Probabilistic Power Flow Research Articles

Operation Scheduling Method in an Off-grid using Optimal Probabilistic Power Flow

Operation Scheduling Method in an Off-grid using Optimal Probabilistic Power Flow

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

Increased investiture towards Distribution Generation (DG) in recent past had prompted for transmutation of existing networks to form microgrids that can operate autonomously or in conjunction with main-grid. In particular, the augmenting zeal for renewable DGs is thriving at recent times; but non-dispatchability and uncertain nature pose specific challenges, which is the primary concern in this work. This paper employs Point Estimate Method (PEM) for handling uncertainties to solve the probabilistic-optimal power flow problem (POPF) with multiple objectives formulated in an islanded microgrid with droop coordinated DGs including uncertainties involved in load, wind and solar PV with suitable probability distributions. The devised POPF problem is proposed to solve for optimal droop parameters by multi-objective ant-lion optimization algorithm; tested and verified on modified 33-bus system. Furthermore, in actuality, spatial correlations among renewables play vital role in devising operational schedule for energy management strategies. This paper deals with PEM compounded with Nataf Transformation for POPF to handle spatial correlations in renewable generations. The dominance of wind correlation over solar PV correlations in POPF problem is highlighted. The robustness of proposed approach is verified with benchmark Monte Carlo Simulation (MCS) to affirm about its accuracy for suitable replacement of proposed approach to MCS.

Planning for resilience in power distribution networks: A multi‐objective decision support

Power grid response against high-impact low-probability (HILP) events could be enhanced by (a) hardening mechanisms to boost its structural resilience and (b) corrective recovery and mitigation analytics to improve its operational resilience. Planning for structural resilience and attempts to find the optimal location of the Tie switches in radially operated power distribution networks that enable harnessing the network topology for maximised resilience against HILP disasters are focussed. This goal is achieved through a novel resilience-oriented multi-objective decision making platform, which employs a k-PEM based probabilistic power flow (PPF) algorithm. The proposed framework offers a decision making analytic embedded with the fuzzy satisfying method (FSM) that characterises the system resilience features, such as robustness, restoration agility, load criticality, and recovered capacity, to assess different network reconfiguration options and select the optimal solution for implementation. The aforementioned resilience features are formulated in nodal level and then aggregated over the entire system to characterise the system-level objective functions. The performance of the suggested framework is analysed on the IEEE 33-Bus test system under a designated HILP event, and the applicability on larger networks has been verified on the IEEE 69-bus test system. The results demonstrate the efficacy and applicability of the proposed framework in boosting the network resilience against future extremes.

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.

Probabilistic optimal power flow considering the dependence of multiple wind farms using pair diffusive kernel copula

As wind farms commonly gather in areas with abundant wind resources, spatial dependence of wind speeds among nearby wind farms should be taken into account when modeling a power system with large-scale wind power. In this paper, we first propose a novel bivariate non-parametric copula, bivariate diffusive kernel (BDK) copula, to formulate the dependence between random variables. Then, we extend the application of the BDK copula to high dimension by pair-copula method and name it as pair diffusive kernel (PDK) copula, which is flexible to formulate the complicated dependence structure of multiple random variables. Besides, a quasi-Monte Carlo (QMC) method, based on the combination of the Sobol sequence and the Rosenblatt transformation of the PDK copula, is elaborated in the sampling procedure to generate correlated wind speed samples. The proposed method is applied to solve the probabilistic optimal power flow (POPF) problems. We first validate the effectiveness of the BDK copula in copula definitions. Then, in order to verify the superior performance of the PDK copula in formulating the dependence structure of wind speeds among different wind farms, three different data sets are used in various goodness-of-fit (GOF) tests. Furthermore, samples obtained from the PDK copula are used to solve POPF problems, which are modeled on three modified IEEE 57-bus power systems. Compared with the Gaussian, T and parametric-pair copulas, the results obtained from the PDK copula are superior to formulate the complicated dependence in solving POPF problems.

Parametric Problems in Power System Analysis: Recent Applications of Polynomial Approximation Based on Galerkin Method

In power systems, there are many uncertainty factors such as power outputs of distributed generations and fluctuations of loads. It is very beneficial to power system analysis to acquire an explicit function describing the relationship between these factors (namely parameters) and power system states (or performances). This problem, termed as parametric problem (PP) in this paper, can be solved by Galerkin method, which is a powerful and mathematically rigorous method aiming to seek an accurate explicit approximate function by projection techniques. This paper provides a review of the applications of polynomial approximation based on Galerkin method in power system PPs as well as stochastic problems. First, the fundamentals of polynomial approximation and Galerkin method are introduced. Then, the process of solving three types of typical PPs by polynomial approximation based on Galerkin method is elaborated. Finally, some application examples as well as several potential applications of power system PPs solved by Galerkin method are presented, namely the probabilistic power flow, approximation of static voltage stability region boundary, and parametric time-domain dynamic simulation.

