On robust optimization of two-stage systems

Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty

Coordinated operations of electricity and district heating networks offer a potential for mitigating inherent variability of renewable energy sources (RES) in the ongoing transition to smart grids. This paper proposes a two-stage distributionally robust optimization (DRO) approach to determine the optimal day-ahead unit commitment in coordinated electricity and district heating networks with variable RES power output. The proposed formulation is to minimize the worst-case expected total cost over an ambiguity set comprising a family of probability distributions with given support and moments of RES power output. As such, the proposed DRO approach can overcome the limitations of stochastic programming in its inherent dependence of exact probability distributions along with a huge computational burden, but also becomes less conservative than classical robust optimization. The pertinent DRO model is eventually reformulated as a tractable mixed-integer second-order cone (SOC) programming after employing linear decision rules and the SOC duality. Simplified affine policies are utilized to further improve computational tractability and performance. Finally, case studies are conducted based on Barry Island electricity and district heating networks. The numerical results demonstrate the decision-making superiority of the proposed method as compared with deterministic, stochastic programming, and robust optimization approaches. They also validate the computational improvement of the proposed approach by employing simplified affine policies.

A novel robust optimization approach for an integrated municipal water distribution system design under uncertainty: A case study of Mashhad

A robust optimization framework for energy management of CCHP users with integrated demand response in electricity market

We present a new methodology to support an energy trading company (ETC) to devise contracting strategies under an optimal risk-averse renewable portfolio. The uncertainty in the generation of renewable energy sources is accounted for by exogenously simulated scenarios, as is customary in stochastic programming. However, we recognize that spot prices largely depend on unpredictable market conditions, making it difficult to capture its underlying stochastic process, which challenges the use of fundamental approaches for forecasting. Under such framework, industry practices make use of stress tests to validate portfolios. We then adapt the robust optimization approach to perform an endogenous stress test for the spot prices as a function of the buy-and-sell portfolio of contracts and renewable energy generation scenarios. The optimal contracting strategy is built through a bilevel optimization model that uses a hybrid approach, mixing stochastic and robust optimization. The proposed model is flexible to represent the traditional stochastic programming approach and to express the ETC's uncertainty aversion in the case where the price distribution cannot be precisely estimated. The effectiveness of the model is illustrated with examples from the Brazilian market, where the proposed approach is contrasted to its stochastic counterpart and both are benchmarked against observed market variables.

Commitment and dispatch of heat and power units via affinely adjustable robust optimization

This study presents a parameter coordination and robust optimization approach based on knowledge network modeling. The method allows multidisciplinary designer to synthetically coordinate and optimize parameter considering multidisciplinary knowledge. First, a knowledge network model is established, including design knowledge from assembly, manufacture, performance, and simulation. Second, the parameter coordination method is presented to solve the knowledge network model, monitor the potential conflicts due to engineering changes, and obtain the consistency solution space corresponding to the given knowledge. Finally, the robust parameter optimization model is established, and genetic arithmetic is used to obtain the robust optimization parameter. A design instance is introduced to show the validity of this method.

Chapter 4 - Data-driven robust stochastic optimization for power systems operations

The economic and low‐carbon operation strategy of multi‐energy microgrids (MEM) has become an important research topic in smart grids. The operation of MEM is affected by uncertain factors from renewable energy and internal load. To handle uncertainties from both source and load sides, the authors propose a novel worst‐case‐and‐probability uncertainty sets and a novel worst‐expectation min‐max‐max‐min two‐stage four‐level robust optimisation (RO) model considering stepped carbon trading for MEM. This four‐level RO model is a more generalised model compared with the traditional two‐stage RO and distributionally robust optimisation based models. First, the unsolvable original four‐level model is decoupled into a master problem and a sub‐problem (SP). Second, Karush–Kuhn–Tucker condition is applied to convert SP into a solvable problem. Third, column and constraint generation (C&CG) algorithm is employed to solve the reformulated four‐level RO. Finally, the effectiveness of the proposed model and solution method is verified by case studies. The results indicate that the convergence behaviour of this solution method is excellent. The MEM operator can select appropriate robustness factors and comprehensive norms to control its conservatism level. Besides, MEM could reduce carbon emissions by participating in carbon trading and installing carbon capture devices.

This research aims to adopt two robust optimisation approaches for a real-world flow shop manufacturing system with batch processing machines, where the processing time and size of job are non-deterministic and uncertain. Each machine can process multiple jobs simultaneously as long as the machine capacity is not exceeded. Two important decisions are required: (1) grouping jobs into batches and (2) scheduling the established batches on machines. A mathematical optimisation model is presented, and then two famous robust optimisation approaches are adopted for the purpose of converting the deterministic model to the robust one. An efficient particle swarm optimisation (PSO) algorithm is developed to solve the problem in a reasonable time. In order to verify the developed model and evaluate the performance of our proposed algorithm, a set of small to large test problems are generated, and a simulation approach and a commercial optimisation solver are used to solve these problems. Analysis of the implementation of two independent robust optimisation methods is performed by the paired t-test on all of the test problems. Furthermore, the Taguchi method, as a statistical optimisation technique, is employed to investigate the appropriate level of PSO parameters.

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

Robust optimization based optimal chiller loading under cooling demand uncertainty

Proton plans are subject to a number of uncertainties which must be accounted for to ensure that they are delivered safely. Misalignment resulting from residual errors in daily patient positioning can result in both a displacement and distortion of dose distributions. This can be particularly important for intensity modulated proton therapy treatments where the accurate alignment of highly modulated fields may be required to deliver the intended treatment. A number of methods to generate plans that are robust to these uncertainties exist. These include robust optimisation approaches which account for the effect of uncertainties on the dose distribution within the optimisation process. However, robustness to uncertainty comes at the cost of plan quality. For this reason, it is important that the uncertainties considered are realistic. Existing approaches to robust optimisation have neglected the role of fractionated treatment deliveries in reducing the uncertainties that result from random setup errors. Here, a method of robust optimisation which accounts for this effect is presented and is evaluated using a 2D planning environment. The optimisation algorithm considers the dose in the estimated upper and lower bounds of the dose distribution under the effect of setup and range errors. A comparison with plans robustly optimised without consideration of the effect of fractionation and conventionally optimised plans is presented. Fractionation incorporated robust optimisation demonstrates a reduced sensitivity to uncertainty compared to conventionally optimised plans and a reduced integral dose compared to robustly optimised plans.

A robust optimization model for multi-product two-stage capacitated production planning under uncertainty

Quay cranes are among the most important resources in port terminals, and their efficient use is crucial for port terminals to remain competitive in the market. The problem of determining the sequence of tasks performed by the quay cranes that minimizes the turnaround time of a vessel is an NP‐hard problem known as the quay crane scheduling problem (QCSP).In this paper, we consider the unidirectional QCSP under uncertain processing times, where the cranes are only allowed to move in a specific direction. We start by presenting the first distributionally robust optimization (DRO) model for this problem and an exact decomposition algorithm to solve it. The proposed DRO model makes it possible to derive a stochastic programming model and a robust optimization model for the unidirectional QCSP by an appropriate choice of the risk‐averse parameter of the model. Through extensive numerical results, we compare these three approaches—stochastic programming, robust optimization, and DRO—to investigate whether significant differences among the solutions obtained exist. Finally, we propose a new method for helping practitioners to determine a representative set of different DRO solutions.