Two-stage Robust Optimization Model Research Articles

Research on joint optimization model and algorithm of multi‐area generation and reserve with considering its availability

The reserve sharing amongst the interconnected power systems will facilitate the improvements on their anti-disturbance capability and operating economy. Based on the design of reserve sharing mechanism across different areas, a multi-area generation-reserve sharing model is constructed. The stochastic fluctuation of load and wind power will cause the power grid to be in an unpredictable random state when the reserve is needed, and the transmission capacity constraint of the power grid may restrict the effective deployment of the reserve. To address these issues, a max-min bi-level optimization model is proposed to check the reserve availability. The joint generation and reserve optimization sub-model, together with the reserve check sub-model, constitutes the two-stage robust optimization model in this paper. The column-and-constraint generation (C&CG), algorithm is adopted to solve the proposed model. The modified IEEE 118-node test case is used to verify the proposed model and algorithm. The test results indicate that the proposed method can realize the multi-area generation-reserve joint optimization under the condition of guaranteeing the reserve availability, and it possesses the advantage of high efficiency in dealing with stochastic scenarios.

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 two-stage 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.

Two-stage robust optimization for perishable inventory management with order modification

In this paper, we study an inventory management problem with perishability and demand uncertainty where the goal is to minimize the sum of ordering, purchasing, holding, shortage, wastage, and modification costs. Our motivation for studying this problem is to more specifically study the efficiency of order modification in an uncertain environment. In this problem, we suppose that the demand belongs to an uncertainty set without specific probability distributions. We formulate the problem as a novel two-stage robust integer optimization model and develop an exact column-and-row generation algorithm to solve it. The importance of using robust optimization for the problem is to immunize the decision maker in the worst-case scenario where vital perishable products such as blood must always be available. Our extensive computational experiments demonstrated the significant efficiency of our robust model compared with a deterministic model and a stochastic model in both risk-neutral and worst-case settings. We concluded that considering order modification results in more reliable decisions in inventory systems.

Two-stage robust optimization model for park integrated energy system based on dynamic programming

The promotion and construction of park integrated energy system (PIES) has strengthened the interaction among electric, gas, heating and cooling systems. However, the static programming strategy (SPS) of PIES results in problems such as lower system economics and poorer reliability of energy supply. Therefore, a suitable planning method is required to promote the economic operation of PIES under multiple uncertainties. Based on the background, this study proposes a two-stage robust optimization model based on dynamic programming strategy (DPS) for PIES. Firstly, the coupled devices and energy networks in the system are analyzed in detail to comprehensively identify the complex interactions among multiple energy carriers. Moreover, a DPS is introduced to improve the economic efficiency of PIES. Then, a two-stage robust model is established to solve the heterogeneous uncertainties in the planning strategy. Specifically, the column and constraint generation (C&CG) algorithm is used to solve the optimization model thereby obtain the equipment configuration scheme for each planning phase. Finally, the effectiveness of the proposed method is verified by comprehensive case studies on a test system.

Two-Stage Robust Optimization Model for Uncertainty Investment Portfolio Problems

Investment portfolio can provide investors with a more robust financial management plan, but the uncertainty of its parameters is a key factor affecting performance. This paper conducts research on investment portfolios and constructs a two-stage mixed integer programming (TS-MIP) model, which comprehensively considers the five dimensions of profit, diversity, skewness, information entropy, and conditional value at risk. But the deterministic TS-MIP model cannot cope with the uncertainty. Therefore, this paper constructs a two-stage robust optimization (TS-RO) model by introducing robust optimization theory. In case experiments, data crawler technology is used to obtain actual data from real websites, and a variety of methods are used to verify the effectiveness of the proposed model in dealing with uncertainty. The comparison of models found that, compared with the traditional equal weight model, the investment benefits of the TS-MIP model and the TS-RO model proposed have been improved. Among them, the Sharpe ratio, Sortino ratio, and Treynor ratio have the largest increase of 19.30%, 8.25%, and 7.34%, respectively.

