Probabilistic Load Flow Method Research Articles

Frequency-Control-Aware Probabilistic Load Flow: An Analytical Method

Probabilistic load flow (PLF) calculation, as a fundamental tool to analyze transmission system behavior, has been studied for decades. Despite a variety of available methods, existing PLF approaches rarely take system control into account. However, system control, as an automatic buffer between the fluctuations in random variables and the variations in system variables (e.g., nodal voltages), has a significant impact on the final PLF result. To consider control actions' influence, this paper proposes the first analytical PLF method for the transmission grid that takes into account primary and secondary frequency controls. This method is based on a high-precision linear power flow model, whose precision is even further improved in this paper by an original correction approach. This paper also proves that if the joint probability distribution (JPD) of random variables is expressed by a Gaussian mixture model (GMM), then the JPD of system variables is an infinite GMM. By leveraging this proposition, the proposed method generates the joint PLF of the whole system at the quasi-steady and steady states. The high accuracy and satisfactory efficiency of this method are verified on multiple test systems with different percentages of renewables.

An improved probabilistic load flow in distribution networks based on clustering and Point estimate methods

Clustering approaches are one of the probabilistic load flow (PLF) methods in distribution networks that can be used to obtain output random variables, with much less computation burden and time than the Monte Carlo simulation (MCS) method. However, a challenge of the clustering methods is that the statistical characteristics of the output random variables are obtained with low accuracy. This paper presents a hybrid approach based on clustering and Point estimate methods. In the proposed approach, first, the sample points are clustered based on the k-means method and the optimal agent of each cluster is determined. Then, for each member of the population of agents, the deterministic load flow calculations are performed, and the output variables are calculated. Afterward, a Point estimate-based PLF is performed and the mean and the standard deviation of the output variables are obtained. Finally, the statistical data of each output random variable are modified using the Point estimate method. The use of the proposed method makes it possible to obtain the statistical properties of output random variables such as mean, standard deviation and probabilistic functions, with high accuracy and without significantly increasing the burden of calculations. In order to confirm the consistency and efficiency of the proposed method, the 10-, 33-, 69-, 85-, and 118-bus standard distribution networks have been simulated using coding in Python® programming language. In simulation studies, the results of the proposed method have been compared with the results obtained from the clustering method as well as the MCS method, as a criterion.

Probabilistic approach to investigate the impact of distributed generation on voltage deviation in distribution system

The increasing penetration of renewable-based generation in the distribution network adds randomness to the bus voltage due to its stochastic characteristics. Consequently, the distribution network faces voltage limit violations at multiple buses. In this scenario, the traditional voltage control method fails to maintain the voltage profile of the network due to its slow response time. A fast and dynamic voltage control technique can be achieved by utilizing the voltage-controlling capability of available DGs in the network with the information about the impact of DG-integrated bus on change in voltage at any bus in the network. Evaluating this information using the traditional probabilistic load flow method is computationally inefficient for real-time applications. Therefore, in this paper, a principal component analysis (PCA)-based novel probabilistic voltage sensitivity index (PVSI) is proposed. It ranks the DG-integrated buses by evaluating the impact of power perturbation at DG-integrated buses on changes in voltage at any bus in the network. The PVSI is analytically derived to reduce the computation time for ranking. Further, the proposed PVSI is validated on the 69-bus and the 141-bus distribution systems by comparing it with the traditional Monte Carlo simulation (MCS) and joint differential entropy (JDE) in terms of accuracy and computation time. From the simulation, it is observed that PVSI gives a similar ranking as both methods in less computation time. Due to the advantage in computation time, the proposed method enables the distribution system operator (DSO) to use this information for fast and dynamic voltage control in a real-time scenario.

Three-Phase Second-Order Analytic Probabilistic Load Flow With Voltage-Dependent Load

This paper proposes a fully analytic second-order probabilistic load flow (PLF) method to realize an accurate and fast three-phase load flow analysis considering unbalanced uncertainties from voltage-sensitive loads and photovoltaic (PV) generation in distribution systems. The load flow equations are modelled by an accurate quadratic expression based on the bus injection model (BIM). To work at the distribution level, the voltage dependence of the load is considered based on the constant impedance, constant current, and constant power (ZIP) model, with the ZIP parameters acting as more realistic random variables. The PV generation is also considered as a constant power model. The uncertainties are modelled in time series as conditional probabilities, reducing the complexity of their probability distribution functions (PDFs). The PLF is modelled in a fully analytic second-order stochastic formulation, which can accurately and easily handle the PDFs of voltage and current by computing the first two moments. The computation is accelerated by an analytical calculation of the quadratic coefficients over the ZIP parameters. Case studies on a practical distribution network show the significance of considering the voltage-dependent load model and the high accuracy of the proposed method.

