Data-driven Uncertainty Set Research Articles

Data-driven robust optimization in the face of large-scale datasets: An incremental learning approach

One of the most significant current discussions in optimization under deep uncertainty is integrating machine learning and data science into robust optimization, which has led to the emergence of a new field called Data-Driven Robust Optimization (DDRO). When creating data-driven uncertainty sets, it considers a dataset’s complexity, hidden information, and inherent form. One of the more practical machine learning algorithms for creating data-driven uncertainty sets is support vector clustering (SVC). This algorithm has no prerequisites for preliminary information to generate uncertainty sets with arbitrary geometry. More scenarios can reduce risk when developing SVC-based uncertainty sets. However, the lack of a systematic way to manage the large number of these scenarios hinders the employment of SVC. This paper puts forward an incremental learning algorithm based on support vector clustering, called Incremental Support Vector Clustering (ISVC), to construct an uncertainty set incrementally and efficiently using large datasets. This approach’s novelty and main contributions include incrementally constructing uncertainty sets and dynamic management of outliers. In order to update the temporarily stored Bounded Support Vectors (BSV) and identify outliers, the idea of BSV-archive is offered, where the revision-and-recycle operation is tailored to do just that. As a result, some of the newly acquired information is preserved. Experiments on large-scale datasets demonstrate that the proposed ISVC approach can create an uncertainty set comparable to that of an SVC-based method while using significantly less time.

Evaluating the reliability of microgrids consisting of renewable energy sources using stochastic scheduling based on the data-driven uncertainty set

The paper addresses challenges arising from the unpredictable and probabilistic nature of wind speed and solar irradiance, leading to variable power production in renewable sources. Power imbalance issues in grids result from the difficulty in accurately estimating these variables. The study introduces an improved data-driven uncertainty set, utilizing a neural network trained with extensive historical data from a multi-microgrid system. This set efficiently identifies strong coupling and extracts characteristic scenarios, ensuring high reliability with a minimal number of scenes. The proposed probabilistic model, implemented in the coordination of pumped storage, wind, and solar power plants, incorporates energy limits and storage characteristics. A four-mode IEEE model for storage units allows quick response to system needs. Simulating storage unit states every hour, considering system load, output power, and energy limitations, optimizes power dispatch based on battery capacity. Additionally, the paper addresses the uncertainty in electric power demand in microgrids, proposing an improved data-driven uncertainty set obtained from a neural network. By eliminating unrealistic worst-case scenarios, the approach accurately captures system characteristics, maintaining high reliability while significantly reducing convergence time. The proposed uncertainty set is applied in a novel two-stage robust optimization dispatch model, transforming the optimization problem into a mixed-integer linear programming problem, and solving it using a column and constraint generation algorithm. Numerical case studies verify the feasibility and superiority of the proposed approach in terms of economy and convergence performance compared to existing robust optimization methods.

Data-driven robust optimization based on position-regulated support vector clustering

The changes experienced in integrating machine learning and data science into mathematical programming over the past decade remain unprecedented. They have created a novel optimization component under deep uncertainty known as “Data-Driven Robust Optimization” (DDRO). It considers a dataset's complexity, hidden information, and inherent form when creating data-driven uncertainty sets. One of the more practical machine learning algorithms for creating data-driven uncertainty sets is support vector clustering (SVC). There are no prerequisites for preliminary information to generate uncertainty sets with arbitrary geometry. Moreover, it can regularize the conservatism level and adjust a balance between the magnitude of the uncertainty set and information loss by a control parameter. Despite being superior and effective, SVC needs help selecting the best value for the trade-off parameter. This parameter impacts the robustness of the solution and the ability of SVC to handle information loss; hence, it must be accurately determined. Irrational data elimination as outliers alters the degree of conservatism, resulting in information loss and suboptimal solutions. This paper seeks to remedy this problem by suggesting position-regulated support vector clustering (PRSVC) to involve the importance of each data point in estimating the trade-off parameter. Experiments on artificial and real-world data collections indicate that, in comparison to the current SVC-based model, the proposed approach offers a more accurate decision boundary.

