Multistage Scenario Research Articles

Kinetics of the domain structure in calcium orthovanadate crystals under application of the series of electric field pulses

The evolution of as-grown domain structure under the action of the series of rectangular electric field pulses at the elevated temperature was studied by in situ optical imaging in calcium orthovanadate single crystals. It was revealed that the domain switching was accompanied by formation of the holes in the electrodes stimulated by conical domains reaching the surface. The multistage scenario of the domain kinetics was considered. It was shown that the evolution of the domain structure represented the independent growth of few large domains by motion of the smooth domain walls in the bulk and at the surface.

Siting and sizing of energy storage for renewable generation utilization with multi-stage dispatch under uncertainty: A tri-level model and decomposition approach

For grids suffering from large-scale renewable generation curtailment, the reasonable allocation of energy storage can smooth renewable generation fluctuation for better utilization. This paper analyzes the optimal non-profit planning of energy storage in a grid rich in renewable generation from the perspective of third-party investors. First of all, to model the prediction errors of wind and solar output, a multi-stage scenario tree of intraday uncertainty is established to describe the stochastic output of wind and photovoltaics. In addition, scenario reduction techniques and nonanticipativity constraints are adopted to realize multi-stage optimization. Second, aimed to improve social welfare with an acceptable cost, a tri-level optimization model is established to describe storage siting and sizing, storage dispatch, and market clearing respectively to reach a high level of renewable generation utilization. Furthermore, Karush–Kuhn–Tucker conditions and linearization techniques are used to reform the tri-level model into a linear bi-level one. Third, a decomposition algorithm and a differential cut are proposed to solve the bi-level problem. The optimum is gradually approached via iterating by adding differential cuts and integer cuts. Finally, the model and method are verified based on a region of the HRP-38 system. The results show that storage investment aimed at social welfare can effectively achieve a much higher goal of renewable generation utilization than individual profit-seeking storage investment.

Scenario reduction and scenario tree generation for stochastic programming using Sinkhorn distance

Scenario-based stochastic programming is a widely used method for optimization under uncertainty. The solution quality of this approach is dependent on the approximation of the underlying uncertainty distribution. Therefore, the optimal generation of scenarios (or scenario trees) is a pertinent research objective in stochastic programming. In this work, we approach the scenario reduction and scenario tree generation problem through the perspective of optimal transport, specifically entropy-regularized optimal transport. The availability of an iterative procedure to compute the optimal entropy-regularized transport plan between support sets, using the Sinkhorn–Knopp algorithm in lieu of conventional linear programming-based optimal transport, is found to decrease solution time appreciably, with a decrease in memory burden as well. We present algorithms for optimal scenario reduction and multistage scenario tree generation, and illustrate their use through two case studies. We show that the proposed approach generates high-quality scenarios whose use in stochastic programming offers solutions with good accuracy.

Multi-stage stochastic planning of regional integrated energy system based on scenario tree path optimization under long-term multiple uncertainties

Planning a regional energy system based on the advantages of different types of energies and building an economic and efficient regional integrated energy system (RIES) are hot research subjects in the energy planning field. The call for using clean energy and attaining carbon neutrality worldwide has also attracted research attention in the field of low-carbon and clean planning. Because of the growth of the energy system construction cycle together with the utilization of various types of energies, multi-stage planning considering the uncertainty and construction time sequence in a long time scale has gradually become the primary research direction for planning RIES. The uncertainty of multi-energy load growth and the energy price fluctuation in a long time scale considerably affects the construction and economy of the final planning scheme. Aiming to solve the problems of incomplete load data, low accuracy of single load forecasting, and weak regularity of the energy price fluctuation, this study proposes a multi-stage scenario tree generation method based on the conditional generative adversarial network–random forest–Markov chain. And a method of energy price upper and lower boundary determination is based on the multi-index artificial neural network method for analyzing and solving energy price fluctuation. A multi-stage stochastic planning model of RIES is built herein to minimize the comprehensive cost, including the construction investment and regional operation costs, energy hub income, and carbon emission reduction. Finally, the Beichen district of Tianjin, China is considered a case for verifying the effectiveness and applicability of the proposed model and method. Results show that the construction time sequence of the energy hub substantially affects the planning scheme. Energy hub construction in advance benefits the energy hub income and carbon emission reduction; however, the initial stage requires more investment, which affects the planning and construction of renewable energy. Compared with a single-stage deterministic planning scheme, the optimized multi-stage stochastic planning scheme reduces the number of idle facilities and benefits cost recovery, energy hub income, coverage scenarios, and carbon emission reduction.

