Data-driven Uncertainty Set Research Articles

A data-driven approach for microgrid distributed generation planning under uncertainties

The increasing demand for power system decarbonization and resilience raises the necessity of incorporating the renewable distributed generation (DG) into the microgrid planning. The complexity of the microgrid renewable DG planning largely roots from the intermittent wind and solar energy and load variations throughout the planning period. This paper proposes a novel two-stage data-driven adaptive robust distributed generation planning (DDARDGP) framework considering both grid-connected and islanded modes of microgrids, wherein the overall system cost is minimized. By leveraging the spatio-temporal property of historical weather and grid information, a compact uncertainty set is developed based on a data-driven Bayesian nonparametric approach. The problem is further solved by a modified column and constraint generation (CC&G) algorithm. In the study, the effectiveness of the proposed framework is demonstrated using a modified IEEE 33-bus test system. The case study considers the optimal generation sizing, allocation and mixtures. The simulation results confirm that the proposed data-driven uncertainty set adapts well to the increase of data dimensions and solves the over-conservatism issue, leading to 34.14% reduction in uncertainty estimation compared with the traditional budget uncertainty set. Accordingly, the total cost can achieve a $23,185 reduction under the proposed DDARDGP framework.

Unit Commitment under Uncertainty using Data-Driven Optimization with Clustering Techniques

This paper proposes a novel robust unit commitment (UC) framework with data-driven disjunctive uncertainty sets for volatile wind power outputs, assisted by machine learning techniques. To flexibly identify the uncertainty space based on wind power forecast error data with disjunctive structures, the uncertainty data are grouped using K-means and density-based spatial clustering of applications with noise following the optimal cluster number determined by the Calinski-Harabasz index. The 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. The problem is formulated into a two-stage adaptive robust UC 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 of the proposed framework is illustrated using an application on the IEEE 39-bus system. The proposed approach can reduce the price of robustness by 8-38 % compared to the conventional “one-set-fits-all” approaches. Benchmarking with stochastic programming indicates that the proposed framework can achieve the same or better economic performance with over 30 % less computational time.

Data-driven adaptive robust optimization for energy systems in ethylene plant under demand uncertainty

The operational optimization of energy systems is of great significance for improving the overall efficiency of industrial processes. Facing new challenges brought by widespread uncertainties, a data-driven adaptive robust industrial multi-type energy systems optimization framework was proposed by bridging robust optimization and machine learning methods in this paper. The industrial data were used to capture the demand uncertainty of the actual process. Hybrid models of units were first developed considering the operational characteristics, and the energy system optimization model was then formed as a mixed-integer nonlinear programming problem. The uncertain parameter set of process power demands was formed by the process models using historical data of a whole operating period. Afterward, the uncertainty set was constructed by applying the robust kernel density estimation method, which can reduce conservatism by considering the distributional information. By integrating the derived data-driven uncertainty set, a two-stage adaptive robust optimization model aiming at minimizing the weighted total energy consumption was developed. The multi-level robust optimization model was reformulated as a tractable single-level model by employing the affine decision rule. A case study on a plant-wide industrial energy system in the ethylene plant was performed, and the minimum optimal energy consumption was 25,350 kg/h, whose price of robustness was only 2.18%. The robust optimization results can guide the operational optimization of energy systems under uncertainty for the operators of the ethylene plant.

Smart greenhouse control under harsh climate conditions based on data-driven robust model predictive control with principal component analysis and kernel density estimation

Efficient greenhouse climate control under harsh climate conditions at locations such as Qatar is a challenge because of the high temperature and high relative humidity. This work presents an application of a data-driven robust model predictive control for intelligent control of greenhouse indoor climate in Qatar. The framework integrates dynamic control models of temperature, CO2 concentration level, and humidity of a greenhouse with a data-driven robust optimization framework that accurately and rigorously captures uncertainty in weather forecast error. A machine learning approach combining principal component analysis (PCA) with kernel density estimation (KDE) is adopted to construct data-driven uncertainty sets for temperature, solar radiation, and humidity from historical data. The optimal control inputs that minimize control costs and state violations are obtained by solving a data-driven robust optimization problem at each time step. The application of controlling a greenhouse growing tomatoes located in Doha, Qatar is presented. The results suggest that the proposed PCA and KDE-based data-driven robust model predictive control approach needs lower total control cost than rule-based control and other model predictive control to maintain the greenhouse climate for supporting crop production under harsh climate conditions.

