Normal Probability Distribution Function Research Articles

Assessing Wet and Dry Periods Using Standardized Precipitation Index Fractal (SPIF) and Polygons: A Novel Approach

In the open literature, there are numerous studies on the normal and extreme (flood and drought) behavior of wet and dry periods based on the understanding of the standard precipitation index (SPI), which provides a series of categorizations by considering the standard normal (Gaussian) probability distribution function (PDF). The numerical meaning of each categorization assessment is quite lacking in terms of future predictions of wet and dry period duration based on historical records. This paper presents a new approach for calculating possible formations of future wet and dry period durations based on historical records through an effective fractal geometric forecasting approach. The essence of the proposed methodology is based on the number of dry periods (steps) of non-overlapping monthly duration along consecutive broken line paths in the SPI classification for wet and dry period durations. It has been observed that the plot of periods on double logarithmic paper falls along a straight line against the number of such periods, implying a power function, which is the essence of fractal geometry. Extending the empirically derived straight line provides the number of periods that may occur in the future over a range of SPI levels. This methodology is referred to as SPI fractal (SPIF), and the classic SPI classification is converted into SPIF wet and dry polygons, which provide additional information about the drought period number within a valid polygonal area, compared to the classic SPI results. The wet and dry period features of any hydro-meteorology time series are constrained in SPIF polygons. The application of the methodology was carried out on monthly rainfall records on the European side of the Istanbul Florya meteorological station in Turkey.

Asymmetrical fixed-bed breakthrough curve modelling: Comparing simplistic, log-modified, fractal-like, and probability distribution function models

Column adsorption studies frequently adopt classical breakthrough curve models such as semi-mechanistic Bohart-Adams and Thomas models. They are logistic functions which simulate symmetrically sigmoidal curves centrosymmetric at 50% breakthrough; consequently, they fail and estimate erroneous results when dealing with asymmetric breakthrough curves. This work analyses and compares the trend of how log-modified, fractal-like modified, and probability distribution function models simulate asymmetric breakthrough curves with varied extents of tailing. We generated the asymmetric breakthrough curves by varying the bed depth: 6 mm, 9 mm, and 12 mm on a glass microcolumn of 100 mm length and 5.5 mm inner diameter at a fixed flow rate of 1 mL/min using 10 mg/L bisphenol-A as adsorbate and polyaniline as adsorbent. The asymmetrical curves were a consequence of polyaniline’s low specific surface area of 6.96 m²/g, heterogeneous particle size distribution, slow intraparticle and surface diffusion. The log-Bohart-Adams and fractal-like Bohart-Adams models perfectly fit the curves regardless of symmetry although log-Bohart-Adams performed subpar with increased tailing. The fractal-like Bohart-Adams model accurately estimated the breakthrough times of 13 min, 26 min, and 90 min for the 6 mm, 9 mm, and 12 mm bed depths with a critical bed depth of 5.65 mm. The normal and Gompertz probability distribution functions were unable to adapt to asymmetry of the curves owing to their fixed inflection points at c/cc = 0.5 and 0.37 respectively. Although the Weibull, log-normal, and log-Gompertz functions have floating inflection points, only log-Gompertz function had a satisfactory fit (Radj2>0.98) regardless of symmetry.

Laterally loaded pile response in spatially variable soil

The spatial variability of soil has long been recognised, yet there is a dearth of research on the response of piles under lateral loading that considers this variability, especially for the lateral load transfer (P-y) curve in spatially variable clay. To address this gap, the study presents a three-dimensional random finite element analysis utilising a non-stationary random field to model soil variability with a linearly increasing trend. The results reveal that the spatial variability significantly affects the lateral capacity of the pile, emphasising the importance of accounting for this variability in pile design. Additionally, the study finds that while the spatial variability affects the failure mechanism, the mechanism remains comparable to that observed in deterministic analyses. The spatial variability has a substantial effect on ultimate soil resistance pressure, which can be fitted using a normal probability distribution function. An adjust factor is suggested to calculate the characteristic value, and from this characteristic value, the fractiles of ultimate soil resistance pressure can be calculated. This study provides an evaluation method for the spatial variability effect of soil, facilitating a cost-effective design of pile foundation.

