Data-driven Wasserstein distributionally robust chance-constrained optimization for crude oil scheduling under uncertainty

Crude oil scheduling optimization is an effective method to enhance the economic benefits of oil refining.But uncertainties,including uncertain demands of crude distillation units(CDUs),might make the production plans made by the traditional deterministic optimization models infeasible.A data-driven Wasserstein distributionally robust chance-constrained(WDRCC)optimization approach is proposed in this paper to deal with demand uncertainty in crude oil scheduling.First,a new deterministic crude oil scheduling optimization model is developed as the basis of this approach.The Wasserstein distance is then used to build ambiguity sets from historical data to describe the possible realizations of probability distributions of uncertain demands.A cross-validation method is advanced to choose suitable radii for these ambiguity sets.The deterministic model is reformulated as a WDRCC optimization model for crude oil scheduling to guarantee the demand constraints hold with a desired high probability even in the worst situation in ambiguity sets.The proposed WDRCC model is transferred into an equivalent conditional value-at-risk representation and further derived as a mixed-integer nonlinear programming counterpart.Industrial case studies from a real-world refinery are conducted to show the effectiveness of the proposed method.Out-of-sample tests demonstrate that the solution of the WDRCC model is more robust than those of the deterministic model and the chance-constrained model.

A dynamical neural network approach for distributionally robust chance-constrained Markov decision process

In this paper,we study the distributionally robust joint chance-constrained Markov decision process.Utilizing the logarithmic transformation technique,we derive its deterministic reformulation with bi-convex terms under the moment-based uncertainty set.To cope with the non-convexity and improve the robustness of the solution,we propose a dynamical neural network approach to solve the reformulated optimization problem.Numerical results on a machine replacement problem demonstrate the efficiency of the proposed dynamical neural network approach when compared with the sequential convex approximation approach.

Distributionally robust optimization based chance-constrained energy management for hybrid energy powered cellular networks

Energy harvesting has been recognized as a promising technique with which to effectively reduce carbon emis-sions and electricity expenses of base stations.However,renewable energy is inherently stochastic and inter-mittent,imposing formidable challenges on reliably satisfying users'time-varying wireless traffic demands.In addition,the probability distribution of the renewable energy or users’wireless traffic demand is not always fully known in practice.In this paper,we minimize the total energy cost of a hybrid-energy-powered cellular network by jointly optimizing the energy sharing among base stations,the battery charging and discharging rates,and the energy purchased from the grid under the constraint of a limited battery size at each base station.In solving the formulated non-convex chance-constrained stochastic optimization problem,a new ambiguity set is built to characterize the uncertainties in the renewable energy and wireless traffic demands according to interval sets of the mean and covariance.Using this ambiguity set,the original optimization problem is transformed into a more tractable second-order cone programming problem by exploiting the distributionally robust optimization approach.Furthermore,a low-complexity distributionally robust chance-constrained energy management algo-rithm,which requires only interval sets of the mean and covariance of stochastic parameters,is proposed.The results of extensive simulation are presented to demonstrate that the proposed algorithm outperforms existing methods in terms of the computational complexity,energy cost,and reliability.

Adjustable and distributionally robust chance-constrained economic dispatch considering wind power uncertainty

This paper proposes an adjustable and distributionally robust chance-constrained(ADRCC) optimal power flow(OPF) model for economic dispatch considering wind power forecasting uncertainty. The proposed ADRCC-OPF model is distributionally robust because the uncertainties of the wind power forecasting are represented only by their first-and second-order moments instead of a specific distribution assumption. The proposed model is adjustable because it is formulated as a second-order cone programming(SOCP) model with an adjustable coefficient.This coefficient can control the robustness of the chance constraints, which may be set up for the Gaussian distribution, symmetrically distributional robustness, or distributionally robust cases considering wind forecasting uncertainty. The conservativeness of the ADRCC-OPF model is analyzed and compared with the actual distribution data of wind forecasting error. The system operators can choose an appropriate adjustable coefficient to tradeoff between the economics and system security.

