Normal Probability Distribution Function Research Articles

Regional Flood Frequency Analysis Through Some Machine Learning Models in Semi-arid Regions

The machine learning models (MLMs), including support vector regression (SVR), multivariate adaptive regression spline (MARS), boosted regression trees (BRT), and projection pursuit regression (PPR) are compared to traditional method i.e. nonlinear regression (NLR) in regional flood frequency analysis (RFFA). In this study, the Karun and Karkheh watersheds, which is located in the southwestern of Iran, with the same climatic and physiographic conditions are considered. Fifty-four hydrometric stations with a period of 21 years (1993–2013) are selected based on the instructions of U.S. Federal Agencies Bulletin 17 B were applied for RFFA. The generalized normal (GNO) probability distribution function (PDF) is selected by the L-moment method among five PDFs, including the GNO, generalized Pareto (GP), generalized logistic (GL), generalized extreme value (GEV) and Pearson type 3 (P ІІІ) to estimate flood discharge for the expected return periods. Twenty-five predictor variables, such as physiographic, climatologic, geologic, soil and land use variables are extracted. Follow land, maximum 24-h rainfall, mean watershed slope, compactness coefficient, mean and maximum watershed elevation variables are recognized as the appropriate combination of input using gamma test (GT). The overall results indicate that the SVR, PPR, and MARS models in comparison to the NLR and BRT models have a better performance to estimate flood discharge with the expected return periods. Future, the SVR model based on radial basis function (RBF) kernel is chosen as the best model in terms of the mean of the Nash-Sutcliff coefficient (M-Ef) and the mean of relative root mean squared error (M-RMSEr) (i.e. 0.94 and 63.93, respectively) for different return periods.

Robust bi-level risk-based optimal scheduling of microgrid operation against uncertainty

The model introduced in this paper is the first to propose a decentralized robust optimal scheduling of MG operation under uncertainty and risk. The power trading of the MG with the main grid is the first stage variable and power generation of DGs and power charging/discharging of the battery are the second stage variables. The uncertain term of the initial objective function is transformed into a constraint using robust optimization approach. Addressing the Decision Maker’s (DMs) risk aversion level through Conditional Value at Risk (CVaR) leads to a bi-level programming problem using a data-driven approach. The model is then transformed into a robust single-level using Karush–Kahn–Tucker (KKT) conditions. To investigate the effectiveness of the model and its solution methodology, it is applied on a MG. The results clearly demonstrate the robustness of the model and indicate a strong almost linear relationship between cost and the DMs various levels of risk aversion. The analysis also outlines original characterization of the cost and the MGs behavior using three well-known goodness-of-fit tests on various Probability Distribution Functions (PDFs), Beta, Gumbel Max, Normal, Weibull, and Cauchy. The Gumbel Max and Normal PDFs, respectively, exhibit the most promising goodness-of-fit for the cost, while the power purchased from the grid are well fitted by Weibull, Beta, and Normal PDFs, respectively. At the same time, the power sold to the grid is well fitted by the Cauchy PDF.

Monthly Flow Duration Curve Model for Ungauged River Basins

Flow duration curve (FDC) is widely used in hydrology to assess streamflow in a river basin. In this study, a simple FDC model is developed for monthly streamflow data. The model consists of several steps including the nondimensionalization and then normalization in case the monthly streamflow data do not fit the normal probability distribution function. The normalized quantiles are calculated after which a back transformation is applied to the normalized quantiles to return back to the original dimensional streamflow data. In order to calculate annual streamflow of the river basin, an empirical regression equation is proposed using the drainage area and the annual total precipitation only as the input. As the final step of the model, dimensional quantiles of FDC are calculated. Ceyhan River basin in southern Turkey is chosen for the case study. Forty-two streamflow gauging stations are considered; two thirds of the gauging stations are used for the model calibration, and one third for validation. The modeled FDCs are compared to the observation and assessed with a number of performance metrics. They are found similar to the observed ones with a relatively good performance; they are good in the mid and high flow parts particularly while the low flow part of FDCs might require further detailed analysis.

