Probability Distribution Of Wind Power Research Articles

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

Strategic optimization operations in the integrated energy system through multitime scale comprehensive demand response

AbstractIn the context of the dual carbon goal strategy, the proportion of new energy generation has increased annually, large‐scale renewable energy integration has been achieved, and the intermittent and uncertain operating characteristics pose an enormous challenge to the complete and stable operation of an integrated energy system (IES), promoting the complexity of IES optimization models. To increase the stability and accuracy of the system and improve the operation efficiency of the system, a Gaussian mixture model is used to fit the probability distribution of wind power and the prediction errors of the photovoltaic output. In addition, the expected maximization method is used to solve a model with hidden variables, the results show that the expectation maximization algorithm can improve the fitting accuracy, reduce the error caused by the subjective setting of the initial value, and make the fitting accuracy of the wind‐solar power prediction error greater. Then, to solve the problem in which the low resolution of the day‐ahead scheduling time leads to large errors, day‐ahead, day‐to‐day multitime scale operation optimization model takes into account the comprehensive demand response according to the differences in wind and solar output and load forecasting accuracy. Finally, simulation validation is conducted in multiple scenarios, and a comparative analysis is performed for single and multitime scale comprehensive demand response scenarios. The simulation results show that compared with the optimization results of no demand response day, the optimization results of this model reduce the total cost by 23.21%, carbon emissions by 13.98%, purchased electricity by 13.87%, and purchased gas by 19.31%, effectively improving the use of gas turbines in the system. The multitime scale scheduling strategy increases the overall operating cost of the IES, modifies the scheduling results, and agrees with real operating scenarios.

Wind Power Probability Density Prediction Based on Quantile Regression Model of Dilated Causal Convolutional Neural Network

Aiming at the wind power prediction problem, a wind power probability prediction method based on the quantile regression of a dilated causal convolutional neural network is proposed. With the developed model, the Adam stochastic gradient descent technique is utilized to solve the cavity parameters of the causal convolutional neural network under different quantile conditions and obtain the probability density distribution of wind power at various times within the following 200 hours. The presented method can obtain more useful information than conventional point and interval predictions. Moreover, a prediction of the future complete probability distribution of wind power can be realized. According to the actual data forecast of wind power in the PJM network in the United States, the proposed probability density prediction approach can not only obtain more accurate point prediction results, it also obtains the complete probability density curve prediction results for wind power. Compared with two other quantile regression methods, the developed technique can achieve a higher accuracy and smaller prediction interval range under the same confidence level.

Reduction method for multi-period time series scenarios of wind power

In the planning and operation of power systems containing wind power, it is of great significance to use a small number of representative wind power time series scenarios to accurately portray the stochastic characteristics of wind power. With the increase of the number of scenarios, how to form representative typical scenarios to balance computational efficiency and accuracy is an urgent problem to be solved. In this study, a bi-directional optimization method is proposed to generate daily wind power time series scenarios based on a single-period optimal scenario generation strategy and a multi-period scenario reduction strategy. First, the Wasserstein probability distance index is used to form a scenario model that is best approximated to the probability distribution of wind power, and the optimal scenarios of each period are generated. Secondly, a wind power scenario reduction strategy that integrates spatial distance and stochastic features is proposed. The improved taboo search method is used to selectively connect the representative scenarios of each period to form the representative daily wind power time series scenarios. Finally, the effectiveness and practicality of the proposed wind power time series scenario generation method are verified by simulations.

Distributionally Robust Joint Chance-Constrained Dispatch for Electricity–Gas–Heat Integrated Energy System Considering Wind Uncertainty

With the increasing penetration of wind power, the uncertainty associated with it brings more challenges to the operation of the integrated energy system (IES), especially the power subsystem. However, the typical strategies to deal with wind power uncertainty have poor performance in balancing economy and robustness. Therefore, this paper proposes a distributionally robust joint chance-constrained dispatch (DR-JCCD) model to coordinate the economy and robustness of the IES with uncertain wind power. The optimization dispatch model is formulated as a two-stage problem to minimize both the day-ahead and the real-time operation costs. Moreover, the ambiguity set is generated using Wasserstein distance, and the joint chance constraints are used to ensure that the safety constraints (e.g., ramping limit and transmission limit) can be satisfied jointly under the worst-case probability distribution of wind power. The model is remodeled as a mixed-integer tractable programming issue, which can be solved efficiently by ready-made solvers using linear decision rules and linearization methods. Case studies on an electricity–gas–heat regional integrated system, which includes a modified IEEE 24-bus system, 20 natural gas-nodes, and 6 heat-node system, are investigated for verification. Numerical simulation results demonstrate that the proposed DR-JCCD approach effectively coordinates the economy and robustness of IES and can offer operators a reasonable energy management scheme with an acceptable risk level.

