Probabilistic Power Flow Analysis Research Articles

Dual-layer optimization planning for offshore wind farms considering power system security

The vigorous development of offshore wind farms (OWFs) has further promoted the utilization of wind energy. Owing to the demand for low-carbon environmental protection, the increase in the scale of OWF constructions has also had a significant impact on the power system, which cannot be ignored. Specifically, the uncertainty and volatility of wind-power generation should be considered in power system optimization planning schemes. Therefore, this study proposes a probabilistic power-flow analysis index construction method that employs the static-security region theory to optimize the OWFs–grid-connection topology, including the number, location, and capacity of the public connection points connected to the onshore power grid. This method can be used to calculate the probability indicators of power grid insecurity for OWFs–grid-connected systems with different network topologies. Furthermore, a dual-layer multi-objective programming model that considers the economics and stability for integrated planning of electrical collector and multiterminal direct current transmission systems is established. The proposed model comprises two inner-layer models and one outer-layer model. The inner-layer models are used to construct the collection and transmission topologies of the OWFs–grid-connected system that minimize the probability of grid insecurity and the planning cost of the outer-layer model. Finally, the effectiveness of the proposed OWFs planning model was verified using an improved IEEE 24-bus system that can be used for topology planning of electrical collector and transmission systems while ensuring grid stability.

Efficient and Robust Power Flow Algorithm for Asynchronous Grids Coupled Through a VSC-MTDC System and Its Probability Analysis

With the interconnection of asynchronous ac grids through a voltage source converter-based multiterminal direct current (AGs/VSC-MTDC) system and the large-scale integration of renewable energy, it is important to resolve the power flow calculation problems efficiently and robustly for secure and economic operation of such system under stochastic context. Therefore, an improved sequential algorithm for deterministic power flow (DPF) of AGs/VSC-MTDC system is proposed first. This enhances the conventional manner of updating power losses of VSCs, which leads to a one-shot DPF solution of the ac/dc grid, thus boosting the efficiency; while the coupling of ac/dc systems is considered to remain high accuracy. In combination with the efficient DPF method for each sample, probabilistic power flow (PPF) analysis of the AGs/VSC-MTDC system is performed by using Zhao's three-point estimate method (Z-3PEM) with much fewer samples, thus achieving a higher efficiency with good accuracy. In addition, the PPF analysis of AGs/VSC-MTDC systems being more prone to the poor robustness in stochastic scenarios is effectively addressed, benefitting from the decreased nonconvergence of improved DPF and fewer extreme samples of Z-3PEM. Finally, the effectiveness and performance of works are verified through the results on an 11-terminal VSC-MTDC system with three asynchronous ac grids modified from different IEEE test systems.

Probabilistic power flow calculation using principal component analysis-based compressive sensing

The increasing scale of the injection of renewable energy has brought about great uncertainty to the operation of power grid. In this situation, probabilistic power flow (PPF) calculation has been introduced to mitigate the low accuracy of traditional deterministic power flow calculation in describing the operation status and power flow distribution of power systems. Polynomial chaotic expansion (PCE) method has become popular in PPF analysis due to its high efficiency and accuracy, and sparse PCE has increased its capability of tackling the issue of dimension disaster. In this paper, we propose a principal component analysis-based compressive sensing (PCA-CS) algorithm solve the PPF problem. The l1-optimization of CS is used to tackle the dimension disaster of sparse PCE, and PCA is included to further increase the sparsity of expansion coefficient matrix. Theoretical and numerical simulation results show that the proposed method can effectively improve the efficiency of PPF calculation in the case of random inputs with higher dimensions.

Physics-Guided Residual Learning for Probabilistic Power Flow Analysis

Physics-Guided Residual Learning for Probabilistic Power Flow Analysis

Implementing Stochastic Response Surface Method and Copula in the Presence of Data-Driven PV Source Models

As PV penetration in the power distribution network is growing on an unprecedented scale, probabilistic power flow analysis is becoming crucial to assess the health of network operation. State-of-the-art probabilistic analysis relies on Stochastic Response Surface Method and Gaussian Copula for including PV sources correlation. However, it has been observed how in the presence of complex statistical distributions of injected PV powers, such a standard approach can provide inaccurate evaluations of the output variable distributions. In this article, we carefully investigate the origins of such a drawback and propose a novel implementation flow that overcomes the problem. In order to check the correctness of the proposed methodology, the results obtained with the Stochastic Response Surface Method are compared with Monte Carlo simulations. Several investigations, such as voltage uncertainties, network health and probability of violating quality constraints are conducted on two test networks.

