Two-stage Robust Model Research Articles

Mitigating Risk and Emissions in Power Systems: A Two-Stage Robust Dispatch Model with Carbon Trading

The large-scale integration of renewable energy sources is crucial for reducing carbon emissions. Integrating carbon trading mechanisms into electricity markets can further maximize this potential. However, the inherent uncertainty in renewable power generation poses significant challenges to effective decarbonization, renewable energy accommodation, and the security and cost efficiency of power system operations. In response to these challenges, this paper proposes a two-stage robust power dispatch model that incorporates carbon trading. This model is designed to minimize system operating costs, risk costs, and carbon trading costs while fully accounting for uncertainties in renewable energy output and the effects of carbon trading mechanisms. This model is solved using the column-and-constraint generation algorithm. Validation of an improved IEEE 39-bus system demonstrates its effectiveness, ensuring that dispatch decisions are both robust and cost-efficient. Compared to traditional dispatch models, the proposed model significantly reduces system risk costs, enhances the utilization of renewable energy, and, through the introduction of a ladder carbon trading mechanism, achieves substantial reductions in carbon emissions during system operation.

Optimal scheduling of virtual power plants with reversible solid oxide cells in the electricity market

Virtual power plants (VPP) can act as an independent entity to aggregate different distributed resources to participate in the electricity market, which is an inevitable trend in building green and low-carbon power systems. To investigate the beneficial effect of bi-directional electric-hydrogen conversion on market trading and optimal scheduling of VPP, this paper presents an optimal scheduling method for VPP with reversible solid oxide cells (RSOC) in the electricity market. Firstly, according to the Butler-Volmer equation and Faraday law, the relationship between the hydrogen flow rate and the electric power is established, and the equivalent physical models of the two RSOC working modes are developed based on the relationship. Then, the optimal scheduling model of the VPP participating in the energy and reserve joint market is proposed based on the two-stage robust optimization theory. The system operator reserve demand is used to describe the uncertainty of the transactions between VPP and reserve market. Finally, the column and constraint generation (C&CG) algorithm is used to solve the two-stage robust scheduling model. The simulation results show that RSOC can effectively promote renewable energy consumption and improve the VPP operating profit. Where, the operating profit can be increased by 46.5 % and the wind power consumption rate can be increased by 3.4 %, compared with the non-hydrogen VPP. Furthermore, compared with the traditional conversion equipment, RSOC can increase the VPP operating profit by 2.3 % and the wind power consumption rate by 1.85 %.

Managing platelets supply chain under uncertainty: A two-stage collaborative robust programming approach

This study investigates the role of horizontal collaboration among blood centers in mitigating the negative impacts of fluctuations in blood platelet supply and demand during normal circumstances and after sudden-onset disasters. Specifically, four scenarios are studied: no collaboration, collaboration among blood centers, collaboration among hospitals, and collaboration among blood centers and hospitals. To formulate these scenarios, we propose a two-stage robust programming model that enables decision-makers to monitor the collaborative behavior of blood centers and hospitals before and after changes in supply and demand occur. In the first stage, a robust optimization approach is developed to cope with supply and demand uncertainties, where collaboration is limited to blood centers. In the second stage, the uncertain parameters (supply and demand) are treated as deterministic. The key decision is how to transfer platelets among hospitals located in a collaborative cluster. This research uses benchmark data from previous studies to test and validate the developed model and solution approach. Comparing the results obtained in different scenarios, we systematically select the most favorable scenario in terms of model performance under normal and emergency situations. The final results indicate that the network achieves the highest performance when blood centers and hospitals participate in the grand coalition.

Decentralized coordinated optimal dispatching of multi-area interconnected electricity-gas-heat system based on inertia Bregman ADMM

The coordinated interconnection of multi-area interconnected electricity-gas-heat systems (MIEGHS) has become a typical feature of the modern energy internet. While enhancing system operational efficiency, it also introduces complexity and uncertainty into system operation. In consideration of these factors, a decentralized coordinated robust optimization scheduling framework for MIEGHS is proposed to facilitate coordinated optimization. Firstly, a two-stage robust stochastic optimal dispatching model for a single-area integrated electricity-gas-heat system (IEGHS) is established, considering the uncertainty of wind power output and load. Secondly, a decentralized self-regulation control model for MIEGHS is proposed based on the principle of consistency constraints. Next, considering both the timeliness and convergence of the traditional alternating direction method of multipliers (ADMM) in solving problems, an improved ADMM method based on inertial Bregman distance is presented for the optimized solution of the proposed model. Finally, the effectiveness and superiority of the presented model and methods are verified by three interconnected IEGHS.