Gaussian Process Learning-Based Probabilistic Optimal Power Flow

In this letter, we present a novel Gaussian Process Learning-based Probabilistic Optimal Power Flow (GP-POPF) for solving POPF under renewable and load uncertainties of arbitrary distribution. The proposed method relies on a non-parametric Bayesian inference-based uncertainty propagation approach, called Gaussian Process (GP). We also suggest a new type of sensitivity called Subspace-wise Sensitivity, using observations on the interpretability of GP-POPF hyperparameters. The simulation results on 14-bus and 30-bus systems show that the proposed method provides reasonably accurate solutions when compared with Monte-Carlo Simulations (MCS) solutions at different levels of uncertain renewable penetration and load uncertainties. The proposed method requires a lesser number of samples and elapsed time. The non-parametric nature of the proposed method is manifested by testing uncertain injections that follow various distributions in the 118-bus system. The low error value results verify the proposed method's capability of working with different types of input uncertainty distributions.

Analyzing Probabilistic Optimal Power Flow Problem by Cubature Rules

Analyzing Probabilistic Optimal Power Flow Problem by Cubature Rules

Joint Planning of Distributed PV Stations and EV Charging Stations in the Distribution Systems Based on Chance-Constrained Programming

Simultaneous deployment of the electric vehicle charging stations (EVCSs) and distributed photovoltaic stations (DPVSs) in the distribution systems is an effective way to reduce greenhouse gas emissions, promote renewable power adoption, and achieve sustainable development in energy utilization. In this context, how to deploy the EVCSs and DPVSs in the distribution systems with a reasonable scheme is of great importance. In this article, a joint planning model is developed to optimize locations and capacities of the EVCSs and DPVSs simultaneously to reduce energy losses in the distribution systems. In the joint planning model, constraints on bus voltage deviations and line currents are both formulated as chance constraints to ensure that the distribution systems are in reasonable operating statues. To quantify these two chance constraints, a scenario-based method is developed to calculate the probabilistic power flow of the distribution systems during a typical planning day, in which random characters of the DPVS generations and the EVCS charging loads are both considered. The joint planning model of the EVCSs and DPVSs developed in this article is difficult to be solved by mathematical optimization methods. Therefore, genetic algorithm (GA) is customized and utilized to solve the joint planning model of the EVCSs and DPVSs. Finally, a case study based on IEEE 33-bus distribution systems validates the joint planning model and its solving algorithm.

Probabilistic Assessment of Impact of Flexible Loads Under Network Tariffs in Low-voltage Distribution Networks

Given the historically static nature of low-voltage networks, distribution network companies do not possess the tools for dealing with an increasingly variable demand due to the high penetration of distributed energy resources (DERs). Within this context, this paper proposes a probabilistic framework for tariff design that minimises the impact of DER on network performance, stabilises the revenue of network company, and improves the equity of network cost allocation. To deal with the lack of customers' response, we also show how DER-specific tariffs can be complemented with an automated home energy management system (HEMS) that reduces peak demand while retaining the desired comfort level. The proposed framework comprises a nonparametric Bayesian model which statistically generates synthetic load and PV traces, a hot-water-use statistical model, a novel HEMS to schedule customers' controllable devices, and a probabilistic power flow model. Test cases using both energy- and demand-based network tariffs show that flat tariffs with a peak demand component reduce the customers' cost, and alleviate network constraints. This demonstrates, firstly, the efficacy of the proposed tool for the development of tariffs that are beneficial for the networks with a high penetration of DERs, and secondly, how customers' HEM systems can be part of the solution.

Probabilistic impact of wind energy integration on distribution transformers

In traditional power grid, the direction of power flow of transformers is usually fixed in one direction. However, with the integration of the renewable energy sources, such as wind farms, the power flow in the grid becomes fluctuating and uncertain, which may cause the reverse power flow passing the transformer. For example, excessive reverse power flow may lead to the increase of transformer power loss and challenge the thermal stability of the transformer. This paper investigates the impact of wind energy integration on the existing power system infrastructure, such as distribution transformers. Probabilistic power flow (PPF) method is used for the analysis. Firstly, a sampling method based on Halton sequence is applied for PPF calculation. Then, to ensure the accuracy of probabilistic power flow calculation, the correlation between wind speeds of different wind farms is considered. A 30-Bus system is used for the probabilistic analysis of the transformer reverse power flow with different level of wind penetration. The results show that the reverse power flow of transformer may occur before 100% wind penetration is achieved. The higher the total wind power capacity in the grid causes the greater probability of transformer reverse power flow. Also, the location of wind farms will also affect the probability of transformer reverse power flow.