Robust collaborative maintenance logistics network design and planning

Maintenance service providers to advanced technical devices are confronted with uncertain demand, high cost of components, and the need for certified operators. Collaboration in terms of sharing scarce resources among different facilities in a maintenance logistics network is expected to reduce the delays in the delivery of repaired devices. This study proposes a two-stage robust optimization model for collaborative design and planning of maintenance networks under demand uncertainty. The goal of this model is to determine the optimal allocation of customers to each maintenance center along with the initial stock level of different components in each facility so as to minimize the cost of late deliveries under worst-case demand scenarios. Component and operator sharing strategies are proposed as the recourse actions in this model to hedge against the demand surge. The proposed approach is compared with a deterministic model by the aid of Monte-Carlo simulation on several test instances inspired by a real case study. Our numerical experiments demonstrate the significance of adopting the proposed collaborative mechanisms among maintenance facilities in terms of cost reduction, especially when the demand fluctuation is relatively high.

Two-stage robust optimization approach for flexible oxygen distribution under uncertainty in integrated iron and steel plants

Optimal oxygen distribution is one of the most important energy management problems in the modern iron and steel industry. Normally, the supply of the energy generation system is determined by the energy demand of manufacturing processes. However, the balance between supply and demand fluctuates frequently, owing to the uncertainty arising in manufacturing processes. In this study, we developed an optimal oxygen distribution strategy considering uncertain demands and proposed a two-stage robust optimization (TSRO) model with a budget-based uncertainty set that protects the initial distribution decisions with low conservativeness. The main goal of the TSRO model is to make “wait-and-see” decisions, maximizing energy profits, and make “here-and-now” decisions, minimizing operational stability and surplus/shortage penalty. To represent the uncertainty set of energy demands, we developed (1) a Gaussian process-based time-series model to forecast the demand intervals for continuous processes, and (2) a capacity-constrained scheduling model to generate multi-scenario demands for discrete processes. We performed extensive computational studies on TSRO and its components using well-synthesized instances from historical data. The results of model validation and analysis are promising and demonstrate that our approach is well adapted to solving industrial cases under uncertainty.

Robust coordination of repair and dispatch resources for post-disaster service restoration of the distribution system

Since extreme weather events usually severely damage the power grid and lead to widespread power outages, coordinating repair and dispatch resources, such as repair crews (RCs), mobile power sources (MPSs), renewable energy sources (RESs) and energy storage systems (ESSs), is an efficient method to restore the power supply for outage loads. However, the uncertainties existing in RES output and load demand pose significant challenges to the service restoration of distribution systems in terms of security constraints. To address these uncertainties, this paper proposes a robust restoration approach to maximize the restored load in a distribution system by coordinating RCs, MPSs, RESs, ESSs and network reconfiguration. The coordination problem is formulated as a two-stage robust optimization model to generate optimal dispatching and operation schemes of multiple resources with guaranteed robustness against uncertainties. Involving some unexpected changes in topologies and distributed sources during post-disaster repair, the robust coordination model is re-optimized to guarantee the efficiency of repair and operation scheme. Simulation results show that the repair and dispatch resources can be coordinated effectively to accommodate the RES output and load demand uncertainties while maximizing the amount of restored load for service restoration of the distribution system.

Adaptive robust optimal dispatch of microgrid based on different robust adjustment parameters

This paper proposes a microgrid adaptive robust optimal dispatch model with different robust adjustment parameters to improve the operating economy and safety of large-scale renewable distributed energy integration into the microgrid system. Through a robust equivalent process, considering the uncertain characteristics of the renewable energy output and load demand in the microgrid, the uncertain parameters are converted into corresponding definite parameters. A two-stage robust optimization model is established to minimize the operating cost of the microgrid under the premise of ensuring the robustness of the microgrid. An improved Benders algorithm is proposed to solve the established optimization model. The different robust adjustment parameters can be obtained adaptively through the optimization program. The optimized adaptive robust adjustment parameters can better reflect the balance between the economy and robustness of the microgrid operation, and are more suitable for the operation of the microgrid. The improved Benders algorithm can effectively speed up the solution and improve the efficiency of the solution. The simulation of the modified IEEE39-bus system verifies the rationality of the proposed optimization model and the advantages of the improved algorithm.