Probabilistic Load Flow of an Islanded Microgrid with WTGS and PV Uncertainties Containing Electric Vehicle Charging Loads

With the emphasis of utilities on decentralized power systems due to the considerable growth of renewable energy resources (RES), microgrids (MGs) as controllable small grids are need of an hour. For an objective analysis of microgrids with large grid parity of renewable energy sources, probabilistic methods are required along with the already established deterministic methods. A Gauss quadrature-based probabilistic load flow (GQPLF) method for microgrid has been proposed along with intermittent generation and uncertain loads, including electric vehicles (EVs). The distributed generation consists of wind turbine generation systems (WTGS) and photovoltaic systems (PVs) connected to 14 bus microgrid systems with six droop-controlled distributed generators. Extensive simulations have been carried out on a 14-bus microgrid with distributed generation and uncertain loads, and results are compared with the benchmark Monte Carlo simulation (MCS) method, which validates the efficacy of the proposed method. The time taken by the proposed GQPLF for the 14-bus microgrid system simulation is 60.9 seconds, which is significantly less compared to the benchmark Monte Carlo simulation method, in which time taken for a 14-bus microgrid system with 10,000 simulations is 9318.7 seconds. Comparing the proposed GQPLF method with the benchmark method validates its efficacy and efficiency.

RETRACTED: Renewable source uncertainties effects in multi-carrier microgrids based on an intelligent algorithm

High volatility and unpredictable behavior of renewable energy sources provide various challenges in the operation of power systems with maintaining the system's reliability. To solve the challenges of grid-connected renewable energy sources, microgrids (MGs) can play an essential role due to their tuning capability and flexibility. The fast growth of power-to-gas technology causes to support renewable energy integration and transforms the extra renewable generation to combined normal gas at a suitable time. The optimal type, size, and commissioning year of the accessible distributed energy resources through the economic application of nominated resources are evaluated to be applied in the suggested multi-carrier microgrid. This study focuses on the simultaneous operation of multi-carrier MGs and their equipment as an optimization problem. Small-scale energy sources and load demand in MGs are modeled to reduce costs of exchange between the main grid and MGs, as well as the load balance between MGs and load demand. Given that it is difficult to solve the desired problem considering various constraints, the developed meta-heuristic whale optimization algorithm (WOA) is used along with adaptive weighting coefficients and a local search operator. Moreover, this model investigates the unstable behavior of loads, renewables, and electricity prices using the probabilistic load flow method (PLF), considering multiple carriers simultaneously to match the real world. In this paper, MGs with multiple-carriers exchange energy with the upstream grid. The energy exchange between MGs is also considered. The results of simulations performed on the grid consisting of three interconnected multi-carrier MGs confirmed the effectiveness of the suggested model. Obtained numerical results prove the validity of the suggested multi-carrier microgrid expansion planning issue by economic, environmental, and technical points.

Comparison of solving probabilistic optimal power flow methods in the presence of wind and solar sources

Power system control and operation studies have been experienced essential changes due to increasing the penetration of uncertain renewable energy resources. One of the most critical issues in this regard is the optimal power flow (OPF). As a result, the deterministic methods do not have the capability of different modeling uncertainties raised in new power systems, and there is a need to investigate the effective models in this regard. This paper focuses on probabilistic optimal power flow (POPF) methods applied to power systems with uncertain wind and Photovoltaic power generations. In this paper, the Monte Carlo simulation (MCS) and analytical methods such as the three-point estimation method (3PEM), unscented transformation (UT), and Interior-Point method (IPM) are applied to solve the probabilistic optimal power flow problem. MCS has been widely applied as a framework to assess the ability of analytical methods. The mentioned techniques are applied to a sample case study extracted from the IEEE 300-bus system. The main contribution of this work is the comparison of analytical methods concluding 3PEM, UT, and IPM, and with MCS as well. The obtained results on the studied networks by the suggested techniques show that in the 3PEM, due to the limited points, the optimal solution is achieved in less calculation time than the other methods. From another perspective, the voltage changes at the buses would be more stable in the IPM. Also, this method is much faster than the MCS method in terms of the convergence rate. To show the effectiveness of the mentioned methods, this paper presents probabilistic load flow method based on the statistical methods to deal with fluctuations because of large-scale renewable energy integration. The proposed methods are validated on the improved industrial 85–bus system of Kermanshah region (in the west of Iran) by adding solar and wind farms.

