Robust Chance Research Articles

Chance‐constrained scheduling considering frequency support from electric vehicles under multiple uncertainties

AbstractWith the share of renewable generation increasing, considerable synchronous generators have been displaced. This trend results in the decline of system inertia and thus raises the frequency secure issue. Concurrently, electric vehicles (EVs) proliferate. Numerous EVs hold a huge potential of frequency regulation in case of contingency as EVs are able to achieve primary frequency responses (PFRs) via vehicle to grid chargers. However, the value of frequency responses from aggregated EVs has not fully exploited. This paper, proposes a chance constrained scheduling model incorporating the ability of PFRs from aggregated EVs. The proposed model characterizes uncertainties associated PFR capacities from aggregated EVs and wind power forecast errors by the Wasserstein‐metric ambiguity sets, which do not rely on distributional assumption, and then formulates uncertainties‐related constraints as distributionally robust (DR) chance constraints. These DR chance constraints are reformulated as tractable linear programs. Consequently, the proposed model leads to a mixed‐integer linear program. Numerical results on a modified IEEE 39‐bus system show that incorporating PFR from EVs can reduce total costs from $4,540,396 to $4,431,233, which is reduced by 2.4%. The proposed model also achieves better reliability compared to traditional methods, demonstrating the distributionally robust optimization in enhancing electricity scheduling and rising renewable integration.

An IGDT-WDRCC based optimal bidding strategy of VPP aggregators in new energy market considering multiple uncertainties

This study addresses the volatility and uncertainty challenges in managing renewable energy within electricity markets, particularly focusing on the role of Virtual Power Plant (VPP) aggregators. Recognizing the risks these uncertainties pose to the revenue and stability of power systems, the paper presents a novel information gap decision theory (IGDT)-Wasserstein metric based distributionally robust chance constraint (WDRCC) approach to devise an optimal bidding strategy for VPP operators. It involves a data-driven distributionally robust optimization framework, leveraging the worst-case scenario from the distributed resource uncertainties, guided by an ambiguity set rooted in the Wasserstein metric. Furthermore, the distributionally robust chance constraint modeling is introduced ensuring that uncertainty constraints of distributed resources meet a predefined risk level. Although this method shows promising out-of-sample performance, it relies on forecasted energy prices, a notable limitation given the price volatility and information inadequacy in the newly-opened market. To address this, the risk-averse bidding strategy, grounded in IGDT, is proposed simulataneously to safeguard the operator’s expected returns against price uncertainties, implementing an advanced piecewise linear approximation technique, ”nf4l,” for linearizing the bi-linear term from IGDT. The effectiveness of this approach is empirically validated through a comprehensive case study and sensitivity analysis.

Distributionally robust optimal configuration of battery energy storage system considering nodal RoCoF security constraints

The large-scale integration of renewable energy source (RES) exacerbates net load fluctuations, reduces system inertia, limits frequency response capabilities, and leads to uneven spatial distribution of inertia resources. To ensure the economic and safe operation of the system, we propose a distributionally robust optimal configuration scheme of battery energy storage system (BESS) considering nodal RoCoF security constraints to address these issues. First, we describe and model the dynamic frequency response characteristics after disturbances, derive the expressions for the system frequency nadir and maximum nodal RoCoF (MN-RoCoF). Second, we address net load uncertainty using the distributionally robust chance constrained (DRCC) approach and incorporate frequency security constraints into the BESS configuration optimization model, embedding the short-term system operation model into the long-term BESS planning. Then, to tackle the highly nonlinear MN-RoCoF constraint, we propose a divide and conquer-based support vector machine (D&C-SVM), to extract the linear relationship between the BESS virtual inertia constant and MN-RoCoF, reformulating the proposed scheme into a mixed-integer second-order cone programming (MISOCP). Finally, we conducted case studies on the IEEE 39-bus and 118-bus test systems. The results verify that the proposed configuration scheme can ensure that the RoCoF at all nodes remains below 1 Hz/s under the preset disturbances. The proposed D&C-SVM demonstrate 99 % accuracy in managing the nonlinear MN-RoCoF constraint. Moreover, the uncertainty in net load is well managed, with all chance constraints satisfied with a confidence level of over 95 %.