Data-driven robust dispatch for integrated electric-gas system considering the correlativity of wind-solar output

The increasing popularity of distributed renewable power generation represented by wind turbines and solar plants and the deployment of gas turbines (GTs) and power-to-gas (P2G) facilities have promoted stronger interdependence between the power grid and the natural gas system, which makes the economic dispatch of the integrated electrical and gas system (IEGS) more challenging. This paper proposes a two-stage dispatch model for distribution-level IEGS based on a data-driven robust optimization (DDRO) method. First, for distribution-level IEGS, a network model of IEGS is established, and the nonconvex constraints are relaxed by the big M method and second-order cone (SOC) relaxation. Considering the correlation between wind and solar output in multiperiod, a wind-solar output ellipsoid uncertain set is constructed through the minimum volume enclosed ellipsoid (MVEE) algorithm to obtain extreme scenarios. Finally, a column-and-constraint generation (C&CG) method based on extreme scenarios is proposed to solve the two-stage robust optimization model. Simulation results show that the proposed DDRO method can reduce the day-ahead and real-time dispatch costs of energy conversion equipment while ensuring the robustness of IEGS system dispatch. In addition, the economics of IEGS system dispatch is improved.

A robust model for scheduling power productions of multiple offshore wind farms using one‐to‐many maintenance services

This paper studies the power production optimization problem of multiple offshore wind farms (OWFs) considering different maintenance demands and requirements under the wind power uncertainty. A decentralized robust operation optimization model is developed, in which OWFs and a maintenance service provider (MSP) are treated as individual stakeholders with different interests but coupled by maintenance resources. Based on the analytical target cascading (ATC) algorithm, the model is decoupled into independent models for MSP and each OWF. The operation and objectives of individual entities can be autonomously optimized only considering their own conditions. For each OWF, the power production planning coupled with maintenance scheduling is formulated as a two-stage robust optimization model and solved by a column-and-constraint generation (C&CG) algorithm to tackle the wind power uncertainty. Numerical experiments demonstrate the effectiveness of the proposed model and the applicability of the integrated solution method to the studied problem. Results illustrate that the developed model shows a better guarantee for individual interests of OWFs, lower communication burden, and better information privacy than the centralized optimization model, especially in large-scale systems. The integrated solution framework can guarantee a quick convergence and optimality. Moreover, it reduces impacts of uncertain factors on power productions and maintenance schemes.

Two-stage robust railway line-planning approach with passenger demand uncertainty

A railway line-planning problem is one of the fundamental problems in the strategic planning of railway operations. Operation zone, frequency, stop schedule, and passenger distribution for each train based on passenger demand are determined to minimize both the operation cost of the railway enterprise and the total travel cost to the passengers. However, passenger demand uncertainty makes balancing transport capacity and fluctuating demand challenging. Therefore, this paper proposes a two-stage robust optimization model wherein the nominal line plan is determined in the first stage (i.e., the case in which passenger demand has no fluctuation) and the rescheduled line plan based on the nominal line plan is determined according to the demand realization in the second stage. A Lagrangian relaxation algorithm and some strengthening techniques are designed. The model and algorithm are tested on real-world instances of the Wuhan–Guangzhou high-speed railway line under uncertain passenger demands. Computational results indicate that the proposed solution algorithm can yield a high-quality solution in 3600 s. In addition, the proposed two-stage robust solution can help minimize both operation and travel costs on average when compared with the nominal solution under uncertainty realization.