Distributionally Robust Newsvendor Model for Fresh Products under Cap-and-Offset Regulation

The cap-and-offset regulation is a practical scheme to lessen carbon emissions.The retailer selling fresh products can adopt sustainable technologies to lessen greenhouse gas emissions.We aim to analyze the optimal joint strategies on order quantity and sustainable technology investment when the retailer faces stochastic market demand and can only acquire the mean and variance of distribution information.We construct a distributionally robust optimization model and use the Karush-Kuhn-Tucker(KKT)conditions to solve the analytic formula of optimal solutions.By comparing the models with and without investing in sustainable technologies,we examine the effect of sustainable technologies on the operational management decisions of the retailer.Finally,some computational examples are applied to analyze the impact of critical factors on operational strategies,and some managerial insights are given based on the analysis results.

Data-driven Distributionally Adjustable Robust Chance-constrained DG Capacity Assessment

Moving away from fossil fuels towards renewable sources requires system operators to determine the capacity of distribution systems to safely accommodate green and distributed generation(DG).However,the DG capacity of a distribution system is often underestimated due to either overly conservative electrical demand and DG output uncertainty modelling or neglecting the recourse capability of the available components.To improve the accuracy of DG capacity assessment,this paper proposes a distributionally adjustable robust chance-constrained approach that utilises uncertainty information to reduce the conservativeness of conventional robust approaches.The proposed approach also enables fast-acting devices such as inverters to adjust to the real-time realisation of uncertainty using the adjustable robust counterpart methodology.To achieve a tractable formulation,we first define uncertain chance constraints through distributionally robust conditional value-at-risk(CVaR),which is then reformulated into convex quadratic constraints.We subsequently solve the resulting large-scale,yet convex,model in a distributed fashion using the alternating direction method of multipliers(ADMM).Through numerical simulations,we demonstrate that the proposed approach outperforms the adjustable robust and conventional distributionally robust approaches by up to 15%and 40%,respectively,in terms of total installed DG capacity.

A Distributionally Robust Optimization Method for Passenger Flow Control Strategy and Train Scheduling on an Urban Rail Transit Line

Regular coronavirus disease 2019(COVID-19)epidemic prevention and control have raised new require-ments that necessitate operation-strategy innovation in urban rail transit.To alleviate increasingly seri-ous congestion and further reduce the risk of cross-infection,a novel two-stage distributionally robust optimization(DRO)model is explicitly constructed,in which the probability distribution of stochastic scenarios is only partially known in advance.In the proposed model,the mean-conditional value-at-risk(CVaR)criterion is employed to obtain a tradeoff between the expected number of waiting passen-gers and the risk of congestion on an urban rail transit line.The relationship between the proposed DRO model and the traditional two-stage stochastic programming(SP)model is also depicted.Furthermore,to overcome the obstacle of model solvability resulting from imprecise probability distributions,a discrepancy-based ambiguity set is used to transform the robust counterpart into its computationally tractable form.A hybrid algorithm that combines a local search algorithm with a mixed-integer linear programming(MILP)solver is developed to improve the computational efficiency of large-scale instances.Finally,a series of numerical examples with real-world operation data are executed to validate the pro-posed approaches.

Distributionally Robust Optimal Dispatch of Virtual Power Plant Based on Moment of Renewable Energy Resource

Virtual power plants can effectively integrate different types of distributed energy resources,which have become a new operation mode with substantial advantages such as high flexibility,adaptability,and economy.This paper proposes a distributionally robust optimal dispatch approach for virtual power plants to determine an optimal day-ahead dispatch under uncertainties of renewable energy sources.The proposed distributionally robust approach characterizes probability distributions of renewable power output by moments.In this regard,the faults of stochastic optimization and traditional robust optimization can be overcome.Firstly,a second-order cone-based ambiguity set that incorporates the first and second moments of renewable power output is constructed,and a day-ahead two-stage distributionally robust optimization model is proposed for virtual power plants participating in day-ahead electricity markets.Then,an effective solution method based on the affine policy and second-order cone duality theory is employed to reformulate the proposed model into a deterministic mixed-integer second-order cone programming problem,which improves the computational efficiency of the model.Finally,the numerical results demonstrate that the proposed method achieves a better balance between robustness and economy.They also validate that the dispatch strategy of virtual power plants can be adjusted to reduce costs according to the moment information of renewable power output.