Optimization model of trade credit and asset-based securitization financing in carbon emission reduction supply chain

As low-carbon products increasingly become popular among consumers, the manufacturers have begun to advocate low-carbon supply chain to meet consumers’ low-carbon preferences. However, low-carbon investments inevitably bring financing constraints to the supply chain. To provide a potential solution to relieve the financial constrain, we established a two-echelon supply chain consisting of a low-carbon product manufacturer and a retailer. Supply chain members can effectively solve their financing constraints by utilizing portfolio financing consisting of the bank loan (BL), trade credit (TC), and asset-based securitization (ABS). We found that under the financial mode of BL and DC (Dual credit refers to the combination of bank loan and trade credit), only when consumers are highly price-sensitive to low-carbon products can tax preference incentivize the manufacturer to reduce carbon emissions. Under DC mode, consumer’s strong low-carbon preference will push up the retail price level of low-cost products; Under portfolio financing with ABS & DC, consumers’ strong low-carbon preference will force up the retail price level of low-carbon products with low price sensitivity. Compared with pure BL and DC, the cash flow under portfolio financing is the tightest. Besides, we took capital demand of the multi-stage scenario into consideration. Moreover, we found that the tax rate and tax deduction ratio of carbon emissions reduction will affect the retail price, wholesale price, and financing decision in the three financial modes when satisfying certain conditions.

Solving a Biorefinery Location Problem Case in Spain: Uncertainty in Strategic Decisions.

This article presents a case study in which a biorefinery has to be sited in Navarre, Spain, considering uncertainty in prices and biomass availabilities. Stochastic optimization is presented as a multistage scenario tree, which is composed of strategic nodes and operational nodes. Furthermore, combinations of strategic nodes conform strategic scenarios while each operational node, which is rooted in a strategic node, represents an operational scenario. Finally, biorefinery stochastic optimization is achieved by solving a mixed integer linear programming model. Promising results are obtained at strategic (location of the plant), tactical (location of warehouses), and operational (biomass purchase management) levels which clearly differ from the deterministic version of the problem.

A soft-computing framework for automated optimization of multiple product quality criteria with application to micro-fluidic chip production

We describe a general strategy for optimizing the quality of products of industrial batch processes that comprise multiple production stages. We focus on the particularities of applying this strategy in the field of micro-fluidic chip production. Our approach is based on three interacting components: (i) a new hybrid design of experiments (DoE) strategy that combines expert- and distribution-based space exploration with model-based uncertainty criteria to obtain a representative set of initial samples (i.e., settings of essential machining process parameters), (ii) construction of linear and non-linear predictive mappings from these samples to describe the relation between machining process parameters and resulting quality control (QC) values and (iii) incorporation of these mappings as surrogate fitness estimators into a multi-objective optimization process to discover settings that outperform those routinely used by operators. These optimized settings lead to final products with better quality and/or higher functionality for the clients. The optimization module employs a co-evolutionary strategy we developed that is able to deliver better Pareto non-dominated solutions than the renowned NSGA-II multi-objective solver. We applied our proposed high-level surrogate-based multi-objective strategy both in a single/late-stage optimization scenario and in a more challenging multi-stage scenario, yielding final optimization results that improved parameter settings and thus product quality compared to standard expert-based production process parameterizations.

Hierarchical Multi-Stage Cyber Attack Scenario Modeling Based on G&E Model for Cyber Risk Simulation Analysis

With the advancement in cyber-defense capabilities, cyber attacks have continued to evolve like living creatures to breach security. Assuming the possibility of various enemy attacks, it is necessary to select an appropriate course of action by proactively analyzing and predicting the consequences of a particular security event. Cyber attacks, especially in large-scale military network environments, have a fatal effect on security; therefore, various experiments and analyses must be conducted to establish the necessary preparations. Herein, we propose a hierarchical multi-stage cyber attack scenario modeling based on the goal and effect (G&E) model and analysis system, which enables expression of various goals of attack and damage effects without being limited to specific type. The proposed method is applicable to large-scale networks and can be utilized in various scenario-based cyber combat experiments.

