Assumption Of Density Research Articles

Uncertainty resulting from constant bulk density assumption when interpreting soil nutrient concentrations

AbstractSoil nutrient concentrations are often expressed as parts per million (ppm) in soil test reports. For incorporation into nutrient management decisions, ppm‐based concentrations have to be converted into pounds per acre, and a conversion factor (multiplier) of 2.0 is typically recommended universally to do so. However, this conversion factor stems from an assumed value of bulk density (ρb) corresponding to silt loam soil and is invariant to any deviation beyond assumed ρb. Here, we quantify and evaluate the potential ramifications of assuming a constant ρb value on calculating soil nitrogen credits. A true dynamic conversion factor that is sensitive to variation in ρb ranges between 1.28 and 2.68 for soils across US cropland. Failure to account for this dynamic conversion factor was shown to result in an underestimation of soil N credits by up to 40%. In addition to spatial variation, management‐induced changes in ρb are also important to incorporate into the conversion factor.

A semi-parametric claims reserving model with monotone splines

AbstractStochastic reserving models used in the insurance industry are usually based on an assumed distribution of claim amounts. Despite their popularity, such models may unavoidably be affected by the misspecification issue given that it is likely that the underlying distribution will be different from that assumed. In this paper, we incorporate monotone splines to ensure the expected monotonically increasing pattern of cumulative development factors (CDFs) to develop a new semi-parametric reserving model that does not require a density assumption. To allow the maximum utilization of available information, we also propose an enhanced sampling approach that greatly increases the size of unbiased CDFs, particularly in later development periods. Based on the enhanced samples, a bootstrap technique is employed in the estimation of monotone splines, from which incurred-but-not-reported (IBNR) reserves and prediction errors can be obtained. Associated technical features, such as the consistency of estimator, are discussed and demonstrated. Our simulation studies suggest that the new model improves the accuracy of IBNR reserving, compared with a range of classic competing models. A real data analysis produces many consistent findings, thus supporting the usefulness of the monotone spline model in actuarial and insurance practice.

Density-based in-orbit collision risk model valid for any impact geometry

Neglecting small fragments in space debris evolutionary models can lead to a significant underestimation of the collision risk for operational satellites. However, when scaling down to the millimetre range, the debris population grows to over a hundred million objects, making deterministic approaches too computationally expensive. On the contrary, probabilistic models provide a more efficient alternative, which however typically works under some simplifying assumptions on the dynamics, limiting their field of applicability. This work proposes an extension of the density-based collision risk models to any orbital dynamics and impact geometry. The impact rate with a target satellite is derived from a multi-dimensional phase space density function in orbital elements, which discretely varies over both phase space and time. The assumption of a bin-wise constant cloud density allows for the analytical transformation of the six-dimensional distribution in orbital elements into the three-dimensional spatial density function, guaranteeing an efficient and accurate evaluation of the fragments flux. The proposed method is applied to the assessment of the collision risk posed by occurred fragmentation events in different orbital regions on a high-risk target object. The effect on the impact rate of the modelling improvements, compared to previous probabilistic formulations, is discussed.

Water dynamics in solutions of linear poly (N-isopropyl acrylamide) studied by 2H NMR field-cycling relaxometry

We use 2H nuclear magnetic resonance to study the dynamics of deuterated water in a solution of linear poly (N-isopropyl acrylamide) (pNIPAM, 4 wt%) across its coil-to-globule transition at a lower critical solubility temperature (LCST) around 32°C. In agreement with previous studies, we find that the 2H spin-lattice (T1) and, in particular, spin-spin (T2) relaxation times abruptly decrease when heating through the LCST, indicating that the polymer collapse causes an emergence of a water fraction with strongly reduced mobility. To quantify the dynamics of this slow water fraction, we exploit the fact that 2H field-cycling relaxometry allows us to measure the spectral density of the water reorientation in a broad frequency range. We find that the slow water fraction is characterised by a broad logarithmic Gaussian distribution of correlation times (σLG = 2.3), which is centred about τLG ≈ 10–9 s near the LCST. Hence, the common assumption of a Debye spectral density does not apply. We argue that a minor water fraction, which is located inside the pNIPAM globules and shows dynamics governed by the disordered polymer matrix, accompanies a major water fraction with bulk-like dynamics above the LCST. The former fraction amounts to about 0.4 water molecules per NIPAM monomer. Several findings indicate fast exchange between these bound and free water fractions on the T1 and T2 time scales.