Risk-Based Contingency-Constrained Optimal Power Flow With Adjustable Uncertainty Set of Wind Power

This article proposes a risk-based contingency-constrained optimal power flow model by leveraging the methods of both adjustable uncertainty set and distributionally robust optimization. In the proposed model, an adjustable uncertainty set of wind power is developed with network contingencies explicitly incorporated. Based on this uncertainty set, the proposed model is capable of securing the network against both wind power fluctuations and contingencies in a probabilistic manner with the optimal balance between operation cost and risk. Meanwhile, a data-driven <i>L</i>1-norm-based ambiguity set is employed so that the proposed model is distributionally robust to the ambiguous probability distribution of wind power and the size of the model remains unchanged as the available wind power data increases. A decomposition-based algorithm is also derived so that the proposed model is solvable by off-the-shelf solvers. Numerical studies on IEEE 14- and 118-bus systems are conducted to verify the effectiveness of the proposed model.

Decentralized Distributionally Robust Coordinated Dispatch of Multiarea Power Systems Considering Voltage Security

To achieve an efficient coordination of multiarea power systems, this article proposes a new coordinated dispatch optimization model for multiarea power systems. In the proposed optimization model, active and reactive powers are jointly optimized in a decentralized manner. Both of conventional generators and wind farms adopt the automatic voltage control strategy to mitigate the risk of undervoltage or overvoltage, and, thus, the nodal voltage security can be ensured in the optimization. At the same time, optimal admissible regions of wind power can be also obtained from decisions, which are optimized by making a tradeoff between the operational cost and renewable utilization in the optimization. Moreover, considering that it is hard to exactly estimate the true probability distribution of wind power due to modeling errors and insufficient historical data, the distributionally robust optimization is also employed in the proposed optimization model to take the uncertainty of wind power probability distributions into account. Besides, to improve the computational efficiency, an efficient algorithm is also developed. Numerical results on two test power systems demonstrate that compared with the traditional optimization models, the proposed decentralized dispatch model can achieve more reliable decisions and promote the renewable utilization.

Utilizing Energy Storage for Operational Adequacy of Wind-Integrated Bulk Power Systems

Renewable energy resources like wind generation are being rapidly integrated into modern power systems. Energy storage systems (ESS) are being viewed as a game-changer for renewable integration due to their ability to absorb the variability and uncertainty arising from the wind generation. While abundant literature is available on system adequacy and operational reliability evaluation, operational adequacy studies considering wind and energy storage have received very little attention, despite their elevated significance. This work presents a novel framework that integrates wind power and energy storage models to a bulk power system model to sequentially evaluate the operational adequacy in the operational mission time. The analytical models are developed using a dynamic system state probability evaluation approach by incorporating a system state probability estimation technique, wind power probability distribution, state enumeration, state transition matrix, and time series analysis in order to quantify the operational adequacy of a bulk power system integrated with wind power and ESS. The loss of load probability (LOLP) is used as the operational adequacy index to quantify the spatio-temporal variation in risk resulting from the generation and load variations, their distribution on the network structure, and the operational strategies of the integrated ESS. The proposed framework is aimed to serve as a guideline for operational planning, thereby simplifying the decision-making process for system operators while considering resources like wind and energy storage facilities. The methodology is applied to a test system to quantify the reliability and economic benefits accrued from different operational strategies of energy storage in response to wind generation and other operational objectives in different system scenarios.

Privacy-preserving distributed parameter estimation for probability distribution of wind power forecast error

Building the conditional probability distribution of wind power forecast errors benefits both wind farms (WFs) and independent system operators (ISOs). Establishing the joint probability distribution of wind power and the corresponding forecast data of spatially correlated WFs is the foundation for deriving the conditional probability distribution. Traditional parameter estimation methods for probability distributions require the collection of historical data of all WFs. However, in the context of multi-regional interconnected grids, neither regional ISOs nor WFs can collect the raw data of WFs in other regions due to privacy or competition considerations. Therefore, based on the Gaussian mixture model, this paper first proposes a privacy-preserving distributed expectation-maximization algorithm to estimate the parameters of the joint probability distribution. This algorithm consists of two original methods: (1) a privacy-preserving distributed summation algorithm and (2) a privacy-preserving distributed inner product algorithm. Then, we derive each WF’s conditional probability distribution of forecast error from the joint one. By the proposed algorithms, WFs only need local calculations and privacy-preserving neighboring communications to achieve the whole parameter estimation. These algorithms are verified using the wind integration data set published by the NREL.

