Probabilistic Load Flow Analysis Research Articles

Probabilistic Load Flow Analysis Using Nonparametric Distribution

In the pursuit of sustainable energy solutions, this research addresses the critical need for accurate probabilistic load flow (PLF) analysis in power systems. PLF analysis is an essential tool for estimating the statistical behavior of power systems under uncertainty. It plays a vital part in power system planning, operation, and dependability studies. To perform accurate PLF analysis, this article proposes a Kernel density estimation with adaptive bandwidth for probability density function (PDF) estimation of power injections from sustainable energy sources like solar and wind, reducing errors in PDF estimation. To reduce the computational burden, a Latin hypercube sampling approach was incorporated. Input random variables are modeled using kernel density estimation (KDE) in conjunction with Latin hypercube sampling (LHS) for probabilistic load flow (PLF) analysis. To test the proposed techniques, IEEE 14 and IEEE 118 bus systems are used. Two benchmark techniques, the Monte Carlo Simulation (MCS) method and Hamiltonian Monte Carlo (HMC), were set side by side for validation of results. The results illustrate that an adaptive bandwidth kernel density estimation with the Latin hypercube sampling (AKDE-LHS) method provides better performance in terms of precision and computational efficiency. The results also show that the suggested technique is more feasible in reducing errors, uncertainties, and computational time while depicting arbitrary distributions of photovoltaic and wind farms for probabilistic load flow analysis. It can be a potential solution to tackle challenges posed by sustainable energy sources in power systems.

Probabilistic load flow in distribution networks: An updated and comprehensive review with a new classification proposal

Due to the increased penetration of distributed energy resources and the presence of uncertainties in power networks, it is essential to evaluate the uncertainty of system performance. As a result, effective technologies for load flow analysis are required. The stochastic properties of power systems can be discovered via probabilistic load flow analysis. To model the uncertainties and construct the parametric tools of probabilistic power flow, several works recently utilize ensembles of probability density functions. Nevertheless, the uncertainties may not fit into any of the conventional classes of probability density functions. Thus, nonparametric tools are required. Various studies on uncertainty modeling techniques in probabilistic load flow have recently been undertaken, although there is no single best uncertainty modeling methodology to date. The load flow problem and the type of uncertain input variables determine the best strategy. There have been positive evaluations in this area, but the main focus of this research is firstly to present an overview and classification of the available load flow strategies in distribution networks from various perspectives and also, investigate the stochastic correlation or dependency of a random variable's variation relative to others. Then, based on a fresh perspective, we present a review, taxonomy, and discussion of the strengths and shortcomings of existing uncertainty modeling techniques in stochastic load flow as probability methods (parametric, semi- and non-parametric approaches) and non-probability methods. Moreover, for first time, a comprehensive comparison and assessment between these uncertainty methods based on accuracy, complexity, and simulation time presents in the uncertainties handling of the power system. The authors suggest this review to engineers, scientists and researchers who are doing their research work in this field.

Normal-transformation-based probabilistic load flow with correlated wind and load forecast errors

With aggressive government policies driving the electric power systems around the globe towards high renewable energy penetration, methodologies for considering forecast uncertainties in power system analysis and planning are gaining relevance. Probabilistic load flow (PLF) is a useful tool to carry out risk-based power system analysis and planning. This article discusses the need and philosophy of Normal-transformations, elaborating on Nataf transformation and polynomial normal transformation (PNT), to solve PLF and presents a unified approach for representation. Through detailed statistical modeling, the Gaussian mixture (GM) model is found to fit the historical wind and load power forecast error data better compared to widely used parametric models. Sampling of correlated marginal distributions, for scenario generation in PLF, requires the quantile function of the GM distribution, which is not available in a closed expression. A strategy, using quantile functions of Gaussian component sampling and interpolation, to approximate the GM’s quantile function is presented. The presented approximation enables the use of GM for solving PLF with correlated marginals, which is not addressed in the literature. This work also contributes to prevailing research questions by two new performance comparisons: between various higher-order PNT embedded to point estimate methods (PEMs) for PLF, and between PEM and Quasi-Monte Carlo (QMC) methods for PLF analysis. When PEMs are employed for PLF, there is no improvement observed in going for higher than third order PNT (3PNT). Results show that estimating the first four moments of the output variables through QMC-PLF is more accurate compared to PEM-PLF. Formulae useful for implementation of various higher-order PNT, for QMC and PEM, are derived and presented for direct use. Finally, the impact of correlated non-normal modeling of inputs on PLF output variables’ distribution is demonstrated.