Distributionally Robust Chance Constrained Maximum Expert Consensus Model with Incomplete Information on Uncertain Cost

Distributionally Robust Chance Constrained Maximum Expert Consensus Model with Incomplete Information on Uncertain Cost

ALSO-X#: better convex approximations for distributionally robust chance constrained programs

ALSO-X#: better convex approximations for distributionally robust chance constrained programs

A novel frequency constrained unit commitment considering VSC-HVDC's frequency support in asynchronous interconnected system under renewable energy Source's uncertainty

The increasing share of renewable energy source (RES) poses a challenge to the frequency security of the power system. Frequency constrained unit commitment (FCUC) serves as an effective measure to address this challenge at the operational level. This paper introduces a novel FCUC model applicable to the asynchronous interconnected system connected by voltage source converter based HVDC (VSCHVDC), fully taking into account the frequency support capability of VSCHVDC in order to reduce the demand for synchronous generators' inertia and reserve while ensuring frequency security, thereby lowering operating costs. Additionally, the uncertainty of RES is also considered. The constraint expressions for three frequency indicators are derived based on a system frequency response model that includes frequency support from VSCHVDC. The proposed model optimizes unit commitment, generation and reserve dispatch and VSCHVDC transmission power and frequency response parameters, while addressing RES uncertainty using the distributionally robust chance constrained approach. In response to the highly nonlinear characteristics of the maximum frequency derivation (MFD) constraints, we propose an intelligent sampling-based support vector machine to convexify the MFD constraints and introduce a two-stage decomposition algorithm for solving the model. The effectiveness of the proposed model is demonstrated based on a modified IEEE RTS-79 system.

Distributionally Robust Chance Constrained Optimization Method for Risk-Based Routing and Scheduling of Shared Mobile Energy Storage System With Variable Renewable Energy

Distributionally Robust Chance Constrained Optimization Method for Risk-Based Routing and Scheduling of Shared Mobile Energy Storage System With Variable Renewable Energy

A frequency-secured planning method for integrated electricity-heat microgrids with virtual inertia suppliers

The integrated electricity-heat microgrid (IEHM) is characterized by low inertia and a high share of renewable generation. Although sector coupling in the IEHM enhances energy efficiency, the integration of electricity and heat systems restricts the primary frequency regulation (PFR) ability of coupling equipment, further threatening frequency security. This paper presents a frequency-secured planning method for virtual inertia suppliers and combined heat and power (CHP) units in the IEHM. First, we examine the PFR in IEHMs from both device and system perspectives. We develop steady-state models for sector-coupled equipment, specifically CHP units and large-scale heat pumps, for PFR and sizing purposes. To describe the system’s frequency response under emergency conditions, we explicitly derive frequency constraints that account for varying system inertia of heterogeneous resources. Furthermore, we conduct a comprehensive analysis of the impact of PFR on heating systems within the IEHM. Second, we propose a frequency-constrained planning model for IEHMs based on the aforementioned modeling framework. This model balances the power supply, heating supply, and PFR reserves deployment of IEHM by properly sizing the regulation resources. It also leverages distributionally robust (DR) chance constraints to address uncertain wind power generation. To improve the tractability of this model, we introduce a well-tailored reformulation approach that handles the nonconvexity of system inertia and DR chance constraints. Case studies conducted on two test systems demonstrate the effectiveness of the proposed method in securing frequency stability while improving the economic performance of IEHM. This method ensures both dynamic and static frequency security across 100% of time steps by optimally deploying system inertia and PFR reserves. Moreover, by coordinating heterogeneous PFR resources with time-varying system inertia, the proposed approach yields superior economic performance, achieving over a 9% reduction in total capacity compared to average benchmarks.

The Terminator: An Integration of Inner and Outer Approximations for Solving Wasserstein Distributionally Robust Chance Constrained Programs via Variable Fixing

We present a novel approach aimed at enhancing the efficacy of solving both regular and distributionally robust chance constrained programs using an empirical reference distribution. In general, these programs can be reformulated as mixed-integer programs (MIPs) by introducing binary variables for each scenario, indicating whether a scenario should be satisfied. Whereas existing methods have focused predominantly on either inner or outer approximations, this paper bridges this gap by studying a scheme that effectively combines these approximations via variable fixing. By checking the restricted outer approximations and comparing them with the inner approximations, we derive optimality cuts that can notably reduce the number of binary variables by effectively setting them to either one or zero. We conduct a theoretical analysis of variable fixing techniques, deriving an asymptotic closed-form expression. This expression quantifies the proportion of binary variables that should be optimally fixed to zero. Our empirical results showcase the advantages of our approach in terms of both computational efficiency and solution quality. Notably, we solve all the tested instances from literature to optimality, signifying the robustness and effectiveness of our proposed approach. History: Accepted by Andrea Lodi/Design & Analysis of Algorithms — Discrete. Funding: This work was supported by Office of Naval Research [N00014-24-1-2066]; Division of Civil, Mechanical and Manufacturing Innovation [2246414]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0299 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0299 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Low‐carbon demand response program for power systems considering uncertainty based on the data‐driven distributionally robust chance constrained optimization

