A Two-Stage Scenario-Based Robust Optimization Model and a Column-Row Generation Method for Integrated Aircraft Maintenance-Routing and Crew Rostering

Motivated by a critical issue of airline planning process,this paper addresses a new two-stage scenario-based robust optimization in operational airline planning to cope with uncertainty and possible flight disruptions.Following the route network scheme and generated flight timetables,aircraft maintenance routing and crew scheduling are critical factors in airline planning and operations cost management.This study considers the simultaneous assignment of aircraft fleet and crew to the scheduled flight while satisfying a set of operational constraints,rules,and regulations.Considering multiple locations for airline maintenance and crew bases,we solve the problem of integrated Aircraft Maintenance Routing and Crew Rostering(AMRCR)to achieve the minimum airline cost.One real challenge to the efficiency of the planning results is the possible disruptions in the initial scheduled flights.Due to the fact that disruption scenarios are expressed discretely with a specified probability,and we provide adjustable decisions under disruption to deal with this disruption risk,we provide a Two-Stage Scenario-Based Robust Optimization(TSRO)model.In this model,here-and-now or first-stage variables are the initial resource assignment.Furthermore,to adapt itself to different disruption scenarios,the model considers some adjustable variables,such as the decision to cancel the flight in case of disruption,as wait-and-see or second-stage variables.Considering the complexity of integrated models,and the scenario-based decomposable structure of the TRSO model to solve it with better computational performance,we apply the column and row generation(CRG)method that iteratively considers the disruption scenarios.The numerical results confirm the applicability of the proposed TSRO model in providing the AMRCR problem with an integrated and robust solution with an acceptable level of computational tractability.To evaluate the proposed TSRO model,which solves the AMRCR problem in an integrated and robust manner,five Key Performance Indicators(KPIs)like Number of delayed/canceled flights,Average delay time,and Average profit are taken into account.As key results driven by conducting a case study,we show the proposed TSRO model has substantially improved the solutions at all indicators compared with those of the sequential/non-integrated and nominal/non-robust models.The simulated instances used to assess the performance of the proposed model and CRG method reveal that both CPLEX and the CRG method exhibit comparable and nearly optimal performance for small-scale problems.However,for large-scale instances the proposed TSRO model falls short in terms of computational efficiency.Conversely,the proposed CRG method is capable of significantly reducing computational time and the optimality gap to an acceptable level.

Research on Regulation Method of Energy Storage System Based on Multi-Stage Robust Optimization

To address the scheduling problem involving energy storage systems and uncertain energy,we propose a method based on multi-stage robust optimization.This approach aims to regulate the energy storage system by using a multi-stage robust optimal control method,which helps overcome the limitations of traditional methods in terms of time scale.The goal is to effectively utilize the energy storage power station system to address issues caused by unpredictable variations in environmental energy and fluctuating load throughout the day.To achieve this,a mathematical model is constructed to represent uncertain energy sources such as photovoltaic and wind power.The generalized Benders Decomposition method is then employed to solve the multi-stage objective optimization problem.By decomposing the problem into a series of sub-objectives,the system scale is effectively reduced,and the algorithm’s convergence ability is improved.Compared with other algorithms,the multi-stage robust optimization model has better economy and convergence ability and can be used to guide the power dispatching of uncertain energy and energy storage systems.

A robust optimization model for demand response management with source-grid-load collaboration to consume wind-power

To accommodate wind power as safely as possible and deal with the uncertainties of the output power of winddriven generators,a min-max-min two-stage robust optimization model is presented,considering the unit commitment,source-network load collaboration,and control of the load demand response.After the constraint functions are linearized,the original problem is decomposed into the main problem and subproblem as a matrix using the strong dual method.The minimum-maximum of the original problem was continuously maximized using the iterative method,and the optimal solution was finally obtained.The constraint conditions expressed by the matrix may reduce the calculation time,and the upper and lower boundaries of the original problem may rapidly converge.The results of the example show that the injected nodes of the wind farms in the power grid should be selected appropriately;otherwise,it is easy to cause excessive accommodation of wind power at some nodes,leading to a surge in reserve costs and the load demand response is continuously optimized to reduce the inverse peak regulation characteristics of wind power.Thus,the most economical optimization scheme for the worst scenario of the output power of the generators is obtained,which proves the economy and reliability of the two-stage robust optimization method.

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.

