Decision-dependent Uncertainty Research Articles

Improve operational flexibility of distribution systems using transportable resources

With the continuous progress of carbon neutrality, the accelerating integration of distributed renewable energy sources (RES) poses tremendous pressure on distribution systems' safe and stable operation. The distribution systems that cannot be upgraded in time, e.g., those in border areas, are hard to adapt to the large-scale integration of RES and the growing energy demand, facing a severe lack of operational flexibility. In light of this, the integrated mobile energy station (IMES), which consists of a battery system and photovoltaic cells, is used to improve operational flexibility and alleviate the need for upgrades. An adaptive robust allocation and dispatch model is introduced to coordinate the operation of the distribution system and IMES under multiple uncertainties. The first stage of the proposed model optimizes the locations of IMES using the forecast information of RES and loads, while the second stage optimizes the dispatch plan that is adaptive to the realization of uncertainties. Since the uncertain photovoltaic power in the IMES depends on its deployment location, the proposed model is an adaptive robust optimization with decision-dependent uncertainty, bringing great challenges to the solution. To solve it efficiently, a warmstart-enhanced column-and-constraint generation algorithm is introduced, in which the bi-level subproblem is solved by the dual method and outer approximation algorithm. Finally, case studies demonstrate the effectiveness of the proposed method. The participation of six IMESs can save 5.23 % on the system's total cost and avoid the load shedding caused by transmission congestion, indicating IMES's potential for economical and flexible operation of the system.

Decision-focused learning under decision dependent uncertainty for power systems with price-responsive demand

Contemporary power systems are experiencing a growing integration of flexible resources on the consumer side. Different from flexible demand that submits specific bids to energy markets, price-responsive demand (PRD) adjusts its power consumption without notice, simply based on the resulting electricity prices as well as internal priorities and limitations. In this paper, we demonstrate how a high penetration of PRD (the behavior of which is invisible to the operator during the day-ahead unit commitment stage) results in systematic inefficiency costs and formulate the so-termed decision-focused learning problem of learning to provide a demand forecast which, once fed as an input to the operator’s economic dispatch optimization problem, results in an efficient dispatch. Interestingly, the prescribed demand forecast affects the resulting prices, which in turn affect the actual demand realization, giving rise to a decision-dependent uncertainty. Motivated by the problem’s hard-to-evaluate objective function, we solve it using Bayesian optimization. The empirical evaluations demonstrate significant savings in the effective real-time system cost, compared to the current practice of using the default demand forecast. Moreover, the method is shown to achieve a system cost that is fairly close to the one achieved by a system that fully integrates PRD into the day-ahead process; but without requiring any change in the operator’s existing dispatch algorithm while avoiding all efforts necessary for the integration of flexible demand, which is a widely pursued field of ongoing research.

Uncertainty reduction in robust optimization

Uncertainty reduction has recently been introduced in the robust optimization literature as a relevant special case of decision-dependent uncertainty. Herein, we identify two relevant situations in which the problem is polynomially solvable. We further provide insights into possible MILP reformulations and the strength of their continuous relaxations.

A robust optimization method for power systems with decision‐dependent uncertainty

AbstractRobust optimization is an essential tool for addressing the uncertainties in power systems. Most existing algorithms, such as Benders decomposition and column‐and‐constraint generation (C&CG), focus on robust optimization with decision‐independent uncertainty (DIU). However, increasingly common decision‐dependent uncertainties (DDUs) in power systems are frequently overlooked. When DDUs are considered, traditional algorithms for robust optimization with DIUs become inapplicable. This is because the previously selected worst‐case scenarios may fall outside the uncertainty set when the first‐stage decision changes, causing traditional algorithms to fail to converge. This study provides a general solution algorithm for robust optimization with DDU, which is called dual C&CG. Its convergence and optimality are proven theoretically. To demonstrate the effectiveness of the dual C&CG algorithm, we used the do‐not‐exceed limit (DNEL) problem as an example. The results show that the proposed algorithm can not only solve the simple DNEL model studied in the literature but also provide a more practical DNEL model considering the correlations among renewable generators.

