Uncertain Model Parameters Research Articles

Tropical Glaciation and Glacio-Epochs: Their Tectonic Origin in Paleogeography

Precambrian tropical glaciation is an enigma of Earth’s climate. Overlooking fundamental difference of land/sea icelines, it was equated with a global frozen ocean, which is at odds with the sedimentary evidence of an active hydrological cycle, and its genesis via the runaway ice–albedo feedback conflicts with the mostly ice-free Proterozoic when its trigger threshold was well exceeded by the dimmer sun. In view of these shortfalls, I put forth two key hypotheses of the tropical glaciation: first, if seeded by mountain glaciers, the land ice would advance on sea level to be halted by above-freezing summer temperature, which thus abuts an open cozonal ocean; second, a tropical supercontinent would block the brighter tropical sun to cause the required cooling. To test these hypotheses, I formulate a minimal tropical/polar box model to examine the temperature response to a varying tropical land area and show that tropical glaciation is indeed plausible when the landmass is concentrated in the tropics despite uncertain model parameters. In addition, given the chronology of paleogeography, the model may explain the observed deep time climate to provide a unified account of the faint young Sun paradox, Precambrian tropical glaciations, and Phanerozoic glacio-epochs, reinforcing, therefore, the uniformitarian principle.

APPLICATION OF GLOBAL SENSITIVITY ANALYSIS FOR IDENTIFICATION OF PROBABILISTIC DESIGN SPACES

The design space (DS) is defined as the combination of materials and process conditions that provides assurance of quality for a pharmaceutical product. A model-based approach to identify a probability-based DS requires costly simulations across the entire process parameter space (certain) and the uncertain model parameter space and material properties space. We demonstrate that application of global sensitivity analysis (GSA) can significantly reduce model complexity and reduce computational time for identifying and quantifying a probabilistic DS by screening out nonimportant uncertain parameters. The novelty of this approach is that the use of an indicator function, which takes only binary values as a model function, allows application of a straightforward GSA based on Sobol' sensitivity indices and avoids using more costly Monte Carlo filtering or GSA for constrained problems. We consider a chemical reaction example to illustrate how this formulation results in a model reduction and a significant decrease of model runs.

Adaptive Proportional-Integral Control Design for a Class of Continuous Stirred Tank Reactors with Uncertain Parameters

Mathematical model of reaction systems contains experimental parameters such as reaction enthalpies, which may be inaccurate and, therefore, severely affect the computation as well as the implementation of feedback control laws. This paper aims to design an adaptive PI-like controller to regulate a chemical reaction system by means of the Lyapunov theory. More precisely, uncertain model parameters are updated online by solving a set of ordinary differential equations while the global asymptotic convergence of closed-loop system trajectories towards the desired equilibrium is ensured by using the proposed adaptive PI-like controller under the assumption of stability of isothermal conditions. The applicability of theoretical developments is illustrated with an irreversible first-order reaction system having multiple steady states and taking place in a non-isothermal continuous stirred tank reactor. Simulation results show that system trajectories initiated at different conditions are asymptotically stabilized at the desired values and the closed-loop system is robust against the uncertainty of heat exchange coefficient and dilution rate.

Introducing the Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations—II: Illustrative Application to Heat and Energy Transfer in the Nordheim–Fuchs Phenomenological Model for Reactor Safety

This work presents an illustrative application of the newly developed “Second-Order Features Adjoint Sensitivity Analysis Methodology for Neural Ordinary Differential Equations (2nd-FASAM-NODE)” methodology to determine most efficiently the exact expressions of the first- and second-order sensitivities of NODE decoder responses to the neural net’s underlying parameters (weights and initial conditions). The application of the 2nd-FASAM-NODE methodology will be illustrated using the Nordheim–Fuchs phenomenological model for reactor safety, which describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted. The representative model responses that will be analyzed in this work include the model’s time-dependent total energy released, neutron flux, temperature and thermal conductivity. The 2nd-FASAM-NODE methodology yields the exact expressions of the first-order sensitivities of these decoder responses with respect to the underlying uncertain model parameters and initial conditions, requiring just a single large-scale computation per response. Furthermore, the 2nd-FASAM-NODE methodology yields the exact expressions of the second-order sensitivities of a model response requiring as few large-scale computations as there are features/functions of model parameters, thereby demonstrating its unsurpassed efficiency for performing sensitivity analysis of NODE nets.

Model uncertainty quantification of a degradation model of miter gates using normalizing flow-based likelihood-free inference

Degradation modeling is essential for failure prognostics of miter gates and subsequent risk-informed maintenance or operational decision-making. Typically, degradation models are formulated based on certain assumptions and simplifications and may not fully capture the complexity of underlying degradation mechanisms. Even minor errors in such models may lead to significant discrepancies in failure prognostics due to error accumulation over time. Aiming to improve the accuracy of the degradation model and ensure reliable failure prognostics, this article proposes a degradation model updating framework for miter gates using normalizing flow-based likelihood-free inference, accounting for both model parameter uncertainty and model form uncertainty (i.e., model bias). The proposed work consists of two phases: the offline training phase and the online updating phase. During the offline training phase, the unknown model bias term is modeled as a random noise with uncertain standard deviation. Synthetic data are then generated using a physical model based on prior knowledge of the uncertain model parameters and model bias. The synthetic data are utilized to train an inference model, which has a summary network for data compression and a conditional invertible neural network for parameter estimation. In the online updating phase, the trained inference model uses strain observations to obtain posterior samples. The Gaussian mixture model and dual particle filter techniques are utilized to recursively update posterior samples, thereby improving the estimation accuracy. Subsequently, model bias is inversely estimated using the degradation model, which uses the updated model parameters and the gap length from the inference model. Following that, a regression model is constructed to correct the biases inherent in the simplified degradation model. This process is implemented iteratively over time to perform continuous model updating and failure prognostics. Results of a case study demonstrate the efficacy of the proposed framework in reducing both model parameter uncertainty and recovering model biases and thereby improving the accuracy of failure prognostics of miter gates.

Positioning anti-sway control method for tower crane based on improved MFAC

A model-free adaptive control (MFAC) based on partial format is proposed for the complex modeling and uncertain model parameters of the tower crane (TC) in this paper. Firstly, considering the complexity of the TC nonlinear system, the data model of the TC is obtained through the partial format dynamic linearization method. Then, a model-free adaptive controller is designed for the TC, incorporating the change rate of the tracking error and its derivative into the objective function to address the inefficiency of traditional model-free controllers in controlling TCs. The convergence and stability of the proposed controller are also proved. To further improve the dynamic performance of the system, a tracking differentiator (TD) is introduced after the input signal. Finally, MATLAB simulation and experiments are conducted to verify that the proposed method can achieve precise trolley positioning while suppressing the load swing angle.

Enchanting dynamics: Juvenile and adult predators with prey model in an intuitionistic fuzzy environment

This study represents an interaction model including juvenile and adult predators along with prey, involving key characteristics like the inability of juvenile predators to reproduce, the impact of adult predators on juvenile growth rates and the transformation of some juveniles to adulthood. Furthermore, it is supposed that adult predators intake prey for sustenance, and the logistic growth of prey is assumed in the non-appearance of adult predators. Intra-species competition among adult predators, together with predation of juveniles and chasing, is conveyed. To tackle the imprecise biological parameters, most parametric values are considered as triangular intuitionistic fuzzy numbers. The model is further processed into an intuitionistic fuzzy model using the Hukuhara derivative and Yager’s ranking method is implemented to make the model crispified. Non-negativity and boundedness of the model are explored, after that, an investigation of possible equilibrium points is discussed. Local and global stability analyses close to these points are supervised. The study accentuates the significant influence of uncertain model parameters on the solution. Especially, uncertain estimates of intra-species competition among adult predators remarkably regulate the model’s stability. Extended uncertainty in intra-species competition for prey, although, is discovered to stabilize the system. In addition, increased consumption of juvenile predators by adult predators is observed to provide the system’s instability.

Propagation of hybrid uncertainty by synthesizing B-spline chaos and augmented change of probability measure

Acquiring engineering data is frequently expensive, resulting in sparse data that may lead to a lack of knowledge for design and analysis. Thus, it is not always feasible to precisely determine the probability density functions (PDFs) of uncertain model parameters. Under such circumstances that involve simultaneous aleatory and epistemic uncertainties, repeated uncertainty propagation (UP) analysis is generally required. In this paper, a novel approach for hybrid UP is proposed by integrating B-spline chaos and augmented change of probability measure (aCOM) for meeting different goals. The B-spline chaos is adopted to represent the complicated computational model as a function of an arbitrary input random variable, while the aCOM is employed to reconstruct the PDF of the model output when the input PDF is changed due to epistemic uncertainty. In the case of small epistemic uncertainty, hybrid UP can be achieved directly by changing the assigned probabilities of existing sample results. While in the case of large epistemic uncertainty, additional samples from an augmenting PDF are generated. The proposed method is compatible with both cases. The numerical algorithm of the proposed method is presented and illustrated by four benchmark problems. Further, the accuracy and efficiency of the proposed method are substantiated by four numerical examples compared with analytical solutions or Monte Carlo simulations. An attempt to enhance the proposed method with the aid of active subspace methods to handle high-dimensional problems is also discussed in this work. The limitations and potential improvements of the proposed approach are outlined as well.

Review of Fourth-Order Predictive Modeling and Illustrative Application to a Nuclear Reactor Benchmark. I. Typical High-Order Sensitivity and Uncertainty Analysis

This work (in two parts) will review the recently developed predictive modeling methodology called “4th-BERRU-PM” and its applicability to nuclear energy systems as exemplified by an illustrative application to the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The acronym 4th-BERRU-PM designates the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” methodology, which uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses to model parameters, while yielding best-estimate results with reduced uncertainties for the first fourth-order moments (mean values, covariance, skewness, and kurtosis) of the optimally predicted posterior distribution of model results and calibrated model parameters. The 4th-BERRU-PM methodology encompasses the scopes of high-order sensitivity analysis (SA), uncertainty quantification (UQ), data assimilation (DA) and model calibration (MC), as will be illustrated in this work by means of the above-mentioned OECD/NEA reactor physics benchmark. This benchmark is modeled using the neutron transport Boltzmann equation involving 21,976 imprecisely known parameters, the solution of which is representative of “large-scale computations”. The model result (“response”) of interest is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. Part 1 of this work illustrates the impact of high-order sensitivities, in conjunction with parameter standard deviations of various magnitudes, on the determination of the expected value and variance of the computed response in terms of the first four moments of the distribution of the uncertain model parameters. Part 2 of this work will illustrate the capabilities of the 4th-BERRU-PM methodology for combining computational and experimental information, up to and including forth-order sensitivities and distributional moments, for producing best-estimate values for the predicted responses and model parameters while reducing their accompanying uncertainties.

Model Parameter Calibration for Vibration Fatigue Analysis by Means of Bayesian Updating and Artificial Neural Network Based Surrogate Models

Abstract In this paper, a methodology for model parameter calibration for vibration fatigue analysis is proposed. It combines Bayesian updating of uncertain model parameters and artificial neural networks (ANNs). The calibrated parameters are used to increase the accuracy of fatigue lifetime calculations for components submitted to vibrational loads. The Bayesian updating uses eigenfrequencies, mode shapes, total mass, and the frequency response functions (FRFs). These quantities are predicted by ANN-based surrogate models to accelerate the Bayesian updating process. A novel strategy for the prediction of the magnitude and phase of FRFs with ANNs is proposed. The frequency is used as an additional input variable, and a schematic selection of significant points of the FRF curves is presented. A high prediction accuracy of the surrogate models could be achieved. The procedure includes the analysis of the relevant frequency range and a sensitivity analysis based on the Morris method to identify appropriate modes and the most-influential parameters. The proposed framework is applied to a current vehicle component subjected to vibrational loads. An experimental modal analysis is used for the calibration and consideration of real parameter uncertainty. First, the accuracy of the surrogate models and Bayesian updating is verified by a nominal reference simulation and then validated with experimental data. The measurable control parameter thickness and component mass are used to examine the calibration accuracy. Finally, a decrease in the dispersion of the vibration fatigue distribution is obtained with the calibrated parameters.

Optimally tuned cascaded FOPI-FOPIDN with improved PSO for load frequency control in interconnected power systems with RES

In the operation and control of power systems, load frequency control (LFC) plays a critical role in ensuring the stability and reliability of interconnected power systems. Modern power systems with significant penetration of highly variable and intermittent renewable sources present new challenges that make traditional control strategies ineffective. To address these new challenges, this paper proposes a novel LFC strategy that employs a cascaded fractional-order proportional integral-fractional-order proportional integral derivative with a derivative filter (FOPI-FOPIDN) as a controller. The parameters of the FOPI-FOPIDN are optimised using a variant of the particle swarm optimization (PSO) in the literature called ADIWACO. The effectiveness and scalability of the proposed strategy are validated by extensive simulations conducted on two- and three-area test systems and performance comparisons with recent LFC control strategies in the literature. The performance metrics used for the evaluation are ITAE values, deviations in the power flows in the tie-lines, and deviations in the frequencies of the control areas with the power systems subjected to diverse load and RES generation disturbances in several experimental scenarios. Governor dead band, communication time delay, and generation rate constraints are considered in one of the scenarios for more realistic evaluation. Again, the controller’s robustness to uncertain model parameters is validated by varying the parameters of the three-area test system by ± 50%. The simulation results obtained confirm the controller’s robustness and its superiority over the comparison LFC strategies in terms of the above performance metrics.

Bayesian Model Updating of Multiscale Simulations Informing Corrosion Prognostics Using Conditional Invertible Neural Networks

Abstract Physics-based multi-scale corrosion simulation plays a vital role in predicting the evolution of pitting corrosion on large civil infrastructure such as miter gates, contributing to a model-informed structural health monitoring (SHM) strategy for risk-based asset health management. The physics-based analysis, however, may not accurately reflect the underlying true physics due to various uncertainty sources and needs to be updated using Bayesian inference methods based on observations to make the prediction closer to field observations. However, traditional Bayesian inference requires the evaluation of a likelihood function, which is often unavailable due to the complex model architecture and various surrogate models used in the analysis. Therefore, likelihood-free inference approaches are required for the updating of the multi-scale corrosion simulation models. This paper meets this need by proposing a conditional invertible neural network (cINN)-based Bayesian model updating method for an existing corrosion simulation model. We first train an cINN model based on simulated observations generated from a high-fidelity forward corrosion analysis model. A convolutional neural network (CNN)-based feature extraction algorithm is then employed to extract key features from corrosion images. After that, the extracted corrosion features from CNN is used as inputs of the cINN model to directly obtain posterior distributions of uncertain corrosion model parameters without evaluating the likelihood function. The results show that the proposed cINN-based model updating approach can provide more accurate inference results with a reduced computational cost in comparison to the classical approximate Bayesian computation (ABC) approach.

Machine Learning Driven Sensitivity Analysis of E3SM Land Model Parameters for Wetland Methane Emissions

AbstractMethane (CH4) is globally the second most critical greenhouse gas after carbon dioxide, contributing to 16%–25% of the observed atmospheric warming. Wetlands are the primary natural source of methane emissions globally. However, wetland methane emission estimates from biogeochemistry models contain considerable uncertainty. One of the main sources of this uncertainty arises from the numerous uncertain model parameters within various physical, biological, and chemical processes that influence methane production, oxidation, and transport. Sensitivity Analysis (SA) can help identify critical parameters for methane emission and achieve reduced biases and uncertainties in future projections. This study performs SA for 19 selected parameters responsible for critical biogeochemical processes in the methane module of the Energy Exascale Earth System Model (E3SM) land model (ELM). The impact of these parameters on various CH4 fluxes is examined at 14 FLUXNET‐ CH4 sites with diverse vegetation types. Given the extensive number of model simulations needed for global variance‐based SA, we employ a machine learning (ML) algorithm to emulate the complex behavior of ELM methane biogeochemistry. We found that parameters linked to CH4 production and diffusion generally present the highest sensitivities despite apparent seasonal variation. Comparing simulated emissions from perturbed parameter sets against FLUXNET‐CH4 observations revealed that better performances can be achieved at each site compared to the default parameter values. This presents a scope for further improving simulated emissions using parameter calibration with advanced optimization techniques.

Modelling and analysis of quench in the 15-kA HTS conductor

High Temperature Superconductors (HTS) are very promising materials for possible application in future fusion magnets, with significant R&D progress on HTS conductors in recent years. However, since geometric and thermo-physical characteristics of HTS and LTS conductors differ significantly, some doubts have arisen if the approaches successfully used in numerical simulations of the thermal–hydraulic behavior of LTS conductors would be sufficient also for HTS, particularly in cases when fast transient processes (such as e.g. quench) are considered. In order to provide data for better understanding of the quench phenomenon in HTS conductors as well as for testing different numerical approaches and proper tuning of the numerical codes, a dedicated experimental campaign (Quench Experiment) was carried out at the SULTAN test facility within the international collaboration between the EUROfusion consortium and China. Our present study is a part of the work on analysis and interpretation of the data collected during this experiment. Simulations of the chosen experimental run were performed using two THEA models with different levels of complexity. Uncertain model parameters (thermal resistances and copper RRR) were explored across a wide range. Our goal was to identify the possibly simple model that accurately reproduces the experimental results.

Improving crop yield estimation by unified model parameters and state variable with Bayesian inference

Data assimilation techniques integrating crop growth models and remote sensing technologies offer a feasible approach for large-scale crop yield estimation. Previous research has primarily focused on either recalibrate the uncertain model parameters or updating model state variables independently using remotely sensed observations. In this study, we developed a two-step inference algorithm that couples the parameter inference and the state update, to solve the joint posterior distribution of uncertain parameters and state variables given the remote sensing observations. An Observing Simulation System (OSS) experiment was first performed based on the WOFOST crop model to validate the effectiveness of the parameter inference method. The results indicate that, the parameter inference method successfully improved the estimation of different types of model parameters and enhanced yield estimation. Furthermore, leaf area index (LAI) retrieved from Sentinel-2 was assimilated into the WOFOST model to simulate winter wheat yield at the plot scale in the northeastern part of Henan Province. The results demonstrated that the proposed two-step inference algorithm can more effectively correct model simulation biases and improve winter wheat yield estimation accuracy (R²=0.58, MAPE=12.75 %, and RMSE=1112 kg·ha⁻¹), outperforming the standard EnKF algorithm (R²=0.51, MAPE=14.62 %, RMSE=1328 kg·ha⁻¹). Overall, attributed to its unified approach to estimating both model parameters and state variables, the proposed two-step inference algorithm shows promising application prospects for data assimilation-based crop yield estimation.

Validation of the thermo-chemical approach to modelling of CO2 conversion in sub-atmospheric pressure microwave gas discharges

The CO 2→ CO + 12 O2 conversion experiment [D’Isa et al 2020 Plasma Sources Sci. Technol. 29 105009] has been compared with thermo-chemical calculations. The experiment is a 2.45 GHz plasma torch with a straight channel in the effluent. The 1.5D model of the CO2/CO/O2/O/C mixture without turbulent transport has been applied with plasma acting only as a prescribed heat source. The parameter range covered is the specific energy input (SEI) 0.3–5 eV/molecule at pressure p = 0.9 bar, and SEI = 0.6–2 eV/molecule at p = 0.5, 0.2 bar. The calculated conversion χ is always close to the experimental values. At the same time, the calculated temperatures T deviate significantly from the experiment, especially for p = 0.2 bar. The calculated T were also found to be sensitive with respect to the uncertain model parameters, but χ is not sensitive. The net conversion in the model is driven to a large extent by the radial diffusion of CO and O from the hot core towards the wall and steep radial temperature gradients. The main factor that reduces the energy efficiency is the re-oxidation of CO at the edge of the hot plasma region and downstream. The 1.5D approximation applied has the principle limitation that the impact of a realistic bulk flow field on the chemical process cannot be studied. Hence, the results must be considered preliminary and have to be confirmed with a more elaborate and accurate model of the vortex-stabilized flow inside the reactor.

Fuzzy Uncertainty Analysis of A Fractional Order HIV Dynamic Model with Type-1 and Interval Type-2 Parameters

Abstract In recent years, the study of mathematical models for the Human Immunodeficiency Virus (HIV) has attracted considerable interest due to its importance in comprehending and combating the propagation of the virus. Typically, the model's governing equations are a system of ordinary differential equations. In order to explain the inheritance behaviour, fractional order HIV models may be more helpful than integer order models. In addition, the presence of uncertainty in real world phenomena can not be avoided, and fuzzy numbers are of great use in these scenarios. In view of the above, the numerical solution of the fuzzy fractional order HIV model is analysed in this paper. The model takes into account the interactions between susceptible, asymptomatic, and symptomatic populations, as well as the effects of fractional order derivatives and fuzzy uncertainty. Here, the differentiation of the fuzzy parameters is considered in granular sense. The uncertain model parameters are addressed with triangular fuzzy numbers (TFNs) and interval type-2 triangular fuzzy numbers (IT2TFNs). The use of interval type-2 fuzzy numbers rather than type-1 fuzzy numbers to express the imprecise parameters may be helpful in some instances where the membership grade is unclear. The generalised modified Euler method (GMEM) is used to derive the corresponding solutions. Lastly, the behaviour of various populations in crisp as well as uncertain environments is also studied using graphical results.

Theory guided Lagrange programming neural network for subsurface flow problems

A deep learning model can perform efficient uncertainty quantification (UQ) for reservoir flow with uncertain model parameters, but usually requires large amounts of training data to ensure accuracy. However, the cost of obtaining large amounts of data is prohibitive, and the performance will deteriorate if sufficient training data is lacking. Alternatively, more interpretable neural networks with embedded physical laws have recently been used to solve partial differential equations as well as to solve UQ problems. This approach has received a lot of attention due to its low data volume requirements and its adherence to the laws of physics during the training process. In this paper, we propose a theory-guided framework based on a bilevel programming model with hard constraints to embed physical meaning in the model. Theory guided Lagrange programming neural network (TGLPNN) combines the method of Lagrange programming neural network approach where physical laws such as stochastic partial differential equations and boundary conditions are incorporated into the training process of a convolutional neural network. At the same time, the upper-level variables are iteratively optimized. The method based on Lagrange programming neural network inherently embeds physical laws in the network. Practical applications have shown that TGLPNN can provide higher prediction accuracy compared to state-of-the-art physics-driven methods and improved efficiency compared to numerical methods.

Data-driven optimization for microgrid control under distributed energy resource variability

The integration of renewable energy resources into the smart grids improves the system resilience, provide sustainable demand-generation balance, and produces clean electricity with minimal leakage currents. However, the renewable sources are intermittent in nature. Therefore, it is necessary to develop scheduling strategy to optimise hybrid PV-wind-controllable distributed generator based Microgrids in grid-connected and stand-alone modes of operation. In this manuscript, a priority-based cost optimization function is developed to show the relative significance of one cost component over another for the optimal operation of the Microgrid. The uncertainties associated with various intermittent parameters in Microgrid have also been introduced in the proposed scheduling methodology. The objective function includes the operating cost of CDGs, the emission cost associated with CDGs, the battery cost, the cost of grid energy exchange, and the cost associated with load shedding. A penalty function is also incorporated in the cost function for violations of any constraints. Multiple scenarios are generated using Monte Carlo simulation to model uncertain parameters of Microgrid (MG). These scenarios consist of the worst as well as the best possible cases, reflecting the microgrid’s real-time operation. Furthermore, these scenarios are reduced by using a k-means clustering algorithm. The reduced procedures for uncertain parameters will be used to obtain the minimum cost of MG with the help of an optimisation algorithm. In this work, a meta-heuristic approach, grey wolf optimisation (GWO), is used to minimize the developed cost optimisation function of MG. The standard LV Microgrid CIGRE test network is used to validate the proposed methodology. Results are obtained for different cases by considering different priorities to the sub-objectives using GWO algorithm. The obtained results are compared with the results of Jaya and PSO (particle swarm optimization) algorithms to validate the efficacy of the GWO method for the proposed optimization problem.

Scenario based optimisation over uncertain system identification models

Models of dynamical systems are often built from observed data using system identification techniques, and the uncertainty in the model parameters is commonly described using confidence regions to which the true model parameters belong with a certain probability. In this paper we propose methods which use the scenario approach to solve robust optimisation problems for model based design where there is uncertainty in the model parameters. The methods draw samples (scenarios) of the uncertain model parameters and solve an optimisation problem only involving the drawn scenarios. Both a Bayesian system identification setting where the model parameters are stochastic, and a non-Bayesian system identification setting where the model parameters are treated as being deterministic are considered. The Bayesian and non-Bayesian settings require different treatment, and whereas standard scenario optimisation theory can be directly utilised in the Bayesian setting, this is not the case in the non-Bayesian setting. For that reason the model class is restricted to linear regression models with deterministic regressors in the non-Bayesian case. Algorithms are proposed for drawing samples of the parameters such that the uncertainties in the system identification models are reflected in the samples. Theoretical results are given bounding the probability that the found solution to the scenario optimisation problem will give a worse performance when applied to the true system than seen on the drawn scenarios. The methods are illustrated in simulation examples showing good performance.