Probabilistic three-phase power flow in a distribution system applying the pseudo-inverse and cumulant method

Abstract A new, analytical approach using the cumulant method is proposed for the three-phase probabilistic power-flow (PPF) analysis. The approach to forming the sensitivity matrix is based on quantifying the pseudo-inverse instead of the inverse jacobian matrix, since it is commonly singular in a distribution power network. The results are compared with those obtained using the point-estimate method (PEM) and the Monte Carlo (MC) method, which is a commonly used reference method for the PPF analysis in a distribution power network.

Probabilistic Power Flow Analysis Based on Partial Least Square and Arbitrary Polynomial Chaos Expansion

This paper presents a new algorithm based on the partial least square (PLS) techniques and the arbitrary polynomial chaos expansion (aPCE) for the probabilistic power flow (PPF) analysis of a power system having many uncertain variables. The proposed method uses the nonlinear PLS to transform a set of random input variables to a smaller number of de-correlated random variables. Then, the aPCE technique is applied to generate the basis polynomial functions and build a surrogate model of the power system response. The algorithm has been implemented and tested with the modified IEEE 118-bus and European 1354-bus systems. The numerical results indicate that, similar to the sparse PCE method and the low-rank approximation technique, the proposed method can be applied to high-dimensional PPF problems. Nonetheless, the proposed approach uses smaller computation time, and estimates statistical characteristics of the response with higher accuracy.

Application of Joint Raw Moments-Based Probabilistic Power Flow Analysis for Hybrid AC/VSC-MTDC Power Systems

A joint raw moments-based analytical method (JRMAM) is first proposed herein to deal with probabilistic power flow (PPF) of hybrid AC/VSC-MTDC (Alternate Current/Voltage Source Control-Multiple Terminal Direct Current) power systems with integration of the offshore wind farms. JRMAM is able to overcome the challenges of some existing combined methods in estimating the high order moments of target outputs, and the dramatic time consumption of Monte Carlo simulation method (MCSM) in performing the PPF analysis particularly due to the complication of its deterministic power flow (DPF) model. The proposed JRMAM has a further superiority on efficiency, even compared to the joint cumulants-based analytical method (JCAM). As a preliminary step, discrete Fourier transformation matrix (DFTM) method is applied to calculate the joint raw moments of random inputs which are required by conducting JRMAM, and the probability density function (PDF) of outputs can be estimated by means of the Gram-Charlier expansions (GCE) based on the results obtained from JRMAM. Two modified IEEE test cases connected with DC subsystems through the VSC devices are adopted to verify the effectiveness and the scalability of JRMAM, where the result of MCSM is treated as a reference, and the properties of the hybrid AC/VSC-MTDC power system with these master-slave controlled VSCs are investigated for a further acceleration on PPF analysis.

Operational Probabilistic Power Flow Analysis for Hybrid AC-DC Interconnected Power Systems with High Penetration of Offshore Wind Energy

With increasing penetration of intermittent renewable energy, power systems are facing great pressures in real-time operation and control. Minutes ahead short-term static risk assessment using the probabilistic power flow (PPF) can help system operators detect potential over-limit risks in advance and take preventive measures in time. However, current probability models of renewable power outputs lack enough adaption and efficiency in operational stage, and the PPF formulation also neglects the effects of control system and tie line power exchange of large power systems. This paper establishes a systematic operational PPF scheme for hybrid AC-DC interconnected power systems with offshore wind energy integration. An adaptive probabilistic modeling and fast update method is proposed to model multiple time-varying offshore wind power outputs. Considering different responses of control systems and AC/DC tie line power exchanges, a novel hybrid PPF is proposed to reflect power flow uncertainty more robustly and accurately. The above techniques are validated on a modified IEEE 30 bus system and the whole scheme is applied to the short-term static risk assessment of a real system in Southern China. The results show that our proposed scheme can well meet online calculation requirements, and can also achieve a more realistic characterization of operational PPF.

Soft clustering based probabilistic power flow with correlated inter temporal events

There is a growing demand to deploy renewable energy sources (RES) and plug-in electric vehicle (PEV) charging station (EVCS) in distribution networks. This, however, will change power grid landscapes and potentially impose uncertainties in short and long term grid planning. The solution to quantify randomness of intermittent events is to implement probabilistic power flow (PPF) analysis. The computation of PPF due to myriad sources of unknown uncertainties is very expensive. This computation analysis becomes more complex when inter-period temporal-spatial correlation and multivariate uncertainties are considered. This paper addresses the issue by resorting to a novel hybrid nonparametric algorithm for short-term PPF analysis in radial distribution networks. The proposed algorithm deals with probabilistic nonlinear loads, unit generators including PV panels, Wind farms, and EVCS charging pattern. Temporal correlation characterized by memory coefficient is used to model inter-event temporal correlation (IETC) over multiple time steps. Detailed uncertainty modeling of PEVs driving and their queues at EVCSs are included. The proposed method is superior over other existing numerical methods as it accounts for the multivariate inter temporal uncertainty events of RES and PEVs charging. Simulation results in a spatio-temporally correlated IEEE 34 node radial distribution system revealed that using IETC coefficients leads to improved prediction accuracy with average error less than 2.5% and improves the computation burden. The proposed method is applied to IEEE 123 node to test the performance of our proposed method. A comparative analysis is done with other well-known analytical/numerical methods. The PPF Computation time of our proposed method is shown to be faster than other methods when IETC is implemented. The results show insensitivity of our proposed method to the size of the system.

