Quasi-Monte Carlo Based Probabilistic Optimal Power Flow Considering the Correlation of Wind Speeds Using Copula Function

Abstract:The traditional cumulant method (CM) for probabilistic optimal power flow (P-OPF) needs to perform linearization on the Karush-Kuhn-Tucker (KKT) first-order conditions, therefore requiring input variables (wind power or loads) varying within small ranges. To handle large fluctuations resulting from large-scale wind power and loads, a novel P-OPF method is proposed, where the correlations among input variables are also taken into account. Firstly, the inverse Nataf transformation and Cholesky decomposition are used to obtain samples of wind speeds and loads with a given correlation matrix. Then, the K-means algorithm is introduced to group the samples of wind power outputs and loads into a number of clusters, so that in each cluster samples of stochastic variables have small variances. In each cluster, the CM for P-OPF is conducted to obtain the cumulants of system variables. According to these cumulants, the moments of system variables corresponding to each cluster are computed. The moments of system variables for the total samples are obtained by combining the moments for all grouped clusters through the total probability formula. Then, the moments for the total samples are used to calculate the corresponding cumulants. Finally, Cornish-Fisher expansion is introduced to obtain the probability density functions (PDFs) of system variables. IEEE 9-bus and 118-bus test systems are modified to examine the proposed method. Study results show that the proposed method can produce more accurate results than traditional CM for P-OPF and is more efficient than Monte Carlo simulation (MCS).

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“…The standardized vector W can be transformed to a linear combination of an uncorrelated vector U by (15):…”

Section: The Methods Of Handling Correlations Among Input Variablesmentioning

confidence: 99%

A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables

Deng

Zhang

2017

Energies

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“…The first is to reduce the number of samples by using a smaller subset to reflect the probabilistic properties of the PPF. The representative sampling methods include importance sampling [11], Latin hypercube sampling [12], Latin supercube sampling [13] and Quasi-Monte sampling [14]. However, even the reduced sample set can still be fairly large since it has to capture complex features in uncertainties associated with renewables and load demands.…”

Section: B Literature Review and Backgroundmentioning

confidence: 99%

“…The difference of the parameter updating between the loss functions in (3) and (5) shows up in (11). From (6) and (7), we can further decompose it into (14) and (17) In equations (16) and (17), Y is the output vector of the PPF; ̂ is the (normalized) output of the DNN; ̂ is the antinormalized value of ̂; and ⊙ is a Hadamard multiplier.…”

Section: B Loss Function Design Based On the Power Flow Equationsmentioning

confidence: 99%

Fast Calculation of Probabilistic Power Flow: A Model-Based Deep Learning Approach

Yang

et al. 2020

IEEE Trans. Smart Grid

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Probabilistic power flow (PPF) plays a critical role in the analysis of power systems. However, its high computational burden makes practical implementations challenging. This paper proposes a model-based deep learning approach to overcome these computational challenges. A deep neural network (DNN) is used to approximate the power flow calculation process, and is trained according to the physical power flow equations to improve its learning ability. The training process consists of several steps: 1) the branch flows are added into the objective function of the DNN as a penalty term, which improves the generalization ability of the DNN; 2) the gradients used in back propagation are simplified according to the physical characteristics of the transmission grid, which accelerates the training speed while maintaining effective guidance of the physical model; and 3) an improved initialization method for the DNN parameters is proposed to improve the convergence speed. The simulation results demonstrate the accuracy and efficiency of the proposed method in standard IEEE and utility benchmark systems. Index Terms-Model-based deep learning, initialization methods, deep neural networks, probabilistic power flow.

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“…where, () l e  is the encoding function of lth DAE, l=1, 2, …, n, and n is the number of DAE in SDAE, ( 1) n e   is the encoding function of top layer.…”

Section: A Structure Of Sdae-based Opfmentioning

confidence: 99%

Fast Calculation of Probabilistic Optimal Power Flow: A Deep Learning Approach

Yang

et al. 2019

2019 IEEE Power &Amp; Energy Society General Meeting (PESGM)

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Probabilistic optimal power flow (POPF) is an important analytical tool to ensure the secure and economic operation of power systems. POPF needs to solve enormous nonlinear and nonconvex optimization problems. The huge computational burden has become the major bottleneck for the practical application. This paper presents a deep learning approach to solve the POPF problem efficiently and accurately. Taking advantage of the deep structure and reconstructive strategy of stacked denoising auto encoders (SDAE), a SDAEbased optimal power flow (OPF) is developed to extract the highlevel nonlinear correlations between the system operating condition and the OPF solution. A training process is designed to learn the feature of POPF. The trained SDAE network can be utilized to conveniently calculate the OPF solution of random samples generated by Monte-Carlo simulation (MCS) without the need of optimization. A modified IEEE 118-bus power system is simulated to demonstrate the effectiveness of the proposed method.Index Terms--Probabilistic optimal power flow (POPF), deep learning, stacked denoising auto encoders (SDAE), Monte-Carlo simulation (MCS).I.

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Quasi-Monte Carlo Based Probabilistic Optimal Power Flow Considering the Correlation of Wind Speeds Using Copula Function

Cited by 189 publications

References 31 publications

A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables

A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables

Fast Calculation of Probabilistic Power Flow: A Model-Based Deep Learning Approach

Fast Calculation of Probabilistic Optimal Power Flow: A Deep Learning Approach

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