Probabilistic Optimal Power Flow Research Articles

Hierarchical Blocking Control for Mitigating Cascading Failures in Power Systems with Wind Power Integration

The increasing uncertainty of wind power brings greater challenges to the control for mitigation of cascading failures. In order to minimize the risk of cascading failures in large-scale wind power systems at a lower economic cost, a multi-stage blocking control model is proposed based on sensitivity analysis. Firstly, the propagation mechanism of cascading failures in power systems with wind power integration is analyzed, and the propagation path of such failures is predicted. Subsequently, sensitive lines that are prone to failure are identified using the power sensitivity matrix, taking into account the effects of blocking control on the propagation path. By constraining the power flow of these sensitive lines, a multi-stage blocking control model for the predicted cascading failure path is proposed with the objective of minimizing the control cost and cascading failure probability. Based on probabilistic optimal power flow calculations, the constraints related to wind power uncertainty are transformed into opportunity constraints. To validate the effectiveness of the proposed model, the IEEE 39-node system is used as an example, and the results show that the obtained control method is able to balance economy and safety. In addition, the control costs for the same initial failure are higher as the wind power penetration rates and confidence levels increase.

Probabilistic Optimal Power Flow Computation Using Stratified Quadrature Rule

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 deviations POPF outputs. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the accuracy and efficiency of the proposed method.

Analysis of probabilistic optimal power flow in the power system with the presence of microgrid correlation coefficients

Analysis of probabilistic optimal power flow in the power system with the presence of microgrid correlation coefficients

Solving probabilistic optimal power flow with renewable energy sources in distribution networks using fire hawk optimizer

The main contribution in this article is proposing a modern meta-heuristic algorithm for solving nonlinear, nonconvex, and complex OPF problems effectively in distribution networks, considering RES. The Fire Hawk Optimizer (FHO) algorithm is proposed in this work. The considered objective function (OF) is the combined minimization of generating cost, active power loss (Plosses), and load bus voltage deviation (VD). The proposed strategy is validated using the standard IEEE 69-bus benchmark. The obtained results are compared with various recent meta-heuristic techniques such as Capuchin Search Algorithm (CSA), Bear Smell Search Algorithm (BSSA), Black Widow Optimization (BWO) Algorithm, Walrus Optimization Algorithm (WaOA), and Prairie Dog Optimization (PDO) Algorithm. One-way ANOVA and the Tukey test, two common statistical tests, are used to compare algorithms effectively. The FHO shows the best and superior performance with minimum mean cost and standard deviation values of 51.2711 and 5.1131 respectively. This demonstrates that FHO results are more consistent and less diverse. The FHO shows a lower computational time of (25.3241 s) as compared with other strategies. The statistical results confirm the proposed approach's reliability, stability, and consistency. The proposed method provided reliable, high-quality results with faster computing times. Additionally, it has a significant convergence that makes it superior to solving OPF problems in distribution networks with variable RES and NRES.

Data-Driven Nonparametric Probabilistic Optimal Power Flow: An Integrated Probabilistic Forecasting and Analysis Methodology

With large-scale integration of renewable energy such as wind power, probabilistic analysis of optimal power flow becomes crucial for the decision-making of power systems. This paper proposes a novel data-driven integrated probabilistic forecasting and analysis (IPFA) methodology for the nonparametric probabilistic optimal power flow (N-POPF), which internalizes the probabilistic forecasting and nonparametric distributional description forms into uncertainty analysis. The proposed IPFA methodology fully utilizes the uncertainty analysis method to guide the model-free nonparametric probabilistic forecasting of wind power, and then conducts the N-POPF analysis effectively based on the uncertainty information contained in historical data. A comprehensive uncertainty evaluation criterion based on point estimate method and differential entropy is proposed to assess both the inherent uncertainty and uncertainty influence of input random variables. Then a model-free multivariate probabilistic forecasting method is established to directly support the solving of N-POPF problems with similar historical measurements. Finally, with deterministic optimal power flow problems corresponding to the selected historical samples, a weighted combination approach for the power flow results is developed to derive the quantiles of the output random variables. Comprehensive experiments on IEEE 24-bus and 118-bus test systems validate the superiority of the proposed IPFA methodology in estimation accuracy and computational efficiency.

