Efficient Probabilistic Load Flow Calculation Considering Vine Copula⁃Based Dependence Structure of Renewable Energy Generation

Correlations among random variables make significant impacts on probabilistic load flow(PLF)calculation results.In the existing studies,correlation coefficients or Gaussian copula are usually used to model the correlations,while vine copula,which describes the complex dependence structure(DS)of random variables,is seldom discussed since it brings in much heavier computational burdens.To overcome this problem,this paper proposes an efficient PLF method considering input random variables with complex DS.Specifically,the Rosenblatt transformation(RT)is used to transform vine copula⁃based correlated variables into independent ones;and then the sparse polynomial chaos expansion(SPCE)evaluates output random variables of PLF calculation.The effectiveness of the proposed method is verified using the IEEE 123⁃bus system.

Probabilistic Load Flow Calculation of Power System Integrated with Wind Farm Based on Kriging Model

Because of the randomness and uncertainty,integration of large-scale wind farms in a power system will exert significant influences on the distribution of power flow.This paper uses polynomial normal transformation method to deal with non-normal random variable correlation,and solves probabilistic load flow based on Kriging method.This method is a kind of smallest unbiased variance estimation method which estimates unknown information via employing a point within the confidence scope of weighted linear combination.Compared with traditional approaches which need a greater number of calculation times,long simulation time,and large memory space,Kriging method can rapidly estimate node state variables and branch current power distribution situation.As one of the generator nodes in the western Yunnan power grid,a certain wind farm is chosen for empirical analysis.Results are used to compare with those by Monte Carlo-based accurate solution,which proves the validity and veracity of the model in wind farm power modeling as output of the actual turbine through PSD-BPA.

Probabilistic Load Flow Algorithm with the Power Performance of Double-Fed Induction Generators

Probabilistic load flow(PLF)algorithm has been regained attention,because the large-scale wind power integration into the grid has increased the uncertainty of the stable and safe operation of the power system.The PLF algorithm is improved with introducing the power performance of double-fed induction generators(DFIGs)for wind turbines(WTs)under the constant power factor control and the constant voltage control in this paper.Firstly,the conventional Jacobian matrix of the alternating current(AC)load flow model is modified,and the probability distributions of the active and reactive powers of the DFIGs are derived by combining the power performance of the DFIGs and the Weibull distribution of wind speed.Then,the cumulants of the state variables in power grid are obtained by improved PLF model and more accurate power probability distributions.In order to generate the probability density function(PDF)of the nodal voltage,Gram-Charlier,Edgeworth and Cornish-Fisher expansions based on the cumulants are applied.Finally,the effectiveness and accuracy of the improved PLF algorithm is demonstrated in the IEEE 14-RTS system with wind power integration,compared with the results of Monte Carlo(MC)simulation using deterministic load flow calculation.

Space Transformation-Based Interdependency Modelling for Probabilistic Load Flow Analysis of Power Systems

Dependence among random input variables affects importantly the results of probabilistic load flow(PLF),system economic operation,and system security.To solve this problem,the main objectiveness of the paper is to analyze the performance of several schemes for simulating correlated variables combined with the point estimate method(PEM).Unlike the existing works that considering one single scheme combined with Monte Carlo simulation(MCS) or PEM,by neglecting the correlation among random input variables,four schemes were presented for disposing the dependence of correlated random variables,including Nataf transformation /polynomial normal transformation(PINT) combined with orthogonal transformation(OT) / elementary transformation(ET).Combining with the 2m+1 approach of PEM,a space transformation-based formulation was proposed and adopted for solving the PLF.The proposed approach is applied in the modified IEEE 30-bus system while considering correlated wind generations and load demands.Numerical results show the effectiveness of the proposed approach compared with those obtained from the MCS.Results also show that the scheme of combining Nataf transformation and ET with PEM provides the best performance.

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 the𝑙-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.

Over-limit risk assessment method of integrated energy system considering source-load correlation

In an integrated energy system,source-load multiple uncertainties and correlations lead to an over-limit risk in operating state,including voltage,temperature,and pressure over-limit.Therefore,efficient probabilistic energy flow calculation methods and risk assessment theories applicable to integrated energy systems are crucial.This study proposed a probabilistic energy flow calculation method based on polynomial chaos expansion for an electric-heat-gas integrated energy system.The method accurately and efficiently calculated the over-limit probability of the system state variables,considering the coupling conditions of electricity,heat,and gas,as well as uncertainties and correlations in renewable energy unit outputs and multiple types of loads.To further evaluate and quantify the impact of uncertainty factors on the over-limit risk,a global sensitivity analysis method for the integrated energy system based on the analysis of covariance theory is proposed.This method considered the source-load correlation and aimed to identify the key uncertainty factors that influence stable operation.Simulation results demonstrated that the proposed method achieved accuracy to that of the Monte Carlo method while significantly reducing calculation time.It effectively quantified the over-limit risk under the presence of multiple source-load uncertainties.