Two-stage robust optimization dispatch for multiple microgrids with electric vehicle loads based on a novel data-driven uncertainty set

The uncertainty in electric power demand associated with the increasing penetration of electric vehicles can profoundly affect the stability of microgrids. Therefore, the present work addresses the need to reduce the operating cost of multi-microgrids and improve the convergence performance of the solution algorithms applied for their optimized electric power dispatch when considering the uncertainties associated with existing loads, renewable energy sources, and electric vehicle usage by proposing a novel double-layer robust optimization dispatch model. The multi-microgrid layer updates the constraints according to the number of real-time electric vehicle insertions and controls multi-type energy units to obtain the lowest operating cost under worst-case scenarios. The electric vehicle aggregator layer achieves the smallest power fluctuations and most ideal charging processes by controlling charging power while avoiding safety violations. Unrealistic worst-case scenarios are eliminated in an improved data-driven uncertainty set that is obtained from a neural network trained using a large volume of historical data. The proposed approach accurately captures the characteristics of the system, and thereby enables the replacement of a large volume of historical data with characteristic system features when formulating the uncertainty set, which simultaneously maintains high reliability and significantly reduces the convergence time. The optimization problem is then transformed into a mixed integer linear programming problem and finally solved by a column and constraint generation algorithm. The feasibility of the proposed approach and its superiority in economy and convergence performance compared to existing robust optimization methods are verified by numerical case studies.

Robust price optimization of multiple products under interval uncertainties

In this paper, we solve a multi-product price optimization problem under interval uncertainty of the price sensitivity parameter in the demand function. The objective of the problem is to maximize the revenue of the firm where the decision variables are the prices of the products supplied by the firm. We propose an approach that yields solutions that remain optimal under different variations of the estimated price sensitivity parameters. We adopt a robust optimization approach by building a data-driven uncertainty set for the parameters, and then construct a deterministic counterpart for the robust optimization model. The numerical results show that two research objectives are fulfilled: the method reflects the uncertainty embedded in parameter estimations, and also an interval is obtained for optimal prices. We also conducted a simulation study to which we compared the results of our approach. The comparisons demonstrate that although robust optimization is deemed to be conservative, the results of the proposed approach indicate little lost revenue compared to those from the simulation.

New York State's 100% renewable electricity transition planning under uncertainty using a data-driven multistage adaptive robust optimization approach with machine-learning

Abstract Power system decarbonization is critical for combating climate change, and handling systems uncertainties is essential for designing robust renewable transition pathways. In this study, a bottom-up data-driven multistage adaptive robust optimization (MARO) framework is proposed to address the power systems’ renewable transition under uncertainty. To illustrate the applicability of the proposed framework, a case study for New York State is presented. Machine learning techniques, including a variational algorithm for Dirichlet process mixture model, principal component analysis, and kernel density estimation, are applied for constructing data-driven uncertainty sets, which are integrated into the proposed MARO framework to systematically handle uncertainty. The results show that the total renewable electricity transition costs under uncertainty are 21%-42% higher than deterministic planning, and the costs under the data-driven uncertainty sets are 2%-17% lower than the conventional uncertainty sets. By 2035, on-land wind and offshore wind would be the major power source for the deterministic planning case and robust optimization cases, respectively.

Data-driven stochastic unit commitment considering commercial air conditioning aggregators to provide multi-function demand response

This paper proposes a two-stage data-driven unit commitment dispatching scheme, which coordinates the multi-function demand response of commercial air conditioning aggregators with thermal units to encounter the variations of load and wind power. In the proposed dispatching scheme, the flexible ramping capability of the thermal units is reflected in the coupling constraints between the ramping capacity and reserve capacity. To improve the flexibility of the whole system, multi-function demand response of commercial air conditioning aggregators is incorporated into the scheduling in both stages. In the day-ahead stage, load-shifting demand response reduces the ramping requirements by reshaping the load profile. In the real-time stage, reserve-capacity demand response releases the ramping capability by providing reserve for the rebalance of wind power fluctuation, which is represented by a data-driven uncertainty set. Since the dispatching scheme is formulated as a min–max-min problem, an accelerating solution method is developed to improve the computational efficiency, which combines the column-and-constraints generation algorithm and the specific property of the model. The simulation results tested on the modified IEEE 118-bus system and Henan power grid in China validate the effectiveness of the dispatching scheme and solution method in increasing flexible ramping capability, reducing wind power curtailment and load shedding, improving economic efficiency, and accelerating computation.