Research on the Probabilistic Load Flow Method of Power System Considering Wind Power Integration

Abstract Due to the uncertainty of renewable energy and load, probabilistic power flow as an effective method to analyse the probabilistic properties of power systems. In this paper, a linearized ac probabilistic power flow model is established, and Monte Carlo simulation method is used to analyze the probabilistic power flow of the power grid containing wind farms. By connecting wind farm to node 13 of IEEE14-bus system, the voltage accumulative distribution curves of node 4, node 5 and node 13 are obtained. The results show that the random output of wind farm increases the fluctuation of node voltage, and the fluctuation phenomenon of node voltage becomes serious as the distance of the access point of the wind farm approaches. By connecting wind farms at node 4 and node 14, the voltage accumulative distribution curves of access nodes were obtained. Through comparison, it is concluded that the access of wind farm has a small effect on the voltage of PQ nodes close to PV nodes, but has great influence on the voltage of PQ nodes away from PV nodes. By adding wind farms at node 13, the active power cumulative distribution curves of branch 2-5, branch 3-4, branch 5-6 and branch 6-13 were obtained. By comparison, the active power flow of different branches of the system is affected differently after access to the wind farm. The fluctuation of the active power flow of the branch becomes serious with the shortening of the distance from the access point of the wind farm. The simulation analysis can provide more comprehensive and useful information for the planning and operation personnel of power system including wind farm.

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.

C-Vine Copula 기반의 풍력 발전 투입을 위한 확률론적 조류 계산

Owing to the increase in renewable energy, the power system is exposed to more variability and uncertainty, which increases the difficulty in analyzing the power systems. Thus, an accurate power system analysis considering the stochastic characteristics of renewable energy is required. In this paper, we propose a Monte-Carlo based probabilistic load flow method that takes into account the statistical dependencies between the power generation from the multiple wind sites. To this end, the vine copula method was employed to capture their dependencies and the marginal distribution of the power generation from the multiple wind sites is modeled using a 3-parameter Weibull distribution. In addition, we suggested the chance-constrained optimization problem for properly allocating a target wind power capacity to the buses. From the numerical experiment conducted on the IEEE 39-bus system, it has been demonstrated that the proposed method can more accurately identify the potential risk with the modeling of the dependencies between the power generation from the multiple wind sites as compared to the deterministic approach or probabilistic load flow based on the random sampling method.

An Improved Integrated Cumulant Method by Probability Distribution Pre-Identification in Power System With Wind Generation

The increase of wind generation (WG) has challenged the conventional way of probabilistic load flow (PLF) calculation. A reliable and efficient PLF method is required to face the stochastic nature of various power systems with WG. Firstly, the paper analyzes several typical cumulant methods (CMs) for PLF, such as Gram-Charlier expansion of type A (GCA), Gram-Charlier expansion of type C (GCC), and maximum entropy (ME). Then, an improved integrated CM by probability distribution pre-identification is proposed for power systems with WG based on doubly fed induction generations (DFIGs). The skewness and kurtosis are used as probability distribution pre-identification indices in the CM framework. Meanwhile, the influence of the DFIG control strategy on reactive power is considered in the load flow model and the moment calculation. Finally, the accuracy and efficiency of the proposed method are validated with the IEEE test system. In various scenarios, suitable CM is selected and applied to the PLF based on pre-identifying distribution characteristics. Results reveal that probabilistic density functions (PDFs) of bus voltages and line flows obtained by the proposed method have both accuracy and efficiency.

Novel fast forecasting method for nodal voltage violations

With the increase penetration of new energy and electric vehicles (EVs), the nodal voltage violations (NVVs) problem is widely concerned. Voltage predictive regulation methods can prevent NVVs and improve the performance of traditional voltage control. A fast and reliable forecasting method for NVVs is a key requirement to support this application. However, it is time-consuming to forecast NVVs using a Monte Carlo simulation-based probabilistic load flow method and the traditional deterministic method based on load forecasting has low reliability due to it not considering the uncertainty of load forecasting errors. A novel fast forecasting method for NVVs with shorter computational time and higher reliability is proposed. In the proposed method, the uncertainty of the load forecasting is taken into account and the impact of active/reactive power change on NVVs can be quantified like the traditional deterministic method. Thus, this method can assist in correcting voltage regulation plans so that voltage constraints can be satisfied as much as possible. Finally, the IEEE 69-bus test system and real historical load series are used to validate the effectiveness of the proposed method.