Robust Airfoil Optimization with Multi-objective Estimation of Distribution Algorithm

A transonic airfoil designed by means of classical point-optimization may result in its dramatically inferior performance under off-design conditions. To overcome this shortcoming, robust design is proposed to find out the optimal profile of an airfoil to maintain its performance in an uncertain environment. The robust airfoil optimization is aimed to minimize mean values and variances of drag coefficients while satisfying the lift and thickness constraints over a range of Mach numbers. A multi-objective estimation of distribution algorithm is applied to the robust airfoil optimization on the base of the RAE2822 benchmark airfoil. The shape of the airfoil is obtained through superposing ten Hick-Henne shape functions upon the benchmark airfoil. A set of design points is selected according to a uniform design table for aerodynamic evaluation. A Kriging model of drag coefficient is constructed with those points to reduce computing costs. Over the Mach range from 0.7 to 0.8, the airfoil generated by the robust optimization has a configuration characterized by supercritical airfoil with low drag coefficients. The small fluctuation in its drag coefficients means that the performance of the robust airfoil is insensitive to variation of Mach number.

Risk-Averse Two-Stage Distributionally Robust Economic Dispatch Model Under Uncertain Renewable Energy

With the participation of large quantities of renewable energy in power system operations,their volatility and intermittence increases the difficulties and challenges of power system economic scheduling.Considering the uncertainty of renewable energy generation,based on the distributionally robust optimization method,a two-stage economic dispatch model is proposed to minimize the total operation costs.In this paper,it is assumed that the fluctuating of renewable power generation follows the unknown probability distribution that is restricted in an ambiguity set,which is established by utilizing the first-order moment information of available historical data.Furthermore,the theory of conditional value-at-risk is introduced to transform the model into a tractable model,which we call robust counterpart formulation.Based on the stochastic dual dynamic programming method,an improved iterative algorithm is proposed to solve the robust counterpart problem.Specifically,the convergence optimum can be obtained by the improved iterative algorithm,which performs a forward pass and backward pass repeatedly in each iterative process.Finally,by comparing with other methods,the results on the modified IEEE 6-bus,118-bus,and 300-bus system show the effectiveness and advantages of the proposed model and method.

Distributionally robust optimization configuration method for island microgrid considering extreme scenarios

The marine climate conditions are intricate and variable. In scenarios characterized by high proportions of wind and solar energy access, the uncertainty regarding the energy sources for island microgrid is significantly exacerbated, presenting challenges to both the economic viability and reliability of the capacity configuration for island microgrids. To address this issue, this paper proposes a distributionally robust optimization (DRO) method for island microgrids, considering extreme scenarios of wind and solar conditions. Firstly, to address the challenge of determining the probability distribution functions of wind and solar in complex island climates, a conditional generative adversarial network (CGAN) is employed to generate a scenario set for wind and solar conditions. Then, by combining k-means clustering with an extreme scenario selection method, typical scenarios and extreme scenarios are selected from the generated scenario set, forming the scenario set for the DRO model of island microgrids. On this basis, a DRO model based on multiple discrete scenarios is constructed with the objective of minimizing the sum of investment costs, operation and maintenance costs, fuel purchase costs, penalty costs of wind and solar curtailment, and penalty costs of load loss. The model is subjected to equipment operation and power balance constraints, and solved using the columns and constraints generation (CCG) algorithm. Finally, through typical examples, the effectiveness of this paper’s method in balancing the economic viability and robustness of the configuration scheme for the island microgrid, as well as reducing wind and solar curtailment and load loss, is verified.

