Probabilistic Optimal Power Flow Computation Using Stratified Quadrature Rule

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).

Wind farms commonly cluster in regions rich in wind resources. Thus, correlation of wind speeds from different wind farms should not be ignored when modeling a power system with large wind energy penetration. This paper proposes a probabilistic optimal power flow (POPF) technique based on the quasi-Monte Carlo simulation (QMCS) considering the correlation of wind speeds using copula functions. In this paper, a copula function is used to model the dependent structure of random wind speeds and their forecast errors. QMCS is employed in the sampling procedure to reduce computation burden. The proposed method is applied in probabilistic power flow (PPF). Furthermore, the PPF is used in the POPF problem that aims at minimizing the expectation and downside risk of fuel cost simultaneously. Simulation studies are conducted on a modified IEEE 118-bus power system with wind farms integrated in two areas, and the results show that the accuracy and efficiency are improved by the proposed method.

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.

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

This paper presents an application of a two-point estimate method (2PEM) to account for uncertainties in the optimal power flow (OPF) problem in the context of competitive electricity markets. These uncertainties can be seen as a by-product of the economic pressure that forces market participants to behave in an “unpredictable” manner; hence, probability distributions of locational marginal prices are calculated as a result. Instead of using computationally demanding methods, the proposed approach needs <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2n$</tex> runs of the deterministic OPF for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> uncertain variables to get the result in terms of the first three moments of the corresponding probability density functions. Another advantage of the 2PEM is that it does not require derivatives of the nonlinear function used in the computation of the probability distributions. The proposed method is tested on a simple three-bus test system and on a more realistic 129-bus test system. Results are compared against more accurate results obtained from MCS. The proposed method demonstrates a high level of accuracy for mean values when compared to the MCS; for standard deviations, the results are better in those cases when the number of uncertain variables is relatively low or when their dispersion is not large.

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.

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.

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.

Probabilistic optimal power flow in islanded microgrids with load, wind and solar uncertainties including intermittent generation spatial correlation

This paper proposes a new optimization methodology to solve the AC optimal power flow (OPF) problem considering renewable energy sources (RES). The formulation of the OPF problem comprises the minimization of power generation costs and gas emissions considering a set of operational and physical constraints. This minimization is achieved through controlling power dispatch generators, position changing of the tap transformers, and controllable reactive shunt compensation. RES and demand uncertainties are modeled using the (2m+1) point-estimate method. The mathematical formulation of the OPF problem is a mixed-integer nonlinear programming multiobjective model. A matheuristic algorithm is proposed to solve this problem efficiently, combining a classic nonlinear OPF model and the Variable Neighborhood Descent (VND) metaheuristic algorithm. The potential of the proposed algorithm is shown through numerical experiments carried out using the IEEE 300-bus systems.

This paper proposes a stochastic framework to enhance the reliability and operability of wind integration using energy storage systems. A genetic algorithm (GA)-based optimization approach is used together with a probabilistic optimal power flow (POPF) to optimally place and adequately size the energy storage. The optimization scheme minimizes the sum of operation and interrupted-load costs over a planning period. Historical wind speed, load and equipment failure data are used to stochastically model the wind generation, load and equipment availability. Using Fuzzy C-Means (FCM) clustering, wind and load samples are grouped into 40 clusters of days with similar sample points to account for seasonal variations. The IEEE 24-bus system (RTS) is used to evaluate the performance of the proposed method and realize the maximum achievable reliability level. A cost-benefit analysis compares storage technologies and conventional gas-fired alternatives to reliably and efficiently integrate different wind penetration levels and determine the most economical design. Storage distribution and its effect on performance enhancement of wind integration are examined for higher wind penetrations.

Summary Integration of wind resources into the power systems has increased the technical and financial concerns in transmission expansion planning. In this paper, a stochastic structure is demonstrated for transmission expansion planning with allocation of fixed series compensation under wind and load uncertainties. Fixed series compensations have the ability to increase the transfer capacity of transmission lines. However, their significance benefit for transmission expansion planning is their higher effectiveness in power dispatching, which results in lower investment costs, compared with planning without fixed series compensations. The proposed planning model simultaneously optimizes targets such as the investment cost, the congestion cost, and the expected energy not supplied in each stage. The presented planning model is a complicated nonlinear optimizing problem. For introducing group of nondominated optimal solutions, multi-objective gray wolf optimizer algorithm based on probabilistic optimal power flow is used. Then, fuzzy approach is used to lead to the best response. For demonstrating the feasibility and susceptibilities of the proposed algorithm, the planning methodology has been illustrated on the IEEE 24-bus and Colombian test systems. The results acquired from the simulations monitor capability of the proposed structure in order to satisfy various planning indices.