Data-driven robust strategies for joint optimization of rail renewal and maintenance planning

We study the problem of rail renewal and maintenance planning. The problem is to determine when and what type of maintenance tasks or rail renewal are required to be performed on different segments to maintain the rail in a safe and reliable condition. This problem is formulated as a Mixed Integer Linear Programming (MILP) model. The model applies Track Quality Index and also defines a new index to represent the current condition of the rail. Maintenance recovery effect is intrinsically uncertain; therefore, we develop data-driven uncertainty set approximation approaches and leverage robust optimization to handle the uncertainty. Data-driven uncertainty sets are constructed by approximating convex hulls of uncertain data points and by adding cutting planes to mix of classic robust uncertainty sets. We also obtained the robust counterpart formulations of the proposed MILP model for constructed uncertainty sets. Furthermore, a heuristic algorithm is developed to facilitate solving large-scale instances. Applicability and efficiency of the proposed approach are demonstrated through an illustrative case study of a Class I freight railroad network in the United States. Our analyses reveal that the proposed approaches introduce efficient strategies to deal with uncertainties in rail networks at the reasonable cost of increasing the budget.

Data-driven robust day-ahead unit commitment model for hydro/thermal/wind/photovoltaic/nuclear power systems

According to the complementary characteristics of various power sources, this paper establishes a data-driven robust day-ahead unit commitment model for a hydro-thermal-wind-photovoltaic-nuclear power system that can be used by the independent system operaters (ISOs). A data-driven robust optimization method based on the robust kernel density estimation (RKDE) is employed to deal with the uncertainties of wind and photovoltaic (PV) power. That is, the distributional information of wind and PV power is extracted by RKDE from the big data, then it is incorporated into the data-driven uncertainty set, and finally a robust optimization model is formed. In view of the facts that the conventional water spillage methods fail to pay equal attention to the benefits of the basin and the individual hydro plants, eight kinds of strategies that can make the water spillages or hydropower curtailments distributed proportionally in each hydro plant are proposed. In addition, the operating model of nuclear power unit involved in peak load regulation is established to promote its operational flexibility. The numerical results and simulation on a modified New England 39-bus system verify the superiority and practicability of the proposed model.

Data Driven Robust Energy and Reserve Dispatch Based on a Nonparametric Dirichlet Process Gaussian Mixture Model

Traditional robust optimization methods use box uncertainty sets or gamma uncertainty sets to describe wind power uncertainty. However, these uncertainty sets fail to utilize wind forecast error probability information and assume that the wind forecast error is symmetrical and independent. This assumption is not reasonable and makes the optimization results conservative. To avoid such conservative results from traditional robust optimization methods, in this paper a novel data driven optimization method based on the nonparametric Dirichlet process Gaussian mixture model (DPGMM) was proposed to solve energy and reserve dispatch problems. First, we combined the DPGMM and variation inference algorithm to extract the GMM parameter information embedded within historical data. Based on the parameter information, a data driven polyhedral uncertainty set was proposed. After constructing the uncertainty set, we solved the robust energy and reserve problem. Finally, a column and constraint generation method was employed to solve the proposed data driven optimization method. We used real historical wind power forecast error data to test the performance of the proposed uncertainty set. The simulation results indicated that the proposed uncertainty set had a smaller volume than other data driven uncertainty sets with the same predefined coverage rate. Furthermore, the simulation was carried on PJM 5-bus and IEEE-118 bus systems to test the data driven optimization method. The simulation results demonstrated that the proposed optimization method was less conservative than traditional data driven robust optimization methods and distributionally robust optimization methods.

Data-Driven Robust Optimization for Greenhouse Temperature Control Using Model Predictive Control

This work proposes a novel data-driven robust model predictive control (DDRMPC) framework for automatic control of greenhouse temperature and CO2 concentration level. The essential concept is to combine dynamic models of greenhouse temperature and CO2 concentration level with data-driven models that identify uncertainty in weather forecast error. By leveraging a machine learning approach, support vector clustering with weighted generalized intersection kernel, data-driven uncertainty sets for ambient temperature and solar radiation are constructed from historical weather data. A training-calibration procedure that tunes the size of uncertainty sets is implemented to ensure that data-driven uncertainty sets attain appropriate performance guarantee. In order to solve the optimization problem in DDRMPC, an affine disturbance feedback policy that provides tractable approximations of optimal control is utilized. A case study of controlling temperature and CO2 concentration level in a greenhouse is carried out. The results show that the proposed DDRMPC framework can prevent the greenhouse climate from becoming harmful to plant and fruit. DDRMPC approach ends up with 20 % less total economic cost than rule-based control strategy. The proposed DDRMPC approach also gives better control performance comparing to certainty equivalent MPC and robust MPC.

