⋯ = ^^^^∗^^^^,^^^^. Also, the states of function^^^^ ^^^^∈ ^^^^ in the normal state can be represented by ^^^^∗, where ^^^^∗in normal states remain equal to

[0078] The following sections will reuse the notations mentioned in this section.

[0079] Signal Features. This section presents the signal features employed in the sensor selection criteria of the disclosed method and system. As the ISFA model represents system signals through variables, the terms "variable" and "signal" are used synonymously in the following explanation.

[0080] Assume that ^^^^, ^^^^, ^^^^, ^^^^ are time series. Let ^^^^ ={^^^^1, … , ^^^^^^^^, … , ^^^^^^^^}, where ^^^^ is the total number of time steps in ^^^^. The expression ^^^^^^^^denotes the value of time series ^^^^ at the time point ^^^^^^^^. Let ^^^^ =� ^^^^1, … , ^^^^^^^^, … , ^^^^^^^^�, where ^^^^ is the total number of time steps in ^^^^. The expression ^^^^^^^^denotes the value of time series ^^^^ at the time point ^^^^^^^^.

[0081] Definition 1.1 Deviation Index. ^^^^(^^^^, ^^^^)describes how time series ^^^^ deviates from time series ^^^^. 1, ^^^^ ^^^^ ^^^^ ^^^^ > ^^^^ ^^^^^^^^ ^^^^( ^^^^, ^^^^) =� 0, ^^^^ ^^^^ ^^^^ ^^^^ = ^^^^ ^^^^, 1 ≤ ^^^^ ≤ ^^^^ ^^^^ ^^^^ ( ^^^^, ^^^^) (7)−1, ^^^^ ^^^^ ^^^^ ^^^^ < ^^^^ ^^^^Attorney docket no.103361-516WO1 T2023-289

[0082] In the formula, if the evaluated value ^^^^^^^^is strictly greater than the referenced value ^^^^^^^^, the deviation index is 1. On the other hand, if the evaluated value ^^^^^^^^is strictly less than the referenced value ^^^^^^^^, the deviation index is -1. The deviation index is 0 when ^^^^^^^^is equal to ^^^^^^^^. In practice, perfect equality may not exist for some physical variables. Therefore, a deviation index is defined which incorporates a tolerance limit.

[0083] Definition 1.2 Deviation Index with Tolerance. ^^^^( ^^^^, ^^^^, ^^^^) describes how time series ^^^^ deviates from time series ^^^^ within a tolerance interval 2 ^^^^. 1, ^^^^ ^^^^ ^^^^ ^^^^ − ^^^^ ^^^^ > ^^^^^^^^ ^^^^( ^^^^, ^^^^, ^^^^) =� 0, ^^^^ ^^^^ − ^^^^ ≤ ^^^^ ^^^^ − ^^^^ ^^^^ ≤ ^^^^, ^^^^ ≥ 0,1 ≤ ^^^^ ≤ ^^^^ ^^^^ ^^^^ ( ^^^^, ^^^^) (8)−1, ^^^^ ^^^^ ^^^^ ^^^^ − ^^^^ ^^^^ < − ^^^^

[0084] When the tolerance ^^^^ is zero, this definition equates to Definition 1.1, ^^^^( ^^^^, ^^^^).

[0085] Definition 1.3 Deviation Vector. ^�^�^^(^^^^, ^^^^)={^^^^1, … , ^^^^ ^^^^ , … , ^^^^ ^^^^}represents thedeviation indices of time series ^^^^ and time series ^^^^ over{^^^^1, … ,An example deviation vector is shown in Fig.5.

[0086] Definition 1.4 Deviation Vector with Tolerance.� ^^�^^ ( ^^^^, ^^^^, ^^^^) ={ ^^^^1, … , ^^^^ ^^^^ , … , ^^^^ ^^^^} represents the deviation indices with tolerance of time series ^^^^ and timeseries ^^^^ over{^^^^1, … , ^^^^ ^^^^, … , ^^^^ ^^^^}.

[0087] Definition 2.1 Variation Index. ^^^^(^^^^)describes how a time series ^^^^ changes over time.

[0088] In the formula, the variation index is 1 when the value of time series ^^^^ at the time step ^^^^^^^^is strictly larger than the value at the time step ^^^^^^^^−1, which is the previous time step. On the other hand, the variation index will be -1 when the value of time series ^^^^ (at the time step ^^^^^^^^) is strictly less than the value at the previous time step. The variation index is 0 if the value of time series ^^^^ does not change. Similarly, since the values of some physical variables cannot be perfectly equal in an ISFA model, a variation index is defined with a tolerance interval.

[0089] Definition 2.2 Variation Index with Tolerance. ^^^^(^^^^, ^^^^)describes how a time series ^^^^ changes over time. The changes are assessed within a tolerance interval 2 ^^^^.Attorney docket no.103361-516WO1 T2023-289 0,1 < ^^^^ ≤ ^^^^ (10)

[0090] Definition 2.3 Variation Vector.�^�^^^ ( ^^^^) = { ^^^^1, … , ^^^^ ^^^^ , … , ^^^^ ^^^^} representsvariation indices of time series ^^^^ over { ^^^^1, … , ^^^^^^^^, … , ^^^^^^^^}. An example variation vector is shown in Fig.6.

[0091] Definition 2.4 Variation Vector with Tolerance.�^�^^^ ( ^^^^, ^^^^) = { ^^^^1, … , ^^^^ ^^^^, … , ^^^^ ^^^^}represents variation indices with tolerance of time series ^^^^ over { ^^^^1, … , ^^^^^^^^, … , ^^^^^^^^}.

[0092] Definition 3.1 Deviation Time Operator. ^^^^( ^^^^) represents the first time point at which the value of time series ^^^^ is different from zero. An example deviation time operator is shown in Fig.7. The values of the variation vector defined above are reused for the following examples. 0,1 ≤ ^^^^ ≤ ^^^^0(11)

[0093] Definition 4.1 Unique Operator. ^^^^( ^^^^) removes the adjacent repeated values in a time series. An example unique operator is shown in Fig.8.

[0094] Definition 5.1 Similarity Index. ^^^^ ^^^^( ^^^^, ^^^^) describes the degree of consistency between the values of time series ^^^^ and time series ^^^^.

[0095] Definition 5.2 Similarity Operator. ^^^^(^^^^, ^^^^)represents the similarity between time series ^^^^ and time series ^^^^ over { ^^^^1, … , ^^^^^^^^, … , ^^^^^^^^}.

[0096] In the equation, the notation ^^^^ represents a time step shift tolerance. Generally, a small value of integer can be used, e.g., ^^^^ = 1,2 … An example similarity operator is shown in Fig.9. In the example, ^^^^ = 1. The length of X is 7 and the length of Y is 7, henceAttorney docket no.103361-516WO1 T2023-289 ^^^^ ^^^^ ^^^^( ^^^^, ^^^^) = 7. The maximum value of ^^^^( ^^^^, ^^^^) can be obtained when ^^^^ = 1, where result, the val( ) 6ue of ^^^^ ^^^^, ^^^^ is 7.