Virtual power plant optimization with demand response and multi-complementary energy planning

Abstract To effectively address the increasing complexity of system operation optimization and resource allocation caused by the continuous enrichment of supply-side resources and the flexible variability of demand-side loads in the energy system, a robust optimization model for virtual power plants considering demand response and multi-energy complementarity is proposed. Handling renewable energy and load uncertainty in virtual power plants by building a two-stage robust optimisation model. The introduction of robust coefficients allows for flexible adjustment of the conservatism of the optimized scheduling. Results indicate that considering demand response and multi-energy complementarity can effectively enhance the system’s robustness and flexibility.

Coordinated operation optimization strategy for multiple microgrids considering uncertainties in renewable energy output

Abstract In the process of constructing a new power system with a focus on new energy sources, microgrids will be vigorously developed as an effective means of accommodating renewable resources. The synergistic operation of multiple microgrids, merging into a multi-stakeholder framework, is a method to strengthen the overall effectiveness of the system. However, existing Nash bargaining models consider relatively fewer influencing factors. To enhance the reliability and economic viability of microgrids, this paper effectively proposes a model that combines Nash bargaining with two-stage robust optimization. Specifically, the optimization model is divided into two stages. In the first stage, equipment configuration is optimized with the objective of system economy. In the second stage, uncertainties arising from renewable energy sources are considered. Subsequently, the two-stage robust model utilizes C&CG to obtain energy exchange values between microgrids. The Nash bargaining model is then addressed using the ADMM algorithm. Finally, through case simulations, the improvement in the benefits of different microgrids is verified to be in the range of 13% to 24%.

Frequency security-constrained unit commitment with fast frequency support of DFIG-based wind power plants

The integration of inverter-based renewable energy sources into power grids presents a unique challenge: a reduction in grid inertia, leading to potential frequency stability issues. This paper introduces an innovative approach to address this challenge, focusing on the dynamic role of wind power plants equipped with dual-fed induction generators (DFIG). Unlike traditional methods that rely on fixed inertia control strategies and wind power droop coefficients, our method offers a dynamic solution tailored to the fluctuating nature of grid frequency dynamics. We propose a novel variable inertia support mechanism, coupled with an optimized power point tracking control and strategic reserve, to maximize the effective use of wind power reserves without waste. Additionally, we account for the inherent uncertainties in wind power generation, enhancing the frequency support capabilities of wind turbines through a fast frequency response strategy. A key feature of our approach is the application of an advanced piecewise linearization fitting method to establish a robust frequency nadir constraint. This is integrated into a two-stage robust frequency security-constrained unit commitment model, enabling optimal real-time decision making for frequency stability in wind power plants. The efficacy and practicality of our proposed method are verified through a detailed case study on a modified IEEE 39-bus system and a hardware-in-the-loop experiment. Compared with the traditional fixed parameter control method, the cost is reduced by approximately 0.42%, while the linearization error at frequency nadirs remains within the range of -4% to 1.5%.

A robust microgrid formation method for risk-resistant service restoration considering subsequent contingency

Service restoration (SR) is essential for resilience enhancement of distribution systems (DSs) when natural disasters occur. However, a restored DS may face potential outage risks caused by subsequent contingency during the prolonged event. To improve the risk-resistance of SR strategies, this paper proposes a microgrid-based SR method to continuously supply critical loads when considering subsequent contingency. The microgrids are proactively formed to mitigate the influence of subsequent contingencies, which can effectively reduce the load shedding and wind power curtailment during the SR process. To deal with uncertainties of subsequent contingency and wind power, a two-stage robust model is established. Moreover, an improved uncertainty set is established to further reduce the conservatism of robust optimization. Specially, the line interruption probability is considered when constructing the Shannon entropy-based uncertainty set of subsequent contingencies, and the spatial–temporal correlations of wind power is considered when formulating the budget-constrained uncertainty set. Then, the robust model is solved by column-and-constraint generation algorithm efficiently. Numerical results demonstrate that the proposed SR method can effectively enhance the risk-resistance of SR strategy and reduce the conservatism while ensuring the robustness of solutions.

