Deterministic Values Research Articles

3D SN and Monte Carlo calculations of the Utah TRIGA reactor core using PENTRAN and MCNP6

We present a systematic and detailed approach to simulate the University of Utah TRIGA Reactor (UTR) in support of core criticality calculations to profile the entire reactor in detail. In performing this work, we utilized both a 3-D Cartesian SN deterministic code, PENTRAN, and a Monte Carlo code, MCNP6, to calculate complimentary, high accuracy 3-D transport derived neutron flux distributions at reactor full power. For deterministic models, our study shows that a 14-group CONTRIBUTON weighted multi-group cross-section library with up-scattering compared extremely well with continuous energy Monte Carlo results. We also modeled results to confirm the activity of an activated arsenic sample placed in the heavy water moderated thermal irradiation chamber of the UUTR (University of Utah TRIGA Reactor).A 138 pcm difference in the system eigenvalue for the full 3-D core PENTRAN discrete ordinates (SN) UUTR model utilizing 2×109 equations, in 14 energy groups, is compared to the reference case of a similar full core MCNP6 Monte Carlo calculation with continuous energy cross-sections. On average, deterministic and Monte Carlo core neutron group flux values differ by less than 1%, with some local maximum relative differences between 10% and 20% for the 3-D core flux distribution in some energy groups are observed. The differences of neutron fluxes from group 3/14 to group 10/14, with energies spanning from 1.11 eV to 13.8 MeV, are comparable to within 5% in the UUTR core active fuel region. For energy groups with energies greater than 13.8 MeV, MCNP6 is not able to resolve the flux distributions with reasonable statistical uncertainties due to the extremely low sampling probability for neutrons in those energies; orders of magnitude more histories would be required to do so. In contrast, the PENTRAN 3D SN calculation shows a very detailed flux distribution for energies larger than 13.8 MeV, highlighting the complimentary utility of applying both methods. Detailed approaches for the PENTRAN calculation, including energy group collapsing, homogenization of fuel + gap + cladding, and analysis are presented in the narrative.

Uncertainty and global sensitivity analysis of levelized cost of energy in wind power generation

Modern wind turbines tend to large scale and capacity, which amplifies the uncertainty of the cost of wind power generation. This paper proposes an analytical framework for uncertainty in the cost of wind power. To do so, the levelized cost of the energy model is improved by considering inflation and the learning curve. On this basis, results from the quasi-Monte Carlo simulation are used for uncertainty analysis that is performed to qualify the uncertainty degree of the levelized cost of energy. Meanwhile, sensitivity analysis based on variance is conducted to study the impact of the uncertainty factors on the levelized cost of energy. Results reveal that through improving the cost model, the levelized cost of energy is changed from 54.11 $/MWh to 37.03 $/MWh in 2018, which is close to the real value of projects built in 2018. Meanwhile, the deterministic values of the levelized cost of energy are similar to P50, which means that only 50% credibility is guaranteed by the deterministic design. To achieve a 95% reliability when considering changes caused by uncertainty factors, the margin should be not less than 38% of the determined value. Finally, sensitivity analysis reveals that the interaction effects from scale parameter, shape parameter and air density make the total effect coefficient increased, while the scale parameter is the most influential factor.

Evaluating the variability of the modulus of elasticity of concrete through the use of different types and batches of aggregate

The modulus of elasticity of concrete is often calculated as a function of the compressive strength, and as a deterministic value. However, variations in the aggregates properties may result in module values different than those estimated, which may lead to excessive deformation and eventual instability of the structure. In this work, the influence of the coarse aggregate batch variation on the variability of the modulus of elasticity of the concrete was investigated. Three different aggregate sources (one of granite origin and two of gneiss origin), three water/cement ratios (w/c) and five different batches of each aggregate were investigated. The compressive strength and static modulus of elasticity of the concretes were determined at 28 days. The analysis of variance (ANOVA) showed that the variable "batch" had a significant influence on the modulus of elasticity of the concrete, indicating that this property is a probabilistic variable indeed. The normality of the distribution of its values was attested, and values of characteristic modulus of elasticity were proposed, which were from 6 to 10% lower than the mean values. In addition, the use of gneissic aggregates led to modulus of elasticity values 30% higher than those of the concretes with granitic aggregates and equivalent compressive strengths, while the reduction of the w/c ratio from 0.71 to 0.46 increased the modulus of elasticity in about 5%.