Analysis of corrosion performance of rebar in Red mud blended concrete

In this paper, electrochemical technique was employed to evaluate the performance of Red mud (RM) on the reinforcement corrosion. This paper also reports the effect of chloride, sulphate and the combined chloride-sulphate solutions on the corrosivity of rebar embedded in Ordinary Portland Cement (OPC) and RM blended concrete (95% OPC & 5% RM) slab specimens. The corrosion performance was evaluated by Linear polarisation resistance sweep test on the reinforced slab specimens made with mix water contaminated with 5% NaCl, 2% MgSO4 and the combination of both salts. The corrosion performance was monitored for every 30 days over a period of 180 days, in terms of corrosion rate values. In order to investigate the overall performance of experimental test results they were analysed as per the specifications given by ASTM G16-13. The analysis of the corrosion data includes the descriptive statistics of the Normal, Weibull lognormal, and Smallest Extreme Value probability distribution functions (Pdfs) and the test of fit significance by the Anderson-Darling (AD) goodness of fit statistics. In addition, the analysis of variance was also carried out to determine the influence of each variable on the corrosion data. Further, the analysis of variance is also carried out to evaluate the influence of cementitious material and type of salt on corrosion data.

Representing financial data streams in digital simulations to support data flow design for a future Digital Twin

Abstract Discrete Event Simulations may be used to forecast detailed production system behaviour under future conditions and support better informed decision making. However, data beyond production metrics, e.g. financial information, is also necessary for most significant decisions. Herein a modelling approach is proposed and demonstrated based on simulating both production and financial transactions. The contribution of this paper is to represent the base production events as well as additional events which trigger financial transactions. The method is demonstrated for an idealised production business, employing Discrete Event Simulation and examining the impact of system labour arrangements on the business cash flow with a variable demand based on a normal probability distribution function (with distinct means and variances for each product). Based on the financial transactions a representation of a detail commercial scenario may be modelled alongside the production events. The presented studies identify the potential sensitivity of analysis to the volume and timing of transactions on accumulated accounting metrics such as profit. Additionally the paper illustrates how such a modelling approach can be used to characterise the system and assess specific control strategies when both production and finance data streams are available.

Online control of combine reel height in presence of farm roughness using an on-off controller

Adjusting the harvesting combine reel height in presence of the farm rough surface reduces the harvest losses, prevents possible damage to the headland and increases gravel harvest. Regarding the importance of the issue, in this study, the dynamics of the reel height control system of the John Deere1055 combine was modeled and the controllability and observability of the obtained transfer function is determined in state space. After that, with the aim of cutting the stem from a suitable point, an on-off controller is designed to keep the reel height from the ground in a permitted range. At this stage, a reduced model system is used as an approximated system to design a controller. The designed controller is simulated in MATLAB. In this simulation, the roughness of the ground was considered as the control loop noise with the normal probability distribution function. The results confirm that the reel height change is kept in the specified range against the farm roughness. The laboratory sample was also constructed using an independent excitation circuit DC motor, an optical distance sensor, and a micro controller. The experimental test shows that the height control system responses to changes in surface height at an appropriate speed without any oscillation in range.

Optimal bidding strategy with the inclusion of wind power supplier in an emerging power market

In polistic electricity market structure, each power producers can maximise its profit through bidding strategy. Also, with the advent of renewable generation mostly wind has shaped a new prospect in the bidding process. Although, the wind power output uncertainty, wind power suppliers facing an inevitable uncertainty problem in an emerging power market. To alleviate the adverse impact of this uncertainty on wind power bidding, Weibull distribution is used to model wind power scenarios and the forward-reduction algorithm is utilised to reduce scenarios. Furthermore, an overestimation and underestimation cost function is modelled to measure the deviation of wind power output. The bidding strategy with the inclusion of wind power is proposed in this study to maximise profit. However, the uncertainty of rival's behaviour affects the bidding process, which minimised by utilising the normal probability distribution function. The proposed problem is tested on the IEEE standard 30-bus and 57-bus systems and solved by the gravitational search algorithm (GSA). The results are obtained without and with wind power and shows that the effects of wind power on market clearing price and bidding strategy. Moreover, GSA gives higher market clearing price and net profit as compared with particle swarm optimisation and genetic algorithm.