A Two-Stage Robust Pricing Strategy for Electric Vehicle Aggregators Considering Dual Uncertainty in Electricity Demand and Real-Time Electricity Prices

To enable the regulation and utilization of electric vehicle (EV) load resources by the power grid in the electricity market environment, a third-party electric vehicle aggregator (EVA) must be introduced. The strategy of EVA participation in the electricity market must be studied. During operation, the EVA faces a double uncertainty in the market, namely, electricity demand and electricity price, and must optimize its market behavior to protect its own interests. To achieve this goal, we propose a robust pricing strategy for the EVA that takes into account the coordination of two-stage market behavior to enhance operational efficiency and risk resistance. A two-stage robust pricing strategy that takes into account uncertainty was established by first considering day-ahead pricing, day-ahead electricity purchases, real-time electricity management, and EV customer demand response for the EVA, and further considering the uncertainty in electricity demand and electricity prices. The two-stage robust pricing model was transformed into a two-stage mixed integer programming by linearization method and solved iteratively by the columns and constraints generation (CCG) algorithm. Simulation verification was carried out, and the results show that the proposed strategy fully considers the influence of price uncertainty factors, effectively avoids market risks, and improves the adaptability and economy of the EVA’s business strategy.

Gate appointment design in a container terminal: A robust optimization approach

Gate congestion is a main challenge faced by container terminals worldwide. In the scenario of gate congestion, the long waiting line of container trucks can block the traffic into the terminal, leading to disrupted terminal operations and unsatisfactory operating efficiency. In this paper, we study a gate appointment design problem that aims to manage the number of trucks allowed to a container terminal over time, so as to minimize the length of the truck waiting line and alleviate gate congestion. To optimize the gate appointment strategy, the truck service plans need also be devised, which involve the decisions for assigning trucks to yard blocks for container transhipment. We develop a two-stage robust optimization model for the problem by taking into account the uncertain service capacities of the yard blocks. Our model considers two decision stages, where the first-stage decision allocates slot capacities for the appointment slots, whereas the second-stage decision designs truck service plans with observed yard service capacity. We solve the two-stage robust model using an adapted column-and-row generation algorithm. In each iteration of the adapted column-and-row generation algorithm, the second-stage decision is generated by means of a pseudo-polynomial-time dynamic programming algorithm, while the first-stage decision is generated by a tailored scenario decomposition method. We test the computational performance of the proposed solution method on instances generated from real terminal operation data, and reveal managerial insights that would inspire terminal operators in the management of truck appointment for gate congestion mitigation.

Two-stage robust planning method for distribution network energy storage based on load forecasting

A two-stage robust planning method for energy storage in distribution networks based on load prediction is proposed to address the uncertainty of active load in energy storage planning. First, considering the uncertainty of active load, a short-term load forecasting model combining the mutual information method and BiLSTM is established based on k-means++ clustering. Second, based on the results of load forecasting, a comprehensive norm-constrained uncertainty set is constructed, and a two-stage robust model for distribution network energy storage planning is established. The first stage aims to minimize the annual investment cost of the energy storage system, while the second stage aims to minimize the daily operating cost of the distribution network. At the same time, a second-order cone relaxation transformation model with non-convex constraints is introduced to ultimately achieve the optimal economy of the distribution network in energy storage planning. Finally, the effectiveness of the proposed method and model is validated on the IEEE 33-node distribution network model using the MATLAB platform.

Two-stage robust programming model for active distribution network with a high proportion of renewable energy

To reduce operating costs and obtain the optimal location and capacity of renewable energy access, a two-stage robust planning model is studied for active distribution networks with a high proportion of renewable energy access. To build a function to minimize annual comprehensive costs, the constraints of power flow, branch power, and node voltage of the active distribution network are studied. Considering the characteristics and properties of power flow safety constraints, the mixed integer second-order cone transformation is used to construct a mixed integer convex programming model, which is convenient for efficiently obtaining the global optimal solution. To avoid the influence of uncertainty, a two-stage robust model is constructed. The explicit maximum value method is used to obtain the solution results that satisfy the planning model of all uncertain problems. Experiments show that the model can obtain the optimal access position and capacity of renewable energy. After application, it can reduce the annual comprehensive cost of the distribution network and ensure the stable and safe operation of the active distribution network.