Delay-Bounded Wireless Network Based on Precise Time Synchronization Using Wireless Two-Way Interferometry

The importance of reliable information transfer in wireless networks, especially regarding communication delay, is drastically increasing to fulfill safe and high-quality communication in the 5G and post-5G era. However, conventional media access control (MAC) protocol for wireless networks, notably Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA), sometimes yields unexpectedly significant delay due to its complex arbitration mechanism assuming asynchronous communication among terminals. As the delay cannot be strictly bounded by a deterministic value, this causes a vulnerability of systems relying on wireless networks. This paper utilizes precise time synchronization achieved by Wireless Two-way Interferometry (Wi-Wi), enabling all terminals to be time-synchronized via wireless signals. We show that by an appropriate periodic assignment of each terminal's data transmission timing, named Arbitration Point (AP), a simple arbitration algorithm obtains a strictly bounded maximum value for the delay while ensuring equalities among all participants. Furthermore, we demonstrate that the total number of terminals manageable in a star-topology wireless network significantly increases by densely packing AP timings, taking into account the spatial geometry information of terminals, which is another feature delivered by Wi-Wi measurement. In the meantime, we experimentally constructed a star-topology wired network where all terminals are time-synchronized via Wi-Wi to confirm the fundamental properties identified in the proposed arbitration protocol. This study paves a new way for future wireless networks where the delay is strictly bounded and provides the basis for ultra-reliable and high-quality information transfer functionalities by utilizing precise time synchronization and space localization (space-time synchronization).

Morfološka raznolikost i odnosi populacija Scardinius dergle i Scardinius plotizza (Actinopteri; Cypriniformes) iz Hrvatske i Bosne i Hercegovine

Genus Scardinius (Cypriniformes, Actinopteri) belongs to the family Leuciscidae, subfamily Leuciscinae, and it includes 10 species. Phylogenetic and morphometric research conducted so far have shown a great taxonomic complexity within this genus. Therefore, the aim of this research was to determine the morphological diversity and contribute to the clarification of taxonomic relationships and status of S. dergle and S. plotizza. For the purpose of this research specimens of S. dergle and S. plotizza were collected from their whole distribution area in Croatia and Bosnia and Herzegovina. On each individual 25 morphometric characters were measured, standardized with allometric conversion and analysed using descriptive and inferential statistics. Results showed significant intra- and interspecific differences in investigated morphometric characters. In addition, distinct grouping of populations of each species and partial overlapping of some populations of both species was recorded, reflecting their possible relationships. Also, investigated meristic characters showed accordance with deterministic values for most investigated populations. Although this research has given indications of relationships and the status of populations of S. dergle and S. plotizza, in order to confirm the results and determine the exact taxonomic status, phylogenetic relationships and genetic diversity of these species, further phylogenetic research is needed.

Cooperative Game Theory Methods in the Analysis of Economic and Political Interaction at the International Level

The main attention is focused on the application of the methods of the theory of cooperative games to the analysis of the relationship between the leading actors in international politics, or, as they say, the centers of power. One of the speci c features of the modern world is the "triple type" of conflicts. Namely, at different levels of relationships, con ict situations with three participants (players) are often observed. Such situations are objectively characterized by the formation of possible paired coalitions, rejecting the third. The main idea of the proposed approach is the transition from cooperative games with deterministic values of characteristic functions to their counterparts with stochastic values. One of the possible concepts of solutions for stochastic cooperative games is associated with the extension of the ideas of the bargaining set to them. Problems of development and interpretation of this concept in the case of a triple conflict of international centers of power. An essential advantage of this approach is the possibility of meaningful interpretations of the significance level, at which the conditions for the rationality of the shares of the players should be ensured, taking into account the non-determinism of their utilities given by the characteristic function.

Probabilistic Settlement Analysis of Strip Footing on Spatially Random Soil

The prediction of settlement in a traditional design approach usually uses a deterministic value of modulus of elasticity (E), which is estimated as an average value by testing the soil at selected locations. However, these deterministic properties of soil may not represent the actual properties of soil and site condition. Due to numerous sources of uncertainty, the properties of soil mass are spatially varying and anisotropic in the natural field condition. In this study, a random finite element method (RFEM) is used to evaluate the reliability of settlement of the strip footing on spatially random soil. Modulus of elasticity is the only considered random parameter. For this purpose, 2000 spatially random realizations of E-field are generated using Monte Carlo Simulation. Each of these realizations of heterogeneous soil profile is passed to FEM to analyze the settlement of footing. The final settlement results measured in all these realizations are then statistically evaluated and compared. The results of analysis show that the mean and standard deviation of the footing settlement are increased with increasing spatial correlation length. The large value of isotropic correlation length led to an increase in the mean settlement value by more than 25% as compared with the deterministic settlement calculations. Also, it is concluded that the rate of increase of settlements for anisotropic correlation length is lower than the one under isotropic condition.