Numerical Probabilistic Load Flow Analysis in Modern Power Systems with Intermittent Energy Sources

Renewable resources integration through distributed generation (DG) affects conventional consideration of power system performance and confronts deterministic load flow (DLF) analysis with serious challenges. The DLF gives a snapshot of the system state neglecting all of the uncertainties arising from intermittent DG driven by variable weather conditions or volatile consumption. Therefore, with the aim of finer tracking and presentation of system variables, a probabilistic load flow (PLF) approach should be adopted. First, this article gives a literature overview of different PLF techniques. It focuses on numerical techniques examining them for simple random and Latin Hypercube sampling, vastly applied in previous works, and proposes a method combining Monte Carlo simulations with Halton quasi-random numbers. Stochastic modelling is performed for solar and wind power output. For method comparison and confirmation of the applicability of suggested PLF method with Halton sequences, different IEEE test cases were used, all modified by attaching DGs. More profound method assessment is conducted through discussing different renewables penetration levels and processing time. The overall simulation outcomes have shown that results of Halton method are of similar precision as the generally used Latin Hypercube method and therefore indicated the relevance of the proposed method and its potential for application in contemporary system analysis.

Review of Charging Load Modeling Strategies for Electric Vehicles: a Comparison of Grid-to-Vehicle Probabilistic Approaches

Objective: In this paper, we review different approaches to how the penetration of electric vehicles (EV) can be modeled in power networks. We also evaluate and compare experimentally the performance of three probabilistic electric vehicle charging load approaches considering four levels of penetration of EV.
 Methodology: We carry out a detailed search of the state-of-the-art in charging load modeling strategies for electric vehicles, where the most representative works on this subject were compiled. A probabilistic model based on Monte Carlo Simulation was proposed and two more methods were implemented. These models take into account the departure time of electric vehicles, the arrival time and the plug-in time, which were conceived as random variables. 
 Results: Histograms of the demand for charging of electric vehicles were obtained for the three models contemplated. Additionally, a similarity metric was calculated to know the distribution that best fits the data of each model. The above was done considering 20, 200, 2000 and 20,000 electric vehicles on average. The results show that if there are a low penetration of electric vehicles, it is possible to model the EV charging demand using a gamma distribution. Otherwise, it is recommended to use a Gaussian or Lognormal distribution if you have a high VE penetration.
 Conclusions: A review of the state of the art of the modeling of electric vehicles under a G2V approach was presented, where three groups are identified: the deterministic approaches, methods that deal with uncertainty and variability, and finally data driven methods were also identified. Additionally, we observed that the EVCP model 3 and the gamma distribution can be appropriate for modeling the penetration of EVs in probabilistic load flow analysis or for stochastic planning studies for active distribution networks.
 Financing: Institución Universitaria Pascual Bravo

Bayesian estimation of copula parameters for wind speed models of dependence

Modelling the uncertainty of wind speed is essential in power flow analysis. Having abundant knowledge of the wind speed in an area is critical. A low volume of data can increase uncertainty in wind speed analysis. Spatial dependencies are often modelled before running probabilistic power flow and load flow analysis. Copulas are a popular way of capturing spatial dependence between multiple wind farms. Using NREL data from seven Northeastern United States wind farm sites, Bayesian inference will be used to determine the copula parameter uncertainty between weekly, daily, and hourly wind speed observations. This approach will be used on elliptical and single parameter Archimedean copulas. For each possible wind farm pair, an uninformative prior will be placed on the copula parameter. The resulting posterior will contain a distribution of copula parameter values based on the prior and the observed wind speed data. The posterior's credible interval is reviewed to determine the uncertainty in parameter estimation. The results show that using a data volume considerably more petite than 8760 hourly data points will result in more uncertainty in parameter estimation and inaccuracies in wind speed forecasting if using non-Bayesian methods for copula parameter estimation.

