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.

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.

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.

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.

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.

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.

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.

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.

Two-stage distributionally robust optimization-based coordinated scheduling of integrated energy system with electricity-hydrogen hybrid energy storage

A coordinated scheduling model based on two-stage distributionally robust optimization(TSDRO)is proposed for integrated energy systems(IESs)with electricity-hydrogen hybrid energy storage.The scheduling problem of the IES is divided into two stages in the TSDRO-based coordinated scheduling model.The first stage addresses the day-ahead optimal scheduling problem of the IES under deterministic forecasting information,while the sec-ond stage uses a distributionally robust optimization method to determine the intraday rescheduling problem under high-order uncertainties,building upon the results of the first stage.The scheduling model also considers col-laboration among the electricity,thermal,and gas networks,focusing on economic operation and carbon emissions.The flexibility of these networks and the energy gradient utilization of hydrogen units during operation are also incor-porated into the model.To improve computational efficiency,the nonlinear formulations in the TSDRO-based coordinated scheduling model are properly linearized to obtain a Mixed-Integer Linear Programming model.The Column-Constraint Generation(C&CG)algorithm is then employed to decompose the scheduling model into a mas-ter problem and subproblems.Through the iterative solution of the master problem and subproblems,an efficient analysis of the coordinated scheduling model is achieved.Finally,the effectiveness of the proposed TSDRO-based coordinated scheduling model is verified through case studies.The simulation results demonstrate that the proposed TSDRO-based coordinated scheduling model can effectively accomplish the optimal scheduling task while consider-ing the uncertainty and flexibility of the system.Compared with traditional methods,the proposed TSDRO-based coordinated scheduling model can better balance conservativeness and robustness.

Distributionally Robust Optimal Dispatch of Offshore Wind Farm Cluster Connected by VSC-MTDC Considering Wind Speed Correlation

Multi-terminal voltage source converter-based highvoltage direct current(VSC-MTDC)transmission technology has become an important mode for connecting adjacent offshore wind farms(OWFs)to power systems.Optimal dispatch of an OWF cluster connected by the VSC-MTDC can improve economic operation under the uncertainty of wind speeds.A two-stage distributionally robust optimal dispatch(DROD)model for the OWF cluster connected by VSC-MTDC is established.The first stage in this model optimizes the unit commitment of wind turbines to minimize mechanical loss cost of units under the worst joint probability distribution(JPD)of wind speeds,while the second stage searches for the worst JPD of wind speeds in the ambiguity set(AS)and optimizes active power output of wind turbines to minimize the penalty cost of the generation deviation and active power loss cost of the system.Based on the Kullback–Leibler(KL)divergence distance,a data-driven AS is constructed to describe the uncertainty of wind speed,considering the correlation between wind speeds of adjacent OWFs in the cluster by their joint PD.The original solution of the two-stage DROD model is transformed into the alternating iterative solution of the master problem and the sub-problem by the column-and-constraint generation(C&CG)algorithm,and the master problem is decomposed into a mixedinteger linear programming and a continuous second-order cone programming by the generalized Benders decomposition method to improve calculation efficiency.Finally,case studies on an actual OWF cluster with three OWFs demonstrate the correctness and efficiency of the proposed model and algorithm.

Distributionally Robust Economic Dispatch Using IDM for Integrated Electricity-heat-gas Microgrid Considering Wind Power

Multi-energy microgrids,such as integrated electricity-heat-gas microgrids(IEHS-MG),have been widely recognized as one of the most convenient ways to connect wind power(WP).However,the inherent intermittency and uncertainty of WP still render serious power curtailment in the operation.To this end,this paper presents an IEHS-MG model equipped with power-to-gas technology,thermal storage,electricity storage,and an electrical boiler for improving WP utilization efficiency.Moreover,a two-stage distributionally robust economic dispatch model is constructed for the IEHSMG,with the objective of minimizing total operational costs.The first stage determines the day-ahead decisions including on/off state and set-point decisions.The second stage adjusts the day-ahead decision according to real-time WP realization.Furthermore,WP uncertainty is characterized through an Imprecise Dirichlet model(IDM)based ambiguity set.Finally,Column-and-Constraints Generation method is utilized to solve the model,which provides a day-ahead economic dispatch strategy that immunizes against the worst-case WP distributions.Case studies demonstrate the presented IEHS-MG model outperforms traditional IEHS-MG model in terms of WP utilization and dispatch economics,and that distributionally robust optimization can handle uncertainty effectively.