Towards robust investment decisions and policies in integrated energy systems planning: Evaluating trade-offs and risk hedging strategies for remote communities

Policy and investment decisions in developing clean energy strategies for remote communities are subject to multiple uncertainties that impact overall strategy outcomes, including those related to environmental emissions and energy costs. In this context, robust modeling approaches are required that can clarify potential outcomes while subject to such uncertainties. This work introduces a novel modeling framework that enables enhanced decision making in energy systems planning for remote communities, which for the first time takes into account context-specific decision-maker attitudes towards multiple inter-related uncertainties and various energy solution philosophies. In particular, multiple energy system configurations are evaluated by simultaneously minimizing the levelised cost of energy and fuel consumption, with a test case for a specific community in the Northwest Territories, Canada. The concept of model robustness and validity together with the stochastic nature of uncertain parameters are combined in a multi-objective optimization framework that elucidates the full spectrum of energy solutions available in such a remote Arctic context. Introducing known uncertainties in renewable energy characteristics was found to reduce overall energy yields from the renewable energy technologies. Specifically, the deterministic renewable energy penetration of 69% from a specific energy system configuration reduced to a mean of 51% after the inclusion of uncertainties via probabilistic simulation. Conversely, diesel fuel consumption increased to 750,000 L/yr (mean) from its initial deterministic value of 447,470 L/yr. Holistic energy solutions which include both supply and demand-side considerations are also analyzed. Specifically, a reduced community domestic heating load of 40% was achieved via retrofit of high performance building fabric enclosures evaluated in conjunction with renewable energy supply options. Ultimately, insights and real-world applications have been synthesized to provide coherent recommendations on strategies to address energy security, energy affordability and environmental sustainability, along with meaningful propositions towards Indigenous community-led energy projects in a range of contexts.

Multi-hazard risk assessment for roads: probabilistic versus deterministic approaches

Abstract. Mountain hazard risk analysis for transport infrastructure is regularly based on deterministic approaches. Standard risk assessment approaches for roads need a variety of variables and data for risk computation, however without considering potential uncertainty in the input data. Consequently, input data needed for risk assessment are normally processed as discrete mean values without scatter or as an individual deterministic value from expert judgement if no statistical data are available. To overcome this gap, we used a probabilistic approach to analyse the effect of input data uncertainty on the results, taking a mountain road in the Eastern European Alps as a case study. The uncertainty of the input data are expressed with potential bandwidths using two different distribution functions. The risk assessment included risk for persons, property risk and risk for non-operational availability exposed to a multi-hazard environment (torrent processes, snow avalanches and rockfall). The study focuses on the epistemic uncertainty of the risk terms (exposure situations, vulnerability factors and monetary values), ignoring potential sources of variation in the hazard analysis. As a result, reliable quantiles of the calculated probability density distributions attributed to the aggregated road risk due to the impact of multiple mountain hazards were compared to the deterministic outcome from the standard guidelines on road safety. The results based on our case study demonstrate that with common deterministic approaches risk might be underestimated in comparison to a probabilistic risk modelling setup, mainly due to epistemic uncertainties of the input data. The study provides added value to further develop standardized road safety guidelines and may therefore be of particular importance for road authorities and political decision-makers.

Derivation and Uncertainty Quantification of a Data-Driven Subcooled Boiling Model

