Search Results (15)

As wind penetration rates continue to increase, the main challenge faced by operators is how to schedule flexible resources, such as traditional generation and storage, in the future to ensure the safe and stable operation of power grids under multiple uncertainties. In this paper, a security-constrained multi-stage robust dynamic economic dispatch model with storage (SMRDEDS) is proposed to address multiple uncertainties of wind power outputs and N-1 contingencies. Compared to the traditional two-stage robust dynamic economic dispatch model, the proposed multi-stage dispatch model yields sequential operation decisions with uncertainties revealed gradually over time. What is more, a combined two-stage Benders’ decomposition and relaxed approximation–robust dual dynamic programming (RA-RDDP) is proposed to handle the computational issue of multi-stage problems due to large-scale post-contingency constraints and the convergence issue of the stochastic dual dynamic programming (SDDP) algorithm. First, a two-stage Benders’ decomposition algorithm is applied to relax the SMRDEDS model into a master problem and sub-problem. The master problem determines the generator output and storage charge and discharge, and the sub-problem determines the total generation and storage reserve capacity to cover all the generator N-1 contingencies. Second, a relaxed approximation–RDDP algorithm is proposed to solve the multi-stage framework problem. Compared to the traditional SDDP algorithm and RDDP algorithm, the proposed RA-RDDP algorithm uses the inner relaxed approximation and outer approximation methods to approximate the upper and lower bounds of the future cost-to-go function, which overcomes the convergence issue of the traditional SDDP algorithm and solution efficiency of the RDDP algorithm. We tested the proposed algorithm on the IEEE-3 bus, IEEE-118 bus, and the German power system. The simulation results verify the effectiveness of the proposed model and proposed algorithm. Full article

In this paper, we introduce a shipping logistics network optimization method for construction waste recycling. In our case, construction waste is transported by a relay mode integrating land transportation, shipping transportation, and land transportation. Under the influence of urban economic life, the quantity (demand) of construction waste is uncertain and stochastic. Considering the randomness of construction waste generation, a two-stage stochastic integer programming model for the design of a shipping logistics network for construction waste recycling is proposed, and an accurate algorithm based on Benders decomposition is presented. Based on an actual case in Shanghai, numerical experiments are carried out to evaluate the efficacy of the proposed model and algorithm. Based on an actual case study in Shanghai, numerical experiments demonstrate that the proposed model can help to reduce transportation costs of construction waste. Sensitivity analysis highlights that factors like the penalty for untransported waste and capacity constraints play a crucial role in network optimization decisions. The findings provide valuable theoretical support for developing more efficient and sustainable logistics networks for construction waste recycling. Full article

Effective supply and demand matching for construction materials is a crucial challenge in large-scale railway projects, particularly in complex and hazardous environments. We propose a two-stage stochastic programing model that incorporates environmental uncertainties, such as natural disasters, into the supply chain optimization process. The first stage determines optimal locations and capacities for material supply points, while the second stage addresses material distribution under uncertain demand. We further enhance the model’s efficiency with Benders decomposition algorithm. The performance of our model is rigorously compared with existing optimization approaches, demonstrating its superior capability in handling environmental uncertainties and complex logistical scenarios. This study provides a novel framework for optimizing supply chains in challenging environments, offering significant improvements over traditional models in both adaptability and robustness. Full article

Collaborative designing of service networks using multiple logistics carriers can bring advantages in both economic and environmental terms, and these carriers have symmetry in their service areas. To enable such a collaborative service network and the corresponding benefits, this study proposes a problem of collaborative service network design (CSND) considering demand uncertainty. This problem is formulated as a two-stage robust optimization model using a budget uncertainty set to handle the uncertain demand. A column-and-constraint generation algorithm is developed to accurately solve the robust model. Numerical experiments show that the proposed algorithm outperforms the Benders decomposition algorithm in terms of solving efficiency and quality. Through comparative experiments, this research validates the advantages of collaborative designing and the robustness of model solutions. In addition, three allocation mechanisms are tested to investigate the importance of allocation in CSND. Full article

