Traditional Robust Optimization Research Articles

A stock border compensation technique for gaseous energy scheduling in steel enterprises under uncertainty

This study proposes a stock boundary compensation method to address the uncertainty of gaseous energy in integrated steel plants. The uncertainty disrupts the balance of temporary storage, resulting in energy waste, environmental pollution, and disturbances in steel production. The method accurately compensates for the impact of gaseous energy uncertainty on storage. Firstly, this paper takes oxygen and byproduct gas systems as the examples to describe the gas energy system and its scheduling model. The storage deviation caused by gaseous energy uncertainty is quantitatively analyzed using formulas. Subsequently, this study formulates an optimization problem to determine the optimal margin for compensating the storage boundary using a data-driven approach. Comparative experiments demonstrate that the proposed method not only significantly enhances the safety of gaseous energy storage subject to uncertainty, but also outperforms traditional robust optimization and stock fluctuation optimization methods in terms of system robustness against uncertainty and operating cost optimality.

Data-driven distributionally robust optimization approach for the coordinated dispatching of the power system considering the correlation of wind power

With the increasing penetration of large-scale wind power into the power grid, it is crucial to develop a precise model to accurately depict the stochasticity and correlation among wind farm outputs, which is highly important for ensuring the safe and efficient utilization of wind energy in grid dispatching. In this article, a data-driven distributionally robust optimization (DDRO) dispatching approach that accounts for spatial correlations among outputs from multiple wind farms is proposed. The proposed approach is applied to a source-network-load-storage grid system to ascertain unit start-stop schedules and resource allocation effectively. First, a truncated spatial correlation model is proposed, enabling a comprehensive representation of spatial correlations and output constraints between distinct wind farms. Second, the ISODATA clustering algorithm is employed to generate typical scenarios, reduce model complexity, and expedite the computation process. Third, a unit commitment model considering the demand response is constructed and solved using the DDRO approach. Finally, the proposed model is applied to the IEEE 30-bus system to test its robustness and cost-effectiveness compared to the traditional robust optimization model. Additionally, it is applied to the IEEE 118-bus system to demonstrate its scalability and stability.

Multidisciplinary robust optimization approach of fan rotors under structural constraints with blade curvature

The robust optimization approach is an important method to solve the problem of fans/compressors performance disparities caused by geometric tolerances. Nonetheless, since the optimum design of turbomachinery blades is multidisciplinary in nature, traditional robust optimization approaches that only consider aerodynamic performance may result in the blade strength exceeding their material limit. Therefore, it is necessary to investigate the multidisciplinary robust optimization approach (MROA) considering both aerodynamic and structural performance. For MROA, there is a big challenge of huge computation cost and runtime for the time-consuming high-fidelity CFD and FEM. In this paper, a new method to reduce the computation cost and runtime of MROA is proposed by using the blade curvature to approximate the evaluation of blade stress. Firstly, data mining by self-organizing mapping method is used to analyze and extract the constraint value of the blade curvature. And the blade curvature constraint value is used as a penalty function instead of the time-consuming high-fidelity FEM, which greatly reduces the computational cost and runtime of the multidisciplinary optimization. Then, Polynomial chaos Kriging (PCK) is then used as a surrogate model for uncertainty quantification (UQ) of aerodynamic performances, and one MROA based on the PCK model-based UQ method and the curvature constraints is developed. The proposed method is verified using a fan rotor as a case study. The results show that the maximum stress of the traditional optimization method was 453 MPa, which exceeded the yield limit of aluminum alloy (420 MPa). In contrast, the maximum stress of the method proposed in this paper was 343 MPa. In terms of aerodynamic performance, compared with the baseline, the mean efficiency of the traditional and the proposed method increased by 2.6% and 2.1%, with a variance reduced by 45.5% and 48.5%, respectively. Therefore, compared with the traditional robust optimization method, the proposed method reduces the maximum stress of the blade by 24.3% with almost the same improvement for aerodynamic performance means and disparities.