Comparison between 4D robust optimization methods for carbon-ion treatment planning

Intensity-modulated particle therapy(IMPT)with carbon ions is comparatively susceptible to various uncertainties caused by breathing motion,including range,setup,and target positioning uncertainties.To determine relative biological effectiveness-weighted dose(RWD)distributions that are resilient to these uncertainties,the reference phase-based four-dimensional(4D)robust optimization(RP-4DRO)and each phase-based 4D robust optimization(EP-4DRO)method in carbon-ion IMPT treatment planning were evaluated and compared.Based on RWD distributions,4DRO methods were compared with 4D conventional optimization using planning target volume(PTV)margins(PTV-based optimization)to assess the effectiveness of the robust optimization methods.Carbon-ion IMPT treatment planning was conducted in a cohort of five lung cancer patients.The results indicated that the EP-4DRO method provided better robustness(P=0.080)and improved plan quality(P=0.225)for the clinical target volume(CTV)in the individual respiratory phase when compared with the PTV-based optimization.Compared with the PTV-based optimization,the RP-4DRO method ensured the robustness(P=0.022)of the dose distributions in the reference breathing phase,albeit with a slight sacrifice of the target coverage(P=0.450).Both 4DRO methods successfully maintained the doses delivered to the organs at risk(OARs)below tolerable levels,which were lower than the doses in the PTV-based optimization(P<0.05).Furthermore,the RP-4DRO method exhibited significantly superior performance when compared with the EP-4DRO method in enhancing overall OAR sparing in either the individual respiratory phase or reference respiratory phase(P<0.05).In general,both 4DRO methods outperformed the PTV-based optimization in terms of OAR sparing and robustness.

Emergency Energy Management of Microgrid in Industrial Park Basedon Robust Optimization

Reducing the impact of power outages and maintaining the power supply duration must be considered in implementing emergency energy dispatching in micro-networks.This paper studies a new emergency energy treatment method based on the robust optimal method and the industrial park micro-network with the optical energy storage system.After controlling the load input,a control strategy of adjusting and removing is proposed.Rolling optimal theory is applied to emergency energy scheduling based on a robust optimal mathematical model.A weighting factor is introduced into the optimal model to balance the importance of reducing and retaining the power supply.Uncertainty is designed to adjust the effect of uncertainty on the problem.The example shows that this method can flexibly set the weight coefficient and uncertainty value according to the actual situation so that the input of the control load can be optimized.

A Metamodeling Method Based on Support Vector Regression for Robust Optimization

Metamodeling techniques have been used in robust optimization to reduce the high computational cost of the uncertainty analysis and improve the performance of robust optimization problems with computationally expensive simulation models. Existing metamodels main focus on polynomial regression（PR）, neural networks（NN） and Kriging models, these metamodels are not well suited for large-scale robust optimization problems with small size training sets and high nonlinearity. To address the problem, a reduced approximation model technique based on support vector regression（SVR） is introduced in order to improve the accuracy of metamodels. A robust optimization method based on SVR is presented for problems that involve high dimension and nonlinear. First appropriate design parameter samples are selected by experimental design theories, then the response samples are obtained from the simulations such as finite element analysis, the SVR metamodel is constructed and treated as the mean and the variance of the objective performance functions. Combining other constraints, the robust optimization model is formed which can be solved by genetic algorithm （GA）. The applicability of the method developed is demonstrated using a case of two-bar structure system study. The performances of SVR were compared with those of PR, Kriging and back-propagation neural networks（BPNN）, the comparison results show that the prediction accuracy of the SVR metamodel was higher than those of other metamodels under uncertainty. The robust optimization solutions are near to the real result, and the proposed method is found to be accurate and efficient for robust optimization. This reaserch provides an efficient method for robust optimization problems with complex structure.

Multi-parameter Sensitivity Analysis and Application Research in the Robust Optimization Design for Complex Nonlinear System

The current research of complex nonlinear system robust optimization mainly focuses on the features of design parameters, such as probability density functions, boundary conditions, etc. After parameters study, high-dimensional curve or robust control design is used to find an accurate robust solution. However, there may exist complex interaction between parameters and practical engineering system. With the increase of the number of parameters, it is getting hard to determine high-dimensional curves and robust control methods, thus it＇s difficult to get the robust design solutions. In this paper, a method of global sensitivity analysis based on divided variables in groups is proposed. By making relevant variables in one group and keeping each other independent among sets of variables, global sensitivity analysis is conducted in grouped variables and the importance of parameters is evaluated by calculating the contribution value of each parameter to the total variance of system response. By ranking the importance of input parameters, relatively important parameters are chosen to conduct robust design analysis of the system. By applying this method to the robust optimization design of a real complex nonlinear system-a vehicle occupant restraint system with multi-parameter, good solution is gained and the response variance of the objective function is reduced to 0.01, which indicates that the robustness of the occupant restraint system is improved in a great degree and the method is effective and valuable for the robust design of complex nonlinear system. This research proposes a new method which can be used to obtain solutions for complex nonlinear system robust design.

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.

Six Sigma Robust Optimization of Fatigue Life for the Passenger Car Battery Hanging Device

In engineering practice, influencing factors including material properties,external load,dimension error and so on,are uncertain to structural fatigue life,and those uncertain factors make the structure fatigue life have a wide dispersion. Aiming at this problem,the parametric model is built in this paper,and it is used to obtain the structural first principal stress in the module of probabilistic design system( PDS). Parameters of P-S-N are added to the parametric model,and then,the fatigue life of chosen points has been calculated automatically under the reliability of 0. 99. The fatigue life response surface model is obtained by simulation of the sampling points using Monte Carlo method. The six sigma robust optimization mathematical model of fatigue life is established with the combination of six sigma robust optimization and fatigue life response surface model. Take a railway passenger car battery hanging device as an example, and the mathematical model of optimization is established. The minimum mean and mean squared of structural fatigue life are obtained under the requirements of design fatigue life. The results show that the material has been saved by the new method,and the robustness of the fatigue life has been improved.

Two-stage robust optimization of power cost minimization problem in gunbarrel natural gas networks by approximate dynamic programming

In short-term operation of natural gas network,the impact of demand uncertainty is not negligible.To address this issue we propose a two-stage robust model for power cost minimization problem in gunbarrel natural gas networks.The demands between pipelines and compressor stations are uncertain with a budget parameter,since it is unlikely that all the uncertain demands reach the maximal deviation simultaneously.During solving the two-stage robust model we encounter a bilevel problem which is challenging to solve.We formulate it as a multi-dimensional dynamic programming problem and propose approximate dynamic programming methods to accelerate the calculation.Numerical results based on real network in China show that we obtain a speed gain of 7 times faster in average without compromising optimality compared with original dynamic programming algorithm.Numerical results also verify the advantage of robust model compared with deterministic model when facing uncertainties.These findings offer short-term operation methods for gunbarrel natural gas network management to handle with uncertainties.

Two-Stage Robust Optimization Under Decision Dependent Uncertainty

In the conventional robust optimization(RO)context,the uncertainty is regarded as residing in a predetermined and fixed uncertainty set.In many applications,however,uncertainties are affected by decisions,making the current RO framework inapplicable.This paper investigates a class of two-stage RO problems that involve decision-dependent uncertainties.We introduce a class of polyhedral uncertainty sets whose right-hand-side vector has a dependency on the here-and-now decisions and seek to derive the exact optimal wait-and-see decisions for the second-stage problem.A novel iterative algorithm based on the Benders dual decomposition is proposed where advanced optimality cuts and feasibility cuts are designed to incorporate the uncertainty-decision coupling.The computational tractability,robust feasibility and optimality,and convergence performance of the proposed algorithm are guaranteed with theoretical proof.Four motivating application examples that feature the decision-dependent uncertainties are provided.Finally,the proposed solution methodology is verified by conducting case studies on the pre-disaster highway investment problem.

Robust optimization for volume variation in timber processing

Volume variation is an uncertainty element which affects timber processing. We studied the volume variation of logs caused by quality defects in traditional timber processing and set up an optimization approach,using a robust optimization method. We used total number of acceptable boards produced to study the relationship between board thickness and raw material logs, using a heuristic search algorithm to control the variation of board volume to improve the output of boards, reduce the quantity of by-products, and lower production costs. The robust optimization method can effectively control the impact of volume variations in timber processing, reduce cutting waste as far as possible using incremental processing and increase profits, maximize the utilization ratio of timber, prevent waste in processing, cultivate the productive type of tree species and save forest resources.

Robust optimization design on impeller of mixed-flow pump

To increase the robustness of the optimization solutions of the mixed-flow pump,the impeller was firstly indirectly parameterized based on the 2D blade design theory.Secondly,the robustness of the optimization solution was mathematically defined,and then calculated by Monte Carlo sampling method.Thirdly,the optimization on the mixed-flow pump′s impeller was decomposed into the optimal and robust sub-optimization problems,to maximize the pump head and efficiency and minimize the fluctuation degree of them under varying working conditions at the same time.Fourthly,using response surface model,a surrogate model was established between the optimization objectives and control variables of the shape of the impeller.Finally,based on a multi-objective genetic optimization algorithm,a two-loop iterative optimization process was designed to find the optimal solution with good robustness.Comparing the original and optimized pump,it is found that the internal flow field of the optimized pump has been improved under various operating conditions,the hydraulic performance has been improved consequently,and the range of high efficient zone has also been widened.Besides,with the changing of working conditions,the change trend of the hydraulic performance of the optimized pump becomes gentler,the flow field distribution is more uniform,and the influence degree of the varia-tion of working conditions decreases,and the operating stability of the pump is improved.It is concluded that the robust optimization method proposed in this paper is a reasonable way to optimize the mixed-flow pump,and provides references for optimization problems of other fluid machinery.

