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

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

This study proposes a Bayesian network model for nonlinear dependence of wind speed, solar irradiation, and load. The probabilistic load flow calculation based on a Bayesian network model can effectively obtain the probability characteristics of probabilistic load flow solutions (such as the probability density function of the bus voltage and branch flow) considering the correlation of random variables. First, the wind speed, solar irradiation, and load time series are converted to random variables by the kernel density estimation method, and the probability values of random variables are obtained. Applying the probability value of a random variable as the input data of the Bayesian network, the correlation model is established through structure learning based on the Monte Carlo Markov chain method and parameter learning based on the maximum-likelihood estimation method. Sampling from the Bayesian network, discrete probability values are obtained, and they are transformed to continuous probability values by interpolation. Then, the correlation samples of random variables are obtained by cumulative probability distribution inverse transformation of continuous probability values. Compared to the C-vine copula method and Latin hypercube sampling with modified alternating projections, the proposed Bayesian network model can better present the nonlinear dependence among wind speed, solar irradiation, and load. Finally, the proposed method is verified by probabilistic load flow calculation of the IEEE 69-bus distribution system.

In order to accurately describe the influence of the random factors in distribution systems on the probabilistic load flow (PLF), a PLF method of distribution system containing wind power and EVCL based on a stochastic response surface method (SRSM) is put forward. The proposed method can sufficiently consider the randomness of wind speed as well as uncertainty of basic load and EVCL. Furthermore, the orthogonal transformation is introduced to deal with the correlated wind speeds among different wind farms. Comparing with the traditional Monte Carlo simulation method with simple random sampling (MSRS), the proposed method can solve problems that they are facing, like more number of calculation times, long simulation time. Simulation results on IEEE14 and IEEE30 bus system demonstrate the validity, accuracy, rapidity and practicability of proposed method. In contrast to the MSRS, the proposed method is of higher computational efficiency and better simulation accuracy.

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.

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.

The multiple solution analysis of probabilistic load flow

In this work we present and apply a new Hyperspectral image classification method based on Dirichlet Process Gaussian mixture models (also known as infinite Gaussian mixture models - IGMMs). Since this approach is a non-parametric Bayesian method, we circumvent the problem of model selection which is unavoidable and often difficult when employing traditional parametric Gaussian mixture models (GMM). In order to infer model parameters from observations, we adopt a Gibbs Sampling method to sample the posterior distributions of these parameters. In the preprocessing step, we use Local Fisher's Discriminant Analysis (LFDA) for dimension reduction since we expect it to preserve the multi-modal, non-Gaussian structure of the hyperspectral data. We compared our proposed IGMM based classification method to the existing state-of-the-art classification methods using popular hyperspectral imagery datasets. To simulate challenging real-world sensing environment, we mix every pixel with background and study the performance of these classification approaches. Our experiments show that the proposed LFDA-IGMM method and GMM method have almost the same performance for most cases, given “pure” data - with mixed pixels, LFDA-IGMM outperformed GMM for sub-pixel classification. Both methods outperformed the other commonly used classification approaches when the training sample size is relatively large. Also, the IGMM based method is less sensitive to pixel mixing than other methods.

Air quality has always been a hot issue of concern to the people, the environmental protection department and the government. Among the massive air quality data, abnormal data can interfere with subsequent experiments and analysis. Therefore, it is necessary to detect abnormal data to improve the accuracy of the data. However, traditional air outlier detection methods require at least one year's data to make inferences about air quality. This paper firstly analyzes the characteristics of air quality big data, and then proposes a framework based on Bayesian non-parametric clustering, namely Dirichlet Process (DP) clustering framework, to realize the outlier detection of air quality. The framework optimizes Gaussian mixture model into infinite Gaussian mixture model according to the results of data analysis, and uses neural network to cluster the data processed by infinite Gaussian mixture model, which effectively improves the clustering accuracy and avoids the need of collecting a large number of training data.

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 ...

Robotic agents that can explore and sample in a completely unsupervised fashion could greatly increase the amount of scientific data gathered in dangerous and inaccessible environments. Our application is imaging the benthos using an autonomous underwater vehicle with limited communication to surface craft. Robotic exploration of this nature demands in situ data analysis. To this end, this paper presents results of using a Gaussian Mixture Model (GMM), a Hidden Markov Model (HMM) filter, an Infinite Gaussian Mixture Model (IGMM) and a Variation Dirichlet Process model (VDP) for the classification of benthic habitats. All of the models are trained using unsupervised methods. Furthermore, the IGMM and VDP are trained without knowing the the number of classes in the dataset. It was found that the sequential information the HMM filter provides to the classification process adds lag to the habitat boundary estimates, reducing the classification accuracy. The VDP proved to be the most accurate classifier of the four tested, and also one of the fastest to train. We conclude that the VDP is a powerful model for entirely autonomous labelling of benthic datasets.

The frequency control of an islanded microgrid is consisted of primary frequency control (PFC) and secondary frequency control (SFC). This paper proposes a novel cooperative application of Battery Energy Storage System (BESS), Photovoltaic (PV) systems and LED lighting loads (LEDLLs) to quickly intercept frequency deviation in the stage of PFC. The slow responding distributed generators handle SFC by restoring the frequency to its nominal value. For participation in PFC, in the case of power shortage, LEDLLs decrease their power consumption, and in the case of power surplus, the PVs decrease their power generation. While PVs and LEDLLs are participating in PFC with their maximum capability, the BESS frequency controller is tuned to inject/absorb enough power in both cases to keep the frequency within safe limits. In this paper for battery sizing, instead of using its nominal power, a modified overloading capability to fast discharge/charge is used to reduce the required battery size for PFC that also prevents damaging it during fast discharge/charge for PFC. The proposed approach is evaluated on the CIGRE low voltage microgrid using MATLAB/Simulink software. Simulation results show that besides the overloading characteristics, the cooperative utilization of BESS, PVs and LEDLLs also reduce the battery size.

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

Advanced Probabilistic Load Flow Technique in Consideration of Non-Gaussianity and Nodal Correlation of Input Variables

In order to explore the influence of multiple wind turbines with correlation on distribution network and reasonably evaluate the operation state of distribution network, a probabilistic load flow calculation method based on orthogonal transformation and cumulant method is proposed. Firstly, an appropriate Copula function is chosen to depict the correlation of wind turbines outputs using rank correlation coefficient and squared Euclidean distance. Secondly, orthogonal transformation is used to transform the correlated power outputs into uncorrelated variables. Finally, the cumulant method combined with Cornish-Fisher expansion is applied to calculate the probability distributions of node voltage and branch current, and the risk assessment of the operating state of distribution network is carried out. The accuracy and rapidity of proposed method are verified by comparison with Monte Carlo method on a typical 34-node system.

To avoid the calculation of limit access capacity of photovoltaic (PV) source is incorrect due to the consideration of the occurrence of a small probability event, this paper proposes a method to calculate the limit access capacity of photovoltaic (PV) source based on the chance onstraints of voltage deviation on the basis of considering the randomness of PV power and load. Maximizing limit access capacity of PV is regarded as the goal, the voltage deviation is regarded as the chance constrained condition, the calculation model of limit access capacity of PV is established. Then, using the LHS and Newton-Raphson to calculate the probabilistic load flow (PLF), the statistical properties of node voltage are obtained. Calculation results of PLF and quantum genetic algorithm are used to solve the model. By comparing the results of optimization with heuristic method, the correctness and validity of the proposed method are verified.