Robust Kernel Density Estimation Research Articles

Data-driven source-load robust optimal scheduling of integrated energy production unit including hydrogen energy coupling

A robust low-carbon economic optimal scheduling method that considers source-load uncertainty and hydrogen energy utilization is developed. The proposed method overcomes the challenge of source-load random fluctuations in integrated energy systems (IESs) in the operation scheduling problem of integrated energy production units (IEPUs). First, to solve the problem of inaccurate prediction of renewable energy output, an improved robust kernel density estimation method is proposed to construct a data-driven uncertainty output set of renewable energy sources statistically and build a typical scenario of load uncertainty using stochastic scenario reduction. Subsequently, to resolve the problem of insufficient utilization of hydrogen energy in existing IEPUs, a robust low-carbon economic optimal scheduling model of the source-load interaction of an IES with a hydrogen energy system is established. The system considers the further utilization of energy using hydrogen energy coupling equipment (such as hydrogen storage devices and fuel cells) and the comprehensive demand response of load-side schedulable resources. The simulation results show that the proposed robust stochastic optimization model driven by data can effectively reduce carbon dioxide emissions, improve the source-load interaction of the IES, realize the efficient use of hydrogen energy, and improve system robustness.

Wind Power Interval Prediction Based on Robust Kernel Density Estimation

Wind power output has a high degree of randomness, so it is difficult to describe it accurately with some typical probability distribution. The extreme values influence the sample’s general non-parametric kernel density estimation method, and the estimated results are relatively conservative. A wind power interval prediction method based on robust kernel density estimation is proposed to improve the compactness and accuracy of interval prediction. In probability estimation, this method will assign a small weight to the extreme sample data to reduce its influence on the probability density estimation accuracy and has better robustness than the general kernel density estimation method. In addition, the bandwidth of robust kernel density estimation is a dynamic parameter, which can be adjusted with the sample data to improve the prediction accuracy and avoid over-conservative prediction.

Privacy-preserving energy trading management in networked microgrids via data-driven robust optimization assisted by machine learning

Trading mechanism design, uncertainty treatment and privacy protection are the main issues in the energy management of networked microgrids (MGs). To address these issues in a comprehensive manner, this paper proposes a data-driven two-level transactive energy management framework, where the upper-level determines the optimal strategies of internal scheduling within MGs and external trading between MGs while the lower-level formulates a Nash bargaining game model for the fair allocation of trading benefits. The uncertainties of renewable energy sources are fully captured by an adjustable data-driven robust optimization approach with an uncertainty set constructed using the robust kernel density estimation (RKDE) as a machine learning technique. The resulting uncertainty set can provide robust scheduling and trading schemes even when the power generation data of wind turbines and photovoltaic systems are contaminated with anomalous samples, whereas conventional sets are not reliable in the case of contaminated data. To preserve the operational independence and information privacy of MGs, the proposed model is solved in a distributed manner by the alternating direction method of multipliers (ADMM) and the augmented Lagrange-based alternating direction inexact Newton (ALADIN) algorithms. ADMM is commonly used in previous studies, but it has low computational efficiency for handling the consensus process among a large number of MGs. This paper applies ADMM and ALADIN to enhance the applicability of the proposed model when the size and complexity of the networks increase. Numerical tests show the effectiveness of the proposed framework and solution methodology in terms of system cost, solution robustness, and convergence speed.

Data-driven two-stage distributionally robust optimization for refinery planning under uncertainty

This work investigates the refinery planning problem under uncertainty in product prices. A novel data-driven Wasserstein distributionally robust optimization framework is proposed for handling uncertainties in the refinery-wide planning operations. A data-driven ambiguity set is constructed based on the Wasserstein metric to model the distributional uncertainty. The robust kernel density estimation technique is adopted to establish the support set to reduce the effect of the potential outliers. Based on the derived support set and ambiguity set, a data-driven two-stage distributionally robust optimization model for refinery planning is developed. Then, the robust counterpart of the proposed model is formulated to make the problem computationally tractable. Finally, a real-world case study on a petroleum refinery is presented to illustrate the effectiveness and applicability of the proposed framework.

Multi-scenario data-driven robust optimisation for industrial steam power systems under uncertainty

In actual industrial production, the deterministic optimisation of the steam power system cannot meet most production scenarios due to the influence of uncertain factors such as product demand and environmental conditions. This paper proposes an operational optimisation method for SPS under uncertainty by combining multi-scenario partition and data-driven adaptive robust optimisation algorithm. A hybrid equipment model was developed to modify the critical equipment models based on industrial data and process mechanisms. Considering that demand uncertainty varies with the different steam quality, the clustering method divides the entire time horizon into several periods, and the uncertainty set is constructed by variable robust kernel density estimation for each period. A multi-scenario data-driven robust optimisation model is developed by incorporating uncertainty sets into deterministic optimisation, and the counterpart model is obtained through the affine decision rules. Furthermore, the proposed framework is applied to the SPS of a coal chemical plant to verify the feasibility. The annual operating costs before and after optimisation are 125 million USD and 123 million USD, respectively, and the system's energy efficiency can be improved by more than 5%.

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.

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.

Diesel blending under property uncertainty: A data-driven robust optimization approach

With the increasing demand for diesel fuel and the strict diesel standards, diesel blending has become an essential technology in response to clean energy. However, in the traditional blending process, the operator does not consider the uncertainty of component oil properties during the recipe optimization, resulting in a sub-optimal or even infeasible solution. This paper proposes a data-driven robust optimization framework to address this issue. First, a hybrid machine learning method combining principal component analysis and robust kernel density estimation is used to construct uncertainty sets to capture uncertain properties. Then, a data-driven diesel blending model is formulated using the derived uncertainty set through the dual operation. Last, an actual case study is implemented to show the effectiveness of the proposed method in handling uncertainties and obtaining a good balance between robustness and optimality of the recipe optimization for diesel blending. Moreover, the parameters of the uncertainty sets are analyzed in detail for providing reasonable parameters to guide the actual diesel blending.