The flexible movement of racks in the mobile-rack picking system (MRPS) significantly improves the picking efficiency of e-commerce orders with the characteristics of “one order multi–items” and creates a challenging problem of how to place racks in the warehouse. This is because the placement of each rack in the MRPS directly influences the distance that racks need to be moved during order picking, which in turn affects the order picking efficiency. To handle the rack location optimization problem (RLOP), this work introduces a novel idea and methodology, taking into account the heat degree and the relevance degree of racks, to enhance the efficiency of rack placements in the MRPS. Specifically, a two-stage solution strategy is implemented. In stage 1, an integer programming model (Model 1) is developed to determine the heat and relevance degree of racks, and it can be solved quickly by the Gurobi. Stage 2 entails developing a bi-objective integer programming model (Model 2) with the objective to minimize the travel distances of robots in both heavy load and no-load conditions, using the rack heat and relevance degree as inputs. In light of the challenge of decision coupling and the vast solution space in stage 2, we innovatively propose two lower bounds by slacking off the distance between storage locations. A matheuristic algorithm based on Benders decomposition (MABBD) is designed, which utilizes Benders-related rules to reconstruct Model 2, introduces an enhanced cut and an improved optimal cut with RLOP characteristics, and designs the warm start strategy and the master variable fixed strategy. Given the substantial size of real-life problems, the Memetic algorithm (MA) is specifically devised to address them. Instances of varying sizes are also employed to validate the science and efficacy of the model and algorithm. Full article

Hydrogen production modules (HPMs) play a crucial role in harnessing abundant photovoltaic power by producing and supplying hydrogen to factories, resulting in significant operational cost reductions and efficient utilization of the photovoltaic panel output. However, the output of photovoltaic power is stochastic, which will affect the revenue of investing in an HPM. This paper presents a comprehensive analysis of HPMs, starting with the modeling of their operational process and investigating their influence on distribution system operations. Building upon these discussions, a deterministic optimization model is established to address the corresponding challenges. Furthermore, a two-stage stochastic planning model is proposed to determine optimal locations and sizes of HPMs in distribution systems, accounting for uncertainties. The objective of the two-stage stochastic planning model is to minimize the distribution system’s operational costs plus the investment costs of the HPM subject to power flow constraints. To tackle the stochastic nature of photovoltaic power, a data-driven algorithm is introduced to cluster historical data into representative scenarios, effectively reducing the planning model’s scale. To ensure an efficient solution, a Benders’ decomposition-based algorithm is proposed, which is an iterative method with a fast convergence speed. The proposed model and algorithms are validated using a widely utilized IEEE 33-bus system through numerical experiments, demonstrating the optimality of the HPM plan generated by the algorithm. The proposed model and algorithms offer an effective approach for decision-makers in managing uncertainties and optimizing HPM deployment, paving the way for sustainable and efficient energy solutions in distribution systems. Sensitivity analysis verifies the optimality of the HPM’s siting and sizing obtained by the proposed algorithm, which also reveals immense economic and environmental benefits. Full article

This paper presents a tutorial on the state-of-the-art software for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this purpose. The methodologies are classified according to the decomposition alternatives and the types of the variables in the problem. We review the fundamentals of Benders decomposition, dual decomposition and progressive hedging, as well as possible improvements and variants. We also present extensive numerical results to underline the properties and performance of each algorithm using software implementations, including DECIS, FORTSP, PySP, and DSP. Finally, we discuss the strengths and weaknesses of each methodology and propose future research directions. Full article