Integrating renewable energy and V2G uncertainty into optimal power flow: A gradient bald eagle search optimization algorithm with local escaping operator

AbstractHere, a new approach is proposed for solving the optimal power flow (OPF) problem in transmission networks using a Gradient Bald Eagle Search Algorithm (GBES) with a Local Escaping Operator (LEO). The method takes into account uncertainty of the renewable energy sources (wind energy and photovoltaic systems) and Vehicle‐to‐Grid (V2G) in the stochastic OPF problem. To improve the efficiency of the proposed technique and enhance its local exploitation capability, the LEO method's selection features are utilized. Monte Carlo methods are employed to estimate the generation costs of the renewable sources and PEVs and study their feasibility. The uncertainty of the renewable sources and PEVs is represented by Weibull, lognormal, and normal probability distribution functions (PDFs). The GBES approach is experimentally compared with well‐known meta‐heuristics using twenty‐three different test functions, and the results indicate its superiority over BES and other recently developed algorithms. Furthermore, the proposed method's effectiveness is evaluated using IEEE 30‐bus test system under various scenarios, and the simulation results demonstrate that it can effectively address OPF issues considering renewable energy sources and V2G, providing superior optimal solutions compared to other algorithms.

Control charts for processes with variable mean

<p>Despite of strong ability and performance of control charts to control and monitor processes, they have some problems in practical applications. If control chart’s limits are not properly designed then we receive false alarms. For example, several observations may be outside the control limits when the mean of process is in-control. Not considering the variation of the process mean at each sampling time may lead to this error. The process may be adjusted at specific mean but different working conditions and different operators may change mean of the process and it may have a small deviation from its predetermined value and this problem can lead to wrong implementation of control charts. In this paper, the effects of variable mean on control charts are analyzed. It is assumed that the mean of observation varies over time but its probability distribution is normal probability distribution function. It is observed that long-term process mean control chart generates false alarms.<strong></strong></p>

Machine Learning Classification and Prediction of Wind Estimation Using Artificial Intelligence Techniques and Normal PDF

Estimating wind energy at a specific wind site depends on how well the real wind data in that area can be represented using an appropriate distribution function. In fact, wind sites differ in the extent to which their wind data can be represented from one region to another, despite the widespread use of the Weibull function in representing the wind speed in various wind locations in the world. In this study, a new probability distribution model (normal PDF) was tested to implement wind speed at several wind locations in Jordan. The results show high compatibility between this model and the wind resources in Jordan. Therefore, this model was used to estimate the values of the wind energy and the extracted energy of wind turbines compared to those obtained by the Weibull PDF. Several artificial intelligence techniques were used (GA, BFOA, SA, and a neuro-fuzzy method) to estimate and predict the parameters of both the normal and Weibull PDFs that were reflected in conjunction with the actual observed data of wind probabilities. Afterward, the goodness of fit was decided with the aid of two performance indicators (RMSE and MAE). Surprisingly, in this study, the normal probability distribution function (PDF) outstripped the Weibull PDF, and interestingly, BFOA and SA were the most accurate methods. In the last stage, machine learning was used to classify and predict the error level between the actual probability and the estimated probability based on the trained and tested data of the PDF parameters. The proposed novel methodology aims to predict the most accurate parameters, as the subsequent energy calculation phases of wind depend on the proper selection of these parameters. Hence, 24 classifier algorithms were used in this study. The medium tree classifier shows the best performance from the accuracy and training time points of view, while the ensemble-boosted trees classifier shows poor performance regarding providing correct predictions.