Two-stage robust optimization strategy for spatially-temporally correlated data centers with data-driven uncertainty sets

Data center buildings (DCBs) in data center parks consume significant and ever-growing amounts of electric power for processing tremendous data workloads in a modern city. The workloads are divided into two types: interactive workloads (IWs) with spatially transferrable characteristics and batch workloads (BWs) with temporally shiftable characteristics. The unique spatially-temporally workloads show great demand response capability, enabling reducing the electricity cost of DCBs. However, the source-load uncertainties such as the number of arriving workloads, and photovoltaic output affect the cost-efficient operation of DCBs in the electricity market. Inadequate server numbers during worst-case scenarios may result in the inability to fulfill user contracts. This paper presents a model for DCBs that incorporates battery energy storage systems, photovoltaic generation, and electrical loads for processing spatially-temporally correlated workloads. Considering multiple uncertainties, a two-stage robust optimization strategy is proposed to coordinate workload and power for DCBs. To accurately determine the boundaries of the uncertainty sets, a data-driven method using the 1-Wasserstein metric is adopted. This method is reformulated into a distributionally robust chance-constrained programming model. The nested column-and-constraint generation (C&CG) method is used to resolve the established two-stage robust optimization problem with mixed-integer recourse. Finally, case studies verify the effectiveness of the proposed method.

Data-driven robust cost consensus model with individual adjustment willingness in group decision-making

In the consensus reaching process, the moderator, as the representative of collective interests, needs to consume certain resources to help decision-makers (DMs) reach a consensus. However, in real life, when the adjustment opinions provided by the moderator are too different from the initial opinions of the DMs, they may not accept the adjustment. On the other hand, it is difficult for the moderator to determine the unit consensus cost of every DM. However, existing consensus methods ignore the above two problems. Therefore, a data-driven robust minimum cost consensus model with individual adjustment willingness (DDR-MCCM-IAW) is proposed. First, an IAW function based on the approximation degree between the adjustment opinion and the original opinion is provided to construct the MCCM-IAW. Subsequently, principal component analysis and kernel density estimation are combined to obtain a data-driven uncertainty set of unit consensus cost, to reduce the conservatism of classical robust cost consensus models. Meanwhile, the tractable DDR-MCCM-IAW is obtained by using linear dual theory. Finally, to test the effectiveness and rationality of the proposed method, some experiments and comparative analysis are presented.

A data-driven strategy for industrial cracking furnace system scheduling under uncertainty

Cyclic scheduling of ethylene cracking furnace system (CSECFS) has a significant impact on raising economic performance of ethylene plants. However, in actual plants, optimal results of deterministic models may be suboptimal or ineffective because of various uncertainties. This paper proposes a novel data-driven adaptive robust optimization (DDARO) strategy that effectively bridges robust optimization and machine learning methods. A mixed-integer nonlinear programming (MINLP) model for CSECFS is developed initially. Second, data-driven uncertainty sets are generated using the historical data of processing flow rates: maximally correlated principal component analysis (MCPCA) is employed to partition the data into two subspaces, which are then delineated by support vector clustering (SVC) and generalized norm approaches, respectively. Third, a multi-stage DDARO model is established using the derived uncertainty sets and afterward transformed as a tractable single-stage model. Finally, a real-world case is performed to exemplify the effectiveness of the proposed framework.