Demand Steering in a Last-Mile Delivery Problem with Home and Pickup Point Delivery Options

To increase the efficiency of last-mile delivery, online retailers can adopt pickup points in their operations. The retailer may then incentivize customers to steer them from home to pickup point delivery to reduce costs. However, it is usually uncertain whether the customer accepts this incentive to switch to pickup delivery. This setup gives rise to a new last-mile delivery problem with integrated incentive and routing decisions under uncertainty. We model this problem as a two-stage stochastic program with decision-dependent uncertainty. In the first stage, a retailer decides which customers to incentivize. However, customers’ reaction to the incentive is stochastic: they may accept the offer and switch to pickup point delivery, or they may decline the offer and stick with home delivery. In the second stage, after customers’ final delivery choices are revealed, a vehicle route is planned to serve customers via the delivery option of their choice. We develop an exact branch-and-bound algorithm and propose several heuristics to improve the algorithm’s scalability. Our algorithm solves instances with up to 50 customers, realizing on average 4%–8% lower last-mile delivery costs compared with the commonly applied approaches in the industry that do not use incentives or offer incentives to all customers. We also develop a benchmark policy that gives very fast solutions with a 2% average optimality gap for small instances and up to 2% average cost increase compared with the heuristic solutions. Funding: This project received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement [Grant 765395]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0287 .

Valuation of distributed predictive information in robust economic dispatch

Robust economic dispatch is an essential way to deal with the uncertainties of renewable generations, but its performance highly depends on the pre-built uncertainty set. This paper proposes a novel robust economic dispatch model in which the operator can buy predictions from distributed forecasters to build a better uncertainty set and enhance its dispatch efficiency. The value of distributed predictive information can be quantified by the operator’s payment for buying it. The proposed model renders a two-stage robust optimisation problem with decision-dependent uncertainty (DDU). By analysing the structure of the uncertainty set, the proposed model is equivalently transformed into a two-stage robust optimisation problem with decision-independent uncertainty (DIU) so that it can be effectively solved by applying the column-and-constraint generation (C&CG) algorithm. Case studies show that the proposed method is effective.

Distributed Energy Resource and Energy Storage Investment for Enhancing Flexibility Under a TSO-DSO Coordination Framework

This paper presents a distributed energy resource and energy storage investment method under a coordination framework between transmission system operators (TSOs) and distribution system operators (DSOs), which simultaneously addresses two main aspects of the flexibility aggregation of DSOs, i.e., flexibility enhancement and dynamic flexibility provision. First, to characterize the key flexibility features of power distribution networks, a multi-port multi-period feasible region formulation is designed using the robust optimization conception while avoiding over-parameterized formulations that require extensive information exchange. Second, a two-stage robust planning model is built to enhance distribution system flexibility by maximizing the volume of feasible regions, in which the uncertainty of dispatch instructions is modeled as a decision-dependent uncertainty (DDU) set and the worst realization is identified for the operational security evaluation. The proposed two-stage robust planning model is solved via a customized column-and-constraint generation algorithm. Finally, a distributed framework for TSO-DSO coordination is proposed to enable the dynamic adjustment of feasible region provision of DSO, given the TSO’s preference, which is then solved by a DDU-based two-stage robust extension of the alternating direction method of multipliers algorithm. Numerical results verify the effectiveness of our proposed models and the scalability of the associated algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The increasing integration of renewable energy resources has stimulated the need for aggregated flexibility provided by power distribution networks (PDNs). However, there are three challenges: i) using the currently feasible region formulation of PDN requires extensive information exchange with the transmission system operator (TSO), thus not compatible with the current structure. ii) the optimal investment of various assets to leverage their cost-effectiveness in feasible region enhancement needs to be addressed. iii) lack of a proper coordination mechanism between TSO and distribution system operators (DSOs) to facilitate the optimal feasible region provision and leverage the spatiotemporal flexibility. To address the above challenges, we design an easy-to-implement feasible region formulation to characterize the key flexibility features of PDNs. Upon this, we propose a feasible region enhancement planning method based on an extended two-stage robust model to address the decision-dependent uncertainty. Furthermore, we facilitate the optimal feasible region provision through the TSO-DSO coordination process in a distributed manner. The numerical results show that the proposed planning method and coordination mechanism could effectively achieve optimal flexibility enhancement and dynamic flexibility provision. In practical application, our proposed planning method can be readily implemented with advanced analytical tools. The coordination mechanism is also compatible with the current TSO-DSO structure and can be easily integrated into it.