Modelling a bi-level multi-objective post-disaster humanitarian relief logistics network design problem under uncertainty

Post-disaster humanitarian relief logistics focuses on managing the supply and shipment of relief items. Owing to the complexity arising from the hierarchical decision relationship and the uncertainties of demand and transportation, it is challenging to develop an effective humanitarian relief logistics network (HRLN). A globalized robust bi-level multi-objective HRLN model involving uncertainties in transportation time, transportation cost and demand is developed to address this problem, in which an upper-level decision-maker focuses on the allocation problem and a lower-level decision-maker considers the supply problem. For tractability, the goal programming approach is employed to trade off conflicting timeliness, satisfaction and operational cost objectives. The proposed globalized robust bi-level HRLN model is reformulated as its globalized robust counterpart formulations under two different perturbation structures. The primal–dual approach is adopted to transform the bi-level models into equivalent mixed-integer conic programs. A case study concerning an Iranian earthquake is conducted to prove the performance of the proposed model. Globalized robust models are more flexible and able to immunize against uncertainty by comparison with traditional robust optimization and deterministic solutions. The managerial insights of using the globalized robust bi-level optimization approach are reported for disaster response management. The design of an emergency aviation network to circumvent the effects of COVID-19 could be considered in future research.

Selecting Robust Multi-Objective Optimization of a MEMS Electro-Thermal Actuator Considering Multi-Source Uncertainties

The traditional robust multi-objective optimization (RMO) is computationally expensive, making it difficult to apply in engineering. To this end, a selecting RMO (SRMO) framework that considers multi-source uncertainties and the adaptive update of the surrogate model is proposed. The SRMO framework is more efficient and has been applied to optimize a micro-electro mechanical system (MEMS) electro-thermal actuator (ETA). The application of sensitivity analysis and surrogate modeling techniques improves efficiency in terms of reducing the problem dimensionality and the computational cost of fitness values, respectively. A multi-objective particle swarm optimization algorithm is adopted to find the better-performing points where time-consuming finite element analysis (FEA) will be performed, thus ensuring the prediction accuracy of the global surrogate (GS) model around them. A greedy algorithm is employed to select design points from the candidate solutions set to train local surrogate (LS) models that perform the robust evaluation taking into account multi-source uncertainties in dimensions, material parameters, and surface topography features. In the optimization of ETA, only 2.9% and 0.5% of the particles require performing FEA to update the GS model and train LS models, respectively. The final solutions show good robustness, and the effectiveness of the SRMO framework is verified by experiments and simulations.

An optimized scheduling strategy combining robust optimization and rolling optimization to solve the uncertainty of RES-CCHP MG

The popularization of combined cooling, heating and power microgrid containing renewable energy sources (RES-CCHP MG) can effectively alleviate the energy crisis and reduce the emission of air pollutants. However, uncertainties in renewable energy generation and load negatively affect the operation of microgrid. To solve this problem, a multi-time scale optimal dispatch strategy combining robust optimization and rolling optimization is proposed. First, the price-based demand response is introduced to establish the RES-CCHP MG model that is more in line with practical application requirements. In the day-ahead dispatching stage, robust adjustment coefficients are introduced to improve the defects of traditional robust optimization which is conservative, and a two-layer robust optimization model is established to improve the ability to resist risks. In the intra-day dispatching stage, in order to reduce the impact of day-ahead prediction error and real-time power fluctuation, the hierarchical rolling optimization model based on model prediction control is established to face the situation that different loads have different response speeds. The simulation results show that compared with the traditional optimal dispatching strategy, the operating cost and the peak-to-valley difference of electric load are reduced by 5.4% and 14.7%, respectively, and the utilization of renewable energy is increased by 4.8%.

A data-driven multi-stage stochastic robust optimization model for dynamic optimal power flow problem

The uncertainty caused by the distributed generations(DG) with inconspicuous patterns has been an essential subject in the optimization scheduling for the distribution network. We propose a novel data-driven approach to deal with the dynamic optimal power flow(DOPF) problem which contains uncertain variables with their unknown probability distribution. The data-driven model is made to learn the joint probability distribution of the uncertain variables and use robust optimization(RO) to solve the multi-stage stochastic linear DOPF by averaging the worst case from each uncertainty set. In contrast to the motivation for traditional RO to find solutions that perform well on the worst-case realization, our proposed approach adds robustness to the historical data as a tool to avoid overfitting as the number of data points tends to infinity. The application verification for the AC OPF problem is presented for the IEEE-33 system. The simulation verifies the feasibility and robustness of the proposed approach and its results are compared with those of other data-driven stochastic optimization methods. We prove that the proposed approach can effectively solve the overvoltage problem caused by the high permeability of photovoltaic generation and achieve a better out-of-sample performance guarantee, and also has obvious economic advantages over other data-driven methods.

