Probabilistic Optimal Power Flow Problem Research Articles

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.

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.

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.

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.

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.

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.

Analyzing probabilistic optimal power flow problem by cubature rules

This paper is devoted to revealing some properties of the probabilistic optimal power flow (POPF) problem. In conjunction with Hermite polynomial model, Nataf transformation is introduced to map POPF problem to the independent standard normal space. Firstly, a multivariate polynomial model is employed to represent the function relationship between POPF inputs and outputs. Then, moment matching equations are derived to characterize the uncertainty effects of POPF inputs on outputs; three cubature rules are derived to calculate statistical moments of POPF outputs. Finally, along with Monte Carlo simulation method, the proposed methods are tested on IEEE 57-bus system and IEEE 118-bus system, whereby it reveals some characteristics of the function relation between POPF inputs and outputs.

Analyzing Probabilistic Optimal Power Flow Problem by Cubature Rules

Analyzing Probabilistic Optimal Power Flow Problem by Cubature Rules

Surrogate-Assisted Multi-Objective Probabilistic Optimal Power Flow for Distribution Network With Photovoltaic Generation and Electric Vehicles

The uncertainties of solar photovoltaics generation, electric vehicle charging demand, and home appliances load are the major challenge of energy management planning in the residential areas. Optimal allocation of battery energy storage systems for distribution networks based on probabilistic power flow (PPF) is an effective solution to deal with these uncertainties. However, the high computational burden is the main obstacle of this method. Therefore, this paper proposes a surrogate-assisted multi-objective probabilistic optimal power flow (POPF) to reduce the expensive computational time. The surrogate model is developed by using a machine learning method namely deep learning which is used for bypassing the deterministic load flow calculation. Zhao’s point estimation method combined with Nataf transformation is selected to handle the PPF analysis considering correlated uncertain input variables. The multi-objective POPF problem is solved using the multi-objective differential evolution. The historical data including solar irradiation, ambient temperature, residential load, and electric vehicle (EV) travel distance is calculated in the low voltage distribution system to demonstrate the potential advantages of the proposed method. Numerical results show that the proposed surrogate assisted multi-objective POPF method provides the optimal solution for operating cost, helps to prolong transformer life and reducing environmental impact. Moreover, the results show that the proposed surrogate-assisted optimization framework gives a better solution when comparing with the conventional surrogate-assisted method.

Data clustering based probabilistic optimal power flow in power systems

Ever increasing use of renewable energies beside other uncertain parameters in power systems makes it necessary to evaluate power system issues, probabilistically. One of the important studies in power systems operating is optimal power flow (OPF), which should be considered as probabilistic OPF (POPF). However, the Monte Carlo simulation (MCS) method can be efficiently used for handling any type of uncertainty but this method suffers from large calculation requirements and cannot be implemented in various studies. Especially, in evolutionary based optimization problems the use of MCS method is completely restricted. Data clustering is an alternative method which keeps the accuracy of the MCS method and requires very less calculation burden. In this study, data clustering is used for probabilistic assessment of power system in POPF problem. The proposed method is very fast, accurate and can easily handle any type of correlation between stochastic input variables. The correlated input variables are generated by the Cholesky decomposition method. The Genetic Algorithm (GA) is used as optimization tool. In order to demonstrate and validate the performance of the proposed method, IEEE 30- and 118-bus standard test systems are studied and the results are compared with a modified version of the two-point estimate method.

The Generalized Cross-Entropy Method in Probabilistic Optimal Power Flow

This paper presents a new powerful density estimator and the application of the generalized cross-entropy method. A framework for probability density function estimation of the probabilistic optimal power flow problem results is presented. Large-scale probabilistic problems have a lot of uncertain parameters, some of which are correlated. Most of the existing methods for probabilistic power system analysis have a large computational burden or rely on asymptotic approximation to mitigate the computational burden, which reduces the precision level. Others that depend on the uncertainties are not applicable in large-scale problems and do not provide precise results in small cases. Compared to conventional methods, the proposed method has several advantages in probabilistic analysis. Reducing the computational burden while maintaining high level of accuracy is one of them. The proposed method can deal with correlated problems. This method is tested on some valid and conventional networks containing renewable energy resources, nonstationary loads, and their correlations. The results are compared against more accurate results obtained from Monte Carlo simulation, which demonstrate the high level of accuracy according to the error indices. Results confirm the high capacity of the proposed method in accuracy and speed response compared to other methods.

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

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.

A bi-level probabilistic transmission planning with intermittent generations based on life cycle cost

With the growing development of intermittent renewable energy sources, such as wind and solar, transmission planners are faced with uncertainly varying generations and resultant probabilistic power flow. A bi-level programming model is proposed to coordinate the process of decision making and reliability assessment. Based on the concept of life cycle cost (LCC), its minimization can be defined as the objective function of a transmission planner. This upper level problem needs to be solved jointly with the lower probabilistic optimal power flow problem of minimizing the load shedding in the system. Hence the bi-level problem is transformed into a Mathematical Programming with Equilibrium Constraints (MPEC) with Karush-Kuhn-Tucker conditions. Due to the nonlinearity nature of MPEC, disjunctive inequalities and Generalized Benders Decomposition methods are used to solve this problem. Results of both Garver’s 6-bus test system and a realistic 63-bus system are used to illustrate the rationality and validity of the proposed method.

Comparing three methods for solving probabilistic optimal power flow

This paper compares three methods for solving probabilistic optimal power flow (P-OPF) problem: Zhao's point estimate method (PEM), Quasi Monte Carlo simulation (QMCS) and Latin hypercube sampling (LHS). With Nataf transformation, P-OPF problem is formulated as a multiple integral over standard normal space. By introducing a differential operator, a mathematical model is developed to compare the performance of QMCS and LHS. Furthermore, a simplified Gaussian mixture model (GMM) is presented to model distributions of P-OPF solutions. Testing on a modified 118-bus system, it is found LHS outperforms PEM and QMCS with a small sample size, but behaves comparably with QMCS for a large sample size. Compared to other statistic models, GMM shows a higher flexibility for data fitting.

Enhancements to the Cumulant Method for Probabilistic Optimal Power Flow Studies

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.

Introduction to Cumulant-Based Probabilistic Optimal Power Flow (P-OPF)

This letter introduces the cumulant method for the probabilistic optimal power flow (P-OPF) problem. The proposed adaptation of the cumulant method to the P-OPF problem is given.