Privacy-Preserving Distributed Probabilistic Load Flow

In a multi-regional interconnected grid, the probabilistic load flow (PLF) of any region cannot be calculated individually but should consider the uncertainties introduced in other areas. Accordingly, the topologies, loads, and generations of every region are needed. Although the renewable generation data could be assumed as publicly known, some regional independent system operators (ISOs) still would not share important parameters with others. This motivates the development of a privacy-preserving distributed (PPD) PLF method. The challenge is to identify the mapping between the regional flows and uncertain power injections across regions without full information about the entire grid. The main idea of this paper is to respectively calculate the coefficient matrix and constant vector of the mapping: for the former, a PPD accelerated projection-based consensus algorithm is proposed; for the latter, a privacy-preserving accelerated average consensus algorithm is leveraged. Consequently, a PLF method is derived for each ISO to analytically obtain its regional joint PLF in a distributed way without sharing parameters - the key contribution of this paper. Experiments on the 118- and 1354-bus systems demonstrate that this method can generate the same results as the corresponding centralized method, and has satisfactory accuracy compared with frequently used PLF methods.

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.

Probabilistic Load Flow Based on Holomorphic Embedding, Kernel Density Estimator and Saddle Point Approximation Including Correlated Uncertainty Variables

Probabilistic Load Flow (PLF) methods have been extensively regarded due to the growth of uncertainty and renewable energies penetration in power systems. Given that, the current study aims to propose two novel probabilistic load flow methods based on holomorphic embedding method. Holomorphic method, unlike other iterative methods, introduces a nonlinear equation solver in an independent class, recursively. To that end, Kernel Density Estimator (KDE) and Saddle Point Approximation (SPA) methods are used to efficiently estimate probabilistic characteristics of load flow outputs. The correlations between random input variables with non-normal/normal probability distributions have been considered. The proposed algorithms have been examined on the modified IEEE 14- and 118-bus systems. Finally, to consider the uncorrelated and correlated conditions, the researcher compared the simulation results of the two proposed methods with some well-known and published probabilistic iterative load flow methods such as Monte Carlo Simulation (MCS), Parzen Window (PW), Diffusion method, and 2n+1 Point Estimate Method (PEM). MCS is always a dependable solution but time-consuming that make it useless for large power systems. In the present study, the accurate results obtained from MCS are regarded as a reference. The comparison of results shows high efficiency and low computational burden of proposed methods, and also the accuracy of these methods in density function estimation of output random variables.

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.

Probabilistic Load Flow Analysis Based on Sparse Polynomial Chaotic Expansion

To aim at the deficiencies of the traditional second-order polynomial chaotic expansion with more input variables and larger calculation quantities applied in power systems probabilistic load flow (PLF) calculation, an improved PLF method is proposed based on sparse polynomial chaotic expansion theory in this paper. In this method, to obtain the sparse expression of the polynomial chaotic expansion (PCE) and reduce calculation quantity, the quadratic cross terms in the second-order PCE are eliminated by Sobol sensitivity analysis. And then, Nataf transformation is used to control the correlation among non-normal input variables. Through comparison on computational efficiency between probabilistic collocation point method and random sampling method in the sparse polynomial chaotic expansion, the probabilistic collocation point method is selected to suit samples allocation. By applying in IEEE-9, IEEE-30, and IEEE-118 standard test systems, respectively, the results show that the proposed method can dramatically reduce the deterministic PLF calculations quantity compared with the traditional stochastic response surface method (SRSM), and possesses the same accuracy as Monte Carlo method, and is an effective solving method for some high-dimensional input variable systems.

Probabilistic Load Flow Based on Bivariate Dimension Reduction and Johnson System

A probabilistic load flow (PLF) is an effective tool that helps describe the uncertainty of power system operation. However, when confronting random variables with non-Gaussian distributions and highly discrete characteristics, existing PLF methods have difficulty balancing efficiency and accuracy. Therefore, a novel approach based on bivariate dimension reduction (BDR) and the Johnson system is proposed herein. BDR is used to estimate the moments of output random variables (ORVs). Because BDR considers the joint effect of input random variables, it significantly reduces the estimation error for high-order moments in particular. In addition, a strategy to improve BDR efficiency is proposed. The Johnson system is used to obtain the probability distributions of ORVs as it has better adaptability and accuracy than the series expansion method. Case studies including comparisons between this approach and others found in the literature were conducted, and the results obtained showed that the proposed method has better performance than previous approaches.

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.