A new hybrid evolutionary algorithm for multi-objective optimal power flow in an integrated WE, PV, and PEV power system

• A novel hybrid multi-objective evolutionary algorithm (MOEA) was developed based on the decomposition and invasive weed optimization (IWO) techniques for solving the MOOPF problem. • The conventional MOOPF problem was modified by taking stochastic WE, PV, and PEV powers into consideration. • Modeling the stochastic nature of WE, PV, and PEV sources was done using probability distribution functions (PDFs) and calculating its uncertainty cost was done with Monte Carlo simulations. • Using an effective constraint handling method (CHM) known as the superiority of feasible solution (SF) was done to address difficult constraints. • The one-way analysis of variance (ANOVA) test was deployed to give a certain level of confidence to our study. This study proposes a novel hybrid multi-objective evolutionary algorithm (MOEA) based on decomposition and invasive weed optimization (IWO) algorithm for the optimal power flow (OPF) problem in transmission networks. The conventional OPF is modified as a stochastic OPF with the integration of WE, PV, and PEV systems uncertainty. This paper proposes a new constraint-handling method (CHM) that adds the penalty adaptively and avoids parameter reliance on penalty computation. The selection features in the IWO method are deployed to improve the diversity of the proposed method. The OPF problem is represented as a multi-objective optimization (MOO) problem using four objectives: generation cost, emission, power loss, and voltage variation. The generation costs of WE, PV, and PEV sources are evaluated using Monte Carlo simulations to mitigate the whole cost and the influence of intermittency of these systems is investigated in terms of affordability and feasibility. Weibull, lognormal, and normal probability distribution functions (PDFs) are employed to define the uncertainty of WE, PV, and PEV sources respectively. The suggested method's viability is evaluated on IEEE 57 and IEEE 118- bus systems under all conceivable scenarios. In addition, one-way ANOVA test, a statistical approach, is used to evaluate the superiority of the suggested algorithm.

A method for using monthly average temperatures in phenology models for grapevine (<i>Vitis vinifera</i> L.)

In recent years, there have been increasing efforts to link phenology models with seasonal climate predictions in so-called Decision Support Systems (DSS) to tailor crop management strategies. However, temporal discrepancies between phenology models with temperature data gathered on a daily basis and seasonal forecasting systems providing predictability on monthly scales have limited their use. In this work, we present a novel methodology to use monthly average temperature data in phenology models. Briefly stated, we modelled the timing of the appearance of specific grapevine phenological phases using monthly average temperatures. To do so, we computed the cumulative thermal time (Sf ) and the number of effective days per month (effd). The effd is the number of days in a month on which temperatures would be above the minimum value for development (Tb). The calculation of effd is obtained from a normal probability distribution function derived from historical weather records. We tested the methodology on four experimental plots located in different European countries with contrasting weather conditions and for four different grapevine cultivars. The root mean square deviation (RMSD) ranged from 4 to 7 days for all the phenological phases considered, at all the different sites, and for all the cultivars. Furthermore, the bias of observed vs predicted comparisons was not significantly different when using either monthly mean or daily temperature values to model phenology. This new methodology, therefore, provides an easy and robust way to incorporate monthly temperature data into grapevine phenology models.

Stochastic-Metaheuristic Model for Multi-Criteria Allocation of Wind Energy Resources in Distribution Network Using Improved Equilibrium Optimization Algorithm