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.

Towards smart transportation: A learning‐based data‐driven optimization approach for electric taxi dispatch problem

Electric taxi dispatch problem (ETDP) is one of the key issues in smart transportation. Existing study in the context of centralized optimization adopts either deterministic optimization, regular stochastic programming (SP) or simulation technique. Nevertheless, in data‐driven environment, the real passenger demands normally follow complicating probability distribution which cannot be described exactly by the parametric approaches. Hence, we propose a novel data‐driven optimization framework that integrates robust kernel density estimation (RKDE) and the two‐stage SP modeling technique. In particular, the probability distributions of customer demands are derived from historical data by RKDE, and the ETDP is formulated as a two‐stage SP model with the input parameters from RKDE. Meanwhile, a Monte Carlo method called sample average approximation is introduced to reformulate and solve the SP model. Finally, the experimental results show that the proposed approach outperforms the deterministic counterpart with the average demands as the input.

Doubly robust kernel density estimation when group membership is missing at random

When there are subjects with subpopulation memberships missing, the kernel density estimates of the subpopulations based on the subjects with verified memberships may not be valid unless the missingness of the memberships satisfies the stringent missing completely at random assumption. Under more plausible missing at random (MAR) assumptions, the inverse probability weighting (IPW) kernel smoothing method can be employed if the missing mechanism is well understood. On the other hand, if the memberships can be well predicted by a prediction model, one may replace the memberships (either all or only the missing ones), with the probabilities for the memberships to deal with the missing values. By integrating the information of the missing mechanism and the prediction model, we develop a doubly robust kernel density estimate which is valid as long as either of the models is correctly specified. Asymptotic properties of the new estimates are derived and the asymptotic orders of the mean integrated square errors (MISE) are compared with existing methods. We prove that the doubly robust estimate can be more efficient than the IPW estimate when the model for the missing mechanism is correctly specified. We also prove that the prediction-model based approaches may have smaller MISEs than the complete data when the prediction model is correctly specified, suggesting that those methods can be applied even when there are no missing values in the memberships. Simulation studies are carried out to assess the performances of methods and a real study example is used for illustrative purposes.

Data-driven robust optimization for crude oil blending under uncertainty

Optimization of crude oil blending helps improve the operating efficiency of refineries. However, widespread uncertainties, such as oil properties, bring difficulty in realizing this task. A data-driven robust optimization (DDRO) approach is proposed to optimize crude oil blending under uncertainty. First, the blending effect model is used to extract uncertainties of oil components from production data by recursive least squares method. Second, the uncertainty set is constructed by combining principle component analysis and robust kernel density estimation based on the historical data of blending effects. A novel data-driven robust model for recipe optimization of crude oil blending is developed by utilizing the obtained uncertainty set. The dual transformation is applied to derive the linear counterpart of the DDRO model. A case study is adopted to illustrate the effectiveness of the proposed 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.

Operational optimization of industrial steam systems under uncertainty using data‐Driven adaptive robust optimization

This article addresses the operational optimization of industrial steam systems under device efficiency uncertainty using a data‐driven adaptive robust optimization approach. A semiempirical model of steam turbine is first developed based on process mechanism and operational data. Uncertain parameters of the proposed steam turbine model are further derived from the historical process data. A robust kernel density estimation method is then used to construct the uncertainty sets for modeling these uncertain parameters. The data‐driven uncertainty sets are incorporated into a two‐stage adaptive robust mixed‐integer linear programming (MILP) framework for operational optimization of steam systems to minimize the total operating cost. Integer variables are introduced to model the on/off decisions of the steam turbines and electrical motors, which are the major energy consumers of the steam system. By applying the affine decision rule, the proposed multilevel optimization model is transformed into its robust counterpart, which is a single‐level MILP problem. The proposed framework is applied to the steam system of a real‐world ethylene plant to demonstrate its applicability. © 2018 American Institute of Chemical Engineers AIChE J, 65: e16500 2019

A data‐driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty

A novel data‐driven approach for optimization under uncertainty based on multistage adaptive robust optimization (ARO) and nonparametric kernel density M‐estimation is proposed. Different from conventional robust optimization methods, the proposed framework incorporates distributional information to avoid over‐conservatism. Robust kernel density estimation with Hampel loss function is employed to extract probability distributions from uncertainty data via a kernelized iteratively reweighted least squares algorithm. A data‐driven uncertainty set is proposed, where bounds of uncertain parameters are defined by quantile functions, to organically integrate the multistage ARO framework with uncertainty data. Based on this uncertainty set, we further develop an exact robust counterpart in its general form for solving the resulting data‐driven multistage ARO problem. To illustrate the applicability of the proposed framework, two typical applications in process operations are presented: The first one is on strategic planning of process networks, and the other one on short‐term scheduling of multipurpose batch processes. The proposed approach returns 23.9% higher net present value and 31.5% more profits than the conventional robust optimization method in planning and scheduling applications, respectively. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4343–4369, 2017

Distributed Channel Allocation Using Kernel Density Estimation in Cognitive Radio Networks

Typical channel allocation algorithms for secondary users do not include processes to reduce the frequency of switching from one channel to another caused by random interruptions by primary users, which results in high packet drops and delays. In this letter, with the purpose of decreasing the number of switches made between channels, we propose a nonparametric channel allocation algorithm that uses robust kernel density estimation to effectively schedule idle channel resources. Experiment and simulation results demonstrate that the proposed algorithm outperforms both random and parametric channel allocation algorithms in terms of throughput and packet drops.