Probabilistic Load Flow Problem Research Articles

Non-parametric probabilistic load flow using Gaussian process learning

The load flow problem is fundamental to characterize the equilibrium behavior of a power system. Uncertain power injections such as those due to demand variations and intermittent renewable resources will change the system’s equilibrium unexpectedly, and thus potentially jeopardizing the system’s reliability and stability. Understanding load flow solutions under uncertainty becomes imperative to ensure the seamless operation of a power system. In this work, we propose a non-parametric probabilistic load flow (NP-PLF) technique based on the Gaussian Process (GP) learning to understand the power system behavior under uncertainty for better operational decisions. The technique can provide “semi-explicit” form of load flow solutions by implementing the learning and testing steps that map control variables to inputs. The proposed NP-PLF leverages upon GP upper confidence bound (GP-UCB) sampling algorithm. The salient features of this NP-PLF method are: i) applicable for power flow problem having power injection uncertainty with an unknown class of distribution; ii) providing probabilistic learning bound (PLB) which further provides control over the error and convergence; iii) capable of handling intermittent distributed generation as well as load uncertainties. The simulation results performed on the IEEE 30-bus and IEEE 118-bus system show that the proposed method can learn the voltage function over the power injection subspace using a small number of training samples. Further, the testing with different input uncertainty distributions indicates that complete statistical information can be obtained for the probabilistic load flow problem with an average percentage relative error of the order of 10−3% on 50,000 test points.

A Coupled Karhunen–Loève and anisotropic sparse grid interpolation method for the probabilistic load flow problem

In the traditional load flow analysis, a key assumption is that the input variables, i.e., generator output and customer demand, are fixed in time and the associated response has no variability. This assumption, however, is no longer valid as the adoption of renewable energy resources add more variability and uncertainty to the modern electrical system. Addressing these concerns is the definition of the Probabilistic Load Flow (PLF) problem. The challenge of the PLF problem lies in handling high-dimensional input uncertainties and the non-linearity of the load flow equations. The most straightforward way to address these problems, but at the cost of computational time, is to perform a Monte Carlo method. This work, however, solves these problems—accuracy, high-dimensionality, and computational time—with a coupled Karhunen-Loève (KL) expansion and Anisotropic Sparse Grid algorithm. The proposed method is implemented and tested on the IEEE 118-bus test system and a modernized version of the IEEE Reliability Test System–1996, the Reliability Test System–Grid Modernization Lab Consortium (RTS-GMLC). Results for the 194-dimensional case show a decrease in computational time when compared to the 10,000 sample Monte Carlo method given a bound on mean and standard deviation error.

An enhanced particle swarm optimization algorithm to solve probabilistic load flow problem in a micro-grid

This paper presents an Enhanced PSO (EPSO) method to address the weaknesses associated with the traditional PSO such as high sensitivity to the initial conditions, fast convergence, decline of solutions’ variety, and trapping in the local optimum. For this, several strategies are suggested including segmentation of search space, modification of solution’s updating rule, accepting poor solutions using an intelligent probabilistic function, searching in the regions with poor solutions, gradually removing the regions with poor solutions, and focusing the solutions on the local search after removing all regions with poor solutions. The performance of the EPSO was investigated using the 30 CEC 2014 test functions, 30 CEC 2017 test functions, 10 standard optimization algorithms, and a challenging optimization problem called Probabilistic Load Flow (PLF) in a distribution network. The results of the Wilcoxon signed-rank test on the 2014 and 2017 test functions revealed the superiority of the EPSO over nearly 80% of cases compared to the other algorithms. The obtained results of solving 10 benchmark functions by the proposed EPSO and the other six improved PSO algorithms indicated advantages of the proposed EPSO compared to the other algorithms in finding the optimal value. Meanwhile, the proposed EPSO took the average time of 0.66 s to solve the 10 test functions; it was the shortest time compared to other improved PSO algorithms. According to the results of implementing the EPSO algorithm and other algorithms with random agents for 61 times, in more than 65% of the test functions, the proposed EPSO could find the global optimal solution in a shorter time than the other algorithms. The results of solving the probabilistic load flow problem indicated 89% similarity of the results of the proposed EPSO to those of the most accurate method i.e. Monte Carlo Simulation (MCS). Comparison of the obtained results with the other algorithms as well as outcomes of several improved versions of the PSO indicated the competitive proficiency of the proposed EPSO in various optimization circumstances.