A two-stage robust approach to integrated station location and rebalancing vehicle service design in bike-sharing systems

A bike-sharing system is a shared mobility mechanism that provides an alternative transportation mode for short trips with almost no added travel speed loss. However, this model’s low usage ratio and high depreciation rate pose a risk to the sustainable development of the bike-sharing industry. Our study proposes a new integrated station location and rebalancing vehicle service design model. This model aims to maximize daily revenue under a given total investment for station locations and bike acquisition. To address demand ambiguity due to possible bias and loss of data, we present a two-stage robust optimization model with a demand-related uncertainty set. The first stage of our model determines the station locations, initial bike inventory, and service areas of rebalancing vehicles. In contrast to the literature, which either simplifies the rebalancing process as an inventory transshipment problem or formulates it as a complex dynamic bike rebalancing problem, we assign each rebalancing vehicle to a service area composed of several specified stations. An approximate maximum travel distance for each rebalancing vehicle is also designed and constrained to ensure that the rebalancing operation can be performed within each period. In the second stage, our model optimizes the daily fleet operation and maximizes the total revenue minus the rebalancing cost. To solve our model, we design a customized row generation approach. Our numerical studies demonstrate that our algorithm can efficiently obtain exact solutions in small instances. For a real-size problem, the nearly optimal solutions of our model also reveal a high-quality worst-case performance with a small loss in mean performance, particularly when the value of the budget ratio (that is, the average number of bikes per station) is at a medium level. Moreover, the distribution of service areas depends on the bike supply and demand level at each station. The optimal fleet rebalancing operation does not have to be confined to one geographical area. Furthermore, our robust model can achieve larger mean and worst-case revenues and a higher revenue stability than a stochastic model with a small data set.

Robustly Coordinated Distributed Voltage Control Through Residential Demand Response Under Multiple Uncertainties

This article proposes a multilevel coordinated voltage control (CVC) strategy for facilitating distributed residential demand response for network voltage optimization. The proposed CVC scheme coordinates multiple home energy management systems (HEMS) connected on a low-voltage radial distribution network through a sensitivity-based approach to minimize voltage deviations in multiple timescales owing to high photovoltaic (PV) penetrations and load variability. In order to mitigate the effects of uncertain PV output and load during actual system operation, a day-ahead two-stage robust optimization (TSRO) model has been developed, which optimizes network operation in the longer timescale, for worst-case realization of uncertainties in shorter timescales. This is sequentially followed by fully distributed hour-ahead CVC which is decomposed into network optimization and energy management subproblems through alternating direction method of multipliers. In order to counteract the effects of real-time uncertainties, local adjustments are performed by redispatching HEMS at individual households to locally adapt to unforeseen variations, thereby minimizing real-time voltage deviations. Extensive simulations are performed on the modified Dutch LV network. Simulation results demonstrate effectiveness of the proposed multilevel CVC approach to robustly handle uncertainties while ensuring reliable operation and maximum user-comfort.

Two-Stage Robust Optimization Model for Fresh Cold Chain considering Carbon Emissions and Uncertainty

Sustainable development is an everlasting theme and lasting strategy in today’s era. Low-carbon economy is an inevitable approach to the implementation of sustainable development. Cold chain logistics has become one of the main sources of carbon emissions. However, in the research on location planning of cold chain logistics, the costs of carbon emissions have not been taken into consideration in previous studies. The two-stage stochastic optimization (TSSO) model was established based on the comprehensive consideration of transportation costs, time penalty costs, and carbon emission costs. In this case, it is extremely difficult to deal with uncertainty in TSSO model. Therefore, this paper constructs a two-stage robust optimization (TSRO) model using data-driven method and robust optimization theory and verifies the validity of this model through an actual case. The application of this method to a cold chain logistics enterprise showed that the service level of logistics cannot be guaranteed by stochastic optimization model. In the TSRO model, the costs increase by 2.18% at the price of robustness, whereas logistics service level shows an upward trend (from 85.83% to 92.75%). In the TSRO model, enterprises are forced to choose a better distribution path when carbon tax increases, which not only helps enterprises save costs but also achieves low-carbon environmental benefits.