Reliability-based Robust Optimization Design of Automobile Components with Non-normal Distribution Parameters

In the reliability designing procedure of the vehicle components, when the distribution styles of the random variables are unknown or non-normal distribution, the result evaluated contains great error or even is wrong if the reliability value R is larger than 1 by using the existent method, in which case the formula is necessary to be revised. This is obviously inconvenient for programming. Combining reliability-based optimization theory, robust designing method and reliability based sensitivity analysis, a new method for reliability robust designing is proposed. Therefore the influence level of the designing parameters’ changing to the reliability of vehicle components can be obtained. The reliability sensitivity with respect to design parameters is viewed as a sub-objective function in the multi-objective optimization problem satisfying reliability constraints. Given the first four moments of basic random variables, a fourth-moment technique and the proposed optimization procedure can obtain reliability-based robust design of automobile components with non-normal distribution parameters accurately and quickly. By using the proposed method, the distribution style of the random parameters is relaxed. Therefore it is much closer to the actual reliability problems. The numerical examples indicate the following: (1) The reliability value obtained by the robust method proposed increases (】0.04%) comparing to the value obtained by the ordinary optimization algorithm; (2) The absolute value of reliability-based sensitivity decreases (】0.01%), and the robustness of the products’ quality is improved accordingly. Utilizing the reliability-based optimization and robust design method in the reliability designing procedure reduces the manufacture cost and provides the theoretical basis for the reliability and robust design of the vehicle components.

Risk Constrained Self-scheduling of AA-CAES Facilities in Electricity and Heat Markets:A Distributionally Robust Optimization Approach

Advanced adiabatic compressed air energy storage(AA-CAES)has the advantages of large capacity,long service time,combined heat and power generation(CHP),and does not consume fossil fuels,making it a promising storage technology in a low-carbon society.An appropriate self-scheduling model can guarantee AA-CAES’s profit and attract investments.However,very few studies refer to the cogeneration ability of AA-CAES,which enables the possibility to trade in the electricity and heat markets at the same time.In this paper,we propose a multimarket self-scheduling model to make full use of heat produced in compressors.The volatile market price is modeled by a set of inexact distributions based on historical data through-divergence.Then,the self-scheduling model is cast as a robust risk constrained program by introducing Stackelberg game theory,and equivalently reformulated as a mixed-integer linear program(MILP).The numerical simulation results validate the proposed method and demonstrate that participating in multienergy markets increases overall profits.The impact of uncertainty parameters is also discussed in the sensibility analysis.

Degree distribution and robustness of cooperative communication network with scale-free model

With the requirements of users enhanced for wireless communication, the cooperative communication will become a development trend in future. In this paper, a model based on complex networks with both preferential attachment is researched to solve an actual network CCN （Cooperative Communication Network）. Firstly, the evolution of CCN is given by four steps with different probabilities. At the same time, the rate equations of nodes degree are presented to analyze the evolution of CCN. Secondly, the degree distribution is analyzed by calculating the rate equation and numerical simulation. Finally, the robustness of CCN is studied by numerical simulation with random attack and intentional attack to analyze the effects of degree distribution and average path length. The results of this paper are more significant for building CCN to programme the resource of communication.

Application of Robust Strategies in Location Selection of Logistics Distribution Center for Fresh Agricultural Products

In view of the uncertainty in the location selection of logistics distribution center for the fresh agricultural products,the present study established a robust model based on the maximization of principal component score taking budget cost parameters as an example.In the process of model solving,the interval form of the uncertain set was used to clarify the constraint conditions,to transform into a certain 0-1 integer linear programming model,so as to solve with the aid of LINGO software.Finally,through studying the location selection of logistics distribution center for fresh agricultural products in the Beijing-Tianjin-Hebei region,it analyzed the application of the robust model and tested the validity of the model.