[0097] Sensor Selection Criteria. This section addresses various capabilities / properties of OLM systems, as identified in previous studies

[0026] ,

[0027] . A summary of these properties, derived from a literature review, is presented in Table 1. Disclosed herein are multiple sensor selection criteria to rank and evaluate different sensor deployment options (SN) considering those capabilities. To prevent any confusion or ambiguity in terminology, this paper adheres to the definitions provided in Table 1, some of which are adapted from

[0027] .

[0098] In the following description of the sensor selection criteria, it is assumed that ^^^^ = { ^^^^1, … , ^^^^^^^^ ^^^^ ^^^^} ⊆is a subset of system variables. The variables in ^^^^ are measured by at least one sensor. Assume that ^^^^ ^^^^(∙) is a function that returns the sensors measuring a system variable ^^^^, ^^^^ ^^^^( ^^^^) or a function that returns the set of sensors measuring a set of systemvariables ^^^^, ^^^^ ^^^^( ^^^^). The set of sensor selection solutions can be defined as ^^^^ ^^^^ ={ ^^^^ ^^^^( ^^^^1), … , ^^^^ ^^^^( ^^^^ ^^^^ ^^^^ ^^^^)} which is a set of all possible sensor deployment configurations.

[0099] Fault Detection and Discrimination (FDD). The Fault Detection and Discrimination (FDD) criterion is designed to identify signals that are most effective in detecting and distinguishing faults within the target system. An effective signal is one that responds promptly to a fault, exhibiting distinct changes for different types of faults. For instance, a signal might increase following fault 1 but decrease after fault 2. The FDD criterion prioritizes signals that show significant and immediate variations when a fault occurs. Additionally, it is important for these signals to have unique responses to different faults to accurately identify the type of fault. Detailed theories for fault detection and discrimination can be found in

[0022] . Therefore, the FDD criterion evaluates how signals react (i.e., increase or decrease) by analyzing their characteristics derived from the ISFA model simulations.

[0100] The response of a signal to faults can be evaluated by comparing the deviationvector�^�^^^� ^^^^ ^^^^^^^^, and the variationbetween the current variable and the othervariables in the monitored system. The following condition needs to be satisfied to detect a fault ^^^^0, i.e., at least one system variable ^^^^ is being monitored which will deviate from its normal values when the fault ^^^^0occurs.Attorney docket no.103361-516WO1 T2023-289

[0101] Furthermore, to pinpoint the specific fault that has occurred, there needs to be at least one monitorable variable whose response (i.e., the deviation or change) to the fault differs from its response to other faults. This requirement can be expressed using the deviation index operator and the similarity operator in the formula provided.

[0102] In practice, the following condition is defined as a constraint that the fault ^^^^0need to be detectable.

[0103] By reusing the deviation index operator, the differences in signal deviation vectors can be evaluated by the following equation.

[0104] In the equation, the similarity between the deviation vectors^^^^ ^^^^)� indicates that the monitored signal's deviation closely resemblesthat of the other signals within the target system. If the monitored signal exhibits a deviation similar to the other signals (e.g., it is either higher or lower than normal values), it is not suitable for detecting faults. Ideally, the signal with the smallest value of^^^^ ^^^^)� should be chosen as the most effective signal for fault detection.

[0105] Similarly, the similarity between the variation vectorsinthe equation indicates that the trend of the monitored signal closely matches that of the other signals. If the monitored signal exhibits a trend similar to the other signals (e.g., an increase or decrease), it is not suitable for detecting faults. The signal with the smallest value ofthe preferred choice. In the equation, the goal is to maximize theexpressions

[0106] The optimization target is to find the set of variables ^^^^ that maximizes the cumulative value of FDD for all the possible faults.Attorney docket no.103361-516WO1 T2023-289

[0107] Risk Reduction (RRD). The Risk Reduction (RRD) criterion assesses system variables based on their capability to predict system degradations and failures. It determines the effectiveness of a system variable in forecasting a degradation or failure by measuring the time delay between the deviation in variables and the occurrence of the degradation or failure of system functions. In the ISFA results, time steps indicate the sequence of changes in variables and functional states. For instance, if a variable's value increases at time step 1 and a functional state shift from operational to failed at time step 2, this indicates that the variable change anticipates the functional failure. The greater the gap between the time steps of these two events, the longer the lead time is. Therefore, for effective system failure prediction, an online system should monitor signals that provide the earliest warning of such failures. The RRD criterion helps identify variables that offer the earliest indication of an impending system failure.

[0108] Moreover, the RDD criterion involves understanding the severity of system function failures to quantify the consequences of these failures. These consequences are characterized by their magnitude and can be quantified using Safety Integrity Levels (SILs)

[0028] , which may relate to downtime, financial loss, environmental impact, etc. (refer to Table 9 for details). Table 3. Degrees of consequence evaluated using SILs and the equivalent functional states

[0109] Let us denote the system functions ^^^^ =… , ^^^^^^^^ ^^^^} and their risk reduction factors as { ^^^^ ^^^^ ^^^^1, … , ^^^^ ^^^^ ^^^^^^^^ ^^^^}. The fault prognosis score can be calculated using the following formula.Attorney docket no.103361-516WO1 T2023-289

[0110] In the equation, ^^^^(. ) is the difference function defined as below.

[0111] In equation (22), the termdenotes the first time step that thestate of the function f is not nominal. The second termdenotes the first timestep that the signal Q deviates from normal values. Hence the−represents the time difference between those time steps.

[0112] The optimization target is to find the set of variables ^^^^ that maximizes the cumulative value of the FPN for all the possible faults.

[0113] Observability (OBV). The Observability criterion focuses on choosing signals that the OLM system utilizes to monitor and deduce the values of physical variables in the system under different states. Essentially, observability is gauged by the maximum number of sensors that can be removed while still maintaining visibility of the variable. This can be determined using specific indicators. Observability in the OLM system is defined by the relationship between variables; if the trend of one variable can be inferred from the trend of a correlated variable, then the system is considered observable for that variable. This definition guides the selection of signals for effective system monitoring.

[0114] The term ^^^^��^^�^^� ^^^^ ^^^^^^^^, ^^^^ ^^^^^^^^�,�^^�^^ ( ^^^^ ^^^^^^^^, ^^^^ ^^^^^^^^)� represents the correlation between thedeviation of the monitored signal in both faulty and nominal cases. The termrepresent the correlation between the variation of the monitored signal infaulty and nominal cases. The optimization target is to find the set of variables ^^^^ that maximizes the value of the ^^^^ ^^^^ ^^^^ for all the possible faults.Attorney docket no.103361-516WO1 T2023-289

[0115] In addition, observability constraints can be defined. The following equation defines that if the variable ^^^^0needs to be at least partially observable, the variable ^^^^0should be one of the monitored variables or at least one of the current monitored variables need to be correlated (i.e., has the same or reverse deviation or variation) with ^^^^0.