Influence of the Hook Position on the Vertical Vibrations of an Automobile Exhaust System:Application of the Robust Optimization Design

A robust optimization design method is proposed to investigate the influence of the hook position on the vertical vibration(bending)of an automobile exhaust system.A block diagram for the robustness analysis of the exhaust system is initially constructed from the major affecting factors.Secondly,the second-order inertia force is set as the vibration excitation source of the exhaust system and the displacement of four hooks of the exhaust system is selected as the variable factor.Then tests are carried out to investigate the resulting vertical bending considering four influencing factors and three levels of analysis.Finally,a variance analysis of the vertical bending is performed.The present study provides a set of guidelines to control the key factors affecting the vibration of vehicle exhaust systems while proposing an effective method to reduce vehicle vibration and improve noise 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.

A robust optimization method for label noisy datasets based on adaptive threshold: Adaptive-k

The use of all samples in the optimization process does not produce robust results in datasets with label noise.Because the gradients calculated according to the losses of the noisy samples cause the optimization process to go in the wrong direction.In this paper,we recommend using samples with loss less than a threshold determined during the optimization,instead of using all samples in the mini-batch.Our proposed method,Adaptive-k,aims to exclude label noise samples from the optimization process and make the process robust.On noisy datasets,we found that using a threshold-based approach,such as Adaptive-k,produces better results than using all samples or a fixed number of low-loss samples in the mini-batch.On the basis of our theoretical analysis and experimental results,we show that the Adaptive-k method is closest to the performance of the Oracle,in which noisy samples are entirely removed from the dataset.Adaptive-k is a simple but effective method.It does not require prior knowledge of the noise ratio of the dataset,does not require additional model training,and does not increase training time significantly.In the experiments,we also show that Adaptive-k is compatible with different optimizers such as SGD,SGDM,and Adam.The code for Adaptive-k is available at GitHub.

From Viewpoint of Reserve Provider:A Day-ahead Multi-stage Robust Optimization Reserve Provision Method for Microgrid with Energy Storage

Energy storage(ES),as a fast response technology,creates an opportunity for microgrid(MG)to participate in the reserve market such that MG with ES can act as an independent reserve provider.However,the potential value of MG with ES in the reserve market has not been well realized.From the viewpoint of reserve provider,a novel day-ahead model is proposed comprehensively considering the effect of the real-time scheduling process,which differs from the model that MG with ES acts as a reserve consumer in most existing studies.Based on the proposed model,MG with ES can schedule its internal resources to give reserve service to other external systems as well as to realize optimal self-scheduling.Considering that the proposed model is just in concept and cannot be directly solved,a multi-stage robust optimization reserve provision method is proposed,which leverages the structure of model constraints.Next,the original model can be converted into a mixed-integer linear programming problem and the model is tractable with guaranteed solution feasibility.Numerical tests in a real-world context are provided to demonstrate efficient operation and economic performance.

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.

Coordinated Dispatch Based on Distributed Robust Optimization for Interconnected Urban Integrated Energy and Transmission Systems

To improve the economic efficiency of urban integrated energy systems(UIESs)and mitigate day-ahead dispatch uncertainty,this paper presents an interconnected UIES and transmission system(TS)model based on distributed robust optimization.First,interconnections are established between a TS and multiple UIESs,as well as among different UIESs,each incorporating multiple energy forms.The Bregman alternating direction method with multipliers(BADMM)is then applied to multi-block problems,ensuring the privacy of each energy system operator(ESO).Second,robust optimization based on wind probability distribution information is implemented for each ESO to address dispatch uncertainty.The column and constraint generation(C&CG)algorithm is then employed to solve the robust model.Third,to tackle the convergence and practicability issues overlooked in the existing studies,an external C&CG with an internal BADMM and corresponding acceleration strategy is devised.Finally,numerical results demonstrate that the adoption of the proposed model and method for absorbing wind power and managing its uncertainty results in economic benefits.