Sampling Scenario Set Partition Dual Bounds for Multistage Stochastic Programs

We consider multistage stochastic programming problems in which the random parameters have finite support, leading to optimization over a finite scenario set. There has been recent interest in dual bounds for such problems, of two types. One, known as expected group subproblem objective (EGSO) bounds, require solution of a group subproblem, which optimizes over a subset of the scenarios, for all subsets of the scenario set that have a given cardinality. Increasing the subset cardinality in the group subproblem improves bound quality, (EGSO bounds form a hierarchy), but the number of group subproblems required to compute the bound increases very rapidly. Another is based on partitions of the scenario set into subsets. Combining the values of the group subproblems for all subsets in a partition yields a partition bound. In this paper, we consider partitions into subsets of (nearly) equal cardinality. We show that the expected value of the partition bound over all such partitions also forms a hierarchy. To make use of these bounds in practice, we propose random sampling of partitions and suggest two enhancements to the approach: sampling partitions that align with the multistage scenario tree structure and use of an auxiliary optimization problem to discover new best bounds based on the values of group subproblems already computed. We establish the effectiveness of these ideas with computational experiments on benchmark problems. Finally, we give a heuristic to save computational effort by ceasing computation of a partition partway through if it appears unpromising.

AC/DC Hybrid Distribution System Expansion Planning Under Long-Term Uncertainty Considering Flexible Investment

This paper presents a novel flexible multistage AC/DC distribution system expansion planning model, where the flexible investment strategy is taken into account to address the long-term development uncertainty of load demand and DG. Uncertainty is modeled through a multistage scenario tree to capture the possible system states across the planning horizon. As uncertain information revealed gradually over the planning horizon, the planning decisions at each stage are made sequentially within limited disposable reservation funds (RFs). The overall problem is formulated as a mixed-integer linear programming model that guarantees the convergence to optimality by using commercially available software. We compare the total costs of the optimal planning solutions within different proportions of RFs and analyze the flexibility value of RFs for long-term planning problems. In addition, the impact of RFs on the adjustment ability of the plans and the investment risks under uncertainty are evaluated. Case studies are carried out on an AC/DC hybrid system to validate the effectiveness of the proposed model to constitute flexible options when facing long-term uncertainty. The obtained results show the importance of reserving a suitable proportion of RFs in the planning stage, which has a significant influence on the adaptability of the planning solution to uncertain factors.

Optimal capacity planning of generation system integrating uncertain solar and wind energy with seasonal variability

This paper presents a generation capacity planning model for integration of utility-scale wind farms and grid-connected solar photovoltaic (PV) generation systems via multi-stage stochastic programming. A multi-stage scenario tree for available wind power, electric load, and solar irradiance is constructed with nine stages for a year. Random samples for wind, load, and solar irradiance are generated using Gaussian copula which represents correlation between random samples. Environmental energy policies to control carbon dioxide (CO2) emissions and increase energy generation from renewable sources are implemented, and the resulting generation capacity mix is investigated. A case study with a modified IEEE 300-bus system is performed. With the presented model, optimal generation capacity satisfying policy constraints in a future time was found within a reasonable amount of time.

Scenario generation in stochastic programming using principal component analysis based on moment-matching approach

In optimization models based on stochastic programming, we often face the problem of representing expectations in proper form known as scenario generation. With advances in computational power, a number of methods starting from simple Monte-Carlo to dedicated approaches such as method of moment-matching and scenario reduction are being used for multistage scenario generation. Recently, various variations of moment-matching approach have been proposed with the aim to reduce computational time for better outputs. In this paper, we describe a methodology to speed up moment-matching based multistage scenario generation by using principal component analysis. Our proposal is to pre-process the data using dimensionality reduction approaches instead of using returns as variables for moment-matching problem directly. We also propose a hybrid multistage extension of heuristic based moment-matching algorithm and compare it with other variants of moment-matching algorithm. Computational results using non-normal and correlated returns show that the proposed approach provides better approximation of returns distribution in lesser time.