Constraining the giant radio galaxy population with machine learning and Bayesian inference

Large-scale sky surveys at low frequencies, such as the LOFAR Two-metre Sky Survey (LoTSS), allow for the detection and characterisation of unprecedented numbers of giant radio galaxies (GRGs, or `giants', of at least $l_ p,GRG Mpc $ long). This, in turn, enables us to study giants in a cosmological context. A tantalising prospect of such studies is a measurement of the contribution of giants to cosmic magnetogenesis. However, this measurement requires en masse radio--optical association for well-resolved radio galaxies and a statistical framework to infer GRG population properties. By automating the creation of radio--optical catalogues, we aim to significantly expand the census of known giants. With the resulting sample and a forward model that takes into account selection effects, we aim to constrain their intrinsic length distribution, number density, and lobe volume-filling fraction (VFF) in the Cosmic Web. We combined five existing codes into a single machine learning (ML)--driven pipeline that automates radio source component association and optical host identification for well-resolved radio sources. We created a radio--optical catalogue for the entire LoTSS Data Release 2 (DR2) footprint and subsequently selected all sources that qualify as possible giants. We combined the list of ML pipeline GRG candidates with an existing list of LoTSS DR2 crowd-sourced GRG candidates and visually confirmed or rejected all members of the merged sample. To infer intrinsic GRG properties from GRG observations, we developed further a population-based forward model and constrained its parameters using Bayesian inference. Roughly half of all GRG candidates that our ML pipeline identifies indeed turn out to be giants upon visual inspection, whereas the success rate is 1 in 11 for the previous best giant-finding ML technique in the literature. We confirm $5,647$ previously unknown giants from the crowd-sourced LoTSS DR2 catalogue and $2,597$ previously unknown giants from the ML pipeline. Our confirmations and discoveries bring the total number of known giants to at least $11,585$. Our intrinsic GRG population forward model provides a good fit to the data. The posterior indicates that the projected lengths of giants are consistent with a curved power law probability density function whose initial tail index $ p,GRG changes by $ 0.3$ over the interval up to $l_ p Mpc $. We predict a comoving GRG number density $n_ GRG Mpc $, close to a recent estimate of the number density of luminous non-giant radio galaxies. With the projected length distribution, number density, and additional assumptions, we derive a present-day GRG lobe VFF $ V GRG-CW (z=0) = 1.4 $ in clusters and filaments of the Cosmic Web. We present a state-of-the-art ML-accelerated pipeline for finding giants, whose complex morphologies, arcminute extents, and radio-emitting surroundings pose challenges. Our data analysis suggests that giants are more common than previously thought. More work is needed to make GRG lobe VFF estimates reliable, but tentative results imply that it is possible that magnetic fields once contained in giants pervade a significant ($ gtrsim 10<!PCT!>$) fraction of today's Cosmic Web.

Williamson MHD nanofluid flow with radiation effects through slender cylinder

The aim of this article is to analyze the Williamson nanofluid flow with magnetohydrodynamics (MHD) and radiation effects through the slender cylinder. The mass and heat transfer are analyzed under the different assumptions of viscosity, density, and thermal conductivity. Conservation of momentum and energy are modeled to exhibit the impact of the problem. The Buongiorno model is used for this analysis. The flow of Williamson nanofluid through a slender cylinder along with magnetohydrodynamics and radiation effects with constant viscosity is not studied yet. Which is the novelty of current research work. Flow governing equations are firstly converted into ordinary differential equations and then demonstrated numerically by using MATLAB bvp4c. The effects of dimensionless numbers on the non-dimensional fields are investigated and shown in graphical and tabular form. We concluded that the velocity profile reveals the decreasing behavior for curvature and buoyancy parameter. Radiation parameter and Prandtl number boost the temperature profile. The thermophoresis parameter decreases the concentration profile while the Brownian parameter increases it. Applications of this specific study in various scientific and engineering fields that ultimately benefit humanity's health, technology, and environment.

