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

In this paper, the probabilistic optimal power flow (POPF) problem is modeled as a black box uncertainty quantification problem. If distribution functions of POPF inputs are unknown, only samples are available, the multivariate Gaussian mixture model is suggested to recover the joint cumulative distribution function of POPF inputs, and a probability selection method is introduced to relate POPF inputs to an independent standard normal vector. Then, moment matching equations are proposed to characterize the uncertainty effects of POPF inputs on outputs, whereby some properties of the POPF problem are revealed, and a stratified quadrature rule (SQR) is proposed for POPF computation by combining a Hadamard matrix based quadrature rule and a discrete sine transformation matrix based quadrature rule. This newly developed SQR has a computational burden linear with the number of POPF inputs, and performs well in estimating means and standard deviation of POPF outputs. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the accuracy and efficiency of the proposed method.

The conventional Power Flow and Optimal Power Flow (OPF) consider the Power System variables as certain quantities. But the uncertain nature of real time power system variables and the increased integration of renewable energy generation lead the modern power systems to account these uncertainties in their planning and operation. This paper presents the handling of uncertainties in solving the Probabilistic Optimal Power Flow (POPF) problem in the context of operating cost minimization, by using Point Estimate Method (PEM). In this work the uncertainties related with load power, Wind and Photovoltaic (PV) are considered. And to analyze the impact of the voltage dependency nature of loads, constant current and constant impedance load models are also considered. The POPF problem is solved for IEEE-9 bus and IEEE-57 bus test systems by using PEM and the results are compared against Monte Carlo Simulation (MCS).

In the active distribution networks (ADNs), the emerging trend of high penetration of sustainable energy sources such as wind turbines, photovoltaics, and the forthcoming integration of energy storage devices and additional dynamic loads like plug‐in‐electric vehicles (PEVs) influences to change the nature of electric distribution networks. Consequently, it deteriorates power quality and reliability in terms of unacceptable voltage rise‐drop, harmonic distortion, voltage unbalance, and excessive power loss. In this context of transition into the active distribution network, the probabilistic optimal power flow (P‐OPF) in the ADN is considered an important tool that helps the distribution grid operators for defining the optimal settings of the system's control variables. However, the optimal solution of P‐OPF becomes highly complex while considering the multiple variables, time‐varying characteristics and operational uncertainties associated with sustainable resources, consumer behaviour and electricity market prices.This review aims at promoting the research and discussion of P‐OPF studies in the context of ADNs characteristics and challenges. First, we perform a data‐driven and visualized scientometric analysis of 1988 quality documents retrieved from the Scopus during the last ten years using VOSviewer software. Second, an in‐depth investigation of the recent and emerging techniques for solving the P‐OPF problem is presented. Third, a set of recommendations and highlights of future challenges for P‐OPF in ADNs are offered. The review findings showed high interest among the scientific community in sustainable energy source integration, given the recent rise in publications and citations. Also, there is still a lack of innovative optimization techniques that can adequately handle the current issues associated with P‐OPF and objectives with fair computational efficiency and accuracy. Finally, this paper offers a guide for P‐OPF researchers to navigate the recent studies considering the sustainable sources, spot the new research frontiers, and identify the most impactful countries and publishers.

This paper introduces two extenstions to the cumulant method (CM) for probabilistic optimal power flow (P-OPF) studies; the first is an enhancement to provide improved handling of limits within the P-OPF problems and the second is a way to include correlated variables. The first enhancement is termed the Limit corrected cumulant method (LCCM) which specifically addresses errors in the existing CM when limits, away from the mean solution, are encountered while solving a P-OPF problem. The LCCM approach relies on the CM to produce multiple probability density functions (PDFs) and then combines these PDFs together to generate final PDFs. The second enhancement for incorporating correlated variables into P-OPF problems is based on the composition of correlated variables from several independent ones. The proposed approaches are verified against Monte Carlo simulations (MCS) consisting of 10 000 samples and demonstrate significant improvements when compared with traditional CM results.

This paper is concerned with the probabilistic optimal power flow (POPF) calculation including wind farms with correlated parameters which contains nodal injections. The two-point estimate method (2PEM) is employed to solve the POPF. Moreover, the correlation samples between nodal injections and line parameters are generated by Cholesky Factorization method. Simulation results show that 2PEM is feasible and effective to solve the POPF including wind farms with correlated parameters, while the 2PEM has higher computation precision and consume less CPU time than Monte Carlo Simulation.

