Probability Distribution Function Research Articles

Application of the Standardised Streamflow Index for Hydrological Drought Monitoring in the Western Cape Province, South Africa: A Case Study in the Berg River Catchment

In many regions around the world, drought has been recurrent, more frequent, and more intense over time. Hence, scientific research on drought monitoring has become more urgent in recent years. The aim of this study was to test the applicability of the Standardised Streamflow Index (SSI) for hydrological drought monitoring in the Berg River catchment (BRC), Western Cape (WC) province, South Africa (SA). Using various methods described in this study, the sensitivity of the SSI to the commonly used Gamma, Log-normal, Log-logistic, Pearson Type III, and Weibull Probability Distribution Functions (PDFs) was tested. This study has found that all the tested PDFs produced comparable results for mild to severe drought conditions. The SSI calculated using the Gamma, Log-Normal, and Weibull PDFs is recommended for the BRC because it consistently identified extreme drought conditions during the 1990–2022 study period and identified the 2015–2018 droughts as the worst during the study period. Although more studies are required to test other PDFs not considered, this study has shown that the SSI can be applicable in the BRC. This study has provided a foundation for more research on the application of the SSI in the BRC and other catchments in SA.

Deformation-dependent permeability of fibrous materials

Fluid flow in fibrous materials depends on volumetric deformation, distortional deformation and fibre reorientation. The latter two effects introduce anisotropy in the hydraulic permeability. Motivated by the study of biological tissues, this work aims to develop an analytical model for evaluating the deformation-dependent permeability of fibrous materials. The material consists of a three-dimensional assembly of long fibres of a single type, with the fluid flowing through the interstices between the fibres. The macroscopic deformations are related to the deformation of a cylindrical representative element of volume (REV), comprised of a fibre surrounded by a cylindrical jacket of fluid. The effect of fibre orientation is modelled using an orientational probability distribution function (OPDF), and the REV permeability tensor is then evaluated using previously developed methods, accounting for the macroscopic deformation. Finally, the overall permeability tensor is calculated via directional averaging of the REV permeability tensor, in which the fibre OPDF serves as the weighing function. The model is validated numerically via a uniaxial contraction (confined compression) test performed for the case of an OPDF that is isotropic in the reference (undeformed) configuration, and is found to qualitatively reproduce experimental findings. The proposed model offers an analytical tool for evaluating the permeability of fibrous materials under large deformations, based on structural information such as fibre volumetric fraction, diameter and spatial orientation.

A BER formula of OTFS modulation with message‐passing receiver

AbstractThe orthogonal time frequency space (OTFS) modulation is a promising multicarrier waveform which can have good performance in high‐mobility scenarios. Among various receivers designed for the OTFS modulation, the message‐passing (MP) receiver has received much attention which can achieve a good bit error rate (BER) performance. However, the BER performance of MP receiver is usually obtained through simulations due to the lack of corresponding BER formula in the literature. This work studies the BER formula of OTFS modulation with MP receiver over doubly selective fading channel. Specifically, equivalent signal models of the MP receiver are first defined and the expression of the corresponding signal‐noise‐ratio (SNR) is derived, then the probability distribution function of the SNR is estimated by assuming the distribution of SNR as a lognormal distribution, then the methods to estimate the parameters of the distribution function are proposed, and finally the formula for calculating the BER of the OTFS modulation with MP receiver is obtained. Simulation results verify that the proposed BER formula is accurate.

Measuring the hot ICM velocity structure function using XMM–Newton observations

ABSTRACT It has been shown that the gas velocities within the intracluster medium (ICM) can be measured by applying the novel XMM–Newton EPIC-pn energy scale calibration, which uses instrumental Cu Kα as reference for the line emission. Using this technique, we have measured the velocity distribution of the ICM for clusters involving AGN feedback and sloshing of the plasma within the gravitational well (Virgo and Centaurus) and a relaxed one (Ophiuchus). We present a detailed study of the kinematics of the hot ICM for these systems. First, we compute the velocity probability distribution functions (PDFs) from the velocity maps. We find that for all sources, the PDF follows a normal distribution, with a hint of a multimodal distribution in the case of Ophiuchus. Then, we compute the velocity structure function (VSF) for all sources in order to study the variation with scale as well as the nature of turbulence in the ICM. We measure a turbulence driving scale of ∼10–20 kpc for the Virgo cluster, while the Ophiuchus cluster VSF reflects the absence of strong interaction between the ICM and a powerful Active Galactic Nucleus (AGN) at such spatial scales. For the former, we compute a dissipation time larger than the jet activity cycle, thus indicating that a more efficient heating process than turbulence is required to reach equilibrium. This is the first time that the VSF of the hot ICM has been computed using direct velocity measurements from X-ray astronomical observations.