In this paper, a stochastic-meta-heuristic model (SMM) for multi-criteria allocation of wind turbines (WT) in a distribution network is performed for minimizing the power losses, enhancing voltage profile and stability, and enhancing network reliability defined as energy not-supplied cost (ENSC) incorporating uncertainty of resource production and network demand. The proposed methodology has been implemented using the SMM, considering the uncertainty modeling of WT generation with Weibull probability distribution function (PDF) and load demand based on the normal PDF and using a new meta-heuristic method named the improved equilibrium optimization algorithm (IEOA). The traditional equilibrium optimization algorithm (EOA) is modeled by the simple dynamic equilibrium of the mass with proper composition in a control volume in which the nonlinear inertia weight reduction strategy is applied to improve the global search capability of the algorithm and prevent premature convergence. First, the problem is implemented without considering the uncertainty as a deterministic meta-heuristic model (DMM), and then the SMM is implemented considering the uncertainties. The results of DMM reveal the better capability of the IEOA method in achieving the lowest losses and the better voltage profile and stability and the higher level of the reliability in comparison with conventional EOA, particle swarm optimization (PSO), manta ray foraging optimization (MRFO) and spotted hyena optimization (SHO). The results show that in the DMM solving using the IEOA, traditional EOA, PSO, MRFO, and SHO, the ENSC is reduced from $3223.5 for the base network to $632.05, $636.90, $638.14, $635.67, and $636.18, respectively, and the losses decreased from 202.68 kW to 79.54 kW, 80.32 kW, 80.60 kW, 80.05 kW and 80.22 kW, respectively, while the network minimum voltage increased from 0.91308 p.u to 0.9588 p.u, 0.9585 p.u, 0.9584 p.u, 0.9586 p.u, and 0.9586 p.u, respectively, and the VSI improved from 26.28 p.u to 30.05 p.u, 30.03 p.u, 30.03 p.u, 30.04 p.u and 30.04 p.u; respectively. The results of the SMM showed that incorporating uncertainties increases the losses, weakens the voltage profile and stability and also reduces the network reliability. Compared to the DMM, the SMM-based problem is robust to prediction errors caused by uncertainties. Therefore, SMM based on existing uncertainties can lead to correct decision-making in the conditions of inherent-probabilistic changes in resource generation and load demand by the network operator.

Innovative standard degree-day indicator (SDI) concept and application for monthly energy consumption control

In practice, monthly temperature records play a major role calculating the heating and cooling degree-days of the air conditioner management in buildings. Classically, heating and cooling degree calculations are based on the comparison of outdoor temperature measurements with a standard level called the base temperature. Truncating the temperature series at a constant base temperature causes heating (deficits) and cooling (surpluses) degrees. Since outdoor temperature records are of random character, future predictions with probabilistic and statistical methods are convenient means for calculations. All previous methodologies in the literature are based on crisp logic with simple mathematical calculations. This paper proposes a new methodological approach by converting the actual temperature record probability distribution function (PDF) into a probabilistic standardized degree indicator (SDI) in normal (Gaussian) PDF form with zero mean and unit standard deviation. Furthermore, as a new evaluation methodology fuzzy logic inference calculations are provided. SDI is divided into seven classes, three for cooling, heating, and a single class for uncooled-unheated states. Heating and cooling degree changes according to SDI values are presented with crisp and fuzzy sets. The application of the methodology is made for Kadıköy urban area on the Asian side of Istanbul, Turkey. In the case of fuzzy set SDI air conditioned operation, it has been determined that there is cooling energy saving of approximately 25% compared to the crisp set. The same is true for the heating degree-day side.

Precipitation and Temperature Climatologies over India: A Study with AGCM Large Ensemble Climate Simulations

This study investigated the precipitation and temperature climatologies over India from large ensemble (100 members) historical climate simulations in two recent past climate periods (1951–1980 and 1981–2010). The main focus was to statistically examine the usefulness of such large historical climate simulations by discussing (1) the precipitation and temperature climatologies and their distribution patterns, (2) the annual cycle of the temperature and precipitation climatologies, and (3) the frequency distributions and potential spatial patterns of climate variability. We applied empirical orthogonal function to understand the characteristics and normal probability distribution function to investigate the frequency. Results indicated good agreements of these large ensembles simulated results with the observation over Indian region. The precipitation amount over many regions of India is decreased and temperature over entire India is increased in 1981–2010 compared to that in 1951–1980. The annual cycle of the precipitations over India indicated a decrease of the precipitation amounts from June through October, while an increase of precipitation for the months from November through January. The annual cycle of the temperature over India indicated an increase of temperature during July through March. The frequency distributions of monthly precipitations and temperatures indicate an overall decrease of precipitation and an overall increase of temperature in recent climate period. The reason of decreased precipitation in recent climate period is attributed to a decrease of relative humidity and cloud together with weaker vertical velocity over Indian region during 1981–2010. Overall study validates the usefulness of these large ensemble climate simulations for the assessment of climate over India and suggests that these datasets may be used for various purposes related to weather and climate over India.