Bayesian deep neural networks for spatio-temporal probabilistic optimal power flow with multi-source renewable energy

Probabilistic optimal power flow (POPF) plays a crucial role in ensuring the economic and secure operation of power systems with multiple fluctuating loads and renewable energy power generations. However, the practical application of POPF faces challenges because of the lack of accurate input scenarios and the heavy computational burden. With the increasing power system uncertainties and frequent changes in topologies, the existing POPF analysis methods struggle to achieve both high precision and efficient training. To address these issues, firstly, this work decomposes the POPF into three stages: 1) the generation of source-load random scenarios; 2) the non-linear mapping derived from the Karush-Kuhn-Tucker conditions; and 3) the non-linear mapping derived from the power flow equations. Secondly, this work proposes two Bayesian deep neural networks to separately solve each stage of the POPF. Multi-collinearity reduction conditional variational autoencoder can generate source-load random scenarios while consistency condition guided Bayesian deep neural network can predict POPF calculation results. The work further conducts numerical simulations on four IEEE benchmark power systems with multiple renewable energy sources, using uncertainty data collected from the Belgian wind power generation, photovoltaic power generation and load dataset. The simulation results demonstrate that: 1) the precision in generating source-load random scenarios has been enhanced, leading to a reduction of up to 128% in the distribution statistical quantities of the accuracy measures; and 2) the mean absolute error of active power injections, and voltage angles as well as the mean absolute percentage error of the voltage amplitudes in the benchmark systems are reduced by at least 0.7% compared to deterministic neural networks.

Probabilistic Optimal Power Flow in Large-Scale Electric Transmission Systems through a Matheuristic Solution Approach

Probabilistic Optimal Power Flow in Large-Scale Electric Transmission Systems through a Matheuristic Solution Approach

Enhanced coati optimization algorithm-based optimal power flow including renewable energy uncertainties and electric vehicles

Power systems now face new issues due to incorporating electric vehicles (EVs) and renewable energy resources (RERs). This paper proposes a novel Enhanced Coati Optimization Algorithm (ECOA) for obtaining the optimal solution of the probabilistic optimal power flow (POPF) problems. The ECOA is a metaheuristic optimization algorithm that is robust and efficient for solving complex problems. It is used to tackle the OPF problem, which considers the stochastic characteristics of RERs. Moreover, EVs are included in the presented power systems in this paper. The novel approach is tested and verified on the IEEE-57 and IEEE-118 networks. The effectiveness of the proposed method is demonstrated by making a comparison with other metaheuristic-based methods. To obtain a practical study, real data of wind speed, solar irradiance, and electric vehicles profile are incorporated in the dynamic analyses. The simulation results show that the ECOA is robust and efficient for solving the OPF problem. It can also improve the performance of power systems with RESs and EVs. The findings of this research demonstrate that the suggested approach is promising for power system optimization problems, including RERs and EVs.

Giant Trevally Optimization Approach for Probabilistic Optimal Power Flow of Power Systems Including Renewable Energy Systems Uncertainty

In this study, the Giant Trevally Optimizer (GTO) is employed to solve the probabilistic optimum power flow (P-OPF) issue, considering Renewable Energy Source (RES) uncertainties, achieving notable cost reduction. The objective function is established to minimize the overall generation cost, including the RES cost, which significantly surpassing existing solutions. The uncertain nature of the RES is represented through the employment of a Monte Carlo Simulation (MCS), strengthened by the K-means Clustering approach and the Elbow technique. Various cases are investigated, including various combinations of PV systems, WE systems, and both fixed and fluctuating loads. The study demonstrates that while considering the costs of solar, wind, or both might slightly increase the total generation cost, the cumulative generation cost remains significantly less than the scenario that does not consider the cost of RESs. The superior outcomes presented in this research underline the importance of considering RES costs, providing a more accurate representation of real-world system dynamics and enabling more effective decision making.