Probabilistic System Availability Model and Prevailing Factors for 33kV Primary Power Distribution Feeders in Nigeria

Outage and complimentary load flow analyses of a selected 33kV primary distribution network in Ughelli Delta State, Nigeria, were carried out to depict feeder availability obtainable in the country. Outage and load flow analyses were carried out using parametric statistics and Newton-Raphson computation algorithm respectively. The results provide model distribution of monthly feeder availability, contributions of outage factors and normal load flow values. It was concluded that: monthly feeder availability represented using normal probability distribution function of; shows that overcurrent and earth faults contribute to outage duration, taking 5% to 12% of the month outage duration respectively; coupled with significant line losses and weak voltage profile. Recommendations proposed include: replacement of weak or failed elements and insulations in the systems with new and modern ones to mitigate especially earth faults in the short term; planning for immediate increase in system capacity to accommodate the power demand effectively for present and forecast periods to eliminate load shedding, overload events, and high technical losses; maintenance practice at the planning and execution stages to mitigate failure events and repair time. Keywords – fault, outage, probability, load flow, availability, distribution network.

Normality of Post-Crack Performance Data for Fiber-Reinforced Concrete

Abstract With the increasing use of fiber-reinforced concrete (FRC) for civil infrastructure projects, challenges have become apparent in quality control (QC) testing for this material. The most common problem is high variability, but a secondary problem is identifying the most appropriate probability distribution function (PDF) for post-crack performance data. This article has examined the second of these challenges. It is widely assumed that the normal PDF most reliably describes the distribution of outcomes for many stochastic processes. Most of the conventions underlying probabilistic modeling of engineering processes are therefore predicated upon a normal distribution of material properties. However, evidence exists that other functions, such as the Weibull, lognormal, or gamma distribution functions, may be more appropriate representations of the distribution of test results for some post-crack performance parameters related to FRC, such as fatigue life. The present investigation has examined a large body of QC data to identify the most reliable function describing the probability distribution of post-crack performance of FRC as obtained from ASTM C1550, Standard Test Method for Flexural Toughness of Fiber Reinforced Concrete (Using Centrally Loaded Round Panel), and ASTM C1609/C1609M, Standard Test Method for Flexural Performance of Fiber-Reinforced Concrete (Using Beam with Third-Point Loading). The results demonstrate that the normal and lognormal functions model the distribution of post-crack performance data obtained for FRC equally well. However, the normal function produces the best fit for data assembled from multiple specimen sets, thereby confirming the validity of the central limit theorem. Moreover, a derivative of the normal function also reliably predicts the distribution of variances for data sets comprising a small number of replicate test results.

Irrigation distribution uniformity analysis on a lateral-move irrigation system

Probability distribution functions (PDF) have been used for decades to model the applied water depth by irrigation equipment; however, there are few studies that evaluate the PDF goodness-of-fit to mechanical lateral-move irrigation system water application, especially with oscillating plate sprinklers. The objectives of this study with a lateral-move irrigation system in the field were as follows: (1) to evaluate the normal, uniform, Weibull, gamma and logistic PDFs’ goodness-of-fit to represent the applied water depth ; (2) to evaluate the influence of wind on the coefficient of variation (CV) of the applied water depth; (3) to evaluate the relationship of the distribution coefficient (H), the availability factor (F), the average deficit (Da) and the water storage efficiency (E) as a function of wind speed. Twenty-nine in-field tests were carried out at the Empresa Brasileira de Pesquisa Agropecuaria (Embrapa). The PDFs’ goodness-of-fit to the applied water depths by the irrigation system was based on the Kolmogorov–Smirnov test at a 5% significance level. The normal, uniform, gamma, Weibull and logistic PDFs fitted 21, 8, 19, 26, and 25 of the 29 in-field tests, respectively; however, application of the Weibull PDF is complicated and the normal distribution is suggested instead. The logistic PDF had good fit and warrants further study. In addition, a negative linear relationship (R2 = 0.73) was observed between the wind speed and the coefficient of variation of the applied water depth by the irrigation system, as well as the distribution coefficient (H), the availability factor (F), and water storage efficiency (E), while the average deficit (Da) showed a positive linear relationship with the wind speed.