Probabilistic load flow analysis using Halton sequences in power systems with renewable resources

Probabilistic load flow analysis using Halton sequences in power systems with renewable resources

Probabilistic load flow analysis of electric vehicle smart charging in unbalanced LV distribution systems with residential photovoltaic generation

Several studies have presented electric vehicle smart charging schemes to increase the temporal matching between photovoltaic generation and electric vehicle charging, including a smart charging scheme with an objective to minimize the net-load variance. This method has proved, through simulations, that the self consumption could be increased, but the benefit of the approach has not been tested on a low voltage distribution system. To increase the quality of grid impact analyses of the smart charging scheme, probabilistic methods that include input and spatial allocation uncertainties are more appropriate. In this study, a probabilistic load flow analysis is performed by modelling the variability of electric vehicle mobility, household load, photovoltaic system generation, and the adoption of photovoltaic system and electric vehicle in society. The results show that the smart charging scheme improves the low voltage distribution system performance and increases the correlations between network nodes. It is also shown that concentrated allocation has more severe impacts, in particular at lower penetration levels. This paper can form the basis for the development of probabilistic impact analysis of smart charging to allow society to integrate more electric vehicles and photovoltaic systems for a more sustainable future.

Distribution network operational risk assessment and early warning considering multi‐risk factors

Under the impact of various uncertainties, the potential operation risk of the distribution network is gradually gaining more and more attention in both security and economy. To comprehensively determine the risk level, an operational risk assessment method considering multiple risk factors is proposed. Firstly, probabilistic models are established including temporal distributions of electric vehicles and uncertainties of renewable energy. Secondly, a time-sequential security risk assessment method with several security indicators including overvoltage risk, undervoltage risk, line-overload risk and load-loss risk is established using probabilistic load flow and dynamical sensitivity weight analysis method. Thirdly, conditional value at risk model is presented for time-varying economic risk considering line-loss risk and operational profit or loss risk of distributed generations. Finally, a risk early warning standard is proposed to realise the visualisation of risk warning by defining the risk level and risk area. Simulations of different scenarios are conducted on the modified IEEE 34-bus radial test system and the results reveal that the proposed strategy is effective and practical.

An improved transformation based probabilistic load flow analysis using appropriate reference variable

Power systems are faced with a huge number of uncertainties, which should be addressed in the operational analyses. In this paper, an improved transformation method is used to consider the uncertainties of random variables in probabilistic load flow (PLF) problems. This method transforms non-normal input random variables with non-linear dependence to appropriate independent reference variables. The proposed transformation overcomes the point estimation method (PEM) limitations in estimating high statistical moments of PLF outputs. The proposed PLF method is examined on IEEE 14-bus and 118-bus test systems, consisting of uncertain loads, wind farms, and PV power plant generation. The results of Monte Carlo Simulation (MCS) are used as a benchmark. The results are compared with Nataf transformation, Rotational transformation, and particle swarm optimization clustering methods to prove the accuracy and efficiency of the proposed method.

A scenario-based analytical method for probabilistic load flow analysis

In recent years, power system uncertainties have increased due to the growing integrations of intermittent renewable energy resources. It is imperative to introduce probabilistic load flow analysis in the study of power system operation and planning to adapt to the ever-increasing uncertainties. This paper proposes a scenario-based analytical method for the probabilistic load flow analysis, which takes advantage of both the scenario analysis method and the cumulant method. This method can not only consider various kinds of correlations among power inputs but also accurately represent the probability distributions of desired outputs with a reasonable computational burden. The performance of this method is evaluated on the IEEE 14-bus and 118-bus test systems. The accuracy and efficiency of the proposed method are validated through quantitative and graphical comparisons with Monte-Carlo simulation.