Bayesian estimation of copula parameters for wind speed models of dependence

Modelling the uncertainty of wind speed is essential in power flow analysis. Having abundant knowledge of the wind speed in an area is critical. A low volume of data can increase uncertainty in wind speed analysis. Spatial dependencies are often modelled before running probabilistic power flow and load flow analysis. Copulas are a popular way of capturing spatial dependence between multiple wind farms. Using NREL data from seven Northeastern United States wind farm sites, Bayesian inference will be used to determine the copula parameter uncertainty between weekly, daily, and hourly wind speed observations. This approach will be used on elliptical and single parameter Archimedean copulas. For each possible wind farm pair, an uninformative prior will be placed on the copula parameter. The resulting posterior will contain a distribution of copula parameter values based on the prior and the observed wind speed data. The posterior's credible interval is reviewed to determine the uncertainty in parameter estimation. The results show that using a data volume considerably more petite than 8760 hourly data points will result in more uncertainty in parameter estimation and inaccuracies in wind speed forecasting if using non-Bayesian methods for copula parameter estimation.

Classified atmospheric states as operating scenarios in probabilistic power flow analysis for networks with high levels of wind power

Large-scale atmospheric circulation patterns are the primary drivers of wind power variability on power networks at timescales of hours to days. This paper proposes a methodology that allows power system operators and planners working on networks with high levels of wind generation, to conduct probabilistic power flow (PPF) analyses by defining network ‘operating scenarios’ – i.e. the probability density functions of generators, and correlations between generators representative of a future system state – based on concurrent classified atmospheric states. The most significant contribution made by this paper is in illustrating how PPF operating scenarios derived from clustering historic generation data as a function of a set of classified atmospheric states reduces simulation uncertainty within a PPF analysis. It is anticipated that the proposed methodology may provide network planners with more appropriate operating scenarios for PPF analyses when compared to an unclustered base state, and may assist network operators in converting wind power point-forecasts into probabilistic forecasts whereby the spatial correlations between generators are incorporated. This methodology is illustrated through a case study considering 11 geographically disperse wind generators on the South African transmission network.

Static Stability Analysis Based on Probabilistic Power Flow Calculation considering P2G Technology

At present, integrated energy systems have received extensive attention, but there is no basic framework for stability analysis of coupled systems. The injection of a large amount of renewable energy also has a great impact on the stability of the system. This paper focuses on how to analyze the static stability of the coupling system with uncertainty, which mainly considers the uncertainty of wind power generation and photovoltaic power generation and also considers the influence of P2G technology on the whole system. Firstly, this paper analyzes the principles of wind power generation and photovoltaic power generation and constructs the probability model of renewable energy power generation power. Then, the three-point estimation method is used to process the data, and the probability distribution of the unknown quantity is obtained by probabilistic power flow analysis. Finally, the probability distribution of each eigenvalue is obtained by analyzing the sensitivity of the characteristic roots to the voltage. Thus, the static stability of the system is judged. The applicability of proposed methodology is demonstrated by analyzing an integrated IEEE 14-bus power system and a Belgian 20-node gas system in this paper.

Arbitrary polynomial chaos expansion and its application to power flow analysis-Fast approximation of probability distribution by arbitrary polynomial expansion

This paper introduces an arbitrary polynomial chaos expansion method for performing probabilistic power flow analysis in power systems. The proposed method is used for uncertainty analysis, expressing the uncertainty of a system as random variables with an arbitrary output distribution based on orthogonal polynomial expansion. This method is advantageous because of its calculation speed and accuracy. This study expresses probabilistic power flow in a power system with many uncertain power sources using linear combination polynomial expansion. The orthogonal polynomial system employed is generated by moment analysis from renewable energy output data, with the polynomial coefficients derived from a collocation method. Simulation of probabilistic power flow using the proposed method is applied to a 29-bus transmission network model including three renewable energies, and the calculation speed and accuracy are evaluated by changing the expansion order of the polynomial. In addition, the influence on the polynomial coefficient is assessed when the system topology is changed due to a line fault. Therefore, since the arbitrary polynomial chaos expansion method can represent complex networks by linear combination of orthogonal polynomial sets, calculation based on it is several hundred times faster than the conventional Monte Carlo method. The results demonstrate that the proposed method is very useful for analyzing the probabilistic power distribution and that third-order expansion is practically appropriate.