An analytical probabilistic analysis of slopes based on limit equilibrium methods

Probabilistic analysis of slopes has been used as an effective tool to evaluate uncertainty that is so prevalent in variables. In this paper, the jointly distributed random variables (JDRV) method is used as an analytical method to compare the reliability of four widely used limit equilibrium methods for slope stability analysis. These methods include the simplified Bishop, simplified Janbu, Morgenstern–Price, and Spencer’s methods. The selected stochastic parameters are angle of shearing resistance (φ), cohesion intercept (c), and unit weight (γ) of soil, which are modeled using a truncated normal probability distribution function. Geometric parameters such as height and angle of the slope relative to the horizontal are regarded as constant parameters. For reliability assessment, the reliability indices of the limit equilibrium methods for the critical surface with minimum factor of safety are determined by the particle swarm optimization (PSO) technique. It is shown that, among the assessed methods, the Janbu and Bishop methods are those with upper and lower probabilities of failure, respectively, in two conditions with and without considering cross correlation between c and φ.

Data on triethylenetetramine effect on steel-rebar corrosion-rate in concrete immersed in 0.5 M H 2 SO 4

Abstract In this article, the dataset on the effect of different triethylenetetramine (TETA: C6H18N4) concentrations on the corrosion-rate of steel-rebar embedment in steel-reinforced concrete immersed in 0.5 M H2SO4 (for simulating industrial/microbial environment) is presented. The corrosion test-data was obtained from weekly monitoring, over seven weeks of steel-reinforced concrete immersion, using linear-polarization-resistance (LPR) measuring instrument. The dataset and its requisite analyses, as per ASTM G16 —13, are presented in graphs and tables. The analyses detailed include descriptive statistics of the Normal, Gumbel and Weibull probability distribution functions (pdf's), and tests-of-fit significance by the Kolmogorov-Smirnov goodness-of-fit statistics. The detailed information from this corrosion tests dataset is useful for further research on the inhibition mechanism and effectiveness of the triethylenetetramine chemical on the corrosion-protection of reinforcing-steel material in steel-reinforced concrete designed for the industrial/microbial service-environment.

Stochastic multi-objective energy management in residential microgrids with combined cooling, heating, and power units considering battery energy storage systems and plug-in hybrid electric vehicles

Abstract Governmental incentives to use clean energy, concerns about high and rising prices of fossil fuels, and environmental issues are the most important motivations for adding distributed energy resources to conventional power systems. In these circumstances, new technologies such as combined cooling, heating, and power systems, energy storage systems including battery and thermal storages, and plug-in hybrid electric vehicles not only can improve system efficiency but also can reduce operation and investment cost as well as emissions. In this paper, a residential microgrid consisting of combined cooling, heating and power, plug-in hybrid electric vehicles, photovoltaic unit, and battery energy storage systems is modeled to obtain the optimal scheduling state of these units by taking into account the uncertainty of distributed energy resources. To achieve this goal, a scenario-based method is used to model uncertainties of electrical market price, electrical and thermal demand, and solar irradiance using Normal, Weibull, and Beta probability distribution functions, respectively. The scenario tree is used to generate scenarios and representative scenarios are selected with scenario reduction techniques. The proposed problem is formulated as a mixed-integer nonlinear programming to minimize objectives of operation cost and total emissions. This multi-objective problem is solved by the Augmented ɛ-constraint method and the best solution on the Pareto front set is determined by a fuzzy approach. In order to maintain system reliability during on-peak periods, the time-of-use demand response program is considered to promote a change in energy consumption patterns. Finally, plug-in hybrid electric vehicles are considered as electrical energy storage instead of batteries in optimal energy and emissions management. Proposed energy management is applied on a residential microgrid and the effectiveness of the model is verified. It is found that uncertainties increase the cost by 12.2%; however, energy storage systems reduce cost and emissions by 4.7% and 23%, respectively.