On seismic gradiometric wave equation inversion for density

SUMMARY Material density remains poorly constrained in seismic imaging problems, yet knowledge of density would provide important insight into physical material properties for the interpretation of subsurface structures. We test the sensitivity to subsurface density contrasts of spatial and temporal gradients of seismic ambient noise wavefields, using wave equation inversion (WEI), a form of seismic gradiometry. Synthetic results for 3-D acoustic media suggest that it is possible to estimate relative density structure with WEI by using a full acoustic formulation for wave propagation and gradiometry. We show that imposing a constant density assumption on the medium can be detrimental to subsurface seismic velocity images. By contrast, the full acoustic formulation allows us to estimate density as an additional material parameter, as well as to improve phase velocity estimates. In 3-D elastic media, severe approximations in the governing wave physics are necessary in order to invert for density using only an array of receivers on the Earth's free surface. It is then not straightforward to isolate the comparatively weak density signal from the influence of phase velocity using gradiometric WEI. However, by using receivers both at the surface and in the shallow subsurface we show that it is possible to estimate density using fully elastic volumetric WEI.

A Unified Phenomenological Model Captures Water Equilibrium and Kinetic Processes in Soil

AbstractSoil water sustains life on Earth, and how to quantify water equilibrium and kinetics in soil remains a challenge for over a century despite significant efforts. For example, various models were proposed to interpret non‐Darcian flow in saturated soils, but none of them can capture the full range of non‐Darcian flow. To unify the different models into one overall framework and improve them if needed, this technical note proposes a theory based on the tempered stable density (TSD) assumption for the soil‐hydraulic property distribution, recognizing that the underlying hydrologic processes all occur in the same, albeit very complex and not measurable at all the relevant scales, soil‐water system. The TSD assumption forms a unified fractional‐derivative equation (FDE) using subordination. Preliminary applications show that simplified FDEs, with proposed hydrological interpretations and TSD distributed properties, effectively capture core equilibrium and kinetic water processes, spanning non‐Darcian flow, water retention, moisture movement, infiltration, and wetting/drying, in the soil‐water system with various degrees and scales of system heterogeneity. Model comparisons and evaluations suggest that the TSD may serve as a unified density for the properties of a broad range of soil‐water systems, driving multi‐rate mass, momentum, and energy equilibrium/kinetic processes often oversimplified by classical models as single‐rate processes.

Review of Smog Chamber Experiments for Secondary Organic Aerosol Formation

In this study, we reviewed smog chamber systems and methodologies used in secondary organic aerosol (SOA) formation studies. Many important chambers across the world have been reviewed, including 18 American, 24 European, and 8 Asian chambers. The characteristics of the chambers (location, reactor size, wall materials, and light sources), measurement systems (popular equipment and working principles), and methodologies (SOA yield calculation and wall-loss correction) are summarized. This review discussed key experimental parameters such as surface-to-volume ratio (S/V), temperature, relative humidity, light intensity, and wall effect that influence the results of the experiment, and how the methodologies have evolved for more accurate simulation of atmospheric processes. In addition, this review identifies the sources of uncertainties in finding SOA yields that are originated from experimental systems and methodologies used in previous studies. The intensity of the installed artificial lights (photolysis rate of NO2 varied from 0.1/min to 0.40/min), SOA density assumption (varied from 1 g/cm3 to 1.45 g/cm3), wall-loss management, and background contaminants were identified as important sources of uncertainty. The methodologies developed in previous studies to minimize those uncertainties are also discussed.

Characterizing Volcanic Ash Density and Its Implications on Settling Dynamics

AbstractVolcanic ash clouds are carefully monitored as they present a significant hazard to humans and aircraft. The primary tool for forecasting the transport of ash from a volcano is dispersion modeling. These models make a number of assumptions about the size, sphericity and density of the ash particles. Few studies have measured the density of ash particles or explored the impact that the assumption of ash density might have on the settling dynamics of ash particles. In this paper, the raw apparent density of 23 samples taken from 15 volcanoes are measured with gas pycnometry, and a negative linear relationship is found between the density and the silica content. For the basaltic ash samples, densities were measured for different particle sizes, showing that the density is approximately constant for particles smaller than 100 μm, beyond which it decreases with size. While this supports the current dispersion model used by the London Volcanic Ash Advisory Centre (VAAC), where the density is held at a constant (2.3 g cm−3), inputting the measured densities into a numerical simulation of settling velocity reveals a primary effect from the silica content changing this constant. The VAAC density overestimates ash removal times by up to 18%. These density variations, including those varying with size beyond 100 μm, also impact short‐range particle‐size distribution measurements and satellite retrievals of ash.