Trapezius activity during personal computer work with progressive addition lenses for general purpose and for computer work in neophytes.

This study analysed the impact of general purpose progressive addition lenses (GP-PALs) and computer progressive addition lenses (PC-PALs) on the activity of the trapezius muscle during computer operation. In this randomised, single-blinded, crossover study, surface electromyography (SEMG) signals were recorded bilaterally from the trapezius muscle during a 30-min computer task performed wearing different presbyopic corrections. The amplitude probability distribution function and its percentiles, gap frequency, muscular rest time and sustained low-level muscle activity periods were analysed in 32 subjects with artificially induced presbyopia. Subjectively perceived differences in vision and postural load between lenses were evaluated using a seven-item questionnaire (non-standardised, visual analogue scale ranged from 1 [bad] to 100 [good]). Considering the SEMG data, no significant difference in the muscular activity of the trapezius muscle was observed when using GP-PALs or PC-PALs for computer operation. However, PC-PALs showed statistically and clinically significantly higher results for subjectively perceived visual quality (78.4-31.3; p < 0.001), spontaneous tolerance (79.2-31.3; p < 0.001) and field of view (75.9-23.5; p < 0.001) compared with GP-PALs. Even though the electromyographic approach did not show a significant differentiation between the lenses, the subjective evaluation was clearly in favour of PC-PALs. Eye care practitioners should always take an occupational history of presbyopes, ask about the workplace situation and consider the use of PC-PALs.

Logarithmic Duality of the Curvature Perturbation.

We study the comoving curvature perturbation R in the single-field inflation models whose potential can be approximated by a piecewise quadratic potential V(φ) by using the δN formalism. We find a general formula for R(δφ,δπ), consisting of a sum of logarithmic functions of the field perturbation δφ and the velocity perturbation δπ at the point of interest, as well as of δπ_{*} at the boundaries of each quadratic piece, which are functions of (δφ,δπ) through the equation of motion. Each logarithmic expression has an equivalent dual expression, due to the second-order nature of the equation of motion for φ. We also clarify the condition under which R(δφ,δπ) reduces to a single logarithm, which yields either the renowned "exponential tail" of the probability distribution function of R or a Gumbel-distribution-like tail.

Error correction for IMERG precipitation estimates based on climatological adjustment combining the dry–wet season division and weight allocation

Error correction to obtain more accurate gridded precipitation estimates enhances the application prospects of the integrated multi-satellite retrievals for global precipitation measurement (IMERG). This study developed a climatological adjustment approach combining dry–wet season division and weight allocation (i.e., the DW approach) to achieve IMERG estimate correction without the need for contemporaneous gauge data. By learning the characteristics of an 18-year bias series from June 1, 2000 to December 31, 2017, through empirical probability calculation (the probability distribution function of every bias value in the dry and wet seasons), the bias field can be obtained without mathematical distribution assumptions for the biases. After recognizing the bias field at each of the 687 gauges in mainland China, the error adjustment can be tailored for every gauge to lower the current bias magnitude. The bias adjustment can be obtained as the product of two components, namely, the adjustment amount (AA) and the scaling factor (SF). The SF was discussed under different value intervals of the IMERG estimates to decide whether the original IMERG estimates should subtract or add the AA to generate the adjusted IMERG estimates. The quantile mapping (QM) approach was designed to be the standard method to examine the DW approach proposed. During the validation period from January 1, 2018, to December 31, 2019, the capability of two error adjustment approaches to the value reduction of evaluation indicators was inspected at gauge, climatic, and topographic region scales. The main conclusions follow: (1) The DW approach performed better than the QM approach regarding the results of all of the evaluation indicators at the gauge scale. (2) The DW approach maintained higher efficacy than the QM approach at the gauge, climatic, and topographic region scales. (3) The proportion of the DW approach as the better-performing adjustment approach was higher than the QM approach both at the gauge scale and in most subregions. (4) In the areas where the IMERG products greatly overestimate the precipitation, the DW approach will be very effective, whereas the QM approach will be very effective in the areas where the precipitation is significantly underestimated by the IMERG data.