A Primal decomposition algorithm for distributed multistage scenario model predictive control

This paper proposes a primal decomposition algorithm for efficient computation of multistage scenario model predictive control, where the future evolution of uncertainty is represented by a scenario tree. This often results in large-scale optimization problems. Since the different scenarios are only coupled via the so-called non-anticipativity constraints, which ensures that the first control input is the same for all the scenarios, the different scenarios can be decomposed into smaller subproblems, and solved iteratively using a master problem to co-ordinate the subproblems. We review the most common scenario decomposition methods, and argue in favour of primal decomposition algorithms, since it ensures feasibility of the non-anticipativity constraints throughout the iterations, which is crucial for closed-loop implementation. We also propose a novel backtracking algorithm to determine a suitable step length in the master problem that ensures feasibility of the nonlinear constraints. The performance of the proposed approach, and the backtracking algorithm is demonstrated using a CSTR case study.

Active Sampling for Open-Set Classification without Initial Annotation

Open-set classification is a common problem in many real world tasks, where data is collected for known classes, and some novel classes occur at the test stage. In this paper, we focus on a more challenging case where the data examples collected for known classes are all unlabeled. Due to the high cost of label annotation, it is rather important to train a model with least labeled data for both accurate classification on known classes and effective detection of novel classes. Firstly, we propose an active learning method by incorporating structured sparsity with diversity to select representative examples for annotation. Then a latent low-rank representation is employed to simultaneously perform classification and novel class detection. Also, the method along with a fast optimization solution is extended to a multi-stage scenario, where classes occur and disappear in batches at each stage. Experimental results on multiple datasets validate the superiority of the proposed method with regard to different performance measures.

Unit commitment under imperfect foresight – The impact of stochastic photovoltaic generation

This paper investigates the impact of uncertain photovoltaic generation on unit commitment decisions for the German rolling planning procedure employing a large-scale stochastic unit commitment electricity market model (stELMOD). A novel approach to simulate a time-adaptive intra-day photovoltaic forecast, solely based on an exponential smoothing of deviations between realized and forecast values, is presented. Generation uncertainty is then incorporated by numerous multi-stage scenario trees that account for a decreasing forecast error over time. Results show that total system costs significantly increase when uncertainty of both wind and photovoltaic generation is included by a single forecast, with more frequent starting processes of flexible plants and rather inflexible power plants mainly deployed at part-load. Including the improvement of both wind and photovoltaic forecasts by a scenario tree of possible manifestations, the scheduling costs could be significantly reduced in representative weeks for spring and summer. In general, stochastic representations increase the need for congestion management as well as more frequent use of storage in the model, leading to a more realistic depiction of the markets.

Integrated scheduling of a hybrid manufacturing and recovering system in a multi-product multi-stage environment with carbon emission

The closed-loop network design of a hybrid manufacturing and recovering system plays an important role on improving both the production efficiency and environmental quality, especially under a multi-product multi-stage scenario. Besides, with the growing awareness of environmental concerns by legislators, decision makers have been forced not only to enhance utilization of the returned products or components but also to control carbon emission of the whole hybrid system. To optimize the closed-loop logistics superstructure design, a multi-objective model integrating the manufacturing and recovering process in a multi-product multi-stage environment is proposed in this research. Both operations cost and environmental influence are regarded as two objective categories. In addition, the carbon emission regulation is considered in the model by integrating and characterizing the energy input and greenhouse output throughout the hybrid system. Finally, a numerical case originated from an electronic assembly plant is presented under different scenarios to illustrate the effectiveness of the proposed approach. A sensitivity analysis is given to compare the impacts of carbon emission and return rate on each sub-objective of the logistics and environmental issues.

Wind-Induced Fatigue and Asymmetric Damage in a Timber Bridge.