Data-driven nested robust optimization for generation maintenance scheduling considering temporal correlation

Traditional power grids are gradually transitioning to smart grids with high penetration of renewable energy, which can realize the efficient utilization of power resources and low carbon emissions. However, the uncertainties of renewable energy (e.g., wind power) and load demand pose considerable challenges to secure operation and cost-effective planning in smart grids, such as generation maintenance scheduling (GMS). In this context, conventional methods including stochastic optimization and robust optimization are adopted to cope with the uncertainties and formulate the GMS plan. Unfortunately, these methods fail to consider the temporal information in uncertain variables, which can introduce extra operational costs brought by the uncertainties. To address this issue, we consider the temporal correlation of the uncertain wind power and load demand, and develop a data-driven two-stage nested robust optimization (NRO) approach for GMS to minimize the total costs of power system operation under uncertain scenarios. In our proposed approach, a temporal correlation Dirichlet process mixture model (TCDPMM) is developed to investigate the temporal information in the wind power and load demand datasets. Then, variational Bayesian inference (VBI) is employed to construct the data-driven uncertainty set, in which the temporal information for the uncertain variables and the correlations between the uncertain variables are considered. Subsequently, combined with this uncertainty set, a two-stage GMS problem is converted to a “min–max-max–min” optimization problem which is solved by the parallel Benders’ decomposition algorithm. The effectiveness and superiority of the proposed approach are demonstrated with a six-bus power system and a practical power system in China.

A data-driven stochastic energy sharing optimization and implementation for community energy storage and PV prosumers

The development of energy sharing community is demanded for the on-site consumption of distributed generations and efficient utilization of renewable energy. This paper proposes a two-stage stochastic energy sharing model considering the photovoltaic (PV) power uncertainties to minimize the social cost of PV prosumers and community energy storage (CES) in community. The proposed optimization model derives the optimal schedule of community under the worst case in the first stage. The regulation model for PV prosumers is formulated in the second stage to address the PV uncertainties including three regulation methods: the demand response, the charging/discharging schedule, and the transactive energy schedule. Particularly, a modeling approach based on the data-driven uncertainty set is introduced to find the worst probability distribution. The formulated two-stage problem can be solved by the column-and-constraint generation algorithm, where the coupled model is decomposed into a main problem and a sub problem and solved by iteration. Numerical simulation results show the effectiveness of the proposed energy sharing optimization model to decrease the energy cost. Experimental results demonstrate the feasibility of multi-ports energy hub used in energy sharing community.

Data-driven worst case model predictive control algorithm for propylene distillation column under uncertainty of top composition

Composition soft sensors have wide application in the distillation process. In this study, considering the limitation of the prediction ability of the composition soft sensor, a data-driven worst case model predictive control of propylene distillation column is proposed to hedge against the uncertainty of top composition. Firstly, based on the compartmental method and the dynamic mechanism model, a linear state space model of the distillation column is constructed. Aiming at dealing with the uncertainty of top propylene content caused by composition soft sensor, the data-driven uncertainty set is constructed by the combination of principal component analysis and kernel density estimation based on the historical data. Then, the certainty equivalent, traditional worst case, data-driven worst case, set-point tracking and offset-free model predictive control algorithm are designed. Finally, a case study of composition control in a propylene distillation column is carried out. Compared with other strategies, the proposed algorithm ensures the quality of the product while achieving small quality surplus and low operating cost.

Robust transmission network expansion planning based on a data-driven uncertainty set considering spatio-temporal correlation

Robust transmission network expansion planning based on a data-driven uncertainty set considering spatio-temporal correlation