Probabilistic optimal power flow in wind energy integrated power system based on the K‐medoids data clustering method considering correlated uncertain variables

AbstractRenewable energy sources have increasingly been integrated into power systems because of their superior environmental, technical, and economic advantages. However, the uncertainty of these sources besides load fluctuations and equipment outages brings new challenges to power system operating and planning studies. Also, the correlation between uncertain variables makes operational decisions more and more complicated. Probabilistic assessment of power systems is essential to evaluate uncertainties’ effects and make reasonable decisions. Methods with high accuracy and low computational burden are efficient for online and fast requirements such as optimal power flow (OPF) problems. A clustering method based on the K‐medoids technique is used for the probabilistic assessment of the power system in the OPF problem. Unlike other clustering techniques the K‐medoids method can manage discrete variables such as equipment outages besides other continuous variables such as load and wind generation. This paper presents an OPF problem considering the uncertainty of renewable energy sources, load fluctuations, and transmission lines outage as well as the correlation among them. The probabilistic assessment is conducted by the K‐medoids method and the optimization problem is solved by the cooperation search algorithm (CSA) as an efficient evolutionary algorithm. Numerical results using the IEEE standard 30 and 118 bus test systems show the efficiency of the proposed method in terms of accuracy and computational burden.

Scalable Probabilistic Optimal Power Flow for High Renewables Using Lite Polynomial Chaos Expansion

This work introduces a novel polynomial chaos expansion (PCE)-based method for uncertainty quantification (UQ) in power systems, addressing probabilistic optimal power flow problem. The proposed approach leverages upon a series of transformations to overcome two main hinders to deploying POPF in practice, i.e., the vast number of random variables (RVs) and their interdependence. The proposed Lite-PCE method requires a small set of optimal power flow samples. The most important transformation is developed based on a Clayton copula to convert substantially large dimensional input variables into a single RV for efficient computing. The transformation allows Lite-PCE to be built on a single RV and offers a significant computational benefit while working with a large number of random renewable injections. The method also handles correlated random injection variables with different types of distributions in a nonparametric fashion. The Lite-PCE method is also used to develop the concept of transmission capacity sensitivity under uncertainty using coefficients of variation. The concept and nonparametric, scalable simulations are demonstrated using various IEEE test cases, up to the 1354-Bus system with 1242 random injection variables. The UQ results illustrate the efficiency and accuracy of the method for systems of considerable size and a high number of renewable injections.

Probabilistic Optimal Power Flow for Balanced Islanded Microgrids

This paper presents a Probabilistic Optimal Power Flow (POPF) for losses minimization in balanced islanded microgrids considering load uncertainties following a normal distribution. The peculiarities of the islanded operation, like the absence of infinite bus and frequency variation in the face of load and loss variations, are adequately considered. The constraints deal with the power balancing, angular reference, droop control mode of generators (P-f e Q-V), and the probability of satisfying the limits of microgrid frequency, nodal voltages, and power generation. As the main contribution, the Differential Evolution method and the Unscented Transformation solve the POPF, reducing the computational time and providing accurate results compared with the Monte Carlo Simulation. The proposed approach is evaluated using a 33-bus islanded microgrid.