Day-Ahead Distributionally Robust Optimization-Based Scheduling for Distribution Systems With Electric Vehicles

In this paper, a day-ahead optimal scheduling strategy for the DSO based on EV aggregation is proposed. Generally, the uncertain parameters of EVs, i.e., arrival and departure times as well as the charging demands, can be effectively modeled by a Gaussian mixture model (GMM) with historical charging records. The empirical data of net load error, i.e., mean values and covariance, can be obtained from historical data, while the explicit probability distribution function may not be fully known. Therefore, a distributionally robust optimization (DRO) model is formulated for the DSO to characterize the net load uncertainty. Then, an EV aggregation-based reservation capacity prearrangement method is proposed to specifically address the net load error. Next, with the reservation capacity from EVs, the DRO model is convexified by transforming into two subproblems with deterministic forms. Finally, simulation and comparison tests are conducted on the 33-bus distribution network to demonstrate that the proposed approach achieves a tradeoff between economic performance and computation efficiency compared with the scenario-based stochastic optimization method and achieves a tradeoff between operation costs and system reliability compared with the chance constraint method.

Euclidpreparation

Recent cosmic shear studies have shown that higher-order statistics (HOS) developed by independent teams now outperform standard two-point estimators in terms of statistical precision thanks to their sensitivity to the non-Gaussian features of large-scale structure. The aim of the Higher-Order Weak Lensing Statistics (HOWLS) project is to assess, compare, and combine the constraining power of ten different HOS on a common set ofEuclid-like mocks, derived fromN-body simulations. In this first paper of the HOWLS series, we computed the nontomographic (Ωm,σ8) Fisher information for the one-point probability distribution function, peak counts, Minkowski functionals, Betti numbers, persistent homology Betti numbers and heatmap, and scattering transform coefficients, and we compare them to the shear and convergence two-point correlation functions in the absence of any systematic bias. We also include forecasts for three implementations of higher-order moments, but these cannot be robustly interpreted as the Gaussian likelihood assumption breaks down for these statistics. Taken individually, we find that each HOS outperforms the two-point statistics by a factor of around two in the precision of the forecasts with some variations across statistics and cosmological parameters. When combining all the HOS, this increases to a 4.5 times improvement, highlighting the immense potential of HOS for cosmic shear cosmological analyses withEuclid. The data used in this analysis are publicly released with the paper.

Diurnal Cycle of Rapid Air Temperature Fluctuations at Jezero Crater: Probability Distributions, Exponential Tails, Scaling, and Intermittency

AbstractWe study the diurnal cycle of rapid thermal fluctuations observed by the Mars Environmental Dynamics Analyzer, onboard the Perseverance rover at Jezero Crater, as a function of local time and season. In this context, rapid refers to periods between 15 min and half a second. Some insight is also provided into wind fluctuations that are the base for most of the existing theories on turbulent flows. The results expand the observations from previous Mars missions, namely Viking, Mars Pathfinder, and Phoenix, and they add to our knowledge of near‐surface fluctuations on Mars. (a) Probability distribution functions of the fluctuations are determined and found to have exponential tails. This means that models that represent the interaction within the environment and with the surface as a stochastic forcing need to account for the sudden events responsible for the exponential tails. (b) Power density spectra are calculated and show several dynamical regimes with different slopes associated to forcing, an intermediate regime, and a higher frequency regime. All change with time of the day. The results imply that the fastest regime is not a universal scenario for the temperature fluctuations near the surface. (c) The scale dependence of the fluctuations confirms the existence of intermittent outbursts associated to the slower fluctuations, possibly associated to the larger scale structures, and explains why the spectral density slopes do not follow Kolmogorov's law. Understanding the role of larger scale structures would help refine scaling theories of the near‐surface Martian atmosphere and its interactions with the surface.