Performance Improvement of Multiobjective Optimal Power Flow-Based Renewable Energy Sources Using Intelligent Algorithm

Producing energy from a variety of sources in a power system requires an optimal schedule to operate the power grids economically and efficiently. Nowadays, power grids might include thermal generators and renewable energy sources (RES). The integration of RES adds complexity to the optimal power flow problem due to intermittence and uncertainty. The study suggests a Multi-Objective Search Group Algorithm (MOSGA) to deal with multi-objective optimal power flow integrated with a stochastic wind and solar powers (MOOPF-WS) problem. Weibull and lognormal probability distribution functions (PDFs) are respectively adopted to describe uncertainties in wind speed and solar irradiance. The MOSGA incorporates crowding distance strategies, fast non-dominated sorting, and an archive selection mechanism to define and preserve the best non-dominated solutions. The total cost, real power loss, and emission were defined as the objectives for the MOOPF-WS problem. In the economic aspect, formulated cost modelling includes both overestimation and underestimation situations related to wind and solar power prediction. Further, uncertainty in load demand is represented by a normal PDF and is considered as a special study case due to its novelty. The effectiveness of MOSGA was validated on the IEEE 30-bus and 57-bus systems considering various combinations of objective functions as well as different loading scenarios. Its performance was comprehensively compared with the other three well-regarded multi-objective optimization algorithms including NSGA-II, MOALO, and MOGOA in terms of Spread metric, Hypervolume metric, and the best compromise solutions for all scenarios. The comparisons showed MOSGA was capable of obtaining well-distributed Pareto fronts and producing better quality solutions compared to the others in all tested scenarios. In addition, MOSGA also obtained better solution quality than significant research in the literature for all the comparable cases. These show the superiority of the MOSGA in dealing with the MOOPF-WS problem.

Statistical modelling of tensile properties of natural fiber yarns considering probability distributions of fiber crimping and effective yarn elastic modulus

Unlike synthetic fiber yarns, due to limited fiber length, natural fiber yarns are made of twisted short natural fibers with various crimping status obeying certain probability distribution. Meanwhile, variations from fiber tensile properties and number of fibers in a yarn also result in scattering of mechanical properties of natural fiber yarns. An analytical model is proposed based on statistic theory to characterize the stochastic tensile properties of natural fiber yarns by defining crimp strain of fibers in a yarn and effective elastic modulus of yarns as random variables. The model is demonstrated on jute fiber yarns. Beta and normal probability distribution functions are selected to describe the statistical distributions of the crimp strain and effective elastic modulus, respectively. Parameters of the model are obtained from tensile testing results of jute yarns through inverse approach. The proposed model can provide accurate prediction bounds for tensile tests of jute yarns. Moreover, the proposed model is extended to predict the tensile response with progressive failure process of unidirectional fabric.

