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

This article 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.

Adaptive robust unit commitment considering distributional uncertainty

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.

This study proposes a novel robust unit commitment (UC) framework with data-driven disjunctive uncertainty sets for volatile wind power generation, which integrates a two-stage adaptive robust UC model with machine learning techniques to flexibly capture the uncertainty space of the wind power forecast errors with disjunctive structures. K-means clustering and density-based spatial clustering of applications with noise (DBSCAN) are applied for clustering, and data-driven disjunctive uncertainty sets are constructed as the union of multiple basic uncertainty sets following the clustering results. 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. A tailored decomposition-based optimization algorithm is developed to facilitate the solution process. A numerical experiment and a UC case study on the IEEE 39-bus system are presented to demonstrate the effectiveness of the proposed approach. In both applications, using the proposed disjunctive uncertainty sets can effectively reduce the price of robustness compared to the conventional "one-set-fits-all" approach. Furthermore, the disjunctive uncertainty sets using DBSCAN tend to provide less conservative solutions than those using K-means, implying that DBSCAN can handle the outliers and noise of the uncertainty data more efficiently.

Unit commitment (UC), one of the critical tasks in the operations of electricity markets, is an optimization problem in power systems that determine the optimal schedule and dispatch of the generating units in the day-ahead market. UC is a challenging problem due to the many sources of uncertainty such as demand, generators’ failures, transmission lines’ outages, and more importantly, intermittent supply of renewable energy. Comparing with the other uncertainty handling approaches for UC, robust optimization is extensively used to address uncertainty in the UC problem. Robust unit commitment (RUC) results in a higher degree of flexibility and provides a stronger layer of protection against uncertainty for decision makers of the power systems. This research delves into the works done in the RUC problems to shed greater light on existing modeling approaches, definitions of uncertainty, and developed solution methods. The review of the literature reveals that the stage-based, the two-stage in particular, is a popular and effective modeling approach in the RUC problems. The stage-based modeling approach is capable of incorporating any source of uncertainty from different components of the electricity markets in its formulation, typically in the form of budgeted uncertainty sets. Furthermore, hybrid decomposition-based algorithms are tested as effective methods for handling the complexity of the RUC problems. Exploring uncertainty estimation techniques that offer more flexibility such as chance-constraint modeling is a promising research direction for the RUC problem, particularly in the presence of higher degrees of uncertainty. Failure of a generating facility could tremendously affect the reliability of the operations of the electricity markets. However, this source of uncertainty is not addressed properly in the RUC literature. In addition, stage-based modeling approach has substantial potential to handle the complexity of the RUC problems. Lastly, when the problem size increases, the complexity and computational time increase substantially, so the (meta)heuristics, either as a standalone solution method or in combination with the exact methods, are promising approaches for handling the complexity of RUC models in a reasonable time.

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

This article proposes a novel data-driven adaptive robust optimization (ARO) framework based on principal component analysis (PCA). By performing PCA on uncertainty data, the correlations among uncertain parameters are effectively captured, and principal components are identified. Uncertainty data are then projected onto each principal component, and distributional information is extracted from the projected uncertainty data using kernel density estimation. To explicitly account for asymmetric uncertainties, we introduce forward and backward deviations into uncertainty sets. The proposed data-driven ARO approach enjoys a less conservative solution compared with conventional robust optimization methods. A numerical example and an application in process network planning are presented to demonstrate the effectiveness of the proposed approach. Some promising extensions are also made within the proposed framework. Specifically, we investigate a data-driven uncertainty set in a low-dimensional subspace, and derive a theoretical bound on the performance gap between ARO solutions due to the dimension reduction of uncertainties.

This paper addresses two vital issues which are barely discussed in the literature on robust unit commitment (RUC): 1) how much the potential operational loss could be if the realization of uncertainty is beyond the prescribed uncertainty set; 2) how large the prescribed uncertainty set should be when it is used for RUC decision making. In this regard, a robust risk-constrained unit commitment (RRUC) formulation is proposed to cope with large-scale volatile and uncertain wind generation. Differing from existing RUC formulations, the wind generation uncertainty set in RRUC is adjustable via choosing diverse levels of operational risk. By optimizing the uncertainty set, RRUC can allocate operational flexibility of power systems over spatial and temporal domains optimally, reducing operational cost in a risk-constrained manner. Moreover, since impact of wind generation realization out of the prescribed uncertainty set on operational risk is taken into account, RRUC outperforms RUC in the case of rare events. Three algorithms based on column and constraint generation (C&CG) are derived to solve the RRUC. As the proposed algorithms are quite general, they can also apply to other RUC models to improve their computational efficiency. Simulations on a modified IEEE 118-bus system demonstrate the effectiveness and efficiency of the proposed methodology

In this paper, economic and environmental objectives are investigated in the unit commitment (UC) problem. Considering forecasting error of wind power, multi-objective stochastic model is formulated as the form of expected values, occurrence probabilities and confidence levels. While deterministic approaches use a point forecast of wind power output, forecasting error uncertainty in the stochastic UC alternative is captured by a confidence interval. The numerical simulation illustrates that the suitable confidence level in operating reserve constraints is related to error distribution of wind power forecast. Moreover, the potential risks of spinning reserve shortage show clearly lower on account of suitable confidence level.