[0116] Functionality (FNC). The functionality criterion covers the capability of achieving the pre-defined functions of sensors, including self-calibration, the ability to perform multiple measurements (multimodal and distributed sensing), or the ability to relay information in real time. Using a sensor performing multiple measurements may decrease the cost of an OLM system. It is considered whether aspects of functionality as they are related to the ability of a sensor to perform multiple measurements.

[0117] In practice, the solution generation function used by the optimization algorithm implements the functionality criterion. For example, when selecting sensors to measure the flow rate and the temperature at a position within the monitored system, the algorithm can choose two sensors which measure these variables respectively or choose one sensor that can measure both variables.

[0118] Integrability (INT). Integrability measures how challenging it is to incorporate a sensor into the system under observation. It takes into account various factors, including the difficulty of accessing the installation site, the space available for the sensor, and the power requirements for the sensor's operation, among others. In this paper, a value with the unit “person hours” is used, which represents the fact that the person hours are required to install a sensor at a position in the target system. This scale helps in evaluating and selecting the most appropriate sites for sensor integration.

[0119] In the equation, the function ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^(∗) returns the position of the variable *, and the function ^^^^ ^^^^ ^^^^(∙) returns the integrability value for the position (∙).

[0120] The optimization target is to find a sensor deployment solution to maximize the integrability value for the sensors used by the OLM system.Attorney docket no.103361-516WO1 T2023-289 ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^( ^^^^)∈ ^^^^ ^^^^^^^^ ^^^^ ^^^^( ^^^^)

[0121] Cost (CST). The cost of incorporating a sensor into an OLM system can be categorized into three main aspects: a) direct costs associated with the sensor itself, such as the purchase costs; b) costs related to the monitoring and support system, which include expenses for monitoring algorithms, computational resources, and, in the case of wireless sensors, network resources; and c) costs incurred from operational modifications or retrofitting. In some examples, CST accounts for the purchasing and installation expenses of the sensors.

[0122] The optimization target is to find a sensor deployment solution to minimize the cost of the sensors used by the target system. ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^( ^^^^)∈ ^^^^ ^^^^^^^^ ^^^^ ^^^^( ^^^^)

[0123] Sensor Failure Tolerance (SFT). Sensors are the inputs of the OLM systems. Sensors’ failures will significantly impact the dependability of the OLM system. Generally, considering the impact of sensors’ failure is a challenging task and is usually ignored during OLM system design. In this study, several representative failure modes of the sensors

[0029] ,

[0030] are contemplated, as shown in Table 4 and Figs.11A-11F. In some examples, the qualitative expressions of sensor failure modes are integrated with the qualitative physical models of the target system when selecting the optimal sensor deployment configurations. Table 4. Failure modes of sensorsAttorney docket no.103361-516WO1 T2023-289

[0124] Sensor Selection Algorithm. For identifying the optimal sensor deployment configurations for the online monitoring system, a multiple objective optimization algorithm is required. In the algorithm, the set of components ^^^^ = { ^^^^1, … , ^^^^^^^^ ^^^^} is defined as the set of possible locations at which sensors can be deployed. The set of variables ^^^^ = { ^^^^1, … , ^^^^^^^^ ^^^^} related to each component determines the types of sensors. For example, ifis an output of ^^^^1, which is a flow rate variable, a flow meter can be possibly deployed at ^^^^1. In the algorithm, each sensor selection criterion may be treated as a separate objective function. In some examples, the multiple objective optimization algorithm is a genetic algorithm. In other examples, the multiple objective optimization algorithm is a Non-dominated Sorting Genetic Algorithm (NSGA-II)

[0024] . In NSGA-II, the algorithm employs crossover and mutationAttorney docket no.103361-516WO1 T2023-289 techniques to create a Non-dominated Pareto Front, representing the best possible sensor deployment configurations. Figure 12 provides a flowchart outlining how the NSGA-II algorithm is used to find optimal solutions for sensor placement.

[0125] As depicted in Fig.12, the NSGA-II algorithm formulates objective functions and constraints for sensor selection based on the criteria established for sensor selection. At each potential location, sensors are considered, and crossover and mutation processes are applied to these sensors. This re-evaluation of objective functions helps in selecting the most suitable solutions, continuing until a pre-set maximum number of generations is reached. This iterative process ensures a thorough exploration of possible sensor configurations to identify the most effective ones.

[0126] Sensor Selection Constraints. Creating an OLM system is a complex task that requires attention to various factors such as budget and available workforce. Utilizing the sensor criteria outlined and the optimization algorithm, it is possible to establish and automatically enforce various constraints in the sensor selection process. This ensures that all sensor deployment configurations suggested by the proposed method adhere to these constraints without any violations. Table 5 summarizes the constraints supported by the proposed method. Table 5. Sensor Selection ConstraintsAttorney docket no.103361-516WO1 T2023-289Examples

[0127] The following examples are put forth so as to provide those of ordinary skill in the art with a complete disclosure and description of how the compounds, compositions, articles, devices and / or methods claimed herein are made and evaluated and are intended to be purely exemplary and are not intended to limit the disclosure. Efforts have been made to ensure accuracy with respect to numbers (e.g., amounts, temperature, etc.), but some errors and deviations should be accounted for. Unless indicated otherwise, parts are parts by weight, temperature is in °C or is at ambient temperature, and pressure is at or near atmospheric.

[0128] Example 1: Sensor Selection Method to a Reactor Cavity Cooling System

[0129] A reactor cavity cooling system (RCCS) from the Modular High-Temperature Gas Reactor (MHTGR) is used as a case study to validate the proposed framework. A key element of this framework is the use of a multi-objective optimization algorithm, which generates a range of sensor deployment options, including the quantity, types, and locations of sensors needed by the OLM system.

[0130] This section details the process of applying the proposed sensor selection method to a reactor cavity cooling system (RCCS), a safety-critical system in the MHTGR.

[0131] System Modeling. The Reactor Cavity Cooling System (RCCS) is a key component of the Modular High-Temperature Gas-Cooled Reactor (MHTGR), situated in the nuclear island

[0031] . It passively removes heat from the reactor cavity using natural convection of air flowing through cooling panels located around the cavity, as shown in Fig.13. These panels form a cylindrical barrier external to the primary coolant pressure boundary, encircling the uninsulated reactor vessel. The RCCS serves two primary functions:

[0132] Its main role is to facilitate energy production processes — including shutdown, refueling, and startup / shutdown activities — by safeguarding the concrete around the reactor module from overheating.