A new data-driven robust optimization approach to multi-item newsboy problems

<p style='text-indent:20px;'>A newsboy problem is a typical stochastic inventory management problem and has extensive applications in the fields of operational research, management sciences and marketing sciences. One of the challenges underlying such problems is to handle the uncertainty of demands. In the existing results, it is often to assume that the demand distribution is given to facilitate solution of the problems. In this paper, a novel data-driven robust optimization model for solving multi-item newsboy problems is proposed by combining the absolute robust optimization with a data-driven uncertainty set, and the latter is leveraged to address the uncertainty of demands. For the single-item situation, a closed-form solution is obtained and influences of parameters on the optimal solutions are analyzed. Owing to complexity of the multi-item situation, a uniform smoothing function is leveraged to smooth the proposed model. Then, an algorithm, called a modified Frank-Wolfe feasible direction algorithm, is developed to solve a series of smooth subproblems. Numerical simulation demonstrates that the proposed model in this paper can reduce over-conservation of robust optimization methods and is more robust than other similar well-established methods in the literature. By numerical simulation and sensitivity analysis, it is concluded that: (1) The proposed method can provide more stable optimal order policy and profits than the existing ones; (2) For a product with a higher unit purchase price, the optimal order quantities are more sensitive to its change; (3) In view of profitability, the newsboy should not to be too risk-averse.</p>

A closed-loop data-driven optimization framework for the unit commitment problem: A [formula omitted]-learning approach under real-time operation

Real-time operation of electric power systems under high penetration of renewable energy generation has to incorporate strategies for managing the uncertainty associated with this type of power sources. In the case of the unit commitment problem, models based on robust optimization have been widely used for the day-ahead computation of power dispatch and reserves schedule. Typical approaches use uncertainty sets with fixed levels of conservativeness, without considering the real-time performance of the solutions. This paper proposes a solution scheme for the adaptive robust unit commitment problem using online adjusted data-driven uncertainty sets. Conservativeness parameters of the uncertainty sets are dynamically calculated as a function of previous operating results and incoming data via reinforcement learning, resulting in an online learning-optimization framework. The paper also develops an experimental framework to simulate real-time operation under the proposed methodology. Out-of-sample experiments illustrate the effectiveness of the proposed scheme against well-known benchmarks with fixed robustness levels. Results show improvements in power generation costs, voltage violations, and use of reserves. Two systems of different sizes are analyzed, illustrating the scalability of the proposed approach.

Strategic retail pricing and demand bidding of retailers in electricity market: A data-driven chance-constrained programming

This paper proposes a novel bi-level optimization model to study the strategic retail pricing and demand bidding problems of an electricity retailer that considers the interactions between demand response and market clearing process. In order to accurately forecast the day-ahead demand bids submitted by the retailer, a novel deep learning framework based on convolutional neural networks and long short-term memory is proposed that can capture both local trends and long-term dependency of the forecasting data. In addition, uncertainties about the retailer’s served demand, rivals’ demand bids, and wind power generation are incorporated using the data-driven uncertainty set constructed from data. We further propose chance-constrained programming that introduces a set of chance constraints to represent the operational risk associated with the market uncertainties. To solve this problem, we first reformulate chance-constrained programming as a tractable second-order conic programming and then convert it into a single-level mathematical program with equilibrium constraints by using its Karush Kuhn Tucker conditions. The scope of the examined case studies is four-fold. First, they evaluate the benefits of the proposed forecasting framework in terms of higher accuracy and expected profit compared to the conventional forecasting methods. Second, they demonstrate how demand flexibility affects the retailer’s strategies and its business cases. Third, they highlight the added value of the proposed bi-level model capturing the market clearing process by comparing its outcomes against the state-of-the-art bi-level model with exogenous market prices. Finally, they analyze the retailer’s strategies and business cases at different confidence levels regarding the imposed chance constraints.

Renewable energy-powered semi-closed greenhouse for sustainable crop production using model predictive control and machine learning for energy management