Data Analysis in Nonstationary State

Climate change impact must be taken into account in any water resources planning management studies, because it does not allow future occurrences to be repeated as a replicate of the past. The stationarity is no longer valid, because the climate change plays significant role on ascending or descending trend components in any hydro-meteorological events such as temperature, precipitation, evaporation, runoff and discharge. The identification of trends can be represented by well-known methodologies, such as the Mann-Kendall and sequential Mann-Kendall. These methodologies require restrictive assumptions such as data length, serial independence and Gaussian (normal) probability distribution function (PDF). On the other hand, Innovative trend analysis (ITA) method proposed by Şen is helpful to identify even visually with direct interpretations without restrictive assumptions. The PDF or cumulative distribution function (CDF) is effective tool for risk level determination but it cannot tell anything about the trend in a given hydro-meteorological data. The PDF (CDF) does not yield any clue about the trend possibility, and hence, it’s alone use in any water resources structure design may lead to erroneous planning studies. In this study, nonstationary nature of a given monthly hydro-meteorological data is examined by trend determination procedures. For the application, monthly averages of maximum daily temperatures are used on Oxford station, UK. It is observed that the temperature values of each month have a positive trend and the nonstationary empirical cumulative frequency curves on first half group match better all data group than the stationary state.

A Bayesian Multiresolution Approach for Noise Removal in Medical Magnetic Resonance Images

Abstract A Bayesian approach using wavelet coefficient modeling is proposed for de-noising additive white Gaussian noise in medical magnetic resonance imaging (MRI). In a parallel acquisition process, the magnetic resonance image is affected by white Gaussian noise, which is additive in nature. A normal inverse Gaussian probability distribution function is taken for modeling the wavelet coefficients. A Bayesian approach is implemented for filtering the noisy wavelet coefficients. The maximum likelihood estimator and median absolute deviation estimator are used to find the signal parameters, signal variances, and noise variances of the distribution. The minimum mean square error estimator is used for estimating the true wavelet coefficients. The proposed method is simulated on MRI. Performance and image quality parameters show that the proposed method has the capability to reduce the noise more effectively than other state-of-the-art methods. The proposed method provides 8.83%, 2.02%, 6.61%, and 30.74% improvement in peak signal-to-noise ratio, structure similarity index, Pratt’s figure of merit, and Bhattacharyya coefficient, respectively, over existing well-accepted methods. The effectiveness of the proposed method is evaluated by using the mean squared difference (MSD) parameter. MSD shows the degree of dissimilarity and is 0.000324 for the proposed method, which is less than that of the other existing methods and proves the effectiveness of the proposed method. Experimental results show that the proposed method is capable of achieving better signal-to-noise ratio performance than other tested de-noising methods.

Approximations to the Normal Probability Distribution Function using Operators of Continuous-valued Logic

In this study, novel approximation methods to the standard normal probability distribution function are introduced. The techniques presented are founded on applications of certain operators of continuous-valued logic. It is demonstrated here that application of the averaging Dombi conjunction operator to two symmetric Sigmoid fuzzy membership functions results in a function that is identical with Tocher’s approximation to the standard normal probability distribution function. Next, an approximation connected with a unary fuzzy modifier operator is discussed. Namely, the so-called Kappa function is applied for constructing a novel probability distribution function. It is shown here that the asymptotic Kappa function is just the Sigmoid function and the proposed Quasi Logistic probability distribution function can be utilized to approximate the standard normal probability distribution function. It is also explained how the new probability distribution function is connected with the generator function of Dombi operators. The proposed approximation formula is very simple as it has only one constant parameter. It does not include any exponential term, but has a good approximation accuracy and fulfills certain requirements that only a few of the known approximation formulas do.