Monte Carlo Simulation and a Clustering Technique for Solving the Probabilistic Optimal Power Flow Problem for Hybrid Renewable Energy Systems

This paper proposes a new, metaheuristic optimization technique, Artificial Gorilla Troops Optimization (GTO), for a hybrid power system with photovoltaic (PV) and wind energy (WE) sources, solving the probabilistic optimum power flow (POPF) issue. First, the selected algorithm is developed and evaluated such that it applies to solve the classical optimum power flow (OPF) approach with the total fuel cost as the objective function. Second, the proposed algorithm is used for solving the POPF, including the PV and WE sources, considering the uncertainty of these renewable energy sources (RESs). The performance of the suggested algorithm was confirmed using the standard test systems IEEE 30-bus and 118-bus. Different scenarios involving different sets of the PV and WE sources and fixed and variable loads were considered in this study. The comparison of the obtained results from the suggested algorithm with other algorithms mentioned in this literature has confirmed the efficiency and performance of the proposed algorithm for providing optimal solutions for a hybrid power system. Furthermore, the results showed that the penetration of the PV and WE sources in the system significantly reduces the total cost of the system.

Solution of Probabilistic Optimal Power Flow Incorporating Renewable Energy Uncertainty Using a Novel Circle Search Algorithm

Integrating renewable energy sources (RESs) into modern electric power systems offers various techno-economic benefits. However, the inconsistent power profile of RES influences the power flow of the entire distribution network, so it is crucial to optimize the power flow in order to achieve stable and reliable operation. Therefore, this paper proposes a newly developed circle search algorithm (CSA) for the optimal solution of the probabilistic optimal power flow (OPF). Our research began with the development and evaluation of the proposed CSA. Firstly, we solved the OPF problem to achieve minimum generation fuel costs; this used the classical OPF. Then, the newly developed CSA method was used to deal with the probabilistic power flow problem effectively. The impact of the intermittency of solar and wind energy sources on the total generation costs was investigated. Variations in the system’s demands are also considered in the probabilistic OPF problem scenarios. The proposed method was verified by applying it to the IEEE 57-bus and the 118-bus test systems. This study’s main contributions are to test the newly developed CSA on the OPF problem to consider stochastic models of the RESs, providing probabilistic modes to represent the RESs. The robustness and efficiency of the proposed CSA in solving the probabilistic OPF problem are evaluated by comparing it with other methods, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and the hybrid machine learning and transient search algorithm (ML-TSO) under the same parameters. The comparative results showed that the proposed CSA is robust and applicable; as evidence, an observable decrease was obtained in the costs of the conventional generators’ operation, due to the penetration of renewable energy sources into the studied networks.

A SCOPF model for congestion management considering power flow controlling devices

With an increase in the share of distributed renewable generation, the power flow patterns are changing, and there is an increasing probability of internal congestion. There is also an increase in the number of power flow controlling devices in the transmission system to provide greater flexibility to the system operator. Thus, there is a need to include these devices and intra-zonal transmission lines for the congestion management analysis. This paper presents a probabilistic security-constrained optimal power flow (SCOPF) model for congestion management based on the non-linear ac formulation. To ensure computational efficiency, a second-order cone (soc) relaxation for power flow controlling devices is proposed. A comparison between the ac and soc formulations has been carried out on the modified IEEE 118-bus AC/DC system. It is observed that with the soc formulation, the computational efficiency of the model is increased while ensuring a minimal gap between the ac and soc results.

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

The modern power distribution systems are vulnerable to natural disasters and malicious attacks, while the uncertainty of a large amount of renewable energy sources (RESs) further increases their operational risk in extreme events. An adaptive probabilistic resilience assessment method is thus necessary to provide risk information and help decision making. To tackle the existing problems in uncertainty handling and computation efficiency of resilience assessment in operational stage, this paper establishes a systematic online probabilistic resilience assessment framework. A novel probabilistic modeling method is proposed to characterize the time-varying short-term uncertainty of RESs considering incomplete measurement information due to disasters-induced communication interruption. Moreover, the probabilistic optimal power flow (POPF) model is newly applied to the resilience assessment problem and solved by the cumulant method (CM), which can calculate the statistic information of load power and resilience indexes efficiently with only one optimization at basic operational point. The case study to a modified IEEE 37 node test feeder validates the advantage of the proposed method in reducing operational uncertainty degree, as well as its enough efficiency and accuracy to be applied to the online probabilistic resilience assessment for distribution systems operation.