Linear three-phase power flow model for autonomous microgrid systems

In this work, the authors propose a linear three-phase power flow model for droop-controlled autonomous microgrids. The proposed linear power flow model adopts a set of linear algebraic equations and can, with small errors, be compared to the detailed nonlinear model, account for the lack of a slack bus, the variability of autonomous microgrid frequency and the operation of distributed generation units in droop control mode. The proposed linear model can as such provide a fast and accurate power flow solution and has great potential in the stochastic planning and online operational applications of autonomous microgrids, as well as in linear or convex optimal power flow and probabilistic load flow analysis. The accuracy and effectiveness of the proposed linear power flow model are validated through different case studies considering balanced and unbalanced test systems.

Joined Probabilistic Load Flow and Sensitivity Analysis of Distribution Networks Based on Polynomial Chaos Method

Due to the statistical uncertainty of loads and power sources found in smart grids, effective computational tools for probabilistic load flow analysis and planning are now becoming indispensable. In this paper, we describe a unified simulation framework that allows quantifying the probability distributions of a set of observation variables as well as evaluating their sensitivity to potential variations in the power demands. The proposed probabilistic technique relies on the generalized polynomial Chaos algorithm and on a regionwise aggregation/description of the time-varying load profiles. It is shown how detailed statistical distributions of some important figures of merit, which includes voltage unbalance factor in distribution networks, can be calculated with a two orders of magnitude acceleration compared to standard Monte Carlo analysis. In addition, it is highlighted how the associated sensitivity analysis is of guidance for the optimal allocation and planning of new loads.

An improved probabilistic load flow simulation method considering correlated stochastic variables

As the increasing integration of large-scale renewable energy sources in power systems, the stochastic characteristics of loads and renewable energy systems become much more complex and impacts power systems much more than ever. Probabilistic load flow analysis is a powerful tool to discover the stochastic characteristics of power systems. There are two important issues for probabilistic load flow analysis based on Monte Carlo simulation: (i) How to generate random samples with the specific distribution and correlation; and (ii) how to make the simulation method to work well even when the correlation matrices are not positive definite. In order to handle the two issues, Nataf transformation combined with Latin hypercube sampling and singular value decomposition method is proposed for solving probabilistic load flow problems with correlated variables in this paper. By using the singular value decomposition (SVD), the proposed method works well even when the correlation matrices are not positive definite. And the twice-permutation technique based on SVD ensures that the samples have the desired correlations. The investigation on modified IEEE 14-bus system and modified IEEE 118-bus system shows that the proposed method is accurate and efficient.

Probabilistic Load Flow Analysis of Power System Network Considering Uncertainty with Generation and Correlated Loads

Probabilistic Load Flow Analysis of Power System Network Considering Uncertainty with Generation and Correlated Loads

Maximum Power Photovoltaic Units Penetration under Voltage Constraints Criteria in Distribution Network Using Probabilistic Load Flow

Nowadays, the connections of Renewable Energy Sources to Distribution Networks are increasing in large number. In order to assess the performance of Distribution Network under normal operating conditions Probabilistic Load Flow analysis is necessary in order to take into consideration the stochastic behavior of Load Demands and Renewable Energy Sources (for example the Photovoltaics Generation Units). Generally, Deterministic Load Flow calculations are typically needed to assess the allowed Distributed Generation penetration level for a given network in order to ensure, for example, that Voltage limits are not exceeded. The present paper deals with the Probabilistic Load Flow using Monte Carlo techniques. The developed program allows probabilistic predictions of Voltages profiles at all buses or nodes of a Distribution Networks. A practical case is also discussed to show the application of the study in order to assess the maximum Photovoltaic power penetration that can be installed into a Distribution Network without violating Voltage Constraints.