Renewable energy consumption in agriculture is ascending, catering to the food needs of the rising population and protecting the environment. Maximizing renewable energy usage efficiency is essential for achieving sustainable development goals. In this work, a nonlinear integrated controlled environment agriculture model is constructed to correlate the impact of weather disturbances, temperature control actuators, humidity control actuators, fertilization, and irrigation to the states of crop production facilities and crop growing conditions. Linearization of the model is performed to reduce the computation time while retaining the accuracy of the nonlinear model. A robust model predictive control framework is developed to maximize renewable energy power usage efficiency and maintain a hospitable sustainable cultivation environment. To improve the robustness of control to hedge against the forecast uncertainties, the disjunctive data-driven uncertainty sets built upon the historical forecast errors are constructed by using machine learning methods, including principal component analysis and kernel density estimation. This work also presents the result of the simulation of controlling a renewable energy-powered semi-closed greenhouse growing tomatoes located in Ithaca, New York. Compared to other model predictive control frameworks, which do not leverage the machine learning approaches, the proposed control framework enables a 0.7%–66.9% reduction in renewable energy usage and a 0.7 wt% to 16.1 wt% increase in crop production. Further analysis of this work reveals that the integrated controlled environment agriculture model can help in increasing renewable energy usage efficiency from 4.7% to 127.5%.

A robust omnichannel pricing and ordering optimization approach with return policies based on data-driven support vector clustering

This work considers an omnichannel retailer selling a product with market demand uncertainties to customers in different regions through an e-store and multiple brick-and-mortar stores. The retailer also manages online product returns by selecting appropriate return policies. A robust omnichannel pricing and ordering optimization model is proposed with two, i.e., a full-refund and a no-refund, return policies. The online demand and the offline demand depend on the prices of the e-store and the brick-and-mortar stores, where the online demand also depends on the refund of the e-store. A linearization technique is adopted to deal with nonlinearity of the model. A data-driven robust optimization approach is used to construct uncertainty sets based on available historical data using support vector clustering to handle demand uncertainties. Furthermore, the proposed model is transformed into an approximate mixed integer linear programming model which can be solved by using commercial software. An electronics retailer in China is used as a case study to illustrate the effectiveness and practicality of the proposed model and the solution method. A comparison with the box uncertainty set reveals that the data-driven uncertainty set is less conservative and performs better by obtaining higher profit for the retailer. Furthermore, sensitive analysis results indicate that return policies and the return rate of the product affect the optimal pricing and ordering decisions and the total profit.

Machine learning-based data-driven robust optimization approach under uncertainty

On the basis of the machine learning ability to analyze massive data, we propose a new concept suitable for data-driven robust optimization, and design two new methods for constructing data-driven uncertainty sets. Partial least squares (PLS) or kernel principal component analysis (KPCA) is selected to capture the underlying uncertainties and correlation of uncertain data, and the projection of uncertain data on each principal component is obtained. Then, the probability distribution information of project data is extracted via robust kernel density estimation (RKDE). Considering the applicability of PLS to linear data for the idea of canonical correlation analysis and the bias of KPCA to nonlinear data due to kernel function, guidelines for selecting an appropriate method are presented in terms of the linear and nonlinear degree between data. The measurement indicators are Pearson correlation coefficient, mutual information and nonlinear coefficient. The induced robust counterpart framework not only alleviates excessive conservatism, but also significantly improves the robustness, which has the advantages of easy implementation and high computational efficiency. Through a numerical example and two application cases, the effectiveness of the proposed framework is illustrated.

Semiclosed Greenhouse Climate Control Under Uncertainty via Machine Learning and Data-Driven Robust Model Predictive Control

This work proposes a novel data-driven robust model predictive control (DDRMPC) framework for automatic control of greenhouse in-door climate. The framework integrates dynamic control models of greenhouse temperature, humidity, and CO₂ concentration level with data-driven robust optimization models that accurately and rigorously capture uncertainty in weather forecast error. Data-driven uncertainty sets for ambient temperature, solar radiation, and humidity are constructed from historical data by leveraging a machine learning approach, namely, support vector clustering with weighted generalized intersection kernel. A training-calibration procedure that tunes the size of uncertainty sets is implemented to ensure that data-driven uncertainty sets attain an appropriate performance guarantee. In order to solve the optimization problem in DDRMPC, an affine disturbance feedback policy is utilized to obtain tractable approximations of optimal control. A case study of controlling temperature, humidity, and CO₂ concentration of a semiclosed greenhouse in New York City is presented. The results show that the DDRMPC approach ends up with 14% and 4% lower total cost than rule-based control and robust model predictive control with L₁-norm-based uncertainty set, respectively. The constraint violation probability, which is the percentage of time that the greenhouse system states violate the constraint throughout the whole growing period, for DDRMPC is only 0.39%. Hence, the proposed DDRMPC framework can prevent the greenhouse climate from becoming harmful to plants and fruits. In conclusion, the proposed DDRMPC approach can improve the greenhouse climate control performance and reduce cost compared with other control strategies.