The fundamental modeling of energy systems through individual unit commitment decisions is crucial for energy system planning. However, current large-scale models are not capable of including uncertainties or even risk-averse behavior arising from forecasting errors of variable renewable energies. However, risks associated with uncertain forecasting errors have become increasingly relevant within the process of decarbonization. The intraday market serves to compensate for these forecasting errors. Thus, the uncertainty of forecasting errors results in uncertain intraday prices and quantities. Therefore, this paper proposes a two-stage risk-constrained stochastic optimization approach to fundamentally model unit commitment decisions facing an uncertain intraday market. By the nesting of Lagrangian relaxation and an extended Benders decomposition, this model can be applied to large-scale, e.g., pan-European, power systems. The approach is applied to scenarios for 2023—considering a full nuclear phase-out in Germany—and 2035—considering a full coal phase-out in Germany. First, the influence of the risk factors is evaluated. Furthermore, an evaluation of the market prices shows an increase in price levels as well as an increasing day-ahead-intraday spread in 2023 and in 2035. Finally, it is shown that intraday cross-border trading has a significant influence on trading volumes and prices and ensures a more efficient allocation of resources. Full article

A model suitable to obtain where and when renewable energy sources (RES) should be allocated as part of generation planning in distribution systems is formulated. The proposed model starts from an existing two-stage stochastic mixed-integer linear programming (MILP) problem including investment and scenario-dependent operation decisions. The aim is to minimize photovoltaic and wind investment costs, operation costs, as well as total substation costs including the cost of the energy bought from substations and energy losses. A new Benders’ decomposition framework is used to decouple the problem between investment and operation decisions, where the latter can be further decomposed into a set of smaller problems per scenario and planning period. The model is applied to a 34-bus system and a comparison with a MILP model is presented to show the advantages of the model proposed. Full article

For the corrective security-constrained optimal power flow (OPF) model, there exists a post-contingency stage due to the time delay of corrective measures. Line overflows in this stage may cause cascading failures. This paper proposes that the thermal inertia of transmission lines can be used to cope with post-contingency overflows. An enhanced security-constrained OPF model is established and line dynamic thermal behaviors are quantified. The post-contingency stage is divided into a response substage and a ramping substage and the highest temperatures are limited by thermal rating constraints. A solving strategy based on Benders decomposition is proposed to solve the established model. The original problem is decomposed into a master problem for preventive control and two subproblems for corrective control feasibility check and line thermal rating check. In each iteration, Benders cuts are generated for infeasible contingencies and returned into the master problem for adjusting the generation plan. Because the highest temperature function is implicit, an equivalent time method is presented to calculate its partial derivative in Benders cuts. The proposed model and approaches are validated on three test systems. Results show that the operation security is improved with a slight increase in total generation cost. Full article

This paper proposes a distance-based distributionally robust energy and reserve (DB-DRER) dispatch model via Kullback–Leibler (KL) divergence, considering the volatile of renewable energy generation. Firstly, a two-stage optimization model is formulated to minimize the expected total cost of energy and reserve (ER) dispatch. Then, KL divergence is adopted to establish the ambiguity set. Distinguished from conventional robust optimization methodology, the volatile output of renewable power generation is assumed to follow the unknown probability distribution that is restricted in the ambiguity set. DB-DRER aims at minimizing the expected total cost in the worst-case probability distributions of renewables. Combining with the designed empirical distribution function, the proposed DB-DRER model can be reformulated into a mixed integer nonlinear programming (MINLP) problem. Furthermore, using the generalized Benders decomposition, a decomposition method is proposed and sample average approximation (SAA) method is applied to solve this problem. Finally, simulation result of the proposed method is compared with those of stochastic optimization and conventional robust optimization methods on the 6-bus system and IEEE 118-bus system, which demonstrates the effectiveness and advantages of the method proposed. Full article