Confidence Interval Based Distributionally Robust Real-Time Economic Dispatch Approach Considering Wind Power Accommodation Risk

This article proposes a confidence interval based distributionally robust real-time economic dispatch (CI-DRED) approach, which considers the risk related to accommodating wind power. In this article, only the wind power curtailment and load shedding due to wind power disturbances are evaluated in the operational risk. The proposed approach can strike a balance between the operational costs and risk even when the wind power probability distribution cannot be precisely estimated. A novel ambiguity set is developed based on the imprecise probability theory, which can be constructed based on the point-wise or family-wise confidence intervals. The worst pair of distributions in the established ambiguity set is then identified, and the original CI-DRED problem is transformed into a determined nonlinear dispatch problem accordingly. By using the sequential convex optimization method and piecewise linear approximation method, the nonlinear dispatch model is reformulated as a mixed integer linear programming problem, for which off-the-shelf solvers are available. A fast inactive constraint filtration method is also applied to further relieve the computational burden. Numerical results on the IEEE 118-bus system and a real 445-bus system verify the effectiveness and efficiency of the proposed approach.

Stochastic Economic Dispatch With Wind Using Versatile Probability Distribution and L-BFGS-B Based Dual Decomposition

This paper focuses on economic dispatch (ED) in power systems with intermittent wind power, which is a very critical issue in future power systems. A stochastic ED problem is formed based on the recently proposed versatile probability distribution of wind power. The problem is then analyzed and proved to be strictly convex. Although such convex optimization is tractable in many cases, it may take a long time to solve due to its large scale. This paper proposes a dual decomposition method to decompose the large problem. Then, two methods are employed to solve the decomposed problem, namely, the subgradient method and a faster method, limited-memory BFGS with box constraints (L-BFGS-B, a quasi-Newton method). Case studies were conducted to verify the efficiency of the dual decomposition and L-BFGS-B method for solving the stochastic ED problem.

A Two-Stage Distributionally Robust Coordinated Dispatch for Integrated Electricity and Natural-gas Energy Systems Considering Uncertainty of Wind Power

Application of gas-fired units and power-to-gas devices promotes coupling between power system and natural gas system, and hence provides an alternative solution for the wind power integration. Also, the traditional stochastic optimization and robust optimization methods may result in uneconomic or conservative decisions in dealing with the uncertain problems. Thus, a data-driven based distributionally robust dispatch model for the integrated electricity and natural gas system is proposed in this paper, which is constructed in a two-stage optimization fashion. The day-ahead total cost for the integrated system is regarded as the optimization objective in the first stage where the forecast wind power information is also taken into consideration while the output adjustment of thermal generation units and the supply regulation of natural gas source are included in the real-time dispatch (second stage in this study). Besides, the norm-1 and norm-inf are simultaneously combined to constrain the confidence set of wind power probability distribution. Then the model is solved by the Column-and-Constraint Generation algorithm. Finally, numerical results verify the effectiveness of the proposed model.

Optimal Available Transfer Capability Assessment Strategy for Wind Integrated Transmission Systems Considering Uncertainty of Wind Power Probability Distribution

Wind power prediction research shows that it is difficult to accurately and effectively estimate the probability distribution (PD) of wind power. When only partial information of the wind power probability distribution function is available, an optimal available transfer capability (ATC) assessment strategy considering the uncertainty on the wind power probability distribution is proposed in this paper. As wind power probability distribution is not accurately given, the proposed strategy can efficiently maximize ATC with the security operation constraints satisfied under any wind power PD function case in the uncertainty set. A distributional robust chance constrained (DRCC) model is developed to describe an optimal ATC assessment problem. To achieve tractability of the DRCC model, the dual optimization, S-lemma and Schur complement are adopted to eliminate the uncertain wind power vector in the DRCC model. According to the characteristics of the problem, the linear matrix inequality (LMI)-based particle swarm optimization (PSO) algorithm is used to solve the DRCC model which contains first and second-order moment information of the wind power. The modified IEEE 30-bus system simulation results show the feasibility and effectiveness of the proposed ATC assessment strategy.