AbstractThe demand response (DR) program is an effective solution to promote the low‐carbon operation of power systems with increasing penetration of renewable energy sources (RESs). This paper proposes a low‐carbon DR program for power systems to enhance both the environmental friendliness and uncertainty resilience of the system operation. The system operator aims to minimize both the system's operation cost and carbon trading cost. To handle the uncertainty associated with stochastic RES generation power and load consumption power, a data‐driven method named the two‐sided distributionally robust chance‐constrained (TS‐DRCC) approach is adopted to enhance the system's uncertainty resilience. A ladder‐type carbon trading scheme is utilized to calculate the carbon emission cost of the system. Comprehensive analyses of case studies have been conducted to validate that the proposed strategy can effectively reduce the total carbon emissions and total operation costs with good uncertainty resilience performance. The proposed low‐carbon DR program is verified to achieve 63.64% more carbon emission reduction compared with the conventional DR program. Besides this, the proposed low‐carbon DR program can also achieve 4.39% carbon‐intensive generation power reduction and 5.52% RES power consumption compared with the conventional DR program.

Isolated industrial micro-grid demand response with assembly process based on distributionally robust chance constraint

In this paper, demand response for isolated industrial micro-grid is investigated, in which the typical manufacturing assembly process, battery, conventional generator and PV generation are contained. The typical manufacturing assembly process is modelled by fully considering the coupling relationships between tasks and buffers. On the other hand, the uncertainty for PV generation is taken into account, where the data-driven distributionally robust chance constraints are designed and the error between day-ahead forecast and real-time operation is dealt with by the reserve capacity of conventional generator. In addition, the distributionally robust chance constraints and the expectation operator are reformulated as linear constraints, which can be solved by commercial solver directly. Finally, a case study using lithium-ion battery factory is presented to prove the superiority and effectiveness of the proposed model.

A column generation approach for the quay crane allocation problem considering uncertain operation efficiency

This study constructs a mixed integer programming model for the integrated quay crane allocation problem to minimize vessels’ waiting time and delay time cost. Taking the uncertainty of quay crane operation efficiency into consideration, this study constructs a stochastic programming model and a distributional robust chance constraint (DRCC) model. In addition, column generation is applied to solve the three proposed models. Moreover, this paper uses real-world data from Shanghai Yangshan Port to verify the effectiveness of proposed model and algorithm. To prove the robustness of the DRCC model, this study conducts an uncertain parameters sensitivity analysis, and analyzes the performance of the models proposed above. The results show that the proposed DRCC model has better anti-interference ability. The conclusion drawn from this study has potential implications for resource allocation within port operations.

A Two-Step Approach to Wasserstein Distributionally Robust Chance- and Security-Constrained Dispatch

This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution under the Wasserstein metric. The chance constraint is imposed for all distributions in the ambiguity set to form the distributionally robust optimization problem. This problem is nonconvex and computationally heavy to solve exactly. We adopt a two-step approach to find an approximate solution. In the first step, we construct a polyhedral set in the space of uncertainty that contains enough mass under all distributions in the ambiguity set. This set is constructed by solving several two-dimensional distributionally robust problems. In the second step, we solve a linear robust optimization problem where the uncertain constraint is imposed for all uncertainty values lying in the polyhedral set. We demonstrate the scalability and robustness of our method using numerical experiments.

Condition-Based Maintenance for Wind Farms Using a Distributionally Robust Chance Constrained Program

Condition-Based Maintenance for Wind Farms Using a Distributionally Robust Chance Constrained Program

A data-driven distributionally robust chance constrained approach for optimal electricity-gas system operation

The extensive installation of gas-fired units and the substantial increase in natural gas consumption have strengthened the interdependence between power system and natural gas system. Therefore, the uncertainty of wind power brings new challenges to the safe and economic operation of the electricity–gas integrated energy system (IES). To optimize the operating costs of the energy supply side, an optimal electricity–gas energy flow (OEGEF) framework considering wind power uncertainty is established to maximize the social welfare (SW) of energy suppliers, energy storage suppliers and flexible users. Under this framework, based on the historical wind power data, a data-driven distributionally robust chance constrained (DRCC) model is proposed and the tractable reformulation form is given. In addition, a distributed manner via alternating direction method of multipliers (ADMM) is adopted to solve the power system sub-problems and the natural gas system sub-problems for protecting privacy data. Compared with the deterministic model, the uncertain model of Gaussian distribution and symmetrical distribution, the proposed model has stronger robustness. The case studies are conducted by two IESs of different scales, the results of numerical examples show that the model is effective for the optimal coordination of uncertain IES.