[0133] It also plays a crucial role in radionuclide control, ensuring the management of radionuclide release, achieved by dissipating heat to the ultimate heat sink, and minimizing personnel radiation exposure through effective shielding

[0031] .Attorney docket no.103361-516WO1 T2023-289

[0134] Fig.13 illustrates various components of the Reactor Cavity Cooling System (RCCS), including inlet pipes, outlet pipes, and cooling panels, among others. It is important to note that while the RCCS comprises four sets of inlet and outlet pipes, the diagram at the bottom of Fig.13 shows only one of these sets for simplicity. A comprehensive list of components analyzed in the case study is provided in Table 6. Within the RCCS, the four pipeline sets are identified as inlet / outlet pipes 1, 2, 3, and 4. These labels for each component are consistently used throughout the case study section for clarity and reference.

[0135] Fig.14 illustrates the Integrated System Failure Analysis (ISFA) component model, specifically the Configuration Flow Diagram (CFD)

[0015] of the Reactor Core Cooling System (RCCS). This model labels system components as defined in Table 6. It includes a division of the RCCS system's cooling panel into four subcomponents: CP1, CP2, CP3, and CP4. Each of these subcomponents is designed to remove heat from the reactor during operation. The primary function of the RCCS is to reduce the reactor's temperature. Overall, the model comprises 28 components and 32 flow paths. Table 6. The important components and their labelsAttorney docket no.103361-516WO1 T2023-289

[0136] Fig.15 displays a simplified version of the ISFA model's source code which models the behavior of an inlet filter. This model uses various variables: Q (mass flow rate), T (temperature), P (pressure), DR (density of radiation), and DP (density of particles). Additionally, the 'cfg' structure variable contains parameters for each simulation run. For example, 'cfg.CLOG_RATIO' tracks the flow rate ratio when a filter is partially clogged, and 'cfg.DR_RATIO' measures the reduction of radiation density. It's important to note that these variables are symbolic, allowing for either specific values, such as 1.0, or a range, such as 0<value<1. Input variable values represent current time step t, while output variable values correspond to the next time step t+1.

[0137] Candidate Sensors. The physical variables of the RCCS taken into account include temperature, flow rate, pressure, particle density, and radiation density of the airflow transferred through the RCCS. According to these types of signals, ten candidate sensor types were selected for the OLM system. The specification of those sensors, which are generated for illustration purposes, are listed in Table 7. The labels shown in the table will be reused in the result section. Table 7. Specification of the candidate sensors

[0138] It is worth noting that the sensor “FS1” can measure two physical variables (flow rate and temperature). This sensor is a multifunctional sensor. As a conservative assumption, we assume that the sensor failure modes listed in Table 4 may possibly affect any of the types of sensors in Table 7.

[0139] Considered Component Faults. A key role of the Online Monitoring (OLM) system is to identify and differentiate faults during the operation of the system. Table 8 details the various failure modes of components that are taken into account in assessing the OLM system's effectiveness. Those failure modes commonly exist in cooling systems with pipelines. It should be highlighted that this study focuses exclusively on scenarios involvingAttorney docket no.103361-516WO1 T2023-289 single faults, meaning that in each simulated scenario used for generating signal features, only one fault is introduced at a time. Nonetheless, the methodology outlined can also be adapted to scenarios involving multiple simultaneous faults. Table 8. Faults of components considered in this paper

[0140] Algorithms and Parameters. At the beginning of the ISFA simulation, all components are initialized in nominal states and the values of system variables under nominal operating status are used. Then, at time step 1, one of the faults listed in Table 8 is injected in each run of the simulation. Also, every sensor failure listed in Table 4 is applied to each run of the simulation, i.e., for each run of the simulation, every sensor failure mode will be applied to every sensor in the deployment configuration. In addition, for the optimization algorithm, constraints defined for the case study system are used and shown in Table 9. In this paper, the constraints are defined for illustration purposes.Attorney docket no.103361-516WO1 T2023-289 Table 9. Constraints defined for the case study system

[0141] Table 10 presents the Risk Reduction Factors (RRFs) allocated to both the overall function and the subfunctions of the system featured in the case study. In addition to the system's main function of cooling the reactor, specific functions are designated to each component listed in Table 6. For instance, the function of "filtering air" is carried out by each inlet and outlet filter within the system. Similarly, the function of "transferring air" is executed by all the pipes, and the role of "transferring heat" is fulfilled by the cooling panels. Table 10. Risk Reduction Factors for the case study system

[0142] Table 11 displays the integrabilities defined for the possible sensor deployment locations and the candidate sensors. For example, the value 31 in the cell of the row “Filters” and the column “FS1” means that installing a “FS1” sensor at a filter location requires 31 person hours. It is worth noting that all the values at the row “Cooling Panels” are 5000 since it is very difficult to install sensors in the cooling panel.Attorney docket no.103361-516WO1 T2023-289 Table 11. Integrabilities defined for the locations and candidate sensors

[0143] Analysis Results. Taking into account the time consumption and effectiveness factors, we set the population size at 1000 and the number of generations at 100 for the NSGA-II algorithm. Table 12 includes an example of the sensor deployment configuration suggested by the proposed method. For instance, a ‘FS2’ sensor is recommended to be deployed at ‘PI1V’, the vertical inlet pipe 1. Specifically, according to the results, the total number of sensors is 24, the cost of the sensor configuration is $55,800, and the integrability of the configuration is 831 person hours. Table 12. An example optimal sensor selection solution

[0144] Figure 16 displays the Non-dominated Pareto Front for the sensor selection criteria, where each point on the Front corresponds to an optimal solution for sensor placement. In Fig.16, besides the dimensions of x, y, and z, which represent the result data for criteria fault diagnostics, risk reduction, and observability, respectively, the cost is represented by the depth of the color, and the integrability is represented by the size of the points.

[0145] Figures 17A-17B illustrates the distribution of sensors selected for various locations after the optimization process and Fig.18 represents the distribution for different types of sensors. Analyzing Figs.17A-17B, it is apparent that the pipes are better places to deploy sensors than the filters and cooling panels. This is possibly because the filters are at the beginning and end of the system flow hence the variables at those positions are not sensitive to the faults of components inside the system. Also deploying sensors at the cooling panels is very difficult according to the values taken by the integrability criterion. As shownAttorney docket no.103361-516WO1 T2023-289 in Fig.18 the flowrate sensor (FS1) is chosen for fewer locations compared to the flowrate sensor (FS2). Thus, FS1 is more sensitive to location and is best suited for specific locations to measure relevant variables. In contrast, FS2 exhibits a more location-agnostic characteristic. This pattern of location sensitivity versus location agnosticism is also noticeable with other sensors in Figs.17A-17B. Some sensors demonstrate high location sensitivity, suitable for limited specific locations, while others are more versatile and can be deployed across a wider range of locations.