A Cumulant-Tensor-Based Probabilistic Load Flow Method

Probabilistic load flow analysis is an important part of grid design, optimization, and operation due to the uncertainties in the power network for both generation and demand, increasingly so for newly integrated technologies including wind power and plug-in vehicles. A reliable, fast, and robust mathematical method for such analyses is a key requirement to help support widespread integration of these new generation and load sources. Conventional deterministic Monte Carlo analyses, though simple in implementation, becomes too slow as networks become more complex. In this paper, a new cumulant-tensor based method is used to assess power flows. Probability distribution functions and reliability indices are generated as final outputs. Furthermore, general correlation between input random variables is included in the analysis. An illustrative 2-bus network is presented along 24-bus IEEE system as case studies, showing the capabilities and increased reliability of the method.

Nonparametric Probabilistic Load Flow With Saddle Point Approximation

Because of uncertainties of renewable generation resources and load, efficient tools are required for load flow analysis in power systems. Many of the existing papers assume a set of given probability density functions (PDFs) to model uncertainties and develop parametric probabilistic load flow tools. However, the uncertainties might not fall in any standard class of PDFs. Thus, nonparametric tools are needed. This paper presents a method for nonparametric probabilistic load flow analysis to determine PDFs of the load flow outputs. The proposed method is based on the mean value first order saddle point approximation. For a system with $\boldsymbol {n}$ random variables, $\boldsymbol {(n+1)}$ load flow calculations are utilized to establish first order Taylor series expansion and then saddle point approximation is adopted to determine the probabilistic characteristic of desired output variables. The proposed nonparametric estimator provides accurate results while needing a reasonable computational effort. The estimator needs only a limited available data set of random variables without requiring information whether the data set belongs to a certain class of parametric distribution functions. Furthermore, both probability and cumulative distribution functions of load flow outputs are directly established without using integral or differential operators. The proposed method is tested on the IEEE 14-bus and the IEEE 118-bus test systems and promising results are obtained.

An Analytic Approach to Probabilistic Load Flow Incorporating Correlation Between Non-Gaussian Random Variables

This paper presents a cumulant-based method for probabilistic load flow (PLF) analysis which incorporates correlation between input random variables. Our approach can approximate non-Gaussian variables of all kinds (e.g. different load profiles or renewable power injections) accurately using the Gaussian mixture model (GMM), which also facilitates the computation of cumulants in a straightforward numerical way. Multiple correlations can be easily handled by transforming correlated variables into a combination of uncorrelated ones. To reduce the deviations introduced by traditional series expansions such as Edgeworth or Cornish-Fisher series, we use C-type Gram-Charlier series instead, which can better predict the probabilistic tail regions and have good convergence property as well. The good performance of the proposed method is verified using the IEEE 30 test system in terms of accuracy and efficiency.DOI: http://dx.doi.org/10.5755/j01.eie.24.3.20980

Assessment of Wind Power Integrated to the Power Systems Using Probabilistic Load Flow Based on PFV with LS

Power flow analysis (PFA) is one of the most important issues of power system analysis and design. This paper aims at developing and applying a high-speed sampling method to produce large-capacity sample set based on pattern feature vector with layer structure (PFV with LS) of IEEE 14-bus system data analytical methods. In this study, sampling based pattern vectors are constructed in three layers. The first and second layer vectors are very low dimensional, and the sampling of the third layer is in parallel for each local area. Samples generated by this sampling method can be applied to the calculation of probabilistic load flow (PLF) and probabilistic static security assessment. Simulation results show that the proposed method can improve efficiency of PLF analysis and probabilistic static security assessment. The advantages of using PLF based on PFV with LS as the power flow limit can minimize the complicated mathematical equations. Furthermore, the algorithm is very simple and accurate, especially when the system connected to the wind power.