Transmission system model in the track-vehicle relationship based on Long Term Evolution technology

Because of degradations of the wired infrastructure connecting the element of railroad control systems, and related primarily to their destruction, the use of the radio transmission medium for independent management of rail-road traffic control devices is increasingly being considered. An undoubted problem in implying such solutions is the security of transmission in such systems. It should be noted that security at a certain level of transmission is currently already offered by the radio transmission systems themselves, which has also been used in the GSM-R standard. The creation of a separate dedicated system for railroads involves huge expenditures for the design, testing, certification and, finally, construction and implementation of such technology. Therefore, in the opinion of the author of this dissertation, it is possible to use public open radio networks for the needs of railroads, which significantly reduces costs, since such a system is based on existing infrastructure. It is necessary to develop a way of transmitting information that meets the requirements of secure transmission in the sense of railroad traffic control systems. The task is to develop a general model for open radio transmission in traffic control systems based on the latest public radio standard, which is LTE, 5G or Future Railway Mobile Communications System (FRMCS). The article will present the concept of data transmission in the track vehicle relationship with the use of LTE (Long Term Evolution) technology. It will show the concept of transmission based on the PN-EN50159:2011 standard and the results of the tests conducted during control trips on the railway route. The main element that will be subject to the research will include the transmission safety and its delay. The impact of a type of transmission encryption on its delay will be estimated with the assumed blocks of data sent during the transmission. A probability distribution and density functions of the transmission delay probability distribution with the message sizes in the range from 16B to 10 kB with four ways of the signal encoding according to the PN-EN50159:2011 standard will be also analysed.

Original

The present study attempts to assess drought in the Tel River basin, the second largest tributary of Mahanadi flowing through one of the most drought-frequented regions, viz. Nabarangpur, Kalahandi, Bolangir, Boudh, and Kandhamal districts of Odisha using Standard Precipitation Index (SPI). The SPI is one of the most widely used indices for monitoring drought based on rainfall data. It expresses the actual rainfall as a standardized departure concerning the rainfall probability distribution function. The values of SPI are expressed in standard deviations, with positive SPI indicating greater than median precipitation and negative values indicating less than median precipitation. A monthly time series climatic dataset is obtained from the NASA power data for 15 meteorological observation sites around the study basin from 1981 to 2021. The long-term data is then fitted to a probability distribution to create a normal distribution to find the cumulative probability of an observed precipitation event for the given month and time scale at the location in question. The cumulative probability H(x) is then converted to the conventional normal random variable z with a mean of zero and a variance of one yielding the SPI value. A drought event occurs when the index reaches -1.0 or below continuously. For three-time scale analysis of three months SPI 3, six months SPI 6, and twelve months SPI, DrinC software and Arc GIS 10.5 are used. The result shows that in the 40 years (1981-2021), 67 per cent of the droughts have been mild droughts, 18 per cent moderate droughts, 10 per cent severe droughts, and 5 per cent extreme droughts in the study basin. The regional distribution at the sub-basin level did not show any significant pattern based on the selected years of drought in the study basin.

A drought index based on groundwater quantity and quality: Application of multivariate copula analysis

In this paper, a new Groundwater Quantity-Quality-based Drought Index (GQQI) is developed based on multivariate Copula analysis of groundwater quantity and quality indicators. For evaluating the developed index, its temporal and spatial distribution is studied and compared to those of some other indices, such as Standardized Salinity Index (SSI), Standardized Groundwater Index (SGI), and Standardized Water level Index (SWI). The proposed index is applied for the temporal and spatial evaluation of the quantity and quality of groundwater in theLake Urmia basin, having 1084 piezometric and 935 groundwater quality monitoring wells. For a more comprehensive analysis of drought, 24 marginal and 26 joint probability distribution functions are driven. Based on the results, the developed GQQI is correlated with drought indices of SSI, SGI, and SWI by 88%, 86%, and 61%, respectively. Moreover, the values of the developed drought index indicates the occurrence of more severe droughts, compared to those detected by other drought indices. This can be resulted from considering the combined effects of groundwater quantity and quality using the proposed index. In addition, the multivariate GQQI can spatially and temporally represent the drought severity over the basin, and identify severe and extreme drought conditions better than other univariate drought indices.

Graphical interface for wind resource estimation in a region, case study: Cuatrocienegas, Coahuila de Zaragoza

Mexico has an estimated wind potential of 70 GW, however, until 2022 only 7.3 GW were used. This is due, among other factors, to the lack of analysis of the wind resource in many of the country's regions, especially in those where the wind potential is not obvious. With the aim of promoting wind power throughout the country, a free application has been developed that analyzes the resource in detail in any region using the annual records of its wind speed and direction. To demonstrate its functionality, data from the year 2022 from Cuatrocienegas, Coahuila de Zaragoza were used, obtaining the average energy density per hour (MED) using two analytical methods and the Weibull probability distribution function (WPDF). The instantaneous power curve, the histogram of the occurrence of the wind and the wind rose that shows the predominant direction of the wind and its trend of change were also presented. In conclusion, the developed application provides the necessary information to evaluate the technical feasibility of installing horizontal axis wind turbines in the analyzed region.

