Adjustable Uncertainty Set Research Articles

Two-stage robust optimization scheduling for integrated energy systems considering ammonia energy and waste heat utilization

To balance the stability and economic benefits of integrated energy systems (IES) during low-carbon optimization, this paper proposes a two-stage robust optimization scheduling method that considers ammonia energy and waste heat utilization. Firstly, this paper analyzes the Power-to-Ammonia (P2A) operational mechanism and the energy input–output transformation relationship and establishes a Power-to-Ammonia equipment model incorporating waste heat utilization. Subsequently, the low-carbon operational characteristics of the integrated energy system with cooperative operation between Power-to-Ammonia and coal-fired units are explored by integrating Power-to-Ammonia with ammonia-coal co-firing technology. Then, the paper uses an adjustable uncertainty set to describe the variability of renewable generation (RG) and constructs a max–min structured integrated energy system with a two-stage robust optimization scheduling model to minimize the worst-case scenario system operating costs. Finally, the Column-and-Constraint Generation (C&CG) algorithm decomposes the two-stage robust optimization issue into a master problem and sub-problems for the iterative solution. Furthermore, the proposed method is simulated in an actual integrated energy system in northern China. Simulation results indicate that with the intervention of renewable generation uncertainty, the model developed in this study reduces operating costs and carbon emissions by 13.98 % and 20.95 %, respectively, effectively enhancing the capability for low-carbon economic operation. This study provides crucial theoretical support for advancing the low-carbon energy conversion of integrated energy systems.

A Novel Weighted Data-Driven Robust Optimization Approach for Creating Adjustable Uncertainty Sets

Robust optimization provides a progressive and efficient approach to dealing with uncertainty which applies to various optimization problems. However, most existing robust optimization approaches generate a unique uncertainty set for each level of conservatism, disregarding supplementary information such as predicted values. This paper proposes a new weighted data-driven robust optimization approach for creating adjustable uncertainty sets. The proposed approach enables the adjustment of the boundary reduction rate of uncertainty sets, effectively pulling uncertainty sets towards areas of higher density or towards the predicted points. Additionally, a multi-stage clustering algorithm is proposed to fully cover the areas around the predicted point. A regularization parameter search algorithm is also developed to tune the conservatism degree. The numerical results indicate that by weighting historical data, adjustable uncertainty sets with the same fraction of data coverage can be created that ensures the feasibility of the model and reduces the extra conservatism.

Robust stochastic optimal dispatching of integrated electricity-gas-heat systems with improved integrated demand response

In recent years, integrated energy systems with deep coupling of power, natural gas, and heat energy have gradually attracted extensive attention. The increasing issue of uncertainty in the generation and load of an integrated electric-gas-heat system (IEGHS) is a growing prominent problem. The effective implementation of demand response (DR) programs is an important way to solve this problem in the IEGHS. In this paper, a robust stochastic optimal dispatching method for an IEGHS with integrated DR (IDR) under multiple uncertainties is proposed. A robust adjustable uncertainty set is adopted to deal with the uncertainty of wind power. The Wasserstein generative adversarial network based on gradient normalization is proposed to generate load-side demand scenarios. Furthermore, an improved IDR model that considers the peak-valley difference cost of the electricity-gas-heat load is proposed. Finally, the simulation analysis successfully demonstrates the efficacy of the proposed model. By utilizing this approach, the system can obtain a scheduling scheme with the lowest operating cost even under worst-case scenarios.

Distributionally robust optimal dispatching method of integrated electricity and heating system based on improved Wasserstein metric

This paper proposed a distributionally robust optimal dispatching (DROD) method of integrated electricity and heating system based on an improved Wasserstein metric, for dispatching problems to manage the risk from uncertain wind power forecasted errors and operating characteristics of electric vehicles (EVs). The ambiguity set employed in the distributionally robust formulation for wind power forecasted error is the Wasserstein ball centered at the empirical distribution and an adjustable uncertainty set of EV operating characteristics is proposed in this model to quantify the uncertainty of EVs more flexibly and intuitively. The proposed framework minimizes the generating cost, EV operating cost, and the expected generation units adjustable cost under the worst-case distribution in the uncertainty set. To improve the computational efficiency of the proposed DROD model, an improved Wasserstein metric based on extreme scenarios DROD (WMES-DROD) model is proposed to construct the ambiguity set of wind power forecasted error. The comparison between the Wasserstein metric-based DROD and WMES-DROD indicates that the proposed WMES-DROD model can obtain optimal results with less time. The out-of-sample analysis with the IEEE 118-bus test system confirms the excellent robustness and computational superiority of the proposed model.