Subcooled flow boiling occurs in many industrial applications where enormous heat transfer is needed. Boiling is a complex physical process that involves phase change, two-phase flow, and interactions between heated surfaces and fluids. In general, boiling heat transfer is usually predicted by empirical or semiempirical models, which are horizontal to uncertainty. In this work, a data-driven method based on artificial neural networks has been implemented to study the heat transfer behavior of a subcooled boiling model. The proposed method considers the near local flow behavior to predict wall temperature and void fraction of a subcooled minichannel. The input of the network consists of pressure gradients, momentum convection, energy convection, turbulent viscosity, liquid and gas velocities, and surface information. The outputs of the models are based on the quantities of interest in a boiling system wall temperature and void fraction. To train the network, high-fidelity simulations based on the Eulerian two-fluid approach are carried out for varying heat flux and inlet velocity in the minichannel. Two classes of the deep learning model have been investigated for this work. The first one focuses on predicting the deterministic value of the quantities of interest. The second one focuses on predicting the uncertainty present in the deep learning model while estimating the quantities of interest. Deep ensemble and Monte Carlo Dropout methods are close representatives of maximum likelihood and Bayesian inference approach respectively, and they are used to derive the uncertainty present in the model. The results of this study prove that the models used here are capable of predicting the quantities of interest accurately and are capable of estimating the uncertainty present. The models are capable of accurately reproducing the physics on unseen data and show the degree of uncertainty when there is a shift of physics in the boiling regime.

Altering Gaussian process to Student-t process for maximum distribution construction

Gaussian process (GP) regression is widely used to find the extreme of a black-box function by iteratively approximating an objective function when new evaluation obtained. Such evaluation is usually made by optimising a given acquisition function. However, for non-parametric Bayesian optimisation, the extreme of the objective function is not a deterministic value, but a random variant with distribution. We call such distribution the maximum distribution which is generally non-analytical. To construct such maximum distribution, traditional GP regression method by optimising an acquisition function is computational cost as the GP model has a cubic computation of training data. Moreover, the introduction of acquisition function brings extra hyperparameters which made the optimisation more complicated. Recently, inspired by the idea of Sequential Monte Carlo method and its application in Bayesian optimisation, a Monte Carlo alike method is proposed to approximate the maximum distribution with weighted samples. Alternative to the method on GP model, we construct the maximum distribution within the framework of Student-t process (TP) which considers more uncertainties from the training data. Toy examples and real data experiment show the TP-based Monte Carlo maximum distribution has a competitive performance to the GP-based method.

Time-of-Failure Probability Mass Function Computation Using the First-Passage-Time Method Applied to Particle Filter-based Prognostics

One of the main challenges in prognostics corresponds to the estimation of a system’s probability density function (PDF) for the time-of-failure (ToF) prior to reach a fault condition. An appropriate characterization of the ToF-PDF will let the user know about the remaining useful life of the system or component, allowing the users to prevent catastrophic failures through optimal maintenance schedules. However, the ToF-PDF estimation is not an easy task because it involves both the computation of long-term predictions of a fault indicator of the system and the definition of the hazard zone. In most cases, the trajectory of the fault indicator is assumed as a trajectory with monotonic behavior, and the hazard zone may be considered as a deterministic or probabilistic threshold. This monotonic behavior of the fault indicator enables assuming that the system will only fail once when this indicator reaches the hazard zone, and the ToF-PDF will be estimated according to mathematical definitions proposed in the state-of-the-art. Nevertheless, not all the fault indicators may be considered with a monotonic behavior due to its nature as a stochastic process or regeneration phenomenon, which may entail to errors in the ToF-PDF estimation. To overcome this issue, this paper presents an approach for the estimation of the ToF-PDF using the first-passage-time (FPT) method. This method is focused on the computation of the FPT-PDF when the stochastic process under analysis reaches a specified threshold for the first time only. Accordingly, this work aims to analyze the impact in the estimation of the ToF-PMF (probability mass function) when particle-filter-based prognostics algorithms are used to perform long-term predictions of the fault indicator and compute the probability of failure considering specific hazard zones (which may be characterized by a deterministic value or by a failure likelihood function). A hypothetical self regenerative degradation process is used as a case study to evaluate the performance of the proposed methods.

Developing the model of reliability of a complex technical system of repeated use with a complex operating mode

Solving the problems of setting requirements to the reliability of complex technical systems for various purposes presupposes their classification according to the features characterizing the purpose, modes of use, etc. According to the modes of use, systems are divided into objects of continuous long-term use, repeated cyclic use, and single-use. The objects of repeated cyclic use include the systems operating in cycles. Durations of the periods of work and pause in the cycle are considered deterministic values. Technological and/or technical maintenance is carried out in pauses between the operation periods. In addition to the known classification, it was proposed to introduce a group of systems of repeated use with a complex operating mode. A complex mode is understood as a mode that includes waiting for a request of the system use and executing the request after it arrives at a random time. An analytical model of reliability of such a system has been developed in the form of a ratio for a non-stationary total coefficient of operational readiness. This model describes the processes of the system functioning in the intervals of waiting and use. In this case, the duration of the intervals of waiting and/or execution of the request are random values. Ratios for this indicator were obtained for three options of specifying the functions of distribution of durations of waiting in a turn-on condition and fulfilling the request for use. The developed model makes it possible to set requirements for reliability and maintainability of the systems with a complex operating mode. The results of modeling the dependences of the operational indicators of reliability on parameters of the functions of distribution of durations of waiting and executing the request were obtained for different distributions. Recommendations were formulated concerning the substantiation of the requirements to reliability and maintainability of the systems under consideration