This paper proposes a robust optimization formulation to solve unit commitment (UC) problems under wind energy uncertainties. Unlike the conventional stochastic programming or chance-constrained methods, this robust approach does not require information on the exact distribution of wind power. Instead, it protects the system against load loss under all possible wind generation scenarios within a straightforward uncertainty set. The level of conservatism of yielded UC decisions can be readily adjusted by the parameters of the uncertainty set. This robust UC is formulated as a two-stage problem, and the linear decision rule technique is applied to approximate the recourse decisions, so that the solution is computationally tractable. Case studies based on the IEEE Reliability Test System are conducted to demonstrate the performance of the proposed method. The results show that this method can well manage the uncertainty of wind power in UC decision-making.

Efficient solution methods for large-scale unit commitment (UC) problems have long been an important research topic and a challenge, especially in market clearing computation. For large-scale UC, the Lagrangian relaxation methods (LR) and the mixed integer programming methods (MIP) are most widely adopted. However, LR usually suffers from slow convergence; and the computational burden of MIP is heavy when the binary variable number is large. In this paper, a variable reduction method is proposed. First, the time-coupled constraints in the original UC problem are relaxed, and many single-period UC problems (s-UC) are obtained. Second, LR is used to solve the s-UCs. Different from traditional LR with iterative subgradient method, the optimal multipliers and the approximate UC solutions of the s-UCs are obtained by solving linear programs. Third, a criterion for choosing and fixing the UC variables in the UC problem is established; hence, the number of binary variables is reduced. Finally, the UC with reduced binary variables is solved to obtain the final UC solution. The proposed method is tested on the IEEE 118-bus system and a 6484-bus system. The results show the method is very efficient and effective.

This paper presents centralized control for a wind farm using model predictive control (MPC) method. In order to solve intra-day unit commitment (UC) problem, the UC problem is solved by using the MPC method with short-term wind power forecasting. We introduce a new dynamics model for the UC with some constraints to utilize the benefits of the MPC method. The objective function considering the operation and maintenance costs is formulated by adding a new variable in order to average the operating time of each wind turbine (WT) within the whole time. The proposed method could solve the UC problems on-line using the prediction time horizon that could be selected flexibly considering time horizon based on wind power forecasting errors. From the simulation study using 10 WTs, we observed that the proposed dynamics model of UC effectively provided the optimal solution to each scenario. Numerical study will validate that the proposed method can be applied to solving the UC problem of a large scale wind farm by aggregating WTs.

With the increasing proportion of wind power connected to grid, power system dispatching is facing more and more challenges from uncertainty. To cope with this uncertainty, robust optimization has been applied in unit commitment (UC) problem. In this paper, a multi‐band uncertainty set considering the temporal correlation (MBUSCTC) of wind/load prediction error is proposed firstly, which has two characteristics: (1) The MBUSCTC rigorously and realistically reflect the distribution characteristics of uncertainties in uncertainty intervals, thereby effectively reducing the conservatism of the traditional singe‐band uncertainty set; (2) the temporal correlation constraints of wind power/load prediction errors in MBUSCTC could limit the realization of uncertainties fluctuating frequently in uncertain intervals, thereby eliminating scenarios with lower probability in uncertainty sets. Then the proposed MBUSCTC is applied to UC problem, leading a robust UC model based on MBUSCTC, which is solved by Benders decomposition method and C&CG method. Finally, case studies based on the modified IEEE‐118 bus system and an actual power system of China demonstrate that the proposed method can effectively reduce the conservativeness of the robust UC model and ensure the robustness of the unit commitment solution.

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

Despite the large number of wind power forecast methods being proposed, forecasting errors are inevitable; thus, an accurate description of wind power forecast error (WPFE) is vital and is the focus of this paper. On a short-term forecasting scale, the distribution shape of the WPFE exhibits asymmetric and leptokurtic characteristics; however, common existing WPFE distribution models, such as the normal distribution, Laplace distribution and beta distribution, cannot fully describe the WPFE. This paper proposed the Johnson system to describe the WPFE, as this system has flexible skewness and kurtosis ranges and easy to implement. Based on actual WPFE data, the performance of the Johnson system is compared with those of the common distribution models proposed in the literature, and the results show that the Johnson system can represent the WPFE of all output levels of wind power and forecasting scale well.