Improved probabilistic load flow method based on D‐vine copulas and Latin hypercube sampling in distribution network with multiple wind generators

In this study, a D-vine copulas modelling based probabilistic load flow (PLF) computation method is proposed, which considers the dependence among multiple wind generators. Furthermore, this method is not restricted by the type of wind speed distribution, i.e. allow random variables to comply with any types of distribution model. Copula theory plays an important role on dependency modelling. However, when high-dimensional correlation is taken into account, standard multivariate copula suffers from the problems of inflexible structure. Vine copula is flexible to build high-dimensional dependence and able to construct complicated dependence structure by applying bivariate copulas. For marginal distributions of wind speed, non-parametric model can provide a better estimation than those parametric models. An improved Latin hypercube sampling based Monte Carlo simulation method is utilised to solve PLF problems. A modified IEEE 33-node distribution system is used to conduct the numerical experiments for the accuracy and efficiency verification of the proposed PLF method, under the MatlabR2016a platform. The simulation results verify the outstanding accuracy, efficiency and robustness of the proposed PLF method.

An Improved Analytical Methodology for Joint Distribution in Probabilistic Load Flow

This paper presents a novel analytical method based on improved Gaussian mixture model (GMM) to solve the probabilistic load flow problem. The proposed method accounts for the uncertai ...

Probabilistic load flow with correlated wind power sources using a frequency and duration method

Probabilistic load flow (PLF) is an important tool in power system planning and operation. One limitation of conventional PLF is that only the probability information of random variables is obtained as a reference for related analyses. Frequency and duration information often plays an important role in power system assessment. In this study, a frequency and duration method for PLF with correlated wind power sources and loads is proposed based on Latin hypercube sampling (LHS), Nataf transformation, and Markov chains. In the proposed method, the spatial and temporal correlations among wind power sources and loads are addressed using Nataf transformation and Markov chains, which are incorporated into the LHS framework to solve the PLF problem with frequency and duration quantities. With the proposed method, not only the probability information but also the frequency and duration information of the random output variables in a PLF problem can be efficiently and accurately obtained. The performance of the proposed PLF method is verified using comparative tests on three modified test systems.

A robust data clustering method for probabilistic load flow in wind integrated radial distribution networks

Data clustering incorporated in Monte Carlo Simulation (MCS) proves efficient in Probabilistic Load Flow (PLF) of the power grids under uncertainty of renewable energy resources. Fixed cluster agents are assumed for the clusters in the investigations reported in literature. This assumption ignores the changeable characteristics of Normal Data Clustering (NDC). This implies that the agents may change in another execution of the NDC. Under such circumstances, providing precise results for the PLF during frequent executions is not practical and there is high error in the executions. This paper presents a robust data clustering (RDC) scheme to overcome the problem arising from varying results of the NDC, and provides closer solutions to MCS. The proposed RDC method obtains the average of solutions by performing numerous NDCs under a so-called normal to robust (N2R) factor so as to solve the PLF problem in wind-integrated radial distribution systems. The proposed method is applied to various IEEE test systems, and the results are discussed. The results demonstrate the efficacy of the proposed RDC method for probabilistic load flow.

Probabilistic direct load flow algorithm for unbalanced distribution networks considering uncertainties of PV and load

The increasing penetration of distributed generations such as rooftop photovoltaics (PVs) brings about great challenges to distribution network planning and operation, due to the great uncertainty and intermittence. Although many probabilistic load flow (PLF) approaches have been proposed to handle uncertainties in power systems, PLFs for unbalanced distribution networks with superior computational performance are still under demand. Therefore, this study presents a novel PLF algorithm for unbalanced distribution networks considering the uncertainties introduced by PVs and loads. Taking PVs and loads as the random inputs while three-phase voltages and network loss as the random outputs, the proposed PLF integrates the cumulant method and the Gram–Charlier expansion with the advanced direct load flow (DLF) approach to effectively address the unbalanced distribution PLF problems. Finally, the effectiveness and superior performance in terms of efficiency, accuracy, and robustness of the proposed PLF algorithm are verified by detailed simulations and compared with the Monte–Carlo Simulation method on a real Australian unbalanced distribution network.

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.

Private investor-based distributed generation expansion planning considering uncertainties of renewable generations

Distributed generation expansion planning (DGEP) is one of the most important types of planning in power distribution networks. Most of the previous studies associated with the DGEP problem have been performed from the viewpoint of network owner, while the investment, sizing and siting of Distributed Generations (DGs) are mostly taken by the private sector. Therefore, DGEP problem in the viewpoint of private investors should be taken into special consideration. In this paper, attempts are made to propose a model for DGEP problem based on the private investor's viewpoint to meet their objectives and also, take the technical constraints of the distribution network into account, at the same time. In this light, the uncertainties of the renewable DGs have also been considered in the problem and the Hong's Two Point Estimate Method (HTPEM) is used to provide a solution for probabilistic load flow problem. The method presented in this paper has been applied to the modified 18-bus distribution network which has been derived from the 30-bus IEEE system.