Apparatus for swelling deformation of compacted bentonite utilizing multi-ring moulds for evaluation of dry density and water content distribution

In the geological disposal of high-level radioactive waste, bentonite buffer materials are required to have self-sealing property, which is the ability to fill gaps by swelling deformation between the buffer material and the waste containment vessel or surrounding bedrock. Self-sealing property was evaluated based on the pressure generated after the gap filling under the assumption of constant dry density and water content in the specimen and buffer material. On the other hand, it takes tens to hundreds of years for the transitional period until the buffer contacts with groundwater, fills the gap, and the internal state converges to a uniform state. To assess changes more accurately in the condition of the buffer material, it is necessary to evaluate time courses in the density and water content of the buffer material because of groundwater infiltration and swelling deformation. From these, the authors developed an experimental apparatus for swelling deformation of bentonite, which consists of a mould with 2 mm-thick stacked rings. It is aimed to evaluate the distribution of dry density, water content, and degree of saturation inside the specimen by cutting the specimen with a 2 mm thick ring after the end of experiment. In this paper, results of preliminary experiments to confirm the applicability of this apparatus to swelling deformation test of bentonite was firstly described. Then, experiments were conducted on compacted bentonite specimens using this apparatus, and changes in the dry density and water content distribution inside the specimens were evaluated in terms of swelling deformation behaviour.

Mathematical analysis of Williamson nanofluid flow under radiation effects through slender cylinder

ABSTRACT Current research work demonstrates Williamson nanofluid flow through a slender cylinder. A slender cylinder has not been discussed before with Williamson nanofluid under thermal radiation effects, which is the novelty of the problem. The mass and heat transfer are analyzed under the different assumptions of viscosity, density, and thermal conductivity. Conservation of momentum and energy is modeled to exhibit the impact of the problem. Flow governing equations are initially transformed into ordinary differential equations and then demonstrated numerically by using MATLAB bvp4c. The significance of the present study is to understand how radiation interacts with Williamson’s fluid inside a slender cylinder, which is vital for optimizing heat transfer processes. This knowledge can be applied to design more efficient heat exchangers, boilers, and cooling systems in industries ranging from power generation to manufacturing. The effects of dimensionless numbers on the non-dimensional fields are investigated and shown graphically. The main findings are that the velocity curve reveals the decreasing behavior for curvature and buoyancy parameters. Radiation parameter and Prandtl number boost the temperature profile. The thermophoresis parameter decreases the concentration profile, while the Brownian parameter increases it. The heat transfer curve goes up, while skin friction and Sherwood number go down under the effects of different parameters.

A statistical mechanics framework for constructing nonequilibrium thermodynamic models

Abstract Far-from-equilibrium phenomena are critical to all natural and engineered systems, and essential to biological processes responsible for life. For over a century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs, among many others, laid the foundation for our understanding of equilibrium processes, scientists and engineers have dreamed of an analogous treatment of nonequilibrium systems. But despite tremendous efforts, a universal theory of nonequilibrium behavior akin to equilibrium statistical mechanics and thermodynamics has evaded description. Several methodologies have proved their ability to accurately describe complex nonequilibrium systems at the macroscopic scale, but their accuracy and predictive capacity is predicated on either phenomenological kinetic equations fit to microscopic data or on running concurrent simulations at the particle level. Instead, we provide a novel framework for deriving stand-alone macroscopic thermodynamic models directly from microscopic physics without fitting in overdamped Langevin systems. The only necessary ingredient is a functional form for a parameterized, approximate density of states, in analogy to the assumption of a uniform density of states in the equilibrium microcanonical ensemble. We highlight this framework’s effectiveness by deriving analytical approximations for evolving mechanical and thermodynamic quantities in a model of coiled-coil proteins and double-stranded DNA, thus producing, to the authors’ knowledge, the first derivation of the governing equations for a phase propagating system under general loading conditions without appeal to phenomenology. The generality of our treatment allows for application to any system described by Langevin dynamics with arbitrary interaction energies and external driving, including colloidal macromolecules, hydrogels, and biopolymers.

Criteria of Distribution Transitions in Dispersed Multiphase Systems Based on an Extended Lattice Model.