Deterministic and Probabilistic Analysis of Dasdhunga Soil Slope along Narayangarh-Mugling Road Section

Instability of slopes is usually governed by a combination of intrinsic and extrinsic factors. The inherent variability of parameters make the problem probabilistic rather than a deterministic one. This research deals with evaluation of stability of slopes with the calculation of the factor of safety of Dasdhunga soil slope along Narayangarh- Mugling road section under different rainfall conditions through the use of coupled finite element and limit equilibrium method in GeoStudio and the determination of probability of failure by sliding, modeled as infinite slopes by using Monte Carlo simulation in R-Studio. Mean, standard deviation, minimum and maximum values of the parameters like- friction angle, cohesion and unit weight were computed from eight samples of the slope. The pore water pressure developed and its corresponding statistical data for different rainfall conditions were computed from FEM based SEEP/W simulation. The above parameters are assumed to follow truncated normal probability distribution function and the geometric parameters like height and slope angle are regarded as constant parameters. It was observed that the safety factors for theslopeis low in high intensity-low duration rainfalls and the probability of failure is high. The tendency to fail increases as the return period of rainfall increases and viceversa. Sensitivity analysis performed in both deterministic and probabilistic methods showed that friction angle is the most sensitive.

Temporal Trend Possibilities of Annual Rainfall and Standardized Precipitation Index in the Central Highlands, Vietnam

This paper employed the Şen’s innovative trend analysis (ITA) method to identify overall and partial trends in annual rainfall records over the Central Highlands of Vietnam. Additionally, the lowest values of Standardized Precipitation Index (SPI) for 3 and 12 months were subjected to the analysis to reveal potential changes in the greatest intensity of meteorological drought. The outcomes from a preliminary assessment by applying the normal (Gaussian) probability distribution functions (PDFs) indicate significant increases in standard deviation values of the second half subseries (current climate), thereby implying greater level of rainfall variability. The results from temporal trend analysis show that there is high domination of neutral and decreasing trends in annual rainfall records, with the estimated ITA slopes varying mainly from − 10 to − 14 mm/year. Furthermore, the low value sub-groups are mainly characterized by significant decreasing trends, which is an evident indication of possible increases in drought intensity. These results were further demonstrated by evaluating the ITA plots for the lowest values of SPI in both 3-month and 12-month timescales. The lowest values of SPI are mainly lower than − 1.5 and − 2.0 corresponding to severe and extreme drought categories, and most of the scatter points in low SPI cluster are accumulated in the lower triangular area of 1:1 line. Generally, it is expected that these findings will contribute various well-founded foundations to manage water–use activities and natural hazards associated with rainfall shortages.

Jump point identification in hydro-meteorological time series by crossing methodology

The climate change impact appears as a decreasing or increasing monotonic trend in hydro-meteorology time series records due to greenhouse gas (GHG) emissions causing to global warming and climate change impacts. On the other hand, there may be abrupt changes in the form of jumps in these series due to natural and engineering activities. Although trend identification methods are rather common in the literature, jump determination conventional methodologies are rather rare and their applications present some restrictive asumptions like the serial independence and normal (Gaussian) probability distribution function (PDF). The methodology presented in this paper is away from each of such assumptions and it depicts the minimum number of upcrossing along horizontal truncation levels within the time series variation domain. The applications of the methodology are given for annual Danube River discharge records, Romania; New Jersey rainfall and temperature records, USA; monthly rainfall records and Van Lake level fluctuations, Turkey.

Dynamic optimal power flow with cross entropy covariance matrix adaption evolutionary strategy for systems with electric vehicles and renewable generators

Modern power systems which integrate renewable energy sources (RESs), such as wind or solar energy sources and plug-in electric vehicles (PEVs), need to carry out the uncertainty due to the access of demanded or injected power. Therefore, it is necessary to consider uncertainty costs in optimal power flow problems. To develop the uncertainty cost functions, log-normal probability distribution function (PDF) for solar irradiance, normal PDF for unloading and loading behavior of PEVs, and Rayleigh PDF for wind speed are considered. In this work, a dynamic optimal power flow (DOPF) model is formulated by considering the uncertainties of RESs and PEVs. A newly hybrid metaheuristic algorithm named cross entropy (CE) covariance matrix adaption evolutionary strategy (CMAES) is proposed to evaluate the DOPF problem in a modified IEEE 118-bus system. The application of adaptive step length in the CMAES method improves the local exploitation capability, while the CE method is used in the initial stage for global exploration due to its fast convergence. Simulation results indicate that the proposed technique can solve the DOPF problems with RESs and PEVs effectively and can provide high-quality solutions compared to different techniques. The conventional statistical method called ANOVA test is performed for comparative analysis of different techniques. The results of this test show the validation of CE-CMAES compared to different optimization techniques. Therefore, the proposed method could be easily and efficiently applied by power system operators in real-time optimal power flow operations.