Improving the Flexibility of the Distribution Network Based on the Robust Chance Constraint Method

While the large-scale access to distributed new energy brings economic and environmental benefits, its inherent uncertainty characteristics also trigger problems such as voltage overruns and fluctuations. In order to effectively solve the adverse effects of large-scale grid connection of distributed new energy, this paper proposes the optimization method of regional autonomy capacity based on the robust chance constraint model. First, the regional autonomy capacity measurement method based on distribution robust optimization is studied, and the regional autonomy robust chance constraint model containing distributed energy is constructed. Second, the active distribution network scheduling model is studied to achieve the optimization of regional self-energy capacity. Finally, simulation examples show that the proposed approach can fully utilize the flexible resources within the distribution network and effectively improve the autonomy of the distribution network park.

Distributionally robust joint chance-constrained programming: Wasserstein metric and second-order moment constraints

In this paper, we propose a new approximate linear reformulation for distributionally robust joint chance programming with Wasserstein ambiguity sets and an efficient solution approach based on Benders decomposition. To provide a convex approximation to the distributionally robust chance constraint, we use the worst-case conditional value-at-risk constrained program. In addition, we derive an approach for distributionally robust joint chance programming with a hybrid ambiguity set that combines a Wasserstein ball with second-order moment constraints. This approach, which allows injecting domain knowledge into a Wasserstein ambiguity set and thus allows for less conservative solutions, has not been considered before. We propose two formulations of this problem, namely a semidefinite programming and a computationally favorable second-order cone programming formulation. The models and algorithms proposed in this paper are evaluated through computational experiments demonstrating their computational efficiency. In particular, the Benders decomposition algorithm is shown to be more than an order of magnitude faster than a standard solver allowing for the solution of large instances in a relatively short time.

Mathematical programs with distributionally robust chance constraints: Statistical robustness, discretization and reformulation

In this paper, we consider mathematical programs with distributionally robust chance constraints (MPDRCC), where the ambiguity set is given by the general moment information. From the contaminated data-driven viewpoint, we first study the qualitative statistical robustness of MPDRCC. Then, motivated by the computational tractability, we investigate the discrete approximation of MPDRCC. The corresponding convergence results of the optimal value and the optimal solution set of the discrete approximation problem are established. After that, a reformulation of the discrete approximation problem is presented under standard assumptions, which is applied to solve MPDRCC approximately according to the above convergence results. Finally, two applications are reported, and some numerical results show that the statistical robustness assertion and the discrete approximation scheme are practical and effective.

Robust optimization for spread quality and shortfall in guaranteed targeted display advertising planning

The problem addressed in this paper is the impression allocation planning of Guaranteed Targeted Display Advertising (GTDA) problem in online display advertising market. In this problem, a publisher desires (1) an even spread of impressions, i.e., ad exposures, across various audience segments, and (2) a maximum delivery of ads, i.e., fulfillment of the advertisers’ required impressions. For the publisher, the allocation of impressions needs to be appropriately planned to guarantee spread quality and delivery fulfillment simultaneously. The impression supplies are commonly assumed to be deterministic. In reality, however, they are stochastic due to the nature of unpredictability beforehand and variability with individual differences and social hotspots. We study a stochastic GTDA problem, and propose a stochastic programming model under uncertain impression supplies, where chance constraints are employed. Facing the challenge of data scarcity, we propose a novel distributional robust optimization formulation, which optimizes the spread quality, measured by Gini-based metrics, together with delivery shortfall minimization, under the worst-case probability distribution of impression supplies. To efficiently solve the problem, the distributionally robust chance constrained formulation is conservatively approximated by a parametric second-order cone program, for which an efficient interactive iteration heuristic approach is proposed. Computational experiments are conducted to show the advantages of our model, and effectiveness and robustness of our algorithm in comparison with several adapted state-of-the-art methods. Further managerial insights are also drawn.

A Wasserstein‐distance‐based distributionally robust chance constrained bidding model for virtual power plant considering electricity‐carbon trading

AbstractVirtual power plant (VPP) can aggregate distributed energy resources (DER) and controllable loads to participate in the bidding of electricity market and carbon trading market. However, the uncertainty of renewable power production brings high transaction risks to VPP. Given this background, this paper proposes a novel Wasserstein‐distance based two‐stage distributionally robust chance constrained (DRCC) bidding model for VPP participating in the electricity‐carbon coupled market. The uncertainties are modelled as an ambiguity set based on Wasserstein distance, in which the two‐sided chance constraints are guaranteed satisfied. In the first stage, VPP's revenue is maximized according to the forecast information. In the second stage, the re‐dispatch measures are determined to hedge against the perturbation of uncertainties under the worst‐case distribution within the ambiguity set. Finally, a reformulation approach based on strong duality theory and conditional value‐at‐risk (CVaR) approximation is proposed to transform the DRCC problem into a tractable mixed‐integer linear programming (MILP) framework. Case studies are carried out on the IEEE 30‐bus system to verify the effectiveness and efficiency of the proposed approach.