A Credibility Theory-Based Robust Optimization Model to Hedge Price Uncertainty of DSO with Multiple Transactions

This paper addresses the deregulated electricity market arising in a distribution system with an electricity transaction. Under such an environment, the distribution system operator (DSO) with a distributed generator faces the challenge of electricity price uncertainty in a spot market. In this context, a credibility theory-based robust optimization model with multiple transactions is established to hedge the uncertain spot price of the DSO. Firstly, on the basis of credibility theory, the spot price is taken as a fuzzy variable and a risk aversion-based fuzzy opportunity constraint is proposed. Then, to exploit the resiliency of multiple transactions on hedging against uncertain spot price, the spot market, option contract and bilateral contract integrating power flow constraints are studied, because it is imperative for DSO to consider the operational constraints of the local network in the electricity market. Finally, the clear equivalence class is adopted to transform the risk aversion constraint into a deterministic robust optimization one. Under the premise of considering the expected cost of the DSO, the optimal electricity transaction strategy that maximizes resistance to uncertain spot price is pursued. The rationality and effectiveness of the model are verified with a modified 15-node network. The results show that the introduction of option contracts and bilateral contracts reduces the electricity transaction cost of DSO by USD 28.5. In addition, under the same risk aversion factor, the cost of the proposed model is reduced by USD 195.18 compared with robust optimization, which avoids the over-conservatism of traditional robust optimization.

Robust dispatching of integrated energy system considering economic operation domain and low carbon emission

The traditional robust optimization algorithm can only provide the scheduling plan in the worst case, but due to the time coupling constraints such as climbing and energy storage SOC(State of Charge) constraints, it is difficult to provide an adjustment scheme for intra-day scheduling plan. In the first stage, the upper and lower bounds of equipment output are taken as decision variables, and the cumulative change of load is introduced to deal with the constraint of demand response, so as to form the economic operation domain of the park. The Column-and-Constraint generation method (CCG) is used to solve the three-layer optimization problem, and the scheduling plan and optimal operating domain in the worst case are obtained. The intra-day scheduling can be adjusted arbitrarily according to the new energy output within the interval. Carbon transaction cost is included in the goal function to account for reduced carbon emissions. Simultaneously, to take use of the benefits of flexible adjustment, FM income is incorporated into the goal function, and the charging and discharging behavior of energy storage is defined by optimizing calculations based on real-time capacity prices during daily scheduling. The calculation results show that the proposed robust optimization algorithm based on economic operation domain can be regarded as an extension of the traditional robust algorithm. Besides, the total cost can be reduced and the economy can be improved by adjusting the scheduling scheme according to the new energy output in real time under the premise of ensuring the time coupling constraint.

Robust dynamic optimization for nonlinear chemical processes under measurable and unmeasurable uncertainties

AbstractWe formulate an integrated framework for the robust dynamic optimization of nonlinear chemical processes under measurable and unmeasurable uncertainties. An affine decision rule is proposed to approximate the causal dependence of the wait‐and‐see decision variables on the gradually revealed measurable uncertainties. To overcome the computational intractability of the proposed model, a linearization technique based on the first‐order Taylor expansion is introduced around the nominal values of uncertainties to derive the robust dynamic counterpart, which can be discretized to a large‐scale nonlinear programming (NLP) formulation. Effects of first discretizing the dynamic models or introducing the affine decision rule are investigated. The proposed framework is also compared with the state‐of‐the‐art re‐optimization and traditional robust optimization approaches. An illustrative example and an industrial semi‐batch 2‐mercaptobenzothiazole production case are involved to demonstrate the advantages and applicability of the proposed framework.

Nash–Cournot power market model with a high penetration of prosumers: A distributionally robust optimization approach

The penetration of renewables on the demand side has recently attracted a large amount of attention. Its dual identity has an important impact on the competition between power producers and market dispatching. This study analyzed the participation of prosumers in the Nash–Cournot model for power market clearing. On this basis, we considered the uncertainty of renewable energy production, under which we used the distributionally robust optimization (DRO) model to produce a market equilibrium. Moreover, the Column-and-Constraint Generation algorithm was employed to solve the model. As a result, the impact of prosumers penetration and their production uncertainty on market equilibria was quantified and analyzed. Numerical simulations on an IEEE-30 bus system investigated the impact of the uncertainty of renewable energy output, the penetration level of wind power, and the demand level on the market equilibria. Test results demonstrate that the social welfare obtained from the proposed DRO model is 6.89% higher than that obtained from traditional robust optimization models.