Sustainable power systems operations under renewable energy induced disjunctive uncertainties via machine learning-based robust optimization

For sustainable and reliable power systems operations integrating variable renewable energy, it is essential to incorporate the uncertain intermittent power outputs. A novel robust unit commitment framework with data-driven disjunctive uncertainty sets is proposed for sustainable energy systems with volatile renewable wind power, assisted by machine learning. The approach can flexibly identify the uncertainty space based on renewable power forecast error data with disjunctive structures. Specifically, the uncertainty data are grouped using K-means and density-based spatial clustering of applications with noise (DBSCAN) following the optimal cluster number determined by the Calinski-Harabasz index. Disjunctive uncertainty sets are constructed accordingly as the union of multiple basic uncertainty sets, including conventional box and budget uncertainty sets, and data-driven uncertainty sets using Dirichlet process mixture model, principal component analysis coupled with kernel density estimation, and support vector clustering. Subsequently, the problem is formulated into a two-stage adaptive robust unit commitment model with data-driven disjunctive uncertainty sets and with a multi-level optimization structure. To facilitate the solution process, a tailored decomposition-based optimization algorithm is developed. The effectiveness and scalability of the proposed framework are illustrated with two case studies investigating sustainable operations scheduling based on IEEE 39-bus and 118-bus systems, considering the integration of intermittent renewable energy. Results show that the proposed framework can reduce the price of robustness by 8–48% compared to the conventional “one-set-fits-all” robust optimization approaches. Benchmarking with stochastic programming indicates that the proposed framework can achieve the same or better economic performance for sustainable operations with over 75% less computational time.

Dynamic optimization with side information

We develop a tractable and flexible data-driven approach for incorporating side information into multi-stage stochastic programming. The proposed framework uses predictive machine learning methods (such as k-nearest neighbors, kernel regression, and random forests) to weight the relative importance of various data-driven uncertainty sets in a robust optimization formulation. Through a novel measure concentration result for a class of supervised machine learning methods, we prove that the proposed approach is asymptotically optimal for multi-period stochastic programming with side information. We also describe a general-purpose approximation for these optimization problems, based on overlapping linear decision rules, which is computationally tractable and produces high-quality solutions for dynamic problems with many stages. Across a variety of multi-stage and single-stage examples in inventory management, finance, and shipment planning, our method achieves improvements of up to 15% over alternatives and requires less than one minute of computation time on problems with twelve stages.

A general data-driven nonlinear robust optimization framework based on statistic limit and principal component analysis

A general robust optimization framework is proposed to solve nonlinear programming under uncertainty. A compact convex data-driven uncertainty set is first conducted by leveraging the combined index statistic limit technique. It can effectively capture the correlations among uncertain variables by using the principal component analysis model, and eliminate the noise in massive uncertainty data. Based on the proposed uncertainty set, linearization is taken to approximate nonlinear optimization with inequality-only constraints by using first-order Taylor approximation. By using the implicit function theorem, it is extended to a general formulation involving both inequality and equality constraints. Due to the potential limitation of first-order Taylor approximation, an iterative algorithm is designed to realize multiple linearization to search for a global robust solution under large perturbation. The efficiency of the proposed approach is verified on simulated numerical experiences, and the proposed method is applied to the industrial process of gold cyanidation leaching.