Probabilistic load flow calculation by using probability density evolution method

This paper presents a novel framework methodology based on the probability density evolution method (PDEM) for solving the probabilistic load flow (PLF) problem. By leveraging a constructed visual stochastic process, the joint probability density evolution equation of a system statement and random inputs is derived based on the principle of preservation of probability. The probability density function of the system statement can then be numerically solved by means of a TVD-based difference scheme. The proposed method is validated through case studies in which the active power and reactive power consumptions of buses are assumed to obey a normal distribution and Weibull distribution, respectively. The cumulative probability functions of the voltage magnitudes of buses and active power branches are computed using the PDEM with 100 samples. The mean, standard deviation, skewness, and kurtosis are also examined. The comparison to Monte Carlo of 10,000 simulations demonstrates the accuracy and efficiency of the proposed approach and verifies its suitability to solve the PLF problem.

Probabilistic Load Flow Solution Considering Optimal Allocation of SVC in Radial Distribution System

This paper proposes a solution procedure for probabilistic load flow problem considering the optimal allocation of Static Var Compensator (SVC) in radial distribution systems. Pareto Envelope-based Selection Algorithm II (PESA-II) with fuzzy logic decision maker is developed to determine the optimal location and size of SVC based on the minimum total power losses and Voltage Deviation (VD). Combined cumulants and gram-chalier expansion are used for solving the probabilistic load flow problem. The proposed algorithm is tested on 33- bus and 69-bus distribution systems. The developed algorithm gives an acceptable solution with low number of iterations and less computation cost compared with the Monte Carlo method.

Probabilistic load flow computation using saddle-point approximation

PurposeThe purpose of this paper is to present a new cumulant-based method, based on the properties of saddle-point approximation (SPA), to solve the probabilistic load flow (PLF) problem for distribution networks with wind generation.Design/methodology/approachThis technique combines cumulant properties with the SPA to improve the analytical approach of PLF calculation. The proposed approach takes into account the load demand and wind generation uncertainties in distribution networks, where a suitable probabilistic model of wind turbine (WT) is used.FindingsThe proposed procedure is applied to IEEE 33-bus distribution test system, and the results are discussed. The output variables, with and without WT connection, are presented for normal and gamma random variables (RVs). The case studies demonstrate that the proposed method gives accurate results with relatively low computational burden even for non-Gaussian probability density functions.Originality/valueThe main contribution of this paper is the use of SPA for the reconstruction of probability density function or cumulative distribution function in the PLF problem. To confirm the validity of the method, results are compared with Monte Carlo simulation and Gram–Charlier expansion results. From the viewpoint of accuracy and computational cost, SPA almost surpasses other approximations for obtaining the cumulative distribution function of the output RVs.

A Modified Nataf Transformation-based Extended Quasi-Monte Carlo Simulation Method for Solving Probabilistic Load Flow

Compared to conventional probabilistic load flow analysis, the extended probabilistic load flow can be stopped by itself when the error restriction satisfied, which is more suitable in practical use. Since the power flow results already obtained can be retained, the computational burdens of extended probabilistic load flow largely depend on the extension of samples. Much of the literature suggests that the quasi-Monte Carlo simulation method is more efficient than Latin hypercube sampling. In this article, the extended technique for quasi-Monte Carlo simulation is proposed. After that, spline reconstruction is utilized to modify the conventional Nataf transformation to obtain correlated samples by the first several orders of moments. Based on the above two techniques, a modified Nataf transformation-based extended quasi-Monte Carlo simulation method is proposed to solve probabilistic load flow problems. Compared to extended Latin hypercube sampling, the proposed method not only has higher computational efficiency but can also extend the samples by arbitrary steps. The results also show that both temporal and spatial correlations can be managed by the proposed method.

Probabilistic load flow calculation using cumulants and multiple integrals

Probabilistic load flow (PLF) is a powerful and widely used tool to address the random nature of renewable energy resources for the planning and operation of power systems. In this study, a new approach using a combination of the multiple integral method (MIM) and the cumulant method (CM) is proposed for PLF studies. The PLF problem with high dimensionality is decomposed into a bi-level problem and solved by MIM and CM to reduce the computational burden significantly while achieving satisfactory accuracy. The proposed method is applied to the IEEE 118-bus system and is compared with CM and point estimate method. Simulation results show that the proposed method achieves great computational efficiency and accuracy.