[0146] Figure 19 shows the average frequency of different sensor types selected in 1000 solution scenarios. This figure clearly indicates that the flow meter (FS2) is the most commonly chosen sensor, while the Radiation density sensor (RS1) and Particle density sensor (DS1) are selected less frequently.

[0147] Figure 20 focuses on the average frequency of sensors chosen for each specific location within the case study system. The figure reveals that the inlet and outlet pipes near the cooling panel are the preferred spots for sensor installation, aligning with the insights of experts from the relevant domain.

[0148] A straightforward application of the outcomes from the proposed framework involves selecting one of the one thousand optimal solutions as the final configuration for sensor deployment in the online monitoring system. The framework has effectively eliminated inefficient and insufficient options, ensuring that all remaining configurations meet the criteria set by the system designers. It is important to note that the framework was designed without imposing strict constraints on the OLM system design, resulting in a wide array of potential solutions. In a practical setting, more stringent constraints would likely be applied, leading to a reduction in the number of feasible solutions.

[0149] Moreover, insights from Figs.17A-17B reveal that certain types of sensors show high sensitivity to their possible locations within the target system. This information is crucial for choosing appropriate sensors for specific system variables or components. Additionally, it aids in determining which sensor types are most compatible with the target system's requirements.

[0150] Model Uncertainty. The issue of model uncertainty arises due to the incomplete or ambiguous nature of qualitative models, a common scenario during the early stages of system design. This uncertainty can hinder the ISFA engine's ability to produce deterministic results, leading to potential ambiguities in inferred system features. When faced with such uncertainties, the framework may fail to deliver meaningful outcomes. To mitigate this issue, system designers can refine the qualitative system models by incorporating moreAttorney docket no.103361-516WO1 T2023-289 specific constants or introducing more concrete propositions based on established physics rules. This enhancement aims to reduce the uncertainty inherent in these models. For instance, narrowing down the range of a leakage flowrate from a broad spectrum (e.g., 0 to 0.1 L / min) to a more precise range (e.g., 0 to 0.01 L / min) or opting for a specific numerical value can be effective strategies in this context.

[0151] Conclusion. This example presents a framework for sensor selection that aids in the design of OLM systems. It employs an ISFA model to analyze the impact of faults and generate signal features. These features are critical in evaluating the performance of an OLM system using defined sensor selection criteria. Seven such criteria and related constraints are established in this example, offering flexibility to be used collectively or individually to assess various sensor deployment configurations. The Reactor Cavity Cooling System in the Modular High-Temperature Gas-Cooled Reactor serves as a case study to demonstrate the efficacy of the proposed method.

[0152] The case study results show that the framework effectively provides optimal sensor deployment solutions for OLM system designers. Addressing the multi-objective nature of sensor selection, the framework generates multiple optimal solutions rather than just one. Designers can either select one of these solutions as their final sensor deployment configuration or base their decision on an aggregate or statistical analysis of all generated solutions.

[0153] The disclosed framework can be extended by incorporating additional sensor selection criteria, such as the reliability of candidate sensors and the uncertainty in sensor measurements. It is contemplated that the method and system may be configured to enhance the scalability and maintainability of the methodology, making it suitable for larger and more complex systems.

[0154] It should be appreciated that the logical operations described above and in the appendix can be implemented in some embodiments (1) as a sequence of computer- implemented acts or program modules running on a computing system and / or (2) as interconnected machine logic circuits or circuit modules within the computing system. The implementation may be a matter of choice dependent on the performance and other requirements of the computing system. Accordingly, the logical operations described herein are referred to variously as state operations, acts, or modules. These operations, acts, and / or modules can be implemented in software, in firmware, in special purpose digital logic, in hardware, and any combination thereof. It should also be appreciated that more or fewerAttorney docket no.103361-516WO1 T2023-289 operations can be performed than shown in the figures and described herein. These operations can also be performed in a different order than those described herein.

[0155] In addition to the various system discussed herein, Fig.21 shows an illustrative computer architecture for a computer system 2100 capable of executing the software components that can use the output of the exemplary method described herein. The computer architecture shown in Fig.21 illustrates an example computer system configuration, and the computer 2100 can be utilized to execute any aspects of the components and / or modules presented herein described as executing on the analysis system or any components in communication therewith, including providing support of TEE as described herein as well as trusted Time, GPS, and Monotonic Counter as noted above.

[0156] In an embodiment, the computing device 2100 may comprise two or more computers in communication with each other that collaborate to perform a task. For example, but not by way of limitation, an application may be partitioned in such a way as to permit concurrent and / or parallel processing of the instructions of the application. Alternatively, the data processed by the application may be partitioned in such a way as to permit concurrent and / or parallel processing of different portions of a data set by the two or more computers. In an embodiment, virtualization software may be employed by the computing device 2100 to provide the functionality of a number of servers that is not directly bound to the number of computers in the computing device 2100. For example, virtualization software may provide twenty virtual servers on four physical computers. In an embodiment, the functionality disclosed above may be provided by executing the application and / or applications in a cloud computing environment. Cloud computing may comprise providing computing services via a network connection using dynamically scalable computing resources. Cloud computing may be supported, at least in part, by virtualization software. A cloud computing environment may be established by an enterprise and / or may be hired on an as-needed basis from a third- party provider. Some cloud computing environments may comprise cloud computing resources owned and operated by the enterprise as well as cloud computing resources hired and / or leased from a third-party provider.

[0157] In its most basic configuration, computing device 2100 typically includes at least one processing unit 2120 and system memory 2130. Depending on the exact configuration and type of computing device, system memory 2130 may be volatile (such as random-access memory (RAM)), non-volatile (such as read-only memory (ROM), flash memory, etc.), or some combination of the two.Attorney docket no.103361-516WO1 T2023-289

[0158] This most basic configuration is illustrated in Fig.21 by dashed line 2110. The processing unit 2120 may be a standard programmable processor that performs arithmetic and logic operations necessary for the operation of the computing device 2100. While only one processing unit 2120 is shown, multiple processors may be present. As used herein, processing unit and processor refers to a physical hardware device that executes encoded instructions for performing functions on inputs and creating outputs, including, for example, but not limited to, microprocessors (MCUs), microcontrollers, graphical processing units (GPUs), and application-specific circuits (ASICs). Thus, while instructions may be discussed as executed by a processor, the instructions may be executed simultaneously, serially, or otherwise executed by one or multiple processors. The computing device 2100 may also include a bus or other communication mechanism for communicating information among various components of the computing device 2100.

[0159] Computing device 2100 may have additional features / functionality. For example, computing device 2100 may include additional storage such as removable storage 2120 and non-removable storage 2150 including, but not limited to, magnetic or optical disks or tapes. Computing device 2100 may also contain network connection(s) 2180 that allow the device to communicate with other devices such as over the communication pathways described herein. The network connection(s) 2180 may take the form of modems, modem banks, Ethernet cards, universal serial bus (USB) interface cards, serial interfaces, token ring cards, fiber distributed data interface (FDDI) cards, wireless local area network (WLAN) cards, radio transceiver cards such as code division multiple access (CDMA), global system for mobile communications (GSM), long-term evolution (LTE), worldwide interoperability for microwave access (WiMAX), and / or other air interface protocol radio transceiver cards, and other well-known network devices. Computing device 2100 may also have input device(s) 2170 such as keyboards, keypads, switches, dials, mice, trackballs, touch screens, voice recognizers, card readers, paper tape readers, or other well-known input devices. Output device(s) 2160 such as printers, video monitors, liquid crystal displays (LCDs), touch screen displays, displays, speakers, etc. may also be included. The additional devices may be connected to the bus in order to facilitate the communication of data among the components of the computing device 2100. All these devices are well known in the art and need not be discussed at length here.

[0160] The processing unit 2120 may be configured to execute program code encoded in tangible, computer-readable media. Tangible, computer-readable media refers to any media that is capable of providing data that causes the computing device 2100 (i.e., aAttorney docket no.103361-516WO1 T2023-289 machine) to operate in a particular fashion. Various computer-readable media may be utilized to provide instructions to the processing unit 2120 for execution. Example tangible, computer-readable media may include, but is not limited to, volatile media, non-volatile media, removable media, and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules, or other data. System memory 2130, removable storage 2140, and non- removable storage 2150 are all examples of tangible, computer storage media. Example tangible, computer-readable recording media include, but are not limited to, an integrated circuit (e.g., field-programmable gate array or application-specific IC), a hard disk, an optical disk, a magneto-optical disk, a floppy disk, a magnetic tape, a holographic storage medium, a solid-state device, RAM, ROM, electrically erasable program read-only memory (EEPROM), flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices.

[0161] In light of the above, it should be appreciated that many types of physical transformations take place in the computer architecture 2100 in order to store and execute the software components presented herein. It also should be appreciated that the computer architecture 2100 may include other types of computing devices, including hand-held computers, embedded computer systems, personal digital assistants, and other types of computing devices known to those skilled in the art. It is also contemplated that the computer architecture 2100 may not include all of the components shown in Fig.21, may include other components that are not explicitly shown in Fig.21, or may utilize an architecture different than that shown in FIG.21.

[0162] In an example implementation, the processing unit 2120 may execute program code stored in the system memory 2130. For example, the bus may carry data to the system memory 2130, from which the processing unit 2120 receives and executes instructions. The data received by the system memory 2130 may optionally be stored on the removable storage 2140 or the non-removable storage 2150 before or after execution by the processing unit 2120.

[0163] It should be understood that the various techniques described herein may be implemented in connection with hardware or software or, where appropriate, with a combination thereof. Thus, the methods and apparatuses of the presently disclosed subject matter, or certain aspects or portions thereof, may take the form of program code (i.e., instructions) embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, orAttorney docket no.103361-516WO1 T2023-289 any other machine-readable storage medium wherein, when the program code is loaded into and executed by a machine, such as a computing device, the machine becomes an apparatus for practicing the presently disclosed subject matter. In the case of program code execution on programmable computers, the computing device generally includes a processor, a storage medium readable by the processor (including volatile and non-volatile memory and / or storage elements), at least one input device, and at least one output device. One or more programs may implement or utilize the processes described in connection with the presently disclosed subject matter, e.g., through the use of an application programming interface (API), reusable controls, or the like. Such programs may be implemented in a high-level procedural or object-oriented programming language to communicate with a computer system. However, the program(s) can be implemented in assembly or machine language, if desired. In any case, the language may be a compiled or interpreted language, and it may be combined with hardware implementations.

[0164] Moreover, the various components may be in communication via wireless and / or hardwire or other desirable and available communication means, systems and hardware. Moreover, various components and modules may be substituted with other modules or components that provide similar functions.

[0165] Some references, which may include various patents, patent applications, and publications, are cited in a reference list and discussed in the disclosure provided herein. The citation and / or discussion of such references is provided merely to clarify the description of the present disclosure and is not an admission that any such reference is “prior art” to any aspects of the present disclosure described herein. In terms of notation, “[n]” corresponds to the nth 10 reference in the list. All references cited and discussed in this specification are incorporated herein by reference in their entireties and to the same extent as if each reference was individually incorporated by reference.

[0166] The construction and arrangement of the systems and methods as shown in the various exemplary implementations are illustrative only. Although only a few implementations have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements may be reversed or otherwise varied, and the nature or number of discrete elements or positions may be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps may be varied orAttorney docket no.103361-516WO1 T2023-289 re-sequenced according to alternative implementations. Other substitutions, modifications, changes, and omissions may be made in the design, operating conditions, and arrangement of the exemplary implementations without departing from the scope of the present disclosure.

[0167] The present disclosure contemplates methods, systems, and program products on any machine-readable media for accomplishing various operations. The implementation of the present disclosure may be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Implementations within the scope of the present disclosure include program products including machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine- readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine- executable instructions or data structures, and which can be accessed by a general purpose or special purpose computer or other machine with a processor.

[0168] When information is transferred or provided over a network or another communications connection (either hardwired, wireless, or a combination of hardwired or wireless) to a machine, the machine properly views the connection as a machine-readable medium. Thus, any such connection is properly termed a machine-readable medium. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general-purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions.

[0169] Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also, two or more steps may be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule-based logic and other logic to accomplish the various connection steps, processing steps, comparison steps and decision steps.

[0170] It is to be understood that the methods and systems are not limited to specific synthetic methods, specific components, or to particular compositions. It is also to beAttorney docket no.103361-516WO1 T2023-289 understood that the terminology used herein is for the purpose of describing particular implementations only and is not intended to be limiting.

[0171] As used in the specification and the appended claims, the singular forms “a,” “an” and “the” include plural referents unless the context clearly dictates otherwise. Ranges may be expressed herein as from “about” one particular value, and / or to “about” another particular value. When such a range is expressed, another embodiment includes from the one particular value and / or to the other particular value. Similarly, when values are expressed as approximations, by use of the antecedent “about,” it will be understood that the particular value forms another embodiment. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint.

[0172] “Optional” or “optionally” means that the subsequently described event or circumstance may or may not occur, and that the description includes instances where said event or circumstance occurs and instances where it does not.