The transformation of a 30 m long timber pedestrian bridge into a wobbly (laterally swaying) bridge with a dramatically reduced first lateral modal frequency has been monitored by seven annual, multi-sensor surveys. This evidence, in combination with analysis of the wind record, observations of local damage and evidence of wind-induced excitations from other bridges, is used to present a multi-stage scenario of the extraordinary structural weakening of our study bridge in only a few years. Our analysis is constrained by observations of asymmetric damage (longitudinal splitting cracks around metallic connections along the south side of the deck, not explained by ordinary, essentially symmetric lateral oscillations) and over-threshold analysis of strong northerly wind events, including gusts. The proposed scenario is that an unexpected for the area icing event took advantage of construction vicissitudes and produced damage that reduced the lateral stiffness of the bridge, especially of the arch superstructure. In addition, strong winds sharing common direction with gusts produced a combination of semi-static lateral bending and of dynamic oscillations, leading to numerous cycles of asymmetric high amplitude lateral deflections producing tensile stress normal to grain, cracks localized in connections, and fatigue. The vertical stiffness of the bridge was only slightly affected.

Improving Scenario Decomposition for Multistage MPC Using a Sensitivity-Based Path-Following Algorithm

This letter proposes a computationally efficient algorithm for robust multistage scenario model predictive control (MPC). In multistage scenario MPC, the evolution of uncertainty in the prediction horizon is represented via a scenario tree. The resulting large-scale optimization problem can be decomposed into several smaller subproblems where, for example, each subproblem solves a single scenario. Since the different scenarios differ only in the uncertain parameters, the distributed scenario MPC problem can be cast as a parametric nonlinear programming (NLP) problem. By using the NLP sensitivity, we do not need to solve all the subproblems as full NLPs. Instead they can be solved exploiting the parametric nature by a path-following predictor-corrector algorithm that approximates the NLP. This results in a computationally efficient multistage scenario MPC framework. Simulation results show that the sensitivity-based distributed multistage MPC provides a very good approximation of the fully centralized scenario MPC.

Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic

In this paper, we present a multistage time consistent Expected Conditional Risk Measure for minimizing a linear combination of the expected mean and the expected variance, so-called Expected Mean-Variance. The model is formulated as a multistage stochastic mixed-integer quadratic programming problem combining risk-sensitive cost and scenario analysis approaches. The proposed problem is solved by a matheuristic based on the Branch-and-Fix Coordination method. The multistage scenario cluster primal decomposition framework is extended to deal with large-scale quadratic optimization by means of stage-wise reformulation techniques. A specific case study in risk-sensitive production planning is used to illustrate that a remarkable decrease in the expected variance (risk cost) is obtained. A competitive behavior on the part of our methodology in terms of solution quality and computation time is shown when comparing with plain use of CPLEX in 150 benchmark instances, ranging up to 711,845 constraints and 193,000 binary variables.

On strategic multistage operational two-stage stochastic 0–1 optimization for the Rapid Transit Network Design problem

The Rapid Transit Network Design planning problem along a time horizon is treated by considering uncertainty in passenger demand, strategic costs and network disruption. The problem has strategic decisions about the timing to construct stations and edges, and operational decisions on the available network at the periods. The uncertainty in the strategic side is represented in a multistage scenario tree, while the uncertainty in the operational side is represented in two-stage scenario trees which are rooted with strategic nodes. The 0–1 deterministic equivalent model can have very large dimensions. So-called fix-and-relax and lazy matheuristic algorithms, which are based on special features of the problem, are proposed, jointly with dynamic scenario aggregation/de-aggregation schemes. A broad computational experience is presented by considering a network case study taken from the literature, where the problem was only treated as a deterministic 0–1 model. 40 nodes in the strategic multistage tree are considered for passenger demand and investment cost and 8 uncertainties are considered for network disruption in each strategic node, in total 320 uncertain situations are jointly considered. For assessing the validity of the proposal, a computational comparison is performed between the plain use of a state-of-the-art optimization solver and the proposals made in this work. The model is so-large (2.6M constraints and 1.6M binary variables) that the solver alone cannot provide a solution in an affordable time. However, a mixture of the both matheuristics provides a solution with a good optimality gap requiring an affordable elapsed time.