Robust Model Predictive Control of Irrigation Systems With Active Uncertainty Learning and Data Analytics

We develop a novel data-driven robust model predictive control (DDRMPC) approach for automatic control of irrigation systems. The fundamental idea is to integrate both mechanistic models, which describe dynamics in soil moisture variations, and data-driven models, which characterize uncertainty in forecast errors of evapotranspiration and precipitation, into a holistic systems control framework. To better capture the support of uncertainty distribution, we take a new learning-based approach by constructing uncertainty sets from historical data. For the evapotranspiration forecast error, the support vector clustering-based uncertainty set is adopted, which can be conveniently built from historical data. As for precipitation forecast errors, we analyze the dependence of their distribution on forecast values, and further design a tailored uncertainty set based on the properties of this type of uncertainty. In this way, the overall uncertainty distribution can be elaborately described, which finally contributes to rational and efficient control decisions. To assure the quality of data-driven uncertainty sets, a training-calibration scheme is used to provide theoretical performance guarantees. A generalized affine decision rule is adopted to obtain tractable approximations of optimal control problems, thereby ensuring the practicability of DDRMPC. Case studies using real data show that DDRMPC can reliably maintain soil moisture above the safety level and avoid crop devastation. The proposed DDRMPC approach leads to a 40% reduction of total water consumption compared to the fine-tuned open-loop control strategy. In comparison with the carefully tuned rule-based control and certainty equivalent MPC, the proposed DDRMPC approach can significantly reduce the total water consumption and improve the control performance.

Data-Driven Adaptive Robust Unit Commitment Under Wind Power Uncertainty: A Bayesian Nonparametric Approach

This paper proposes a novel data-driven adaptive robust optimization (ARO) framework for the unit commitment (UC) problem integrating wind power into smart grids. By leveraging a Dirichlet process mixture model, a data-driven uncertainty set for wind power forecast errors is constructed as a union of several basic uncertainty sets. Therefore, the proposed uncertainty set can flexibly capture a compact region of uncertainty in a nonparametric fashion. Based on this uncertainty set and wind power forecasts, a data-driven adaptive robust UC problem is then formulated as a four-level optimization problem. A decomposition-based algorithm is further developed. Compared to conventional robust UC models, the proposed approach does not presume single mode, symmetry, or independence in uncertainty. Moreover, it not only substantially withstands wind power forecast errors, but also significantly mitigates the conservatism issue by reducing operational costs. We also compare the proposed approach with the state-of-the-art data-driven ARO method based on principal component analysis and kernel smoothing to assess its performance. The effectiveness of the proposed approach is demonstrated with the six-bus and IEEE 118-bus systems. Computational results show that the proposed approach scales gracefully with problem size and generates solutions that are more cost effective than the existing data-driven ARO method.

Data‐driven multi‐time scale robust scheduling framework of hydrothermal power system considering cascade hydropower station and wind penetration

A data-driven multi-time scale robust scheduling model for a wind–hydro–thermal power system is presented in this study, according to the characteristic that wind power prediction accuracy increases with the decrease of the time scale. In day-ahead scheduling, a generation plan is formulated with the target of minimising the total operating cost, and a data-driven robust optimisation method based on the robust kernel density estimation (RKDE) is employed to deal with the uncertainty of wind power. That is, the distributional information of wind power is extracted by the RKDE from the big data, then the distributional information is incorporated into a data-driven uncertainty set, and finally, a robust optimisation model is formed. During the intraday scheduling stage, the objective is to minimise the total water spillage in cascade hydropower stations and the adjustment cost of thermal units, and the task is to readjust the outputs of units based on the base outputs obtained by the day-ahead scheduling, combined with the rolling forecast data of wind power and load. The real-time scheduling is aimed at satisfying the power balance with minimum power adjustment. Finally, a test system is carried out to verify the efficiency and practicability of the proposed framework.

A data-driven robust optimization approach to scenario-based stochastic model predictive control

Stochastic model predictive control (SMPC) has been a promising solution to complex control problems under uncertain disturbances. However, traditional SMPC approaches either require exact knowledge of probabilistic distributions, or rely on massive scenarios that are generated to represent uncertainties. In this paper, a novel scenario-based SMPC approach is proposed by actively learning a data-driven uncertainty set from available data with machine learning techniques. A systematical procedure is then proposed to further calibrate the uncertainty set, which gives appropriate probabilistic guarantee. The resulting data-driven uncertainty set is more compact than traditional norm-based sets, and can help reducing conservatism of control actions. Meanwhile, the proposed method requires less data samples than traditional scenario-based SMPC approaches, thereby enhancing the practicability of SMPC. Finally the optimal control problem is cast as a single-stage robust optimization problem, which can be solved efficiently by deriving the robust counterpart problem. The feasibility and stability issue is also discussed in detail. The efficacy of the proposed approach is demonstrated through a two-mass-spring system and a building energy control problem under uncertain disturbances.

Reprint of: Data-driven robust optimization under correlated uncertainty: A case study of production scheduling in ethylene plant

To hedge against the fluctuations generated from continuous production processes, practical solutions can be obtained through robust optimization induced by the classical uncertainty sets. However, uncertainties are sometimes correlated in industrial scheduling problems because of the connected process and various random factors. To capture and enrich the valid information of uncertainties, copulas are introduced to estimate the joint probability distribution and simulate mutual scenarios for uncertainties. Cutting planes are generated to remove unnecessary uncertain scenarios in the uncertainty sets, and then robust formulations induced by the cut set are proposed to reduce conservatism and improve the robustness of scheduling solutions. A real-world process of ethylene plant is introduced as the numerical case, and high-dimensional data-driven uncertainty sets are illustrated in detail. The proposed models are proved to control the fluctuation of consumed fuel gas below a lower level of conservatism.

Robust optimization: Lessons learned from aircraft routing

Building robust airline scheduling models involves constructing schedules and routes with reduced levels of flight delays as well as fewer passenger and crew disruptions. In this paper, we study different classes of models to achieve robust airline scheduling solutions, with a focus on the aircraft routing problem. In particular, we compare one domain-specific approach and two general paradigms of robust models, namely, (i) an extreme-value based or robust optimization-based approach, and (ii) a chance-constrained or stochastic optimization-based approach. Our modeling and solution approach demonstrates the creation of data-driven uncertainty sets for aircraft routing using domain-specific knowledge and develops a completely data-driven simulation-based validation and testing approach. We first demonstrate that additional modeling, capturing domain knowledge, is required to adapt these general robust modeling paradigms to the aircraft routing problem, in order to meaningfully add robustness features specific to aircraft routing. However, we show that these models in their naive forms, still face issues of tractability and solution quality for the large-scale networks which are representative of real-world airline scheduling problems. Therefore, we develop and present advanced models that address these shortcomings. Our advanced models can be applied to aircraft routing in multiple ways, through varied descriptions of the uncertainty sets; and moreover, are generally applicable to linear and binary integer programming problems. Through our detailed computational results, we compare the performance of solutions arising from these different robust modeling paradigms and discuss the underlying reasons for their performance differences from a data-driven perspective.

Data-driven decision making under uncertainty integrating robust optimization with principal component analysis and kernel smoothing methods

This paper proposes a novel data-driven robust optimization framework that leverages the power of machine learning and big data analytics for decision-making under uncertainty. By applying principal component analysis to uncertainty data, correlations between uncertain parameters are effectively captured, and latent uncertainty sources are identified. These data are then projected onto each principal component to facilitate extracting distributional information of latent uncertainties using kernel density estimation techniques. To explicitly account for asymmetric distributions, we introduce forward and backward deviation vectors into the data-driven uncertainty set, which are further incorporated into novel data-driven static and adaptive robust optimization models. The proposed framework not only significantly ameliorates the conservatism of robust optimization, but also enjoys computational efficiency and wide-ranging applicability. Three applications on optimization under uncertainty, including model predictive control, batch production scheduling, and process network planning, are presented to demonstrate the applicability of the proposed framework.

Robust Optimization in High-Dimensional Data Space with Support Vector Clustering

Data-driven robust optimization has attracted immense attentions. In this work, we propose a data-driven uncertainty set for robust optimization under high-dimensional uncertainty. We propose to first decompose the high-dimensional data space into the principal subspace and the residual subspace by employing principal component analysis, and then adopt support vector clustering and classic polyhedral uncertainty set to describe the intricate geometry in the principal subspace and the tiny variations in the residual subspace, respectively, giving rise to a new data-driven uncertainty set. Similar to classic uncertainty sets, the proposed data-driven uncertainty set can also preserve the tractability of robust optimization problems. In addition, we establish the probabilistic guarantee theoretically by further calibrating the uncertainty set with an independent dataset, which ensures that the data-driven uncertainty set covers a portion of uncertainty with a given confidence level. Numerical results show the effectiveness of the proposed uncertainty set in reducing conservatism of robust optimization problems as well as the fidelity of the established probabilistic guarantee.