Frequency Reserve Allocation of Large-Scale RES Considering Decision-Dependent Uncertainties

Frequency Reserve Allocation of Large-Scale RES Considering Decision-Dependent Uncertainties

Profit-based unit commitment models with price-responsive decision-dependent uncertainty

Highlighting the increasing importance of demand elasticity in electricity markets and its impact on the revenues of power generating companies, this paper proposes new profit-based unit commitment models that effectively capture the uncertainty in the willingness-to-pay the price set for the elastic demand. To develop a new revenue scheme for power generating companies, we use a coupling function to model the willingness-to-pay response of the elastic demand as a decision-dependent source of uncertainty. The coupling function reflects how power generating companies’ pricing decisions may influence the market appeal (i.e., the buyer’s willingness-to-pay) and how it affects their revenues. The optimization models are stochastic mixed-integer nonlinear problems with nonconvex continuous relaxations and are not amenable to a numerical solution in their original forms. We devise a convexification reformulation method and derive valid inequalities to strengthen the formulation. We propose a learning framework to parameterize the willingness-to-pay functions and the concept of the value of the decision-dependent solution to quantify the value of the uncertainty modeling approach. Numerical tests on power systems of various sizes, demand portfolios, and price elasticity levels show (i) how the valid inequalities speed up the solution process, (ii) the benefits of properly modeling decision-dependent uncertainty and demand elasticity, and (iii) how the incorporation of decision-dependent uncertainty in demand elasticity can change the power generating companies’ decisions and revenue estimation.

Chance-Constrained Generic Energy Storage Operations Under Decision-Dependent Uncertainty

Compared with large-scale physical batteries, aggregated and coordinated generic energy storage (GES) resources provide low-cost, but uncertain, flexibility for power grid operations. While GES can be characterized by different types of uncertainty, the literature mostly focuses on decision-independent uncertainties (DIUs), such as exogenous stochastic disturbances caused by weather conditions. Instead, this manuscript focuses on newly-introduced decision-dependent uncertainties (DDUs) and considers an optimal GES dispatch that accounts for uncertain available state-of-charge (SoC) bounds that are affected by incentive signals and discomfort levels. To incorporate DDUs, we present a novel chance-constrained optimization (CCO) approach for the day-ahead economic dispatch of GES units. Two tractable methods are presented to solve the proposed CCO problem with DDUs: (i) a robust reformulation for general but incomplete distributions of DDUs, and (ii) an iterative algorithm for specific and known distributions of DDUs. Furthermore, reliability indices are introduced to verify the applicability of the proposed approach with respect to the reliability of the response of GES units. Simulation-based analysis shows that the proposed methods yield conservative, but credible, GES dispatch strategies and reduced penalty cost by incorporating DDUs in the constraints and leveraging data-driven parameter identification. This results in improved availability and performance of coordinated GES units.

Robust Optimization with Continuous Decision-Dependent Uncertainty with applications to demand response management

Robust Optimization with Continuous Decision-Dependent Uncertainty with applications to demand response management

Robust Generation Dispatch With Strategic Renewable Power Curtailment and Decision-Dependent Uncertainty

As renewable energy sources replace traditional power sources (such as thermal generators), uncertainty grows while there are fewer controllable units. To reduce operational risks and avoid frequent real-time emergency controls, a preparatory schedule of renewable generation curtailment is required. This paper proposes a novel two-stage robust generation dispatch (RGD) model, where the preparatory curtailment threshold is optimized in the pre-dispatch stage. The curtailment threshold will then influence the variation range of real-time renewable power outputs, resulting in a decision-dependent uncertainty (DDU) set. In the re-dispatch stage, the controllable units adjust their outputs within their reserve contributions to maintain power balancing. To overcome the difficulty in solving the RGD with DDU, an adaptive column-and-constraint generation (AC&CG) algorithm is developed. We prove that the proposed algorithm can generate the optimal solution in a finite number of iterations. Numerical studies show the advantages of the proposed model and algorithm and validate their practicability and scalability.

Modeling the Reserve Capacity of Wind Power and the Inherent Decision-Dependent Uncertainty in the Power System Economic Dispatch

The ever-increasing penetration of renewable generation is demanding additional flexibility for the operation of power systems. In this sense, it has been proved that wind power generators (WPGs) can provide part of the required operating reserve through pitch angle control (PAC) under a deloaded strategy. Unlike conventional operating reserve sources, however, the operating reserve provided by WPGs can't always be considered reliable, which renders the power system dispatch more challenging. In this research, the reserve capacity of the WPGs is modeled considering the decision-dependent uncertainty (DDU). The random wind speed introduces uncertainty to the available reserve capacity of the WPGs while the decisions on the deloading degrees of the WPGs directly affect their coefficients between wind speed and usable wind power, making the uncertainty decision-dependent. Moreover, the operational technical limits of the WPGs and the multiple possible wind speed scenarios are also considered in the wind power reserve capacity modeling. Then, a two-stage reliability constrained stochastic economic dispatch model with WPGs' reserve provision (SED-WRP) is developed to jointly dispatch the generation and reserve. A group-based multi-cut Benders decomposition method is applied to improve the convergence performance of the solution algorithm. Case studies are solved to quantify the impact of considering the reserve provision of WPGs and the inherent DDU in power system dispatch decisions.

Coordinated Planning of Wind Power Generation and Energy Storage With Decision-Dependent Uncertainty Induced by Spatial Correlation

Coordinated Planning of Wind Power Generation and Energy Storage With Decision-Dependent Uncertainty Induced by Spatial Correlation

Stochastic Wind Farm Expansion Planning With Decision-Dependent Uncertainty Under Spatial Smoothing Effect

The global experience on wind farm development reveals that the wind power uncertainty is affected by large-scale wind farm sizes due to the spatial smoothing effect. The dependence on wind farm sizes features the wind power uncertainty as decision-dependent uncertainty (DDU) during the wind farm expansion process. This paper proposes a stochastic expansion planning model for large-scale wind farms considering the DDU and addresses the coupling relationship between expansion decisions and DDU. Specifically, to explore how the structural features of the expansion model change with DDU in different forms, the proposed expansion model with DDU is established based on Point Estimate Method (PEM). Via the explicit expression between decisions and decision-dependent statistical moments of uncertain parameters in the PEM-based model, an in-depth discussion of model features is implemented. An iterative solution method with convergence analysis is proposed based on the explored model features to tackle the DDU. Besides, a modified Benders decomposition method is adopted in each iteration for the PEM-based two-stage optimization model potentially containing both min-min and min-max terms. The effects of DDU on wind farm expansion schemes are analyzed. Case studies verify the proposed expansion model and the solution method.

Transmission Defense Hardening Against Typhoon Disasters Under Decision-Dependent Uncertainty

Transmission defense hardening (TDH) is an effective practice to ensure the safe operation of the power system during and after natural disasters, such as typhoons. However, the coupling among the typhoon disaster, hardening decision, and transmission line status is not addressed in the previous research works. Besides, the enhancement measures for different types of transmission line failures have not yet been assessed. This paper proposes a comprehensive framework to improve power transmission system resilience against typhoon disasters. Firstly, the motion path and the wind field of typhoons are simulated using Monte Carlo sampling to quantify the spatiotemporal impacts of wind speed on the transmission line status. The decision-dependent uncertainty (DDU), which reflects the effects of hardening decisions on the transmission lines, is addressed by sampling according to the corresponding failure probabilities. Then, the scenario simulation is integrated into the proposed two-stage stochastic mixed-integer programming (SMIP) model considering various enhancement measures for different failures. Since the way to handle the DDU needs a large number of scenarios, it leads to high computation complexity. Hence, the sample average approximation (SAA) algorithm is introduced to cope with it. Finally, the proposed framework and its solution algorithm are carried out on the modified IEEE RTS-79 system and the modified IEEE 118-Bus system. The significant cost saving and resilience improvement demonstrate the effectiveness of the proposed framework.

Chance‐constrained co‐expansion planning for power systems under decision‐dependent wind power uncertainty

AbstractVariability and uncertainty in wind resources pose significant challenges to the expansion planning of wind farms and associated flexible resources. In addition, the spatial smoothing effect, indicating the impact of wind farm scale on aggregated wind power prediction errors, further aggravates the challenge. This paper proposes a chance‐constrained co‐expansion planning method considering the spatial smoothing effect, where the expansion of wind farm capacity, batter energy storage capacity, and power transmission lines are co‐optimized. Specifically, a decision‐dependent uncertainty (DDU) model is established capturing the dependency of wind power uncertainties on wind farm expansion decisions under the spatial smoothing effect. Unlike traditional optimization diagram where decisions are made under only decision‐independent uncertainty (DIU) with fixe properties, properties of decision‐dependent uncertain parameters would be inversely altered by decisions. To effectively tackle the coupling relation between decisions and DDU, DDU‐based chance constraints are formulated in an analytical manner, where the decisions and decision‐dependent uncertain parameters are expressed in a closed form. Eventually, with piecewise linearization of the DDU model and the polynomial approximation of cumulative distribution function of uncertain parameters, the proposed chance‐constrained optimization model with DDU is converted into a mixed‐integer second‐order cone program (MISOCP). Case studies verify the effectiveness of the proposed method.

Portfolio optimization of generic energy storage-based virtual power plant under decision-dependent uncertainties

Distributed energy resources (DERs) can be integrated into a smart and aggregated entity, namely a virtual power plant (VPP). This integration is beneficial to facilitate their flexibility utilization and optimally allocate their capacity to participate in competitive electricity markets. While various types of uncertainties exist in the market operation of VPP, such as the stochastic power of renewable energy resources (RES), etc., the literature generally overlooked the decision-dependent uncertainties (DDUs) with the operation of generic energy storage (GES). To be risk-hedging, this manuscript proposes a novel approach for risk-averse portfolio optimization of VPP with special attention to DDUs. To achieve tractable optimization, we propose a two-stage chance-constrained stochastic optimization model and three reformulations of different types of uncertainties. Furthermore, a modified and iterative progressive hedging algorithm is further proposed to handle the conditional value at risk (CVaR) and DDU items and efficiently solve the problem with large-scale GES portfolios. Exhaustive case studies based on real-world data demonstrate that VPP has to face the consequence of a significant real-time penalty (6.36 % loss in average profit) and losses in GES power (about 6.5 MW) when overlooking the DDUs' consequences. Moreover, it is more beneficial for greedy decision-makers to incorporate the DDUs' impact into their less risk-aversion strategy. Additionally, comparative results prove the validity of the proposed algorithm and its superiority in solving problems with large-scale GES portfolios and high-performance computing systems (over 30 % CPU time saved).

A Decision-Dependent Stochastic Approach for Joint Operation and Maintenance of Overhead Transmission Lines After Sandstorms

A sandstorm is one of the extreme weather events causing extensive damage to overhead transmission lines (OTLs). Due to potential postsandstorm insulator flashover, OTLs face high failure probabilities, and implementing in-time maintenance on OTLs is crucial to mitigate postsandstorm losses. Aiming at determining an optimal maintenance sequence of targeted OTLs, this article proposes a decision-dependent stochastic approach for the joint operation and maintenance of OTLs. First, considering the inherent dependency of the uncertain availability of OTLs on maintenance decisions, the multiperiod maintenance process of OTLs is modeled as a stochastic process with decision-dependent uncertainty (DDU). Second, a two-stage stochastic model with DDU is formulated, where the maintenance sequence and unit commitment decisions are made in the first stage and the second stage comprises scenariowise operation. Then, to tackle the coupling relation between decisions and DDU, a unique modeling transformation technique is adopted to convert the established decision-dependent stochastic model into a computationally efficient form. Case studies verify the effectiveness of the proposed method for postsandstorm maintenance scheduling.

Reliability assessment and improvement of distribution system with virtual energy storage under exogenous and endogenous uncertainty

Exploiting flexibility of low-cost but uncertain demand-side resources attracts great attention in the recent research of the distribution system with special respect to reliability assessment and improvement. While various kinds of uncertainties and risks exist, such as the stochastic power of renewable energy resources (RES), etc., the literature overlooked the exogenous and endogenous uncertainties along with the utilization of these resources denoted as virtual energy storage (VES). Instead, this manuscript proposes a novel approach for reliability assessment and improvement of the distribution system in special consideration of the endogenous uncertainties in VES operations, also denoted as decision-dependent uncertainties (DDUs). The modified and unified probabilistic model of VES is first presented to reveal the dynamic process and human behavior of VES. Additionally, we extend our recent work on probabilistic optimization under DDUs to the reliability assessment problem which addresses the optimal load curtailment with the support of VES. Moreover, reliability indices are newly introduced to (1) calculate the response losses of VES and the consequence of overlooking these uncertainties, and (2) explain the difference between theoretical and practical reliability. Case studies based on the ground truth data verify that the available capacity of VES will gradually reduce (maximum of 40 %–60 %) because of DDUs and it reveals the overlooked risks by using “greedy” strategies in the previous models, while the best practical reliability performance (increased 5 %) is witnessed via the proposed methods. We further investigate four key factors affecting the reliability performance, i.e., probability level of chance constraints, DDUs levels, dispatch time period, and locations of VES. Eventually, reasonable suggestions are provided for better flexibility utilization of VES, and the system's reliability is further improved by around 30 %.