[0173] Disclosed are components that can be used to perform the disclosed methods and systems. These and other components are disclosed herein, and it is understood that when combinations, subsets, interactions, groups, etc. of these components are disclosed that while specific reference of each various individual and collective combinations and permutation of these may not be explicitly disclosed, each is specifically contemplated and described herein, for all methods and systems. This applies to all aspects of this application including, but not limited to, steps in disclosed methods. Thus, if there are a variety of additional steps that can be performed it is understood that each of these additional steps can be performed with any specific embodiment or combination of implementations of the disclosed methods.

[0174] Although example embodiments of the present disclosure are explained in some instances in detail herein, it is to be understood that other embodiments are contemplated. Accordingly, it is not intended that the present disclosure be limited in its scope to the details of construction and arrangement of components set forth in the following description or illustrated in the drawings. The present disclosure is capable of other embodiments and of being practiced or carried out in various ways.

[0175] In describing example embodiments, terminology will be resorted to for the sake of clarity. It is intended that each term contemplates its broadest meaning as understood by those skilled in the art and includes all technical equivalents that operate in a similar manner to accomplish a similar purpose. It is also to be understood that the mention of one or more steps of a method does not preclude the presence of additional method steps orAttorney docket no.103361-516WO1 T2023-289 intervening method steps between those steps expressly identified. Steps of a method may be performed in a different order than those described herein without departing from the scope of the present disclosure. Similarly, it is also to be understood that the mention of one or more components in a device or system does not preclude the presence of additional components or intervening components between those components expressly identified. Exemplary Aspects

[0176] Exemplary aspect 1. A method to optimize sensor selection and placement of a target system, the method comprising: generating a model of the target system; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors.

[0177] Exemplary aspect 2. The method of exemplary aspect 1, wherein generating a plurality of solutions comprises: defining an objective function based on the plurality of sensor selection criteria; defining constraints for sensor selection at different locations to simulate different variables; simulating, using the model, different variables based on sensor selection at each of the different locations; applying an objective optimization algorithm to the different variables; iteratively evaluating the objective functions for sensor selection at different locations; selecting an optimal solution (e.g. Pareto front); repeating the steps until a predetermined number of solutions are generated.

[0178] Exemplary aspect 3. The method of any one of exemplary aspects 1-2, wherein the model comprises a qualitative physical model.

[0179] Exemplary aspect 4. The method of any one of exemplary aspects 1-3, wherein input to the model comprises two or more values or equations, wherein the two or more values or equations express target system components, functions of the target system components, input and output variables from target system components, failure modes, behavioral rules, function failure logics, and / or initial conditions.

[0180] Exemplary aspect 5. The method of exemplary aspect 4, wherein the behavioral rules comprise qualitative models representing the behaviors of target system components under nominal and faulty states.

[0181] Exemplary aspect 6. The method of exemplary aspect 4, wherein the model infers a chronological order of input / output variable values from target system components.Attorney docket no.103361-516WO1 T2023-289

[0182] Exemplary aspect 7. The method of exemplary aspect 4, wherein the failure modes relate to abnormal states of target system components during target system operation (e.g. pipe leakage).

[0183] Exemplary aspect 8. The method of exemplary aspect 4, wherein function failure logics comprise logic expressions mapping qualitative variable values to functional states, wherein functional states comprise operating, degraded, and lost.

[0184] Exemplary aspect 9. The method of any one of exemplary aspects 1-8, wherein the model outputs a set of time series of signal features, wherein signal features comprise state functions representing qualitative values of target system variables.

[0185] Exemplary aspect 10. The method of any of exemplary aspects 1-9, wherein the sensor selection criteria are mathematical functions.

[0186] Exemplary aspect 11. The method of any one of exemplary aspects 1-10, wherein the sensor selection criteria comprise one or more of fault detection and discrimination, risk reduction, sensor failure tolerance, observability, functionality, integrability, and cost.

[0187] Exemplary aspect 12. The method of any one of exemplary aspects 1-11, wherein a sensor selection constraint comprises a desired solution to the sensor selection criteria.

[0188] Exemplary aspect 13. The method of any one of exemplary aspects 1-12, wherein each of the plurality of solutions comprises a set of sensors at respective locations in the target system and related values of sensor selection criteria for the set of sensors.

[0189] Exemplary aspect 14. The method of any one of exemplary aspects 2-13, wherein the objective optimization algorithm comprises a genetic algorithm.

[0190] Exemplary aspect 15. The method of exemplary aspect 14, wherein the genetic algorithm is applied with predetermined generation rate, predetermined crossover rate, and predetermined mutation rate.

[0191] Exemplary aspect 16. The method of any one of exemplary aspects 1-15, wherein the target system comprises a safety-critical system comprising one of a reactor system, a cooling system, an electronics system, a construction system, an energy production system, an energy conversion system, an environmental monitoring system, a manufacturing system, a telecommunications system, a transportation system, a motor vehicle system, or any system comprising a plurality of sensors in one or more locations.

[0192] Exemplary aspect 17. A system to optimize sensor selection and placement of a target system, the system comprising: a computing system comprising one or moreAttorney docket no.103361-516WO1 T2023-289 processors and one or more storage devices, wherein the one or more storage devices have instructions stored thereon, that when executed by the one or more processors, causes the one or more processors to perform a method, the method comprising: generating a model of the target system; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors.

[0193] Exemplary aspect 18. The system of exemplary aspect 17, wherein generating a plurality of solutions comprises: defining an objective function based on the plurality of sensor selection criteria; defining constraints for sensor selection at different locations to simulate different variables; simulating, using the model, different variables based on sensor selection at each of the different locations; applying a multiple objective optimization algorithm to the different variables; iteratively evaluating the objective functions for sensor selection at different locations; selecting an optimal solution (e.g. Pareto front); repeating the steps until a predetermined number of solutions are generated.

[0194] Exemplary aspect 19. The system of any one of exemplary aspects 17-18, wherein each of the plurality of solutions comprises a set of sensors at respective locations in the target system and related values of sensor selection criteria for the set of sensors.

[0195] Exemplary aspect 20. The system of any one of exemplary aspects 17-19, wherein one of the plurality of solutions is deployed in the target system.

[0196] The following patents, applications, and publications, as listed below and throughout this document, are hereby incorporated by reference in their entirety herein. References [1] L. Guo, N. Li, F. Jia, Y. Lei, and J. Lin, “A recurrent neural network based health indicator for remaining useful life prediction of bearings,” Neurocomputing, vol.240, pp.98– 109, May 2017, doi: 10.1016 / j.neucom.2017.02.045. [2] M. Kim and K. Liu, “A Bayesian deep learning framework for interval estimation of remaining useful life in complex systems by incorporating general degradation characteristics,” IISE Transactions, vol.53, no.3, pp.326–340, Mar.2021, doi: 10.1080 / 24725854.2020.1766729.Attorney docket no.103361-516WO1 T2023-289 [3] M. Kim, J.-R. C. Cheng, and K. Liu, “An adaptive sensor selection framework for multisensor prognostics,” Journal of Quality Technology, vol.53, no.5, pp.566–585, Oct. 2021, doi: 10.1080 / 00224065.2021.1960934. [4] X. Zhang, L. Tang, and J. Decastro, “Robust Fault Diagnosis of Aircraft Engines: A Nonlinear Adaptive Estimation-Based Approach,” IEEE Trans. Contr. Syst. Technol., vol.21, no.3, pp.861–868, May 2013, doi: 10.1109 / TCST.2012.2187057. [5] R. Xiong, R. Yang, Z. Chen, W. Shen, and F. Sun, “Online Fault Diagnosis of External Short Circuit for Lithium-Ion Battery Pack,” IEEE Trans. Ind. Electron., vol.67, no. 2, pp.1081–1091, Feb.2020, doi: 10.1109 / TIE.2019.2899565. [6] D. M. Capiluppi, “Fault Tolerance in Large Scale Systems: hybrid and distributed approaches”. [7] J. Lunze, “Two methods to find analytical redundancy relations for fault diagnosis,” at - Automatisierungstechnik, vol.65, no.4, pp.219–232, Apr.2017, doi: 10.1515 / auto-2016- 0121. [8] J.-F. Condotta, F. Le Ber, G. Ligozat, and L. Travé-Massuyès, “Qualitative Reasoning,” in A Guided Tour of Artificial Intelligence Research, P. Marquis, O. Papini, and H. Prade, Eds., Cham: Springer International Publishing, 2020, pp.151–183. doi: 10.1007 / 978-3-030-06164-7_5. [9] J. Top, J. Akkermans, and P. C. Breedveld, “Qualitative reasoning about physical systems: an artificial intelligence perspective,” Journal of the Franklin Institute, vol.328, no. 5–6, pp.1047–1065, 1991.

[0010] X. Mao, “Semantic and Qualitative Physics-Based Formal Reasoning for Functional Decomposition in Mechanical Design”.

[0011] R. Baldoni, E. Coppa, D. C. D’Elia, C. Demetrescu, and I. Finocchi, “A Survey of Symbolic Execution Techniques.” arXiv, May 02, 2018. Accessed: Nov.20, 2023. [Online]. Available: http: / / arxiv.org / abs / 1610.00502

[0012] C. Barrett and C. Tinelli, Satisfiability modulo theories. Springer, 2018.

[0013] X. Diao, Y. Zhao, M. Pietrykowski, Z. Wang, and ..., “Fault propagation and effects analysis for designing an online monitoring system for the secondary loop of the nuclearAttorney docket no.103361-516WO1 T2023-289 power plant portion of a hybrid energy system,” Nuclear …, no. Query date: 2023-09-07 10:31:28, 2018, doi: 10.1080 / 00295450.2018.1426963.

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Claims

Attorney docket no.103361-516WO1 T2023-289 Claims What is claimed is:

1. A method to optimize sensor selection and placement of a target system, the method comprising: generating a model of the target system; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors.

2. The method of claim 1, wherein generating a plurality of solutions comprises: defining an objective function based on the plurality of sensor selection criteria; defining constraints for sensor selection at different locations to simulate different variables; simulating, using the model, different variables based on sensor selection at each of the different locations; applying an objective optimization algorithm to the different variables; iteratively evaluating the objective functions for sensor selection at different locations; selecting an optimal solution (e.g. Pareto front); repeating the steps until a predetermined number of solutions are generated.

3. The method of any one of claims 1-2, wherein the model comprises a qualitative physical model.

4. The method of any one of claims 1-3, wherein input to the model comprises two or more values or equations, wherein the two or more values or equations express target system components, functions of the target system components, input and output variables from target system components, failure modes, behavioral rules, function failure logics, and / or initial conditions.

5. The method of claim 4, wherein the behavioral rules comprise qualitative models representing the behaviors of target system components under nominal and faulty states.Attorney docket no.103361-516WO1 T2023-289 6. The method of claim 4, wherein the model infers a chronological order of input / output variable values from target system components.

7. The method of claim 4, wherein the failure modes relate to abnormal states of target system components during target system operation (e.g. pipe leakage).

8. The method of claim 4, wherein function failure logics comprise logic expressions mapping qualitative variable values to functional states, wherein functional states comprise operating, degraded, and lost.

9. The method of any one of claims 1-8, wherein the model outputs a set of time series ofsignal features, wherein signal features comprise state functions representing qualitative valuesof target system variables.

10. The method of any of claims 1-9, wherein the sensor selection criteria are mathematical functions.

11. The method of any one of claims 1-10, wherein the sensor selection criteria comprise one or more of fault detection and discrimination, risk reduction, sensor failure tolerance, observability, functionality, integrability, and cost.

12. The method of any one of claims 1-11, wherein a sensor selection constraint comprises a desired solution to the sensor selection criteria.

13. The method of any one of claims 1-12, wherein each of the plurality of solutions comprises a set of sensors at respective locations in the target system and related values of sensor selection criteria for the set of sensors.

14. The method of any one of claims 2-13, wherein the objective optimization algorithm comprises a genetic algorithm.

15. The method of claim 14, wherein the genetic algorithm is applied with predetermined generation rate, predetermined crossover rate, and predetermined mutation rate.

16. The method of any one of claims 1-15, wherein the target system comprises a safety- critical system comprising one of a reactor system, a cooling system, an electronics system, a construction system, an energy production system, an energy conversion system, an environmental monitoring system, a manufacturing system, a telecommunications system, aAttorney docket no.103361-516WO1 T2023-289 transportation system, a motor vehicle system, or any system comprising a plurality of sensors in one or more locations.

17. A system to optimize sensor selection and placement of a target system, the system comprising: a computing system comprising one or more processors and one or more storage devices, wherein the one or more storage devices have instructions stored thereon, that when executed by the one or more processors, causes the one or more processors to perform a method, the method comprising: generating a model of the target system; generating, using the model, signal features of a plurality of sensors and placement of said plurality of sensors; generating a plurality of solutions using a plurality of sensor selection criteria and the signal features; and providing the plurality of solutions for the plurality of sensors and placement of said plurality of sensors.

18. The system of claim 17, wherein generating a plurality of solutions comprises: defining an objective function based on the plurality of sensor selection criteria; defining constraints for sensor selection at different locations to simulate different variables; simulating, using the model, different variables based on sensor selection at each of the different locations; applying a multiple objective optimization algorithm to the different variables; iteratively evaluating the objective functions for sensor selection at different locations; selecting an optimal solution (e.g. Pareto front); repeating the steps until a predetermined number of solutions are generated.

19. The system of any one of claims 17-18, wherein each of the plurality of solutions comprises a set of sensors at respective locations in the target system and related values of sensor selection criteria for the set of sensors.

20. The system of any one of claims 17-19, wherein one of the plurality of solutions is deployed in the target system.

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