Dispersed multiphase systems are ubiquitous in biological systems, energy industries, and medical science. The distribution transition of the dispersed phase is critical to the properties and functions of the multiphase systems, among which the agglomeration, adsorption, and extraction processes are of most significance due to their impact on the colloidal stability, interface modification, and particle synthesis. To reveal fundamental correlations between the macroscopic particle distributions and the microscopic interactions, general thermodynamic models of the dispersed multiphase systems are needed. Here, based on Meyer's model, which is restricted to binary isotropic mixtures, we propose a novel extended lattice model that can be applied to multicomponent anisotropic mixtures with interfaces considered. For agglomeration, adsorption, and extraction processes, the approximate free energy differences between the initial distribution and the final distribution are obtained. Based on the minimum free energy principle, the above free energy differences are used to derive three criteria for the prediction of the preferable distribution of the system with given interparticle interaction potentials. While the quasi-uniform number density assumption is still required for all the previous lattice models, the long-range interactions neglected by previous lattice models are incorporated. The validity of our model and criteria is verified by many-body dissipative particle dynamics (mDPD) simulations. By tuning the interaction coefficients between mDPD particles, the simulated distribution transitions for all the agglomeration, adsorption, and extraction cases perfectly match the predictions from the three criteria. The good agreement between the theoretical predictions and the mDPD simulation results shows the great potential of our model for applications in various dispersed multiphase systems.

Monitoring the annual geodetic mass balance of Bordu and Sary-Tor glaciers using UAV data

Abstract The geodetic mass balance of a glacier corresponds to glacier-wide volume changes, converted to mass changes using density assumptions. It is typically calculated by differencing multi-temporal digital elevation models. In this study, we show how the annual geodetic mass balance of a glacier can be derived from uncrewed aerial vehicle (UAV) data. The presented workflow is applied to two small- to medium-sized glaciers in the Kyrgyz Tien Shan (Central Asia): Bordu glacier and Sary-Tor glacier. The obtained geodetic mass balance is compared with the glaciological mass balance derived from a network of ablation stakes and snow pits. A previously calibrated mass-balance model is used to correct for the difference in acquisition dates. The results show that the determined geodetic mass balance matches closely with the glaciological mass balance. Besides, for both glaciers the geodetic mass balance does not seem to be particularly sensitive to the assumptions regarding volume-to-mass conversion. Therefore, our results demonstrate that UAVs can serve as a valuable instrument to quantify the annual geodetic mass balance and to validate the glaciological mass balance. The conventional glaciological mass-balance estimation often relies on interpolation and extrapolation methods, whereas UAVs offer the potential for direct data acquisition over the entire glacier surface.

Adsorption of CO2 on Gold Surfaces: Adsorbate Density Assumption Investigated Using Molecular Dynamics Simulations

Adsorption of CO₂ on Gold Surfaces: Adsorbate Density Assumption Investigated Using Molecular Dynamics Simulations

Flavor composition of neutrinos from choked gamma-ray bursts

The nature of the astrophysical sources responsible for producing the observed high energy neutrinos have yet to be unveiled. Choked gamma-ray bursts (CGRBs) are sources that have been proposed as being capable of generating the flux detected by IceCube, since no accompanying gamma-ray signal is expected from them, as required by observations. We focus on obtaining the neutrino flux and flavor composition corresponding to CGRBs under different assumptions for the target photon density and the magnetic field of the emission region. We consider the injection of both electrons and protons into the internal shocks of CGRBs, and using a steady-state transport equation, we account for all the relevant cooling processes. In particular, we include as a target forpγinteractions the usually adopted background of soft photons, which is a fraction of the thermalized emission originated at the shocked jet head. Additionally, we consider the synchrotron photons emitted by the electrons co-accelerated with the protons at the internal shocks in the jet. We also obtain the distribution of charged pions, kaons, and muons using the transport equation to account for the cooling effects due not only to synchrotron emission but also interactions with the soft photons in the ambient. We integrate the total diffuse flux of neutrinos of different flavors and compute the flavor ratios to be observed on Earth. As a consequence of the losses suffered mainly by pions and muons, we find these ratios to be dependent on the energy: for energies above ∼(105 − 106) GeV (depending on the magnetic field, proton-to-electron ratio, and jet power), we find that the electron flavor ratio decreases and the muon flavor ratio increases, while the tau flavor ratio increases only moderately. Our results are sensitive to the mentioned key physical parameters of the emitting region of CGRBs. Hence, the obtained flavor ratios are to be contrasted with cumulative data from ongoing and future neutrino instruments in order to assess the contribution of these sources to the diffuse flux of astrophysical neutrinos.