Dispatchability Maximization for Co-Optimized Energy and Reserve Dispatch With Explicit Reliability Guarantee

In this paper, we consider dispatchability as the set of all admissible nodal wind power injections that will not cause infeasibility in real-time dispatch (RTD). Our work reveals that the dispatchability of the affine policy based RTD (AF-RTD) is a polytope whose coefficients are linear functions of the generation schedule and the gain matrix of affine policy. Two mathematical formulations of the dispatchability maximized energy and reserve dispatch (DM-ERD) are proposed. The first one maximizes the distance from the forecast to the boundaries of the dispatchability polytope subject to the available production cost or reserve cost. Provided the forecast value and variance of wind power, the generalized Gauss inequality (GGI) is adopted to evaluate the probability of infeasible RTD without the exact probability distribution of wind power. Combining the first formulation and the GGI approach, the second one minimizes the total cost subject to a desired reliability level through dispatchability maximization. Efficient convex optimization based algorithms are developed to solve these two models. Different from the conventional robust optimization method, our model does not rely on the specific uncertainty set of wind generation and directly optimizes the uncertainty accommodation capability of the power system. The proposed method is also compared with the affine policy based robust energy and reserve dispatch (AR-ERD). Case studies on the PJM 5-bus system illustrate the proposed concept and method. Experiments on the IEEE 118-bus system demonstrate the applicability of our method on moderate sized systems and its scalability to large dimensional uncertainty.

Distributionally Robust Solution to the Reserve Scheduling Problem With Partial Information of Wind Power

A distributionally robust optimization (DRO) model is presented for the reserve schedule decision-making problem with partial information of wind power, aiming to find a robust solution to the uncertainty of wind power probability distribution. The stochastic problem can be converted into an equivalent deterministic bilinear matrix inequality (BMI) problem. Numerical results verify the effectiveness of the proposed method.

Markov chain modeling for very-short-term wind power forecasting

A Wind power forecasting method based on the use of discrete time Markov chain models is developed starting from real wind power time series data. It allows to directly obtain in an easy way an estimate of the wind power distributions on a very short-term horizon, without requiring restrictive assumptions on wind power probability distribution. First and Second Order Markov Chain Model are analytically described. Finally, the application of the proposed method is illustrated with reference to a set of real data.

Optimal Dispatch of Electric Vehicles and Wind Power Using Enhanced Particle Swarm Optimization

In this paper, an economic dispatch model, which can take into account the uncertainties of plug-in electric vehicles (PEVs) and wind generators, is developed. A simulation based approach is first employed to study the probability distributions of the charge/discharge behaviors of PEVs. The probability distribution of wind power is also derived based on the assumption that the wind speed follows the Rayleigh distribution. The mathematical expectations of the generation costs of wind power and V2G (vehicle to grid) power are then derived analytically. An optimization algorithm is developed based on the well-established particle swarm optimization (PSO) and interior point method to solve the economic dispatch model. The proposed approach is demonstrated by the IEEE 118-bus test system.

Theoretical derivation of wind power probability distribution function and applications

The instantaneous wind power contained in the air current is directly proportional with the cube of the wind speed. In practice, there is a record of wind speeds in the form of a time series. It is, therefore, necessary to develop a formulation that takes into consideration the statistical parameters of such a time series. The purpose of this paper is to derive the general wind power formulation in terms of the statistical parameters by using the perturbation theory, which leads to a general formulation of the wind power expectation and other statistical parameter expressions such as the standard deviation and the coefficient of variation. The formulation is very general and can be applied specifically for any wind speed probability distribution function. Its application to two-parameter Weibull probability distribution of wind speeds is presented in full detail. It is concluded that provided wind speed is distributed according to a Weibull distribution, the wind power could be derived based on wind speed data. It is possible to determine wind power at any desired risk level, however, in practical studies most often 5% or 10% risk levels are preferred and the necessary simple procedure is presented for this purpose in this paper.

The Probability Distribution of Wind Power From a Dispersed Array of Wind Turbine Generators

A method is presented for estimating the probability distribution of wind power from a dispersed array of wind turbine sites where the correlation between wind speeds at distinct sites is less than unity. The distribution is obtained from a model for the joint probability distribution of wind speeds. This is able to incorporate arbitrary inter-site correlations. It is shown that this joint distribution reduces in the single site case to a wind speed distribution closely approximating the widely used Weibull; the multiple site power distribution is also shown to fit adequately to data on wind speeds from four sites in Western Australia. Results presented in graphical and tabular form for a range of representative cases show that a significant reduction in the variability of total wind power output may result from dispersion of aerogenerator sites; a quantitative guide to the magnitude of these effects is also provided.