Heuristic probabilistic power flow algorithm for microgrids operation and planning

One of the basic components of future distribution networks is renewable energy resources (RER). The uncertainty in power production of renewable resources such as wind and solar as well as load is another characteristic of such networks. Conventional power flow methods may not be suitable for active distribution networks such as microgrids. In this study, a heuristic load flow method considering the effects of intermittent behaviour of RERs and load is modelled in probabilistic load flow (PLF) algorithm. The method is suitable for both radial and weakly meshed distribution networks with RER for operation and planning of microgrids. Imperialist competitive algorithm (ICA) as heuristic-based optimisation algorithm is applied to solve the PLF. Based on PLF technique, calculated parameters of the system such as bus voltages and feeders’ current are extracted as random variables. A modified version of IEEE 33-bus test system with RER is used to evaluate efficiency and capability of the algorithm. Results are compared with Monte Carlo simulation method. The probability density function and cumulative distribution function (CDF) of some network variable are compared. Based on the results, the presented approach can solve the PLF problem regardless of the type of distribution network.

Probabilistic load flow computation using univariate dimension reduction method

Hong's point estimate method shows computational advantages in evaluating the mean and standard deviation of the probabilistic load flow output variable, but it is limited to independent input random variables and behaves poorly for assessing higher order moments. This paper attempts to address these shortcomings. The Johnson system is introduced to map the probabilistic load flow problem into the independent standard normal space. The univariate dimension reduction method is employed to assess the first four moments of the output variable, whereby the cumulative distribution function is established. Testing on the Institute of Electrical and Electronics Engineers 118-bus system, the proposed method provides more accurate results than Hong's point estimate method, and requires no extra computation. Copyright © 2013 John Wiley & Sons, Ltd.

Probabilistic Load Flow in Correlated Uncertain Environment Using Unscented Transformation

As a matter of course, the unprecedented ascending penetration of distributed energy resources, mainly harvesting renewable energies, is a direct consequence of environmental concerns. This type of energy resource brings about more uncertainties in power system operation and planning; consequently, it necessitates probabilistic analyses of the system performance. This paper develops a new approach for probabilistic load flow (PLF) evaluation using the unscented transformation (UT) method. The UT method is recognized as a powerful approach in assessing stochastic problems with/without correlated uncertain variables. The capability of the UT method in modeling correlated uncertain variables is very appealing in the power system context, in which noticeable inherent correlation exists. The salient features of the UT method in probabilistic applications have been well proven in other engineering aspects. Following adaptation of the UT method for the PLF problem, three dimensionally different case studies are examined in order to inspect the performance of the proposed methodology. The obtained results are then compared with those of the Monte Carlo simulation as well as two-point estimation method with regards to both accuracy and execution time criteria.

Probabilistic load flow for voltage assessment in radial systems with wind power

Exploitation of renewable energy sources in power networks, especially wind energy, is considered an alternative for power generation as a non-polluting source with a low environmental impact and low operational cost. Hence, renewable energy has motivated the increment of penetration of Distributed Generation (DG) in distribution networks and the development of methodologies to evaluate their effects in the electrical system, which is characterized by uncertainties at all levels (generation, network and load). In this paper, a new analytical approach is explored to formulate and solve the Probabilistic Load Flow (PLF), which shows the voltage profile of a network including uncertainties of power injections and consumption. This approach is based on the linearization of load flow (LF) equations for Radial Distribution Systems (RDSs) using the probabilistic arithmetical laws. The solution is compared to Monte-Carlo Simulation (MCS), which is the most common simulation used to solve the PLF problem.

Probabilistic load flow with non-Gaussian correlated random variables using Gaussian mixture models

This study proposes the use of Gaussian mixture models to represent non-Gaussian correlated input variables, such as wind power output or aggregated load demands in the probabilistic load flow problem. The algorithm calculates the marginal distribution of any bus voltage or power flow as a sum of Gaussian components obtained from multiple weighted least square runs. The number of trials depends on the number of Gaussian components used to model each input random variable. Monte Carlo simulations are used to compare the approximations. The effect of correlation between variables is taken into consideration in both formulations. The main advantage of the Gaussian components method is that the probability density functions of any variable is directly obtained. Test results in the 14-bus system and the 57-bus system provide a broad explanation of the advantages and constraints of the approximations, particularly in presence of correlated variables.