With the deregulation of electricity market, generation companies must take part in strategic bidding by offering its bidding quantity and bidding price in a day-ahead electricity wholesale market to sell their electricity. This paper studies the strategic bidding of a generation company with thermal power units and wind farms. This company is assumed to be a price-maker, which indicates that its installed capacity is high enough to affect the market-clearing price in the electricity wholesale market. The relationship between the bidding quantity of the generation company and market-clearing price is then studied. The uncertainty of wind power is considered and modeled through a set of discrete scenarios. A scenario-based two-stage stochastic bidding model is then provided. In the first stage, the decision-maker determines the bidding quantity in each time period. In the second stage, the decision-maker optimizes the unit commitment in each wind power scenario based on the bidding quantity in the first stage. The proposed two-stage stochastic optimization model is an NP-hard problem with high dimensions. To tackle the problem of “curses-of-dimensionality” caused by the coupling scenarios and improve the computation efficiency, a modified Benders decomposition algorithm is used to solve the model. The computational results show the following: (1) When wind power uncertainty is considered, generation companies prefer higher bidding quantities since the loss of wind power curtailment is much higher than the cost of additional power purchases in the current policy environment. (2) The wind power volatility has a strong negative effect on generation companies. The higher the power volatility is, the lower the profits, the bidding quantities, and the wind power curtailment of generation companies are. (3) The thermal power units play an important role in dealing with the wind power uncertainty in the strategic bidding problem, by shaving peak and filling valley probabilistic scheduling. Full article

The rapid incorporation of wind power resources in electrical power networks has significantly increased the volatility of transmission systems due to the inherent uncertainty associated with wind power. This paper addresses this issue by proposing a transmission network expansion planning (TEP) model that integrates wind power resources, and that seeks to minimize the sum of investment costs and operation costs while accounting for the costs associated with the pollution emissions of generator infrastructure. Auxiliary relaxation variables are introduced to transform the established model into a mixed integer linear programming problem. Furthermore, the novel concept of extreme wind power scenarios is defined, theoretically justified, and then employed to establish a two-stage robust TEP method. The decision-making variables of prospective transmission lines are determined in the first stage, so as to ensure that the operating variables in the second stage can adapt to wind power fluctuations. A Benders’ decomposition algorithm is developed to solve the proposed two-stage model. Finally, extensive numerical studies are conducted with Garver’s 6-bus system, a modified IEEE RTS79 system and IEEE 118-bus system, and the computational results demonstrate the effectiveness and practicability of the proposed method. Full article

In this paper, a two-stage stochastic programming modelling is proposed, to design a multi-period, multistage, and single-commodity integrated forward/reverse logistics network design problem under uncertainty. The problem involved both strategic and tactical decision levels. The first stage dealt with strategic decisions, which are the number, capacity, and location of forward and reverse facilities. In the second stage, tactical decisions, such as base stock level as an inventory policy, were determined. The generic introduced model consisted of suppliers, manufactures, and distribution centers in forward logistic and collection centers, remanufactures, redistribution, and disposal centers in reverse logistic. The strength of the proposed model is its applicability to various industries. The problem was formulated as a mixed-integer linear programming model and was solved by using Benders’ Decomposition (BD) approach. In order to accelerate the Benders’ decomposition, a number of valid inequalities were added to the master problem. The proposed accelerated BD was evaluated through small-, medium-, and large-sized test problems. Numerical results confirmed that the proposed solution algorithm improved the convergence of BD lower bound and the upper bound, enabling to reach an acceptable optimality gap in a convenient time. Full article

Active distribution networks characterized by high flexibility and controllability are an important development mode of future smart grids to be interconnected with large scale distributed generation sources including intermittent energies. However, the uncertainty of intermittent energy and the diversity of controllable devices make the optimal operation of distribution network a challenging issue. In this paper, we propose a stochastic optimal operation strategy for distribution networks with the objective function considering the operation state of the distribution network. Both distributed generations and flexible loads are taken into consideration in our strategy. The uncertainty of the intermittent energy is considered in this paper to obtain an optimized operation and an efficient utilization of intermittent energy under the worst scenario. Then, Benders decomposition is used in this paper to solve the two-stage max-min problem for stochastic optimal operation. Finally, we test the effectiveness of our strategy under different scenarios of the demonstration project of active distribution network located in Guizhou, China. Full article