Actual Precipitation Index (API) for Drought Classification

The Standard Precipitation Index (SPI) is a widely used statistical technique for the characterization of droughts. It is based on a probabilistic standardization procedure, which converts a Gamma-type probability distribution function (PDF) into a normal (Gaussian) standard series with zero mean and unit standard deviation. Drought classification based on SPI indicates dry and wet spell characteristics, provided that the hydro-meteorological records abide by normal (Gaussian) PDF only, otherwise the results will be biased. Therefore, in this paper, the actual precipitation index (API) method is presented, which provides drought classification and information regardless of the underlying PDFs. The main purpose of this paper is to explain the main differences between SPI and API and to prove that the use of API is the more reliable solution for classification of droughts into five categories described as “Normal dry”, “Slightly dry”, “Medium dry”, “Very dry” and “Extremely dry”. The application of the methodology is presented for two sets of precipitation data; one with exponential PDF monthly precipitation records from Istanbul City, Turkey and one for New Jersey, USA with almost normal (Gaussian) PDF based on annual precipitation records. The comparisons indicate that API is applicable regardless of the underlying PDF of the hydro-meteorology data. It produces real drought classification from the original data without recourse to standard normal PDF conversion.

Trend Analyses Methodologies in Hydro-meteorological Records

In recent years, global warming and climate change impacts on hydro-meteorological variables and water resources triggered extensive focus on trend analyses. Especially, in historical records and climate change model scenario projections, trend feature searches help for better predictions prior to mitigation and adaptation activities. Each trend identification technique has a set of restrictive assumptions and limitations, but they are not cared for by many researchers. The major problem with trend research is that the researchers do not care for the basic assumptions of any methodology but use ready software to solve their problems. Among these assumptions, the most significant ones are the normal (Gaussian) probability distribution function (PDF) and serially independent structure of a given time series. It is the main objective of this review paper to present each trend identification methodology including classical ones with the new alternatives so that any researcher in need of trend analysis can have concise and clear interpretations for the choice of the most convenient trend method. In general, parametric, non-parametric, classical and innovative trend methods are explained comparatively including the linear regression, Mann–Kendall (MK) trend test with Sen slope estimation, Spearman’s rho, innovative trend analysis (ITA), partial trend analysis (PTA) and crossing trend analysis (CTA). Pros and cons are given for each methodology. In addition, for improvement of serial independence requirement of the classical trend analyses, methods are introduced briefly by pre- and over-whitening processes. Finally, a set of recommendations is suggested for future research possibilities.

Economic dispatching of microgrid considering renewable energy uncertainty and demand side response

Nowadays, the uncertainty of renewable energy and demand side response have become a significant issue in microgrid dispatch. To optimize the dispatching, it is usually a common way to establish the probability distribution functions of the renewables and the associated load model. Though there have been various modeling approaches, few of them consider the customer satisfaction with uncomplicated parameter solving algorithms. To settle this problem, this paper proposes an optimal dispatching strategy for a thermoelectric coupled microgrid. A parameter simplification approach is firstly presented to deal with the renewable energy uncertainty, by introducing normal distribution probability distribution function and reserve capacity allocation cost. Secondly, as for the demand side response, the effect of transferable and reducible loads on customer satisfaction has been fully considered, and the load tracking ability on renewables has also been taken account. The effectiveness of the proposed strategy and its superiority in reducing the operating cost of the microgrid have been verified by computer simulations and comparisons.