Distributionally Robust Energy Consumption Scheduling of HVAC Considering the Uncertainty of Outdoor Temperature and Human Activities

Achieving a low or zero carbon target is to reduce energy demand and improve energy efficiency of electricity consumers.One of the main electricity consumers in power systems is heating,ventilation,and air conditioning systems(HVACs),which cost around 30%of the total usage in commercial buildings.This paper investigates the scheduling problem of HVAC energy consumption taking into account two uncertainties:outdoor temperature and human activities.The distributionally robust optimization approach(DROA)is extended to deal with these two uncertainties which are modeled by the proposed disjointed layered ambiguity sets according to historical data.Based on the proposed DROA method,the distributionally robust chance constraints(DRCCs)will be formulated as a nonlinear optimization problem,converted into a linear optimization problem using duality theorem and solved using SeDuMi solver.Simulation results are used to compare with existing methods,which shows the proposed DROA can decrease 2.81%and 0.14%of the electricity cost in comparison with traditional RO method and DROA based on a nest layered ambiguity set,respectively.Also,the proposed DROA decreases the number and maximum of violations from the comfort level of users.The multi-zone HVAC system model is used in the case study to verify the proposed DROA with the disjointed ambiguity set.The consecutive simulation results illustrate the proposed DROA approach can provide stable performance in a three-day scheduling period.

A Mixed-integer SDP Solution to Distributionally Robust Unit Commitment with Second Order Moment Constraints

A power system unit commitment(UC)problem considering uncertainties of renewable energy sources is investigated in this paper,through a distributionally robust optimization approach.We assume that the first and second order moments of stochastic parameters can be inferred from historical data,and then employed to model the set of probability distributions.The resulting problem is a two-stage distributionally robust unit commitment with second order moment constraints,and it can be recast as a mixed-integer semidefinite programming(MI-SDP)with finite constraints.The solution algorithm of the problem comprises solving a series of relaxed MI-SDPs and a subroutine of feasibility checking and vertex generation.Based on the verification of strong duality of the semidefinite programming(SDP)problems,we propose a cutting plane algorithm for solving the MI-SDPs;we also introduce an SDP relaxation for the feasibility checking problem,which is an intractable biconvex optimization.Experimental results on the IEEE 6-bus system are presented,showing that without any tuning of parameters,the real-time operation cost of distributionally robust UC method outperforms those of deterministic UC and two-stage robust UC methods in general,and our method also enjoys higher reliability of dispatch operation.

Distributionally Robust Co-optimization of Transmission Network Expansion Planning and Penetration Level of Renewable Generation

Transmission network expansion can significantly improve the penetration level of renewable generation.However,existing studies have not explicitly revealed and quantified the trade-off between the investment cost and penetration level of renewable generation.This paper proposes a distributionally robust optimization model to minimize the cost of transmission network expansion under uncertainty and maximize the penetration level of renewable generation.The proposed model includes distributionally robust joint chance constraints,which maximize the minimum expectation of the renewable utilization probability among a set of certain probability distributions within an ambiguity set.The proposed formulation yields a twostage robust optimization model with variable bounds of the uncertain sets,which is hard to solve.By applying the affine decision rule,second-order conic reformulation,and duality,we reformulate it into a single-stage standard robust optimization model and solve it efficiently via commercial solvers.Case studies are carried on the Garver 6-bus and IEEE 118-bus systems to illustrate the validity of the proposed method.