Distributed Energy Resource and Energy Storage Investment for Enhancing Flexibility Under a TSO-DSO Coordination Framework

This paper presents a distributed energy resource and energy storage investment method under a coordination framework between transmission system operators (TSOs) and distribution system operators (DSOs), which simultaneously addresses two main aspects of the flexibility aggregation of DSOs, i.e., flexibility enhancement and dynamic flexibility provision. First, to characterize the key flexibility features of power distribution networks, a multi-port multi-period feasible region formulation is designed using the robust optimization conception while avoiding over-parameterized formulations that require extensive information exchange. Second, a two-stage robust planning model is built to enhance distribution system flexibility by maximizing the volume of feasible regions, in which the uncertainty of dispatch instructions is modeled as a decision-dependent uncertainty (DDU) set and the worst realization is identified for the operational security evaluation. The proposed two-stage robust planning model is solved via a customized column-and-constraint generation algorithm. Finally, a distributed framework for TSO-DSO coordination is proposed to enable the dynamic adjustment of feasible region provision of DSO, given the TSO’s preference, which is then solved by a DDU-based two-stage robust extension of the alternating direction method of multipliers algorithm. Numerical results verify the effectiveness of our proposed models and the scalability of the associated algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The increasing integration of renewable energy resources has stimulated the need for aggregated flexibility provided by power distribution networks (PDNs). However, there are three challenges: i) using the currently feasible region formulation of PDN requires extensive information exchange with the transmission system operator (TSO), thus not compatible with the current structure. ii) the optimal investment of various assets to leverage their cost-effectiveness in feasible region enhancement needs to be addressed. iii) lack of a proper coordination mechanism between TSO and distribution system operators (DSOs) to facilitate the optimal feasible region provision and leverage the spatiotemporal flexibility. To address the above challenges, we design an easy-to-implement feasible region formulation to characterize the key flexibility features of PDNs. Upon this, we propose a feasible region enhancement planning method based on an extended two-stage robust model to address the decision-dependent uncertainty. Furthermore, we facilitate the optimal feasible region provision through the TSO-DSO coordination process in a distributed manner. The numerical results show that the proposed planning method and coordination mechanism could effectively achieve optimal flexibility enhancement and dynamic flexibility provision. In practical application, our proposed planning method can be readily implemented with advanced analytical tools. The coordination mechanism is also compatible with the current TSO-DSO structure and can be easily integrated into it.

A resilient battery electric bus transit system configuration

Electric mobility is fundamental to combat climate change and attaining the United Nations Sustainable Development Goals (SDG-11). However, electric mobility necessitates a seamless integration between power and transportation systems, as the resiliency of both systems is becoming far more interdependent. Here, we focus on disruption to Battery Electric Bus (BEB) transit system charging infrastructure and offer a resilient BEB transit system planning model. The proposed model optimizes the BEB system costs while ensuring the system’s robustness against simultaneous charging station failures. In our case study, a single charging station failure would lead to up to 34.03% service reduction, and two simultaneous failures would reduce the service by up to 58.18%. Our proposed two-stage robust model addresses this issue with a relatively small added cost (3.26% and 8.12% higher than the base model). This cost enables uninterrupted BEB system operation during disruption, ensuring personal mobility, social interaction, and economic productivity.

Two-stage robust scheduling and real-time load control of community microgrid with multiple uncertainties

With the increasing popularity of wind and solar generation, the stable and economic operation of the community microgrid system become more challenging. This paper is concerned with the robust scheduling and real-time control of demand response (DR) air conditioning (AC) loads for community microgrid. Firstly, a two-stage robust scheduling model coupled by the day-ahead dispatch and real-time dispatch is established by considering the uncertainties of wind/photovoltaic power generation and loads. Moreover, the incorporation of multiple interval uncertain sets is introduced to characterize the uncertainties related to wind power, photovoltaics, and load, thereby reducing the conservativeness of the model. Secondly, the real-time control of AC loads by adjusting the on/off actions and temperature setpoints with limited impact on user comfort so as to track the optimized electricity consumption curve. Finally, the simulation results show the effectiveness and robustness of the proposed optimization dispatch and real-time control algorithm.

Robust low-carbon energy and reserve scheduling considering operational risk and flexibility improvement

With the increase in wind power integration, significant uncertainty leads to enormous requirements for the flexibility and robustness of system operation. To enhance the flexibility of the power system, this paper proposes a robust low-carbon optimization method for energy and reserve scheduling. The system flexibility is improved by various flexible resources including conventional thermal units (CTUs), carbon capture plants (CCPs), energy storage systems (ESSs), and demand response (DR). The operation and regulation characteristics of multiple types of flexible resources are jointly formulated. Moreover, the carbon-capturing mechanism of CCPs is considered to reduce carbon emissions. The operational risk is measured by the conditional value-at-risk (CVaR) to coordinate the wind power accommodation and operational economy. A two-stage robust scheduling model with a flexible uncertainty set is established, which optimizes the energy and reserve scheme in the first stage and checks the feasibility of re-dispatch in the second stage. The column-and-constraint generation (C&CG) algorithm is utilized to solve the proposed robust model. Simulation results on two different scale systems verify the feasibility and effectiveness of the proposed method.