In-orbit cross-calibration of millimeter conically scanning spaceborne radars

Abstract. The planned and potential introduction in global satellite observing systems of conically scanning Ka- and W-band atmospheric radars (e.g., the radars in the Tomorrow.IO constellation, https://www.tomorrow.io/space/, last access: 1 June 2022, and the Wivern (WInd VElocity Radar Nephoscope) radar, https://www.wivern.polito.it, last access: 1 July 2022) calls for the development of methodologies for calibrating and cross-calibrating these systems. Traditional calibration techniques pointing at the sea surface at about 11∘ incidence angle are in fact unfeasible for such fast rotating systems. This study proposes a cross-calibration method for conically scanning spaceborne radars based on ice cloud reflectivity probability distribution functions (PDFs) provided by reference radars like the Global Precipitation Measurement (GPM) Ka-band radar or the W-band radars planned for the ESA-JAXA EarthCARE or for the NASA Atmosphere Observing System missions. In order to establish the accuracy of the methodology, radar antenna boresight positions are propagated based on four configurations of expected satellite orbits so that the ground-track intersections can be calculated for different intersection criteria, defined by cross-over instrument footprints within a certain time and a given distance. The climatology of the calibrating clouds, derived from the W-band CloudSat and Ka-band GPM reflectivity records, can be used to compute the number and the spatial distribution of calibration points. Finally, the mean number of days required to achieve a given calibration accuracy is computed based on the number of calibration points needed to distinguish a biased reflectivity PDF from the sampling-induced noisiness of the reflectivity PDF itself. Findings demonstrate that it will be possible to cross-calibrate, within 1 dB, a Ka-band (W-band) conically scanning radar like that envisaged for the Tomorrow.io constellation (Wivern mission) every few days (a week). Such uncertainties are generally meeting the mission requirements and the standards currently achieved with absolute calibration accuracies.

Sub-regional phase error compensation based ona probability distribution function and gammaprecoding.

In this paper, a phase error compensation method based on a probability distribution function (PDF) is proposed to improve the accuracy of phase extraction, which is helpful for three-dimensional (3D) reconstruction. First, the relationship between the gamma and the gray values is established to segment the projection regions. Then a new method based on a PDF is designed to represent the variation degree of phase error, which fits the precoded gamma value in the minimum range of the phase error. After that, the error compensation method is applied to the self-built system and packaged independently from the 3D reconstruction system to unwrap phases with high precision. The experimental results show that the proposed method can reduce the standard deviation of the phaseerror by 46.9% compared without phase error compensation, and decrease the standard deviation of thephase error by 30% compared with the whole precoding. Generally, our method can effectively avoid overcompensation or under-compensation caused by single global gamma precoding correction, and better reduce the phase error and improve the 3D reconstruction accuracy in the fringe projection system.

Heavy inertial particles in rotating turbulence: Distribution of particles in flow and evolution of Lagrangian trajectories.

We revisit the problem of heavy particles suspended in homogeneous box turbulence flow subjected to rotation along the vertical axis, which introduces anisotropy along the vertical and horizontal planes. We investigate the effects of the emergent structures due to rotation, on the spatial distribution and temporal statistics of the particles. The distribution of particles in the flow are studied using the joint probability distribution function (JPDFs) of the second and third principle invariants of the velocity gradient tensor, Q and R. At high rotation rates, the JPDFs of Lagrangian Q-R plots show remarkable deviations from the well-known teardrop shape. The cumulative probability distribution functions for times during which a particle remains in vortical or straining regions show exponentially decaying tails except for the deviations at the highest rotation rate. The average residence times of the particles in vortical and straining regions are also affected considerably due to the addition of rotation. Furthermore, we compute the temporal velocity autocorrelation and connect it to the Lagrangian anisotropy in presence of rotation. The spatial and temporal statistics of the particles are determined by a complex competition between the rotation rate and inertia of the particle.