Efficient Robust Production Optimization with Reduced Sampling

Summary Model-based robust production optimization is performed by propagating the uncertainty in the reservoir model description through the nonlinear fluid flow dynamics to predict the stochastic production response and the objective function of interest. Propagation of uncertainty is often accomplished through Monte Carlo simulation by sampling from the distribution of uncertain input parameters and computing the corresponding value of the objective function after performing reservoir simulation. The computational burden of performing hundreds of reservoir simulation runs to characterize the distribution of the objective function (or its point statistics) over the sampled input parameters makes the process prohibitive for large-scale deployment. To reduce the computational load of robust production optimization under geologic uncertainty, we propose an efficient stochastic production optimization algorithm based on a reduced sampling strategy. In this approach, at each iteration of the optimization procedure, instead of evaluating the objective function and its gradient over a large number of fixed realizations, we approximate them by using a small subset of model realizations that are selected randomly. This approximation leads to solving a series of “small” subproblems that require significantly fewer reservoir simulation runs, thereby reducing the computational complexity of the robust optimization workflow. Because the samples are selected randomly, over several iterations of the optimization process, a large number of samples are included. For a fixed set of realization, the proposed approach results in each sample being used multiple times during the optimization process. The proposed method is applied to several numerical examples, including a field-scale reservoir model, to show its computational efficiency over traditional robust production optimization under geologic uncertainty.

Robust optimal dispatch of integrated energy system considering with coupled wind and hydrogen system

Hydrogen production from renewable energy will become one of the important features of the integrated energy system in the future. By considering the coupled wind and hydrogen system, it can not only take the advantages of fast power regulation of hydrogen energy, but improve the economy of wind and hydrogen system. In view of the high volatility of the wind power, this paper proposes a robust optimal scheduling model for integrated energy system considering with coupled wind and hydrogen system. Firstly, it establishes a comprehensive energy system model with wind and hydrogen system, forming an efficient electric-gas-electric energy closed-loop flow model with fuel cells. Then, based on improved multi-interval uncertainty sets, the model of wind turbines, electrical load and hydrogen load are described respectively, which can reduce the conservatism compared with the traditional robust optimization. And the two-stage robust optimal scheduling model is established and it can be solved by adopting Column and Constraint Generation method (C&CG). Finally, an example is given to verify the advantages of the proposed method.

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.

Multi-energy microgrid robust energy management with a novel decision-making strategy

Uncertainties in renewable energy sources and load demand have become a consequential issue which has led to a significant effect on the microgrid operation. In this paper, a novel cumulative relative regret decision-making strategy is proposed for the optimal energy management on a grid-connected multi-energy microgrid considering these uncertainties. The proposed strategy can ensure the robustness of the microgrid and reduce the conservatism of microgrid operation as compared with the traditional robust optimization method. Furthermore, the typical optimization model of microgrid energy management is improved by taking the demand response of the thermal load into account. The delay of heat transfer and the fuzziness of heating comfort are also investigated in order to obtain a more economic microgrid scheduling plan. Simulation results using different decision-making strategies are provided to verify the effectiveness of the proposed cumulative relative regret based robust optimization approach.

A Distributionally Robust Power Dispatch Model for Active Distribution Network

The uncertainty of distribution network operation is increasing with the integration of large-scale renewable distributed generation (DG) units. To reduce the conservativeness of traditional robust optimization (RO) solutions, a data-driven robust optimal approach, which incorporates the superiority of both stochastic and robust approaches, is employed to solve the dispatch model in this paper. Firstly, a deterministic optimal dispatching model is established with the minimum total operation cost of distribution network; secondly, a two-stage distributed robust dispatching model is constructed based on the historical data of renewable-generators output available. The first stage of the model aims at finding optimal values under the basic prediction scenario. In the second stage, the uncertain probability distribution confidence sets with norm-1 and norm-∞ constraints are integrated to find the optimal solution under the worst probability distribution. The model is solved by column-and-constraint generation (CCG) algorithm. Numerical simulation on the IEEE 33-bus system has been performed. Comparisons with the traditional stochastic and robust approaches demonstrate the effectiveness of the proposal.