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 deduced, and its influence on line loss was analyzed. Second, a nonparametric kernel density estimation with an adaptive bandwidth was used to accurately describe the probability distribution of the PV and load, and correlation samples of the PV and load were obtained by the mixed copula function. Then an optimization model that considers system loss and static security was established, and a fast nondominated sorting genetic algorithm based on the elite strategy (NSGA-II) was used to calculate the multi-objective probability optimal power flow of the AC/DC system. Finally, a case study was performed on the modified IEEE39 bus system using measured PV and load data. We verified 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.

A probabilistic resilience assessment method for distribution system operation with probabilistic optimal power flow

Probabilistic optimal power flow in correlated hybrid wind-PV power systems: A review and a new approach

The issue of applying high performance computing (HPC) techniques to computation-intensive probabilistic optimal power flow has not been well discussed in literature. In this paper, the probabilistic convex AC OPF based on second order cone programming (P-SOCPF) is formulated. The application of P-SOCPF is demonstrated by accounting uncertainties of loads. To estimate the distributions of nodal prices calculated from P-SOCPF, two point estimation method (2PEM) is deployed. By comparing with Monte Carlo (MC) method, the accuracy of 2PEM is proved numerically. The computation efficiency of 2PEM outperforms MC significantly. In the context of large scale estimation, we propose to apply high performance computing (HPC) to P-SOCPF. The HPC accelerated P-SOCPF is implemented in GAMS grid computing environment. A flexible parallel management algorithm is designed to assign execution threads to different CPUs and then collect completed solutions. Numerical results from IEEE 118-bus and modified 1354pegase case network demonstrate that grid computing is effective means to speed up large scale P-SOCPF computation. The speed up of P-SOCPF computation is approximately equal to the number of cores in the computation node.

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 multi-slack-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.

A multivariate dimension-reduction method for probabilistic power flow calculation

Probabilistic harmonic forecasting of the distribution system considering time-varying uncertainties of the distributed energy resources and electrical loads

Most existing carbon emission management strategies only control the total carbon emission without focusing on both the regional carbon emission and the stochastic properties of the system. Correlated regional loads and unpredictable renewable energies in the power system make regional carbon emission management (RCEM) increasingly challenging and necessary. A complex multi-objective RCEM model based on probabilistic power flow (PPF) considering correlated variables is contributed in this paper. The three objective functions to be minimized are 1) the cost of electricity generated, 2) the total carbon emission, and 3) the carbon emission difference among regions which reflects the regional carbon emission imbalance from the supply side. A new clonal selection algorithm (CSA) coupled with a fuzzy satisfying decision method and an extended 2 m +1 point estimate method (PEM) is proposed to solve this multi-objective RCEM model. The proposed method is illustrated through IEEE 30-bus, IEEE 118-bus and simplified Shanghai case studies. The proposed model can help reduce the total carbon emission, control regional carbon emission, prevent probabilistic congested lines from overloading, and choose the most suitable region for wind farms (WFs).

Based on Monte Carlo and interior point method, this paper presents a probabilistic optimal power flow analysis method for the virtual power plant (VPP) containing photovoltaic (PV) generation, which can take into account both the randomness and correlation of PV generators' power outputs and loads at different locations. An optimal power flow model of VPP is firstly given to minimize the operation cost of VPP. The inverse transformation method and Cholesky decomposition technique are used to simulate the correlations between PV power outputs and loads, which obey Beta and Normal distributions respectively. Finally, the modified IEEE 30-bus test system is used to demonstrate the effectiveness and application of the presented method in the probabilistic optimal power flow analysis of VPP.

As a matter of course, the unprecedented ascending penetration of distributed energy resources (DERs), mainly harvesting renewable energies (REs), is concomitant with environmentally friendly concerns. This type of energy resources are innately uncertain and bring about more uncertainties in the power system, consequently, necessitates probabilistic analyses of the system performance. Moreover, the uncertain parameters may have a considerable level of correlation to each other, in addition to their uncertainties. The two point estimation method (2PEM) is recognized as an appropriate probabilistic method in small scale or even medium scale problems. This paper develops a new methodology for probabilistic optimal power flow (P-OPF) studies for such problems by modifying the 2PEM. The original 2PEM cannot handle correlated uncertain variables but the proposed method has been equipped with this ability. In order to justify the impressiveness of the method, two case studies namely the Wood & Woollenberg 6-bus and the Mathpower 30-bus test systems are examined using the proposed method, then, the obtained results are compared against the Monte Carlo simulation (MCS) results. Comparison of the results justifies the effectiveness of the method in the respected area with regards to both accuracy and execution time criteria.