Robust and Efficient Reliability Estimation for Exponential Distribution

In modeling reliability data,the exponential distribution is commonly used due to its simplicity.For estimating the parameter of the exponential distribution,classical estimators including maximum likelihood estimator represent the most commonly used method and are well known to be efficient.However,the maximum likelihood estimator is highly sensitive in the presence of contamination or outliers.In this study,a robust and efficient estimator of the exponential distribution parameter was proposed based on the probability integral transform statistic.To examine the robustness of this new estimator,asymptotic variance,breakdown point,and gross error sensitivity were derived.This new estimator offers reasonable protection against outliers besides being simple to compute.Furthermore,a simulation study was conducted to compare the performance of this new estimator with the maximum likelihood estimator,weighted likelihood estimator,and M-scale estimator in the presence of outliers.Finally,a statistical analysis of three reliability data sets was conducted to demonstrate the performance of the proposed estimator.

Adaptive Sparse Group Variable Selection for a Robust Mixture Regression Model Based on Laplace Distribution

The traditional estimation of Gaussian mixture model is sensitive to heavy-tailed errors;thus we propose a robust mixture regression model by assuming that the error terms follow a Laplace distribution in this article. And for the variable selection problem in our new robust mixture regression model, we introduce the adaptive sparse group Lasso penalty to achieve sparsity at both the group-level and within-group-level. As numerical experiments show, compared with other alternative methods, our method has better performances in variable selection and parameter estimation. Finally, we apply our proposed method to analyze NBA salary data during the period from 2018 to 2019.

On Robustness of a Sequential Test for Scale Parameter of Gamma and Exponential Distributions

The main aim of the present paper is to study the robustness of the developed sequential probability ratio test (SPRT) for testing the hypothesis about scale parameter of gamma distribution with known shape parameter and exponential distribution with location parameter. The robustness of the SPRT for scale parameter of gamma distribution is studied when the shape parameter has undergone a change. The similar study is conducted for the scale parameter of exponential distribution when the location parameter has undergone a change. The expressions for operating characteristic and average sample number functions are derived. It is found in both the cases that the SPRT is robust only when there is a slight variation in the shape and location parameter in the respective distributions.

Robust Variance Components Estimation in the PERG Mixed Distributions of Empirical Variances—PEROBVC Method

A mixed distribution of empirical variances, composed of two distributions the basic and contaminating ones, and referred to as PERG mixed distribution of empirical variances, is considered. In the paper a robust inverse problem solution is given, namely a (new) robust method for estimation of variances of both distributions—PEROBVC Method, as well as the estimates for the numbers of observations for both distributions and, in this way also the estimate of contamination degree.

考虑移动氢能存储的港口多能微网两阶段分布鲁棒优化调度

为有效应对海上风电固有的间歇及波动性给港口多能微网带来的不确定性风险,提出一种考虑移动氢能存储的港口多能微网两阶段分布鲁棒优化调度模型。首先,结合Wasserstein距离实现风电出力概率分布模糊集的精确刻画,并通过非参数核密度估计拟合海上风电预测误差概率分布,获得不同置信水平下风电出力区间及场景。其次,分析氢能船舶、汽车等移动氢能存储资源对间歇性风电出力的能源存储潜力,并结合用能心理、交通属性差异,将两者分别建模为激励型、价格型需求响应,实现港口移动氢能存储灵活性资源的高效聚合。再次,针对含移动氢能存储的港口多能微网,构建基于概率分布模糊集的日前-日内两阶段分布鲁棒优化调度模型,并运用线性决策规则与强对偶理论将其转换为混合整数线性规划模型求解。最后,基于海上风电实测数据进行仿真验证。结果证明,移动氢能存储可显著提升港口多能微网的低碳灵活性,所提模型在兼顾港口多能微网经济性的同时,可进一步保证风电不确定性风险下的鲁棒性。