Robust Optimal Operation of Active Distribution Network Based on Minimum Confidence Interval of Distributed Energy Beta Distribution

With the gradual increase of distributed energy penetration, the traditional optimization model of distribution network can no longer guarantee the stable and efficient operation of the distribution network. In order to deal with the inevitable uncertainty of distributed energy, a new robust optimal operation method is proposed for active distribution network (ADN) based on the minimum confidence interval of distributed energy Beta distribution in this paper. First, an ADN model is established with second-order cone to include the energy storage device, capacitor bank, static var compensator, on-load tap changer, wind turbine and photovoltaic. Then, the historical data of related distributed energy are analyzed and described by the probability density function, and the minimum confidence interval is obtained by interval searching. Furthermore, via taking this minimum confidence interval as the uncertain interval, a less conservative two-stage robust optimization model is established and solved for ADN. The simulation results for the IEEE 33-bus distribution network have verified that the proposed method can realize a more stable and efficient operation of the distribution network compared with the traditional robust optimization method.

Distributionally robust last-train coordination planning problem with dwell time adjustment strategy

Aiming to increase successful transfers at stations for late-night passengers, we first propose an efficient dwell time adjustment strategy for the last-train coordination planning problem under transfer-passenger flows uncertainty. Unlike the traditional robust optimization model, we present a novel distributionally robust chance-constrained program to model this problem, where the probability distributions of the uncertain parameters are only partially available. By introducing a pessimistic ambiguous chance constraint, the proposed distributionally robust model guarantees that the probability of satisfying the service-oriented objective, i.e., maximum successful transfer-passenger flows in the whole subway system is larger than a predetermined confidence level in the worst case. We then draw the connection of the distributionally robust model with the traditional robust optimization model, and show that the proposed model can be interpreted as a generalized version of the robust optimization model. We further propose a safe tractable approximation method to reformulate the original model as a mixed-integer second-order conic programming under the bounded-perturbation ambiguous set, which can be solved to optimality on only small instances by the CPLEX. Hence, we develop a tabu search heuristic algorithm to obtain high-quality solutions for large-sized instances. We also use local search as a baseline algorithm to observe the improvements of the tabu search algorithm. Finally, we illustrate the superiority of the developed model on the Nanjing and Beijing subway networks and compare the performance of the proposed algorithms.

Distributionally robust coordination optimization scheduling for electricity-gas-transportation coupled system considering multiple uncertainties

The advance of the dynamic wireless charging technique for electric vehicles makes the electrified transportation system to be the development trend. On the other hand, the interdependency between electricity and natural gas systems has been intensified increasingly with the expansion of renewable energy. This creates a critical motivation to formulate the coordination operation model for the integrated electricity, gas and transportation system. In this context, a distributionally robust optimization (DRO) model is proposed considering multiple uncertainties comprehensively for the multi-energy coupled system. Specifically, the traffic flow uncertainty is transformed as the charging load uncertainty while the gas consumption uncertainty by gas-fired units is regarded as the reserve capacity configuration of units. Furthermore, the uncertainties of wind power and charging load are described as an ambiguity set incorporating the distribution information. Then the master-subproblem framework is developed, and a combination of the Benders decomposition and the transformation technique for bi-level sub-problem is implemented for solving this model. Simulation results indicate that DRO has saved the operation cost by 5.06% and 5.11% for the 6-bus and 24-bus systems compared with the traditional robust optimization model, which is beneficial for system decision-makers to achieve a balance between the reliability and economy in practice.

A Historical-Correlation-Driven Robust Optimization Approach for Microgrid Dispatch

Robust optimization (RO) has been a proficient approach for uncertainty dispatch in microgrids. In view of the strong conservativeness of traditional RO models, this article proposes a historical-correlation-driven (HCD) RO method for the MG dispatch problem. Firstly, the distinct spatiotemporal correlations in renewable energy (RE) generation are confirmed based on the historical data series of RE power. To effectively extract the correlation characteristics, the new intervals that envelop all correlation data points of similar days are formulated via the line-fitting approach, and the interval boundaries are added as liner constraints into the polyhedral uncertainty set that actively integrates the spatiotemporal correlations and avoids unreasonable scenarios. Secondly, a gradient-approximating decomposition algorithm with equilibrium constraints is put forward to address the nonlinear RO problem caused by non-independent uncertainties. The alternative optimization procedure ensures the gradient ascent of the objective, thus achieving the fast solution convergence of the Max-Min model with binary recourse variables. Besides, the equilibrium constraints realize the precise linearization of bilinear terms, as well as avoid the introduction of numerous binaries. Case tests verify the effectiveness and superiority of the proposed HCD-RO method comparing with existing models and algorithms.