Dependence of ocean wave return levels on water depth and sampling length: A focus on the South Yellow Sea

A deterministic Extreme Value Analysis method is of particular importance in an engineering-oriented context. In this paper, three extreme value analysis (EVA) methods, in which annual maxima, monthly maxima or the Peaks-Over-Threshold are fitted into Generalized Extreme Value distribution (GEV) function or Generalized Pareto distribution (GP) function, are used to estimate return values of significant wave height. Sensitivity of return levels on water depth and sampling length is vigorously investigated, based on a 40-year long and high-resolution wave hindcast for the South Yellow Sea (SYS). A spatially-and-temporally varied POT sampling method combined with GP function produces most conservative and confident estimates of return levels for a large return period, but produces wider confidence level than the other two EVA approaches based on GEV function for short return period. In the SYS, the return level estimates are significantly reduced with a longer sample length. However, we find that the reduction is closely related to the long-term trend in extreme wave heights, rather than due to the sampling effect. From deep to shallow waters, spatial inhomogeneity of return levels increases, which should be considered in the engineering practice.

Probabilistic design factors for pipes used for hydrogen transport

Probabilistic design factors for pipes used for hydrogen transport are proposed for 3 locations: urban, peri-urban and rural areas.The used method involves a risk equation defined as the product of probability of leakage after failure, probability to have a gas flow greater than a prescribed value, probability of ignition, probability of lethal effects greater than a threshold value and probability of the presence of a person corrected by an environment factor and a risk reduction coefficient. The results obtained are less conservative than the deterministic values provided by ASME code.

Design optimization of hybrid uncertain structures with fuzzy-boundary interval variables

In some special engineering cases, the lower and upper bounds of uncertain parameters are appropriately quantified as fuzzy variables instead of deterministic values. To address such cases, a possibility-based robust design optimization (PBRDO) framework is suggested for the hybrid uncertain structures with fuzzy-boundary interval (FuBI) variables. Firstly, an optimization model with FuBI variables is established where FuBI uncertainties exist in both the objective and constraint functions. The so-called dual robust design is presented and it is adopted to create the optimization objective. The first robust design aims to handle fuzziness while the second one attends to tackle interval property. The failure possibility is employed to construct the optimization constraints in possibilitic context. Then, the fuzzy-boundary interval Taylor series-central difference method (FITS-CDM) is derived to manage FuBI uncertainties and calculate the optimization objective efficiently. Next, the target performance approach (TPA) is employed to process the possibilistic constraints and the simplified constraints can be easily solved by FITS-CDM. The nested-loop PBRDO with FuBI variables can be simplified to a single-loop one based on FITS-CDM and TPA. Finally, the effectiveness of the proposed optimization approach on dealing with FuBI uncertainties is demonstrated by three examples.

Investigation of impact of submarine landslide on pipelines with large deformation analysis considering spatially varied soil

In-situ soil properties exhibit natural spatial variability. In this study, the Karhunen-Loève (K-L) expansion method and the Coupled Eulerian-Lagrangian (CEL) method are integrated to simulate large deformation behavior of slopes with spatially varied shear strengths. The impact on pipelines induced by spatially varied landslides is investigated. The mean value of the maximum impact force is significantly larger than the deterministic value obtained for a homogeneous slope. The proportion of the maximum impact force from stochastic analysis that exceeds the deterministic value is much greater than 50%. This result shows that the deterministic study can significantly underestimate the damage of pipeline by a landslide if the spatial variability is not considered. The uncertainty of the impact force increases with the coefficient of variation (COV) of the undrained shear strength and decreases with the reduction of horizontal correlation length. By considering buckling failure and plastic yielding failure of pipelines, the failure probabilities are assessed and a pipeline design based on stochastic analysis for spatially varied slope is presented.