Using spectral modeling to break light-curve degeneracies of type II supernovae interacting with circumstellar material

A large fraction of red-supergiant stars seem to be enshrouded by circumstellar material (CSM) at the time of explosion. Relative to explosions in a vacuum, this CSM causes both a luminosity boost at early times as well as the presence of symmetric emission lines with a narrow core and electron-scattering wings typical of type IIn supernovae (SNe). For this study, we performed radiation-hydrodynamics and radiative transfer calculations for a variety of CSM configurations (i.e., compact, extended, and detached) and documented the resulting ejecta and radiation properties. We find that models with a dense, compact, and massive CSM on the order of 0.5 M⊙ can match the early luminosity boost of type II-P SNe but fail to produce type IIn-like spectral signatures (also known as “flash features”). These only arise if the photon mean free path in the CSM is large enough (i.e., if the density is low enough) to allow for a radiative precursor through a long-lived (i.e., a day to a week), radially extended unshocked optically thick CSM. The greater radiative losses and kinetic-energy extraction in this case boost the luminosity even for modest CSM masses – this boost comes with a delay for a detached CSM. The inadequate assumption of high CSM density, in which the shock travels essentially adiabatically, overestimates the CSM mass and associated mass-loss rate. Our simulations also indicate that type IIn-like spectral signatures last as long as there is optically-thick unshocked CSM. Constraining the CSM structure therefore requires a combination of light curves and spectra, rather than photometry alone. We emphasize that for a given total energy, the radiation excess fostered by the presence of CSM comes at the expense of kinetic energy, as evidenced by the disappearance of the fastest ejecta material and the accumulation of mass in a dense shell. Both effects can be constrained from spectra well after the interaction phase.

Near surface sediments introduce low frequency noise into gravity models

3D geologic modeling and mapping often relies on gravity modeling to identify key geologic structures, such as basin depth, fault offset, or fault dip. Such gravity models generally assume either homogeneous or spatially uncorrelated densities within modeled rock bodies and overlying sediments, with average densities typically derived from surface and drill-hole sampling. The noise contributed to the gravity anomaly by these density assumptions is zero in the homogeneous case and typically <200 μGal in the uncorrelated case. Rock bodies and sediments, however, show both a range of densities and spatial correlation of these densities, in both surface and drill-hole samples, and this correlation causes an increase in power in the low frequency content of the resulting gravity anomaly. Spatial correlation of densities can be modeled as a Gaussian random field (GRF), with the random field parameters derived from drill-hole and geologic map data. Data from alluvial fan sediments in southern Nevada indicate correlation lengths of up to 300 m in the vertical dimension and kilometers in the horizontal dimension. The resulting GRF density models show that the noise contributed to the measured gravity anomaly is of low frequency and can be several mGal in amplitude, contradicting the common attribution of lower frequencies to deeper sources. This low-frequency noise increases in power with an increase in sediment thickness. Its presence increases the ambiguity of interpretations of subsurface geologic body shape, such as basin analyses that attempt to quantify concealed basement fault depths, offsets, and dip angles. In the southwestern United States, where basin analyses are important for natural resource applications, such ambiguity increases the uncertainty of subsequent process modeling.

Data-driven non-parametric chance-constrained model predictive control for microgrids energy management using small data batches

This paper presents a stochastic model predictive control approach combined with a time-series forecasting technique to tackle the problem of microgrid energy management in the face of uncertainty. The data-driven non-parametric chance constraint method is used to formulate chance constraints for stochastic model predictive control, while removing the dependency on probability density assumptions of uncertain variables and retaining the linear structure of the resulting optimization problem. The proposed approach is suitable for implementation on systems with limited computational power or limited memory storage, thanks to its simple linear structure and its ability to provide accurate results within pre-defined confidence levels, even when using small data batches. The proposed forecasting and stochastic model predictive control approaches are applied on a numerical example featuring a small grid-connected microgrid with PV generation, a battery storage system, and a non-controllable load, showing the ability to reduce costs by reducing the confidence level, and to satisfy pre-defined confidence levels.