Aridity and Risk Calculations in Saudi Arabian Wadis: Wadi Fatimah Case

BackgroundInfrequent rainfall and rare flash floods are the primary sources of water in semi-arid and arid regions and are the key aspects in the management of water supply. In these regions, rainfall varies temporally and spatially with infrequent high intensities and sporadic occurrences, making it impossible to systematically record measurements as a proper time series.PurposeThe main purpose of this paper is to present temporal (risk assessment) and spatial (isohyet maps) methods for the evaluation of monthly rainfall water volumes and infiltration rates. Additionally, an aridity index based on rainfall and temperature measurements is proposed for use in semi-arid and arid regions.MethodThe limited data available for semi-arid and arid regions are treated using effective probabilistic and statistical methods. Calculation of the volume of rainfall is achieved through a simple hydrological procedure which involves examining monthly isohyet maps based on a set of rainfall measurements from weather stations in scattered locations. Calculation of rainfall and infiltration risks are performed according to the logarithmic normal probability distribution function. Furthermore, the potential monthly rainfall and infiltration rates are calculated using a set of pre-selected risk percentages.Results Aridity index, rainfall volume, and risk values are presented for Wadi Fatimah, which is located in the western Saudi Arabia. Potential rainfall and infiltration rates are calculated on a monthly scale using a set of pre-selected risk percentages. These simple calculations provide a sound basis for future studies on groundwater resource storage strategy, operation, and management.ConclusionsWater management studies can be best advanced through simple but effective assessment methods, as presented in this paper with specific applications to the Wadi Fatimah drainage basin that joins the Red Sea.

Multiobjective Symbiotic Search Algorithm Approaches for Electromagnetic Optimization

Optimization metaheuristics is a powerful way to deal with many electromagnetic optimization problems. Their main advantages are that they don't require gradient computation, they are more likely to give a global optimum solution and have a higher degree of exploration and exploitation ability. Recently, the symbiotic organisms search (SOS) algorithm was proposed to solve single-objective optimization problems. SOS mimics the symbiotic relationship among the living beings. In order to extend the classical mono-objective SOS algorithm approach, this paper proposes a new multiobjective SOS (MOSOS) based on nondominance and crowding distance criterion. Furthermore, an improved MOSOS (IMOSOS) based on normal (Gaussian) probability distribution function also was proposed and evaluated. Results on a multiobjective constrained brushless direct current (dc) motor design show that the MOSOS and IMOSOS present promising performance.

Methodology to Backcalculate Individual Speed Data Originally Aggregated by Road Detectors

Highway and traffic engineers collect vehicular speed data with detectors based on a variety of fixed and mobile device technologies, to support analysis and design activities. Most acquisition units aggregate speed data into speed classes for ease of management and storage. An unfortunate result of this practice is a significant loss of content associated with individual speed data. Moreover, the use of individual speeds is often necessary to support road safety analysis and speed management decisions. For bridging of this gap, this paper introduces an algorithm that disaggregates speed data collected with automatic road detectors that can measure speed frequency only in intervals. The objective is to obtain backcalculated individual speeds that operate with continuous distribution functions rather than discrete ones. This information allows the derivation of more robust, basic descriptive measures (average, variance, and percentiles) according to normal, lognormal, and gamma probability distribution functions. Therefore, the information produced is more useful than that calculated from standard aggregated speed reports. In this investigation, individual speed data collected from video cameras were used to derive reference distributions and descriptive measures on the same road sections where inductive double-loop detectors were installed. Comparisons of the backcalculated individual speeds and those collected from video cameras support the validity of the proposed algorithm.

Assessing wind uncertainty impact on short term operation scheduling of coordinated energy storage systems and thermal units

Renewable resources, especially wind power, are widely integrated into the power systems nowadays. Managing uncertainty of the large scale wind power is often known as one of the most challenging issues in the power system operation scheduling. Additionally, energy storage systems (ESSs) have been widely investigated in the power systems owing to their valuable applications, especially renewable energy smoothing and time shift. In this paper, a stochastic unit commitment (UC) model is proposed to assess the impact of the wind uncertainty impact on ESSs and thermal units schedule in UC problem. Wind uncertainty is modeled based on the two measures. First, the wind penetration level is changed with respect to the basic level. Second, the wind forecasting error is modeled through a normal probability distribution function with different variances. The ESSs are modeled based on several technical characteristics and optimally scheduled considering different levels of the wind penetration and forecasting accuracies. The proposed formulation is a stochastic mixed integer linear programming (SMILP) and solved using GAMS software. Simulation results demonstrate that the wind uncertainty have a considerable impact on operation cost and ESSs schedule while proposed optimum storage scheduling through the stochastic programming will reduce the daily operational cost considerably.