A value function-based approach for robust surgery planning

We present a value function-based formulation for solving a two-stage robust optimization model for surgery-to-OR allocation planning where only upper and lower bounds on surgery duration are known. We solve this formulation using a constraint and column generation algorithm and interpret it as an iterative bi-level interdiction game. We derive valid inequalities based in this interpretation and compare the computational performance of this algorithm with the only other exact solution approach in the literature using the data from an academic medical center. The value function-based formulation yields better or similar quality solutions on difficult problem instances with low or intermediate number of surgeries.

Coordination of Preventive Dispatch and Black-Startunder Extreme Operational Conditionsusing Robust Optimization

Extreme operational conditions, e.g., wind storms and ice storms, have been posing great challenges for power grid operation. The extreme conditions are likely to cause widespread blackout, therefore a black-startprocedure is needed to restore the non-blackstart generating untis and transmission assets. To falicilate the balck-start procedure, the coordination between post-event preventive dispatch and post-event black-start dispatch is established, considering the uncertainty of forced transmission line outage scenarios. A two-stage robust optimization model is built to coordinate the preventive dispatch and black-start procedure. An iterative decompostion algorithm is applied to solve the proposed model. A synthesized Chinese 372-bus system is used to illustrate the proposed method. It is shown that, in the worst-case blackout scenario, the coordinated black-start procedure can accelerate the entire process by 15% and reduce the black-start capacity requirements.

Robust transmission network expansion planning considering non-convex operational constraints

This paper proposes a two-stage robust optimization model for the transmission network expansion planning problem. Long-term uncertainties in the peak demand and generation capacity are modeled using confidence bounds, while the short-term variability of demand and renewable production is modeled using a set of representative days. As a distinctive feature, this work takes into account the non-convex operation of conventional generating units and storage facilities, which results in a two-stage robust optimization model with a discrete recourse problem. The resulting problem is solved using a nested column-and-constraint generation algorithm that guarantees convergence to the global optimum in a finite number of iterations. An illustrative example and a case study are used to show the performance of the proposed approach. Numerical results show that neglecting the non-convex operation of conventional generating units and storage facilities leads to suboptimal expansion decisions.

An improved two-stage robust optimization model for CCHP-P2G microgrid system considering multi-energy operation under wind power outputs uncertainties

The combined cooling, heating and power (CCHP) microgrid has the advantages of promoting cleaner production and improving energy utilization efficiency. With the development of renewable energy sources (RES), it is more and more significant to study the optimal operation of CCHP based on the uncertainties of RES outputs. This paper proposes a CCHP-P2G microgrid system, which combined the power-to-gas (P2G) device with traditional CCHP microgrid. And a Data-driven Set based robust optimization (DSRO) model considering the uncertainties of wind power and multiple demand response programs (DRPs) have been presented, which consists of two stages: Day-ahead dispatching stage and Real-time adjusting stage. Simulations are delivered to show the following outcomes: (1) the P2G device can improve the electricity-gas coupling in the CCHP-P2G system, enhancing the system’s stability and economy of the system operation; (2) the multiple DRPs in the DSRO model can play the role of peak shaving and valley filling of electrical load, heating load and cooling load, so as to further reduce the operating cost of the system; (3) the DSRO model can resist the interferences of uncertain wind power outputs and keep both the conservativeness and computational complexity in relatively low levels.

Robust optimization of microgrid based on renewable distributed power generation and load demand uncertainty

The uncertainty of renewable distributed energy (photovoltaic, wind power, etc.) and load demand in the microgrid poses challenges to the economy and safety of microgrid operation. This paper proposes a robust optimization model of microgrid considering uncertainty to take into account the economy and robustness of microgrid operation. A two-stage robust optimization model is established to find a balance between the economy and robustness of microgrid operation. Through the optimization procedure, the robust adjustment parameters for microgrid operation can be obtained. The optimized can effectively balance the economy and robustness. The Benders dual algorithm is used to solve the established two-stage robust optimization model. The CPLEX solver is used to simulate the IEEE39-bus system to verify the feasibility and effectiveness of the method. The simulation results show that the robustness of the system can be achieved by solving the robust adjustment parameters, meanwhile the operating cost can be reduced as much as possible no matter in the buying electricity scenario or in the selling electricity scenario.