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 power flow analysis based on arbitrary polynomial chaos expansion of bus voltage phasor

Probabilistic power flow analysis based on arbitrary polynomial chaos expansion of bus voltage phasor

Learning-Based Probabilistic Power Flow Calculation Considering the Correlation Among Multiple Wind Farms

Wind power generation provides a new route for the sustainable development of energy. However, with the large scale integration of wind farms, the volatile wind energy and the correlation among various wind speeds bring a considerable quantity of uncertainty for the power system operation. In this paper, a probabilistic power flow (PPF) analysis model for the power system incorporating the correlation among various wind speeds is proposed. As distinct from existing studies, in this paper, we introduce the relevance vector machine (RVM) into correlation modeling of wind speeds via historical learning samples to construct bivariate joint distribution. Compared with the conventional parameter estimation methods, the proposed method has higher flexibility and computational efficiency. On this basis, the regular vine copula approach is adopted further to build the multivariate joint distribution model of wind speeds. To calculate the PPF of power system with the integration of wind power, we employ the three-point estimation method (3PEM) while the Rosenblatt Transformation technique is proposed to transform the input variables into independent variables. The effectiveness of the proposed calculation framework is examined through simulation studies, and the obtained results illustrate the advantages of the proposed method.

Fast Probabilistic Voltage Control for Distribution Networks With Distributed Generation Using Polynomial Surrogates

Renewable distributed generation will be a key component of future power distribution networks. In order to control the voltage conditions for distribution networks integrated with distributed generation (DG) units, it is vital to quantify the impacts of control variables on voltage magnitudes under consumption and generation uncertainties. To do so, we need to run, for each control action, several power flow simulations for various consumption and generation realizations. This is computationally infeasible for systems with many uncertain inputs. In this work, we address this challenge by developing surrogates, or metamodels, that analytically estimate the random voltage as a function of input variables which include random parameters (consumption and generation levels) and control actions (power factors). Specifically, we propose a model reduction method for building these surrogates, which reduces the number of simulations needed for training. This method identifies and includes only the consumption and generation variables that are influential on the voltage at a given bus. Using this `reduced' surrogate, we then develop a sensitivity-based approach for probabilistic voltage control. We demonstrate the computational efficacy of the control approach on a IEEE 69-bus system with a large number of correlated input parameters. The highlights of computational efficiency in this case study include (1) accurate probabilistic power flow analysis using surrogates constructed by only 500 training simulations for a system with more than 150 random parameters, and (2) successful surrogate-based voltage control approach which only requires 150 additional simulation samples, as opposed to the conventional perturb-and-observe voltage control which needs more than 500,000 samples.

An efficient Nataf transformation based probabilistic power flow for high-dimensional correlated uncertainty sources in operation

This paper proposes and introduces a novel approach for probabilistic power flow (PPF) analysis on large-scale power systems with such Pearson correlated high-dimensional uncertainty sources existing in normal operation. The proposed method is holistic since it covers the whole procedure of PPF analysis and improves its efficiency from each aspect. As one key step, the combination of Gauss-Hermite quadrature and simplified Newton-Raphson method based Nataf transformation is presented to accelerate the process of probabilistic modeling for wind speed, solar radiation, tidal current speed and load consumption, in particular the correlation relationship amongst these uncertainty sources. Besides, dimension adaptive sparse grid interpolation (DASGI) method and Monte Carlo simulation (MCS) method have been put to joint application to further boost the efficiency of PPF calculation, with the Pearson correlation issue handled in this procedure as the first time. In order to highlight the superiority of the proposed method, several existing advanced PPF analysis methods are used for comparison test, in which simple random samples based MCS method i.e. the crude MCS is assigned as the reference. The performance of the proposed approach is verified in IEEE 118-bus system involving with 114 random variables.

Probabilistic power flow analysis with correlated wind speeds

In this paper, a polynomial transformation model is proposed to fit the quantile function of wind speed, a Laplace copula is developed to model the dependence structure of wind speeds at multiple sites, whereby the probabilistic power flow (PPF) problem can be mapped to the independent standard normal space. Based on a D−dimensional cubature rule and Kronecker product, a new multivariate quadrature rule is developed to calculate statistical moments of power flow solutions. Finally, the performance of the polynomial model and Laplace copula is checked using historical wind speed data, a case study is conducted on a modified IEEE 118-bus system to compare the proposed quadrature rule with the point estimate method for PPF computation.