Flexible job-shop scheduling problem with job precedence constraints and interval grey processing time

In the practical production scheduling of complex products, the processing time always fails to be accurately estimated as a deterministic value due to the fluctuating manufacturing environment caused by unsteady operating conditions of machines and diverse proficiency levels of workers. Besides, scheduling jobs become restricted by precedence constraints on account of the hierarchical dependencies between manufactured parts defined by Bills-Of-Materials of final products. Hence, this study concentrates on the flexible job shop scheduling problem with job precedence constraints and interval grey processing time (G-FJSP-JPC) under the objective of minimizing the interval grey makespan. The uncertain processing time as well as other time data are represented as interval grey numbers with lower and upper boundaries. Specific operators for dealing with interval grey time numbers are devised to acquire precise grey schedules. A two-vector encoding scheme with job precedence repair mechanism is employed and a heuristic active grey decoding scheme is proposed in which the machine assignment is dynamically determined and the uncertain grey schedule is constructed. A novel multi-micro-swarm leadership hierarchy-based optimization algorithm (M2SLHO) is proposed to tackle G-FJSP-JPC. This algorithm divides search agents into multiple independent micro-swarms in which each one conducts a leadership hierarchy-based search and communicates with others by specific strategies. Extensive computational experiments validate that M2SLHO provides highly promising results for the addressed problem.

Uncertainty quantification and global sensitivity analysis of double-diffusive natural convection in a porous enclosure

In this paper, detailed uncertainty propagation analysis (UPA) and variance-based global sensitivity analysis (GSA) are performed on the widely adopted double-diffuse convection (DDC) benchmark problem of a square porous cavity with horizontal temperature and concentration gradients. The objective is to understand the impact of uncertainties related to model parameters on metrics characterizing flow, heat and mass transfer processes, and to derive spatial maps of uncertainty and sensitivity indices which can provide physical insights and a better understanding of DDC processes in porous media. DDC simulations are computationally expensive and UPA and GSA require large number of simulations, so an appropriate strategy is developed to reduce the computational burden. The approach is built on two pillars: (a) an efficient numerical simulator based on the Fourier series method that generates training data, and (b) polynomial chaos expansion (PCE) meta-models that are trained using the simulator data, and then replace the numerical model in UPA and GSA. Assuming that the Rayleigh number (Ra), the solutal to thermal buoyancy ratio (Nb) and the Lewis number (Le) are the uncertain input variables, the results of UPA show that the zones of high temperature and concentration variability are located in the regions where the flow is mainly driven by the buoyancy effects. GSA indicates that Nb is the most influential parameter affecting the temperature and concentration fields, followed respectively by Ra and Le. For the heat-driven flow case (Nb>−1), the concentration field is more influenced by Le than Ra. For deeper understanding of uncertainty propagation, we estimate the bias introduced by replacing uncertain parameters by deterministic values. The resulting spatial maps of the difference between deterministic output and stochastic mean show that a deterministic approach leads to different zones where the temperature, concentration and velocity fields can be either overestimated or underestimated. The conclusions drawn in this work are likely to be helpful in different applications involving DDC in porous enclosures leading to convective circulation cells.

Added-value of ensemble prediction system on the quality of solar irradiance probabilistic forecasts

Abstract Accurate solar forecasts is one of the most effective solution to enhance grid operations. As the solar resource is intrinsically uncertain, a growing interest for solar probabilistic forecasts is observed in the solar research community. In this work, we compare two approaches for the generation of day-ahead solar irradiance probabilistic forecasts. The first class of models termed as deterministic-based models generates probabilistic forecasts from a deterministic value of the irradiance predicted by a Numerical Weather Prediction (NWP) model. The second type of models denoted by ensemble-based models issues probabilistic forecasts through the calibration of an Ensemble Prediction System (EPS) or from information (such as mean and variance) derived from the ensemble. The verification of the probabilistic forecasts is made using a sound framework. A numerical score, the Continuous Ranked Probability Score (CRPS), is used to assess the overall performance of the different models. The decomposition of the CRPS into reliability and resolution provides a further detailed insight into the quality of the probabilistic forecasts. In addition, a new diagnostic tool which evaluates the contribution of the statistical moments of the forecast distributions to the CRPS is proposed. This tool denoted by MC-CRPS allows identifying the characteristics of an ensemble that have an impact on the quality of the probabilistic forecasts. The assessment of the different models is done on several sites experiencing very different climatic conditions. Results show a general superior performance of ensemble-based models as the gain in forecast quality measured by the CRPS ranges from 4% to 16% depending on the site.