Distributionally Robust Transmission Expansion Planning Considering Uncertainty of Contingency Probability

The outage of power system equipment is one of the most important factors that affect the reliability and economy of power system.It is crucial to consider the influence of contingencies elaborately in planning problem.In this paper,a distributionally robust transmission expansion planning model is proposed in which the uncertainty of contingency probability is considered.The uncertainty of contingency probability is described by uncertainty interval based on the outage rate of single equipment.An epigraph reformulation and Benders decomposition are applied to solve the proposed model.Finally,the feasibility and effectiveness of the proposed model are illustrated on the IEEE RTS system and the IEEE 118-bus system.

A Data-driven Distributionally Robust Operational Model for Urban Integrated Energy Systems

A multi-energy conversion can effectively increase the utilization of renewable energy in the urban integrated energy system(UIES).Meanwhile,the uncertainties of renewable energy resources(e.g.,wind energy)also bring increased challenges to the operation of UIES.In this study,a typical two-stage datadriven distributionally robust operation(DDRO)model based on finite scenarios is proposed for UIES including power,gas and heat networks to obtain a salient strategy from both an economic and robustness perspective.In the first stage,the forecasted information for wind power is especially included to improve the economic aspect of robust decisions.The worst probability distribution for the selected known real-time wind power scenarios can be identified in the second stage where the power differences caused by the real-time uncertainties of wind power can be mitigated by flexible regulation of energy purchasing and coupling units(such as gas turbine,power to gas equipment,electric boiler and gas boiler).Moreover,norm-1 and norm-inf co-constraints are utilized to construct a confidence set for the probability distributions of uncertain wind power.The whole two-stage model is solved by the column-and-constraint generation(CCG)algorithm.Finally,case studies are conducted to show the performance of the proposed model and various approaches.Index Terms-Data-driven methods,distributionally robust optimization,urban integrated energy system,wind power.

Distributionally robust optimization of home energy management system based on receding horizon optimization

This paper investigates the scheduling strategy of schedulable load in home energy management system(HEMS)under uncertain environment by proposing a distributionally robust optimization(DRO)method based on receding horizon optimization(RHO-DRO).First,the optimization model of HEMS,which contains uncertain variable outdoor temperature and hot water demand,is established and the scheduling problem is developed into a mixed integer linear programming(MILP)by using the DRO method based on the ambiguity sets of the probability distribution of uncertain variables.Combined with RHO,the MILP is solved in a rolling fashion using the latest update data related to uncertain variables.The simulation results demonstrate that the scheduling results are robust under uncertain environment while satisfying all operating constraints with little violation of user thermal comfort.Furthermore,compared with the robust optimization(RO)method,the RHO-DRO method proposed in this paper has a lower conservation and can save more electricity for users.

Convex Reformulation for Two-sided Distributionally Robust Chance Constraints with Inexact Moment Information

Constraints on each node and line in power systems generally have upper and lower bounds,denoted as twosided constraints.Most existing power system optimization methods with the distributionally robust(DR)chance-constrained program treat the two-sided DR chance constraint separately,which is an inexact approximation.This letter derives an equivalent reformulation for the generic two-sided DR chance constraint under the interval moment based ambiguity set,which does not require the exact moment information.The derived reformulation is a second-order cone program(SOCP)formulation and is then applied to the optimal power flow(OPF)problem under uncertainty.Numerical results on several IEEE systems demonstrate the effectiveness of the proposed SOCP formulation and show the differences with other DR chance-constrained OPF approaches.

Data-driven Stochastic Programming with Distributionally Robust Constraints Under Wasserstein Distance:Asymptotic Properties

Distributionally robust optimization is a dominant paradigm for decision-making problems where the distribution of random variables is unknown.We investigate a distributionally robust optimization problem with ambiguities in the objective function and countably infinite constraints.The ambiguity set is defined as a Wasserstein ball centered at the empirical distribution.Based on the concentration inequality of Wasserstein distance,we establish the asymptotic convergence property of the datadriven distributionally robust optimization problem when the sample size goes to infinity.We show that with probability 1,the optimal value and the optimal solution set of the data-driven distributionally robust problem converge to those of the stochastic optimization problem with true distribution.Finally,we provide numerical evidences for the established theoretical results.