Adaptive First-Crossing Approach for Life-Cycle Reliability Analysis

Abstract Life-cycle reliability analysis can effectively estimate and present the changes in the state of safety for structures under dynamic uncertainties during their lifecycle. The first-crossing approach is an efficient way to evaluate time-variant reliability-based on the probabilistic characteristics of the first-crossing time point (FCTP). However, the FCTP model has a number of critical challenges, such as computational accuracy. This paper proposes an adaptive first-crossing approach for the time-varying reliability of structures over their whole lifecycle, which can provide a tool for cycle-life reliability analysis and design. The response surface of FCTP regarding input variables is first estimated by performing support vector regression. Furthermore, the adaptive learning algorithm for training support vector regression is developed by integrating the uniform design and the central moments of the surrogate model. Then, the convergence condition, which combines the raw moments and entropy of the first-crossing probability distribution function (PDF), is constructed to build the optimal first-crossing surrogate model. Finally, the first-crossing PDF is solved using the adaptive kernel density estimation to obtain the time-variant reliability trend during the whole lifecycle. Examples are demonstrated to specify the proposed method in applications.

Fatigue assessment of ship structures based on equivalent wave probability (EWP) concept (1st report): Proposal of EWP concept and its verification by 8600TEU container ship's onboard hull monitoring

A long-term fatigue assessment method based on EWP concept is proposed. ‘Equivalent wave probability (EWP)’ is the fictitious (HS, Tm)'s joint probability distribution function (JPDF), for which the frequency distribution of the stress variance R2, f(R2), calculated by spectral fatigue assessment agrees with the observed one. By choosing probability function p(R2) to fit f(R2), the R2's statistical model (R2SM) which represents the relation between the EWP parameters and R2's population parameters is developed, and the Bayesian inference, which can estimate the EWP parameters from the measured R2 data is developed. The EWP at the reference position (RP) can be determined by Bayesian inference from the measured R2 through the R2SM at RP. To accurately estimate the measured f(R2) at the target position (TP) from the EWP at RP, an R2SM correction factor at TP, denoted by αTP, is introduced in the process of assimilating R2SM. The resulting R2SM, which has been assimilated by Bayesian inference using measured data, is referred to as data-assimilated R2SM (DAR2SM). The fatigue assessment using EWP at RP as the input of DAR2SM at TP is called Bayes-EWP-DAR2SM analysis. The validity of Bayes-EWP-DAR2SM analysis is verified by using the long-term (about four years) multi(12)-position hull monitoring (HM) data of an 8,600TEU container ship. The fatigue damages estimated by Bayes-EWP-DAR2SM based solely on the stress history of a single sensor are in agreement with measurements with sufficient accuracy, independent of the chosen data assimilation period. This demonstrates that the multi-position fatigue assessment solely through HM at one RP based on EWP concept is realized.

Stochastic Multi-Objective Optimal Reactive Power Dispatch with the Integration of Wind and Solar Generation

The exponential growth of unpredictable renewable power production sources in the power grid results in difficult-to-regulate reactive power. The ultimate goal of optimal reactive power dispatch (ORPD) is to find the optimal voltage level of all the generators, the transformer tap ratio, and the MVAR injection of shunt VAR compensators (SVC). More realistically, the ORPD problem is a nonlinear multi-objective optimization problem. Therefore, in this paper, the multi-objective ORPD problem is formulated and solved considering the simultaneous minimization of the active power loss, voltage deviation, emission, and the operating cost of renewable and thermal generators. Usually, renewable power generators such as wind and solar are uncertain; therefore, Weibull and lognormal probability distribution functions are considered to model wind and solar power, respectively. Due to the unavailability and uncertainty of wind and solar power, appropriate PDFs have been used to generate 1000 scenarios with the help of Monte Carlo simulation techniques. Practically, it is not possible to solve the problem considering all the scenarios. Therefore, the scenario reduction technique based on the distance metric is applied to select the 24 representative scenarios to reduce the size of the problem. Moreover, the efficient non-dominated sorting genetic algorithm II-based bidirectional co-evolutionary algorithm (BiCo), along with the constraint domination principle, is adopted to solve the multi-objective ORPD problem. Furthermore, a modified IEEE standard 30-bus system is employed to show the performance and superiority of the proposed algorithm. Simulation results indicate that the proposed algorithm finds uniformly distributed and near-global final non-dominated solutions compared to the recently available state-of-the-art multi-objective algorithms in the literature.