Probabilistic Optimal Power Flow Considering the Dependence of Multiple Wind Farms Using Pair Diffusive Kernel Copula

As wind farms are commonly installed in areas with abundant wind resources,spatial dependence of wind speed among nearby wind farms should be considered when modeling a power system with large-scale wind power.In this paper,a novel bivariate non-parametric copula,and a bivariate diffusive kernel(BDK)copula are proposed to formulate the dependence between random variables.BDK copula is then applied to higher dimension using the pair-copula method and is named as pair diffusive kernel(PDK)copula,offering flexibility to formulate the complicated dependent structure of multiple random variables.Also,a quasi-Monte Carlo method is elaborated in the sampling procedure based on the combination of the Sobol sequence and the Rosen-blatt transformation of the PDK copula,to generate correlated wind speed samples.The proposed method is applied to solve probabilistic optimal power flow(POPF)problems.The effectiveness of the BDK copula is validated in copula definitions.Then,three different data sets are used in various goodness-of-fit tests to verify the superior performance of the PDK copula,which facilitates in formulating the dependence structure of wind speeds at different wind farms.Furthermore,samples obtained from the PDK copula are used to solve POPF problems,which are modeled on three modified IEEE 57-bus power systems.Compared to the Gaussian,T,and parametric-pair copulas,the results obtained from the PDK copula are superior in formulating the complicated dependence,thus solving POPF problems.

Combined Cumulant and Gaussian Mixture Approximation for Correlated Probabilistic Load Flow Studies:A New Approach

In this paper,a probabilistic load flow analysis technique that combines the cumulant method and Gaussian mixture approximation method is proposed.This technique overcomes the incapability of the existing series expansion methods to approximate multimodal probability distributions.A mix of Gaussian,non-Gaussian,and discrete type probability distributions for input bus powers is considered.Probability distributions of multimodal bus voltages and line power flows pertaining to these inputs are precisely obtained without using any series expansion method.At the same time,multiple input correlations are considered.Performance of the proposed method is demonstrated in IEEE 14 and 57 bus test systems.Results are compared with cumulant and Gram Charlier expansion,cumulant and Cornish Fisher expansion,dependent discrete convolution,and Monte Carlo simulation.Effects of different correlation cases on distribution of bus voltages and line power flows are also studied.

Unified probabilistic gas and power flow

The natural gas system and electricity system are coupled tightly by gas turbines in an integrated energy system. The uncertainties of one system will not only threaten its own safe operation but also be likely to have a significant impact on the other. Therefore, it is necessary to study the variation of state variables when random fluctuations emerge in the coupled system. In this paper, a multislack-bus model is proposed to calculate the power and gas flow in the coupled system. A unified probabilistic power and gas flow calculation, in which the cumulant method and Gram–Charlier expansion are applied, is first presented to obtain the distribution of state variables after considering the effects of uncertain factors. When the variation range of random factors is too large, a new method of piecewise linearization is put forward to achieve a better fitting precision of probability distribution. Compared to the Monte Carlo method, the proposed method can reduce computation time greatly while reaching a satisfactory accuracy.The validity of the proposed methods is verified in a coupled system that consists of a 15-node natural gas system and the IEEE case24 power system.

Cumulant-based correlated probabilistic load flowconsidering photovoltaic generation and electric vehiclecharging demand

This paper applies a cumulant-based analytical method for probabilistic load flow (PLF) assessment in transmission and distribution systems. The uncertainties pertaining to photovoltaic generations and aggregate bus load powers are probabilistically modeled in the case of transmission systems. In the case of distribution systems, the uncertainties pertaining to plug-in hybrid electric vehicle and battery electric vehicle charging demands in residential community as well as charging stations are probabilistically modeled. The probability distributions of the result variables (bus voltages and branch power flows) pertaining to these inputs are accurately established. The multiple input correlation cases are incorporated. Simultaneously, the performance of the proposed method is demonstrated on a modified Ward-Hale 6-bus system and an IEEE 14-bus transmission system as well as on a modified IEEE 69-bus radial and an IEEE 33-bus mesh distribution system. The results of the proposed method are compared with that of Monte-Carlo simulation.

Probabilistic Optimal Power Flow of an AC/DC System with a Multiport Current Flow Controller

To evaluate the impact of the randomness and correlation of photovoltaic(PV)and load on AC/DC systems with a multiport current flow controller(M-CFC),this paper proposes a probabilistic optimal power flow calculation for AC/DC systems,based on a nonparametric kernel density estimation.First,according to the M-CFC model,the DC power flow calculation method with M-CFC was inferred,and its influence on line loss was analyzed.Second,a nonparametric kernel density estimation with an adaptive bandwidth is used to accurately describe the probability distribution of the PV and load,and correlation samples of the PV and load are obtained by the mixed copula function.Then an optimization model that considers system loss and static security is established,and a fast nondominated sorting genetic algorithm based on the elite strategy(NSGA-II)is used to calculate the multi-objective probability optimal power flow of the AC/DC system.Finally,a case study is performed on a modified IEEE39 bus system using measured PV and load data.We verified that the nonparametric kernel density estimation with an adaptive bandwidth can better adapt to random component uncertainty,and M-CFC can improve the static security of the system.