Reliable dynamic wireless charging infrastructure deployment problem for public transport services

Dynamic wireless power transfer (DWPT) technology offers the promise of eliminating the limited driving range of electric buses (EBs), providing greater convenience and safety benefits through the ability to charge EBs in motion. It may be possible to effectively circumvent the drawbacks of EBs such that it becomes mainstream in the future EB market. In this paper, we study the robust DWPT deployment problem for an EB system, aiming to determine the DWPT facility deployment, battery size, and en-route charging schedule simultaneously. We first formulate the deterministic problem as a novel mixed-integer linear programming (MILP) model. Next, we consider the energy consumption uncertainty and travel time uncertainty to develop a two-stage robust model. We further analyze the structural properties of our proposed model, and implement the column-and-constraint generation (C&CG) method to solve it optimally. Finally, a series of experiments based on a test bus network and the National University of Singapore internal bus network are carried out to verify the efficiency and effectiveness of our models and algorithms. Some managerial insights are also provided that public transit (PT) operators could use to enhance the efficiency of the EB system in the future.

Location and inventory pre-positioning problem under uncertainty

In humanitarian logistics, the location and inventory pre-positioning problem (LIPP), making decisions on facility location, inventory pre-positioning, and allocation, is critical to post-disaster rescue efficiency. Since post-disaster conditions are complex and uncertain, it is challenging for planners to make LIPP decisions and perform rescue activities. To reduce operational costs while also guaranteeing service levels to customers, it is necessary to take uncertainties into account when making decisions. This study explores a service-oriented LIPP under two types of uncertainties, i.e., uncertain customer demand and third-party supply. We propose a two-stage robust optimization framework to solve the problem, considering the service-level constraints and minimizing the total cost. We develop two algorithms to solve the problem. Specifically, a column-and-constraint generation algorithm is utilized to solve the original two-stage robust model, and an improved row generation algorithm is developed to tackle the affinely adjustable robust counterpart model. Simulation results show that both algorithms can provide satisfying solutions. A case study based on a power grid company in China is conducted to perform sensitivity analysis and compare models, producing several managerial insights.

Two-Stage Robust Design of Resilient Active Distribution Networks Considering Random Tie Line Outages and Outage Propagation

This paper proposes a two-stage method for the robust design of resilient active distribution networks (ADNs) against high-impact and low-probability (HILP) events. The line hardening and the deployment of remote-controlled switches (RCSs) are considered as two powerful measures for resilience enhancement. Especially, the hardening of tie lines and the deployment of bilateral tie switches are emphasized as part of the resilient design. A novel progressive detection mechanism (PDM) is devised to estimate the potential propagation of outages and identify surviving nodes outside of the minimum outage area after intentional islanding. The proposed PDM method considers potential RCS locations among regular lines and tie lines on outage to calculate the global optimal design scheme. The two-stage robust design model is formulated as a mixed-integer linear programming (MILP). The nested column-and-constraint generation (nested C&CG) algorithm is customized to solve the proposed model. Numerical results on a modified IEEE 33-node distribution system demonstrate the effectiveness and the superiority of the proposed resilience enhancement method.

Hierarchical Two-Stage Robust Planning for Demand-Side Energy Storage with Dynamic Carbon Incentive Mechanism

Demand-side energy storage is an important foundation for enhancing load flexibility to accommodate renewable energy. With the widespread adoption of renewable energy, demand-side energy storage planning, and its incentive mechanism have also attracted the attention of a large number of scholars. However, there are still few studies on incentives from a carbon perspective. To fill the gap, a dynamic carbon incentive mechanism is proposed in this study. In addition, a hierarchical two-stage robust planning model for demand-side energy storage that incorporates the proposed carbon incentive mechanism is developed. At the first level, the economic dispatch is performed, and bus carbon intensities are calculated based on the carbon emission flow theory. The second level is a two-stage robust planning model to obtain the optimal capacities of demand-side energy storage, which is solved based on the nested column and constraint generation algorithm. The proposed model is implemented and evaluated on the MATLAB/YALMIP platform using IEEE 24-bus power systems. The results validate the efficacy of the model in promoting carbon-oriented demand-side energy storage planning, leading to a substantial reduction of carbon emissions by 8.44%. Notably, when compared to existing incentive mechanisms, the proposed carbon incentive mechanism exhibits distinct advantages in achieving carbon reduction with less both subsidy costs and fixed investments.