A Probabilistic Optimal Power Flow in Wind-Thermal Coordination Considering Intermittency of the Wind

This article specifically aims to prove the superiority of the proposed moth swarm algorithm (MSA) in view of wind-thermal coordination. In the present article, a probabilistic optimal power flow (POPF) problem is formulated to reflect the probabilistic nature of wind. Modelling of doubly fed induction generator (DFIG) is included in the proposed POPF to represent the wind energy conversion system (WECS). To reduce DFIG imposed deviation of bus voltage ancillary reactive power support is considered. Moreover, three different optimization techniques, namely, MSA, biogeography-based optimization (BBO), and particle swarm optimization (PSO) are independently applied for the minimization of active power generation cost for wind-thermal coordination, considering different instances in case of IEEE 30-bus and IEEE 118-bus system. From the simulation results, it is confirmed and validated that the proposed MSA performs considerably better than BBO and PSO.

A clustering-based analytical method for hybrid probabilistic and interval power flow

Various probabilistic power flow (PPF) and interval power flow (IPF) methods have been developed to deal with random and interval variables in power systems, respectively. However, the co-existence of these two types of variables poses great challenges to PPF and IPF calculations. To cope with this issue, we propose a clustering-based analytical method for hybrid probabilistic and interval power flow (HPIPF) calculation. The uncertainties of load demands and wind power outputs are treated as random and interval variables, respectively. The remarkable feature of this method is to propose an assumption called the unified optimal scenarios of wind power. On this basis, HPIPF calculation is transformed into IPF and PPF calculations, which can be solved by the optimal-scenarios method and the cumulant method, respectively. The accuracy and efficiency of the proposed method are validated on the IEEE 14-bus and 118-bus test systems through the comparisons with the double-layer Monte-Carlo simulation. Furthermore, the impacts of correlated interval variables are analyzed. The simulations indicate that the estimations of output variables may be conservative without considering the correlations of interval variables.

Probabilistic Approaches to the Security Analysis of Smart Grid with High Wind Penetration: The Case of Jeju Island’s Power Grids

As the importance of renewable generating resources has grown around the world, South Korea is also trying to expand the proportion of renewable generating resources in the power generation sector. Among the various renewable energy sources, wind generating resources are emerging as a key alternative to conventional power generations in the electricity sector in Korea accounted for 17.7 GW of total capacity by 2030. As wind generating resources are gradually replacing traditional generating resources, the system security and reliability are negatively affected because of the variability, due to intermittent outputs. Therefore, existing power grids will need to be correctly re-measured to cover the large scale of renewable energy, including wind generation. To expand the grid, we must understand the characteristics of renewable energy and the impact of its adoption in the grid. In this paper, we analyze various characteristics of wind power generation, and then we propose a probabilistic power output modeling method to consider the uncertainty of wind power generation. For the probabilistic approach, Monte-Carlo simulation is used in the modeling method. The modeled wind power outputs can help planning for the reinforcement and expansion of power systems to expand the capacity for large-scale renewable energy in the future. To verify the proposed method, some case studies were performed using empirical data, and probabilistic power flow calculation was performed by integrating large-scale wind power generation to the Jeju Island power system. The probabilistic method proposed in this paper can efficiently plan power system expansion and play a key strategy of evaluating the security of the power system through the results of stochastic power flow calculation.

Model-Driven Architecture of Extreme Learning Machine to Extract Power Flow Features

Probabilistic power flow (PPF) calculation is an important power system analysis tool considering the increasing uncertainties. However, existing calculation methods cannot simultaneously achieve high precision and fast calculation, which limits the practical application of the PPF. This article designs a specific architecture of the extreme learning machine (ELM) in a model-driven pattern to extract the power flow features and therefore accelerate the calculation of PPF. ELM is selected because of the unique characteristics of fast training and less intervention. The key challenge is that the learning capability of the ELM for extracting complex features is limited compared with deep neural networks. In this article, we use the physical properties of the power flow model to assist the learning process. To reduce the learning complexity of the power flow features, the feature decomposition and nonlinearity reduction method is proposed to extract the features of the power flow model. An enhanced ELM network architecture is designed. An optimization model for the hidden node parameters is established to improve the learning performance. Based on the proposed model-driven ELM architecture, a fast and accurate PPF calculation method is proposed. The simulations on the IEEE 57-bus and Polish 2383-bus systems demonstrate the effectiveness of the proposed method.

Correction to “Convex Relaxations of Probabilistic AC Optimal Power Flow for Interconnected AC and HVDC Grids” [Jul 19 2706-2718

Correction to “Convex Relaxations of Probabilistic AC Optimal Power Flow for Interconnected AC and HVDC Grids” [Jul 19 2706-2718

A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques

Because the penetration level of renewable energy sources has increased rapidly in recent years, uncertainty in power system operation is gradually increasing. As an efficient tool for power system analysis under uncertainty, probabilistic power flow (PPF) is becoming increasingly important. The point-estimate method (PEM) is a well-known PPF algorithm. However, two significant defects limit the practical use of this method. One is that the PEM struggles to estimate high-order moments accurately; this defect makes it difficult for the PEM to describe the distribution of non-Gaussian output random variables (ORVs). The other is that the calculation burden is strongly related to the scale of input random variables (IRVs), which makes the PEM difficult to use in large-scale power systems. A novel approach based on principal component analysis (PCA) and high-dimensional model representation (HDMR) is proposed here to overcome the defects of the traditional PEM. PCA is applied to decrease the dimension scale of IRVs and eliminate correlations. HDMR is applied to estimate the moments of ORVs. Because HDMR considers the cooperative effects of IRVs, it has a significantly smaller estimation error for high-order moments in particular. Case studies show that the proposed method can achieve a better performance in terms of accuracy and efficiency than traditional PEM.

Dimension Reduction Method for Probabilistic Power Flow Computation Considering Wind Farms

With the large-scale integration of renewable energy sources into the power system, a new source of uncertainty is added to the operation planning problem. In this paper, the rank correlation coefficient is introduced to characterize the dependency among random variables in power flow equations, and Nataf transformation is used to map the probabilistic power flow (PPF) problem to the independent standard normal space. Dimension reduction model is introduced to approximation the function relationship between PPF inputs and outputs. Gauss-Hermite quadrature is used to obtain the statistical moments of the univariate function, whereby the statistical moments of outputs of power flow equations are obtained. Testing on an IEEE-118 system, the dimension reduction method is compared with Hong’s point estimate method, it is found the dimension reduction method can improve the accuracy without extra computational burden.

Dynamic Reconfiguration Method of Distribution Network Containing Microgrid

The application of distributed power in the system has changed many characteristics of the traditional distribution network. Based on the consideration of the randomness of the load and the fluctuation and randomness of the output of the distributed generation, a new reconfiguration model of the distribution network with microgrid is established in this paper. Dynamic reconstruction is not a simple superposition of reconstructions in sub-periods. The influence of the number of switching operations on the operation mode between adjacent sub-periods and the overall optimality of the objective function must also be considered. The mathematical probability models of distributed generations and load are constructed in this paper. The dynamic reconfiguration is completed on the basis of probabilistic power flow. The objective function is calculated by stochastic power flow. The optimization algorithm is applied to the reconstruction process. The simulation results verify the effectiveness of the proposed model and algorithm in dynamic reconstruction.

Statistical Machine Learning Model for Stochastic Optimal Planning of Distribution Networks Considering a Dynamic Correlation and Dimension Reduction

Distributed generation and reactive power resource allocation will affect the economy and security of distribution networks. Deterministic scenario planning cannot solve the problem of network uncertainties, which are introduced by intermittent renewable generators and a variable demand for electricity. However, stochastic programming becomes a problem of great complexity when there is a large number of scenarios to be analyzed and when the computational burden has an adverse effect on the programming solution. In this paper, statistical machine learning theories are proposed to quickly solve the optimal planning for capacitors. Various technologies are used: Markov chains and copula functions are formulated to capture the variability and correlation of weather; consumption behavior probability is involved in the weather-sensitive load model; nearest neighbor theory and nonnegative matrix decomposition are combined to reduce the dimensions and scenario scale of stochastic variables; the stochastic response surface is used to calculate the probabilistic power flow; and probabilistic inequality theory is introduced to directly estimate the objective and constraint functions of the stochastic programming model. The effectiveness and efficiency of the proposed method are verified by comparing the method with the scenario reduction algorithm and the Monte Carlo method in a 33-bus distribution system.