Probabilistic Optimal Power Flow Solution Using a Novel Hybrid Metaheuristic and Machine Learning Algorithm

This paper proposes a novel hybrid optimization technique based on a machine learning (ML) approach and transient search optimization (TSO) to solve the optimal power flow problem. First, the study aims at developing and evaluating the proposed hybrid ML-TSO algorithm. To do so, the optimization technique is implemented to solve the classical optimal power flow problem (OPF), with an objective function formulated to minimize the total generation costs. Second, the hybrid ML-TSO is adapted to solve the probabilistic OPF problem by studying the impact of the unavoidable uncertainty of renewable energy sources (solar photovoltaic and wind turbines) and time-varying load profiles on the generation costs. The evaluation of the proposed solution method is examined and validated on IEEE 57-bus and 118-bus standard systems. The simulation results and comparisons confirmed the robustness and applicability of the proposed hybrid ML-TSO algorithm in solving the classical and probabilistic OPF problems. Meanwhile, a significant reduction in the generation costs is attained upon the integration of the solar and wind sources into the investigated power systems.

Advanced studies for probabilistic optimal power flow in active distribution networks: A scientometric review

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.

Nonparametric Probabilistic Optimal Power Flow

With the increasing penetration of renewable energy, accurate and efficient probabilistic optimal power flow (POPF) calculation becomes more and more important to provide decision support for secure and economic operation of power systems. This paper develops a novel nonparametric probabilistic optimal power flow (N-POPF) model describing the probabilistic information by quantiles, which avoids any parametric probability distribution assumptions of random variables. A novel critical region integral method (CRIM) which combines the multiparametric programming theory and discrete integral is proposed to efficiently solve the N-POPF problem. In the CRIM, the critical region partitioning algorithm is firstly introduced into the POPF model to directly establish the mapping relationship from wind power to optimal solutions of the POPF problem. Besides, a discrete integral method is developed in the CRIM to achieve the probability convolution calculation based on quantiles. Comprehensive numerical experiments verify the superior performance of the proposed CRIM in estimation accuracy and computational efficiency, and demonstrate that N-POPF model significantly improves the accuracy of uncertainty analysis. In general, the proposed N-POPF model and CRIM form a new framework of POPF problem for power system analysis and operation.

Comparison of solving probabilistic optimal power flow methods in the presence of wind and solar sources

Power system control and operation studies have been experienced essential changes due to increasing the penetration of uncertain renewable energy resources. One of the most critical issues in this regard is the optimal power flow (OPF). As a result, the deterministic methods do not have the capability of different modeling uncertainties raised in new power systems, and there is a need to investigate the effective models in this regard. This paper focuses on probabilistic optimal power flow (POPF) methods applied to power systems with uncertain wind and Photovoltaic power generations. In this paper, the Monte Carlo simulation (MCS) and analytical methods such as the three-point estimation method (3PEM), unscented transformation (UT), and Interior-Point method (IPM) are applied to solve the probabilistic optimal power flow problem. MCS has been widely applied as a framework to assess the ability of analytical methods. The mentioned techniques are applied to a sample case study extracted from the IEEE 300-bus system. The main contribution of this work is the comparison of analytical methods concluding 3PEM, UT, and IPM, and with MCS as well. The obtained results on the studied networks by the suggested techniques show that in the 3PEM, due to the limited points, the optimal solution is achieved in less calculation time than the other methods. From another perspective, the voltage changes at the buses would be more stable in the IPM. Also, this method is much faster than the MCS method in terms of the convergence rate. To show the effectiveness of the mentioned methods, this paper presents probabilistic load flow method based on the statistical methods to deal with fluctuations because of large-scale renewable energy integration. The proposed methods are validated on the improved industrial 85–bus system of Kermanshah region (in the west of Iran) by adding solar and wind farms.