Robust Optimal Control With Inexact State Measurements and Adjustable Uncertainty Sets

The efficacy of robust optimal control with adjustable uncertainty sets is verified in several domains under the perfect state information setting. This paper investigates constrained robust optimal control for linear systems with linear cost functions subject to uncertain disturbances and state measurement errors that are both residing in adjustable uncertainty sets. We first show that the class of affine feedback policies of state measurements are equivalent to the class of affine feedback policies of estimated disturbances in terms of their conservativeness. Then, we formulate and solve a robust optimal control problem with adjustable uncertainty sets by considering the disturbance feedback policies. In contrast to the conventional robust optimal control, where uncertainty sets are fixed and known a priori, the uncertainty sets themselves are regarded as decision variables in our design. In particular, given the metrics for evaluating the optimal size/shape of the polyhedral uncertainty sets, a bilinear optimization problem is formulated to decide the optimal size/shape of uncertainty sets and a corresponding optimal control policy to robustly guarantee that the system will respect its constraints for all admissible uncertainties. In addition, we introduce a convex approximation for the proposed scheme to provide a computationally efficient inner approximation of the original problem. The proposed scheme is illustrated by numerical simulation of a building temperature control problem to demonstrate its effectiveness.

Distributionally robust unit commitment with an adjustable uncertainty set and dynamic demand response

To address the uncertainty of renewable energy in unit commitment, an adjustable uncertainty set of renewable energy is introduced and unit commitment is scheduled to guarantee the safe operation of the power system in this set. However, the operational risk arises when renewable energy falls out of the adjustable uncertainty set and this risk should be evaluated to determine the adjustable uncertainty set. In this paper, a distributionally robust optimization approach is proposed to calculate the risks of load shedding and renewable energy curtailment at the lower and upper bounds of the adjustable uncertainty set. After the evaluation of the operational risk, the day-ahead unit commitment with the adjustable uncertainty set is determined together with the demand response reserve in the reduction of the operational risk. In real time, the dynamic demand response is proposed to further reduce the operational risk. The proposed distributionally robust method with the dynamic demand response for dealing with the uncertainty of renewable energy is verified on the IEEE 6-bus, 30-bus, and 118-bus systems. Simulation results show that the proposed method reduces the cost of the unit commitment and the operational risk.

Risk-Based Contingency-Constrained Optimal Power Flow With Adjustable Uncertainty Set of Wind Power

This article proposes a risk-based contingency-constrained optimal power flow model by leveraging the methods of both adjustable uncertainty set and distributionally robust optimization. In the proposed model, an adjustable uncertainty set of wind power is developed with network contingencies explicitly incorporated. Based on this uncertainty set, the proposed model is capable of securing the network against both wind power fluctuations and contingencies in a probabilistic manner with the optimal balance between operation cost and risk. Meanwhile, a data-driven <i>L</i>1-norm-based ambiguity set is employed so that the proposed model is distributionally robust to the ambiguous probability distribution of wind power and the size of the model remains unchanged as the available wind power data increases. A decomposition-based algorithm is also derived so that the proposed model is solvable by off-the-shelf solvers. Numerical studies on IEEE 14- and 118-bus systems are conducted to verify the effectiveness of the proposed model.

An almost robust model for minimizing disruption exposures in supply systems

This paper studies two-stage disruption exposure minimization problems, motivated by the supply disruption issues in the energy and water supply systems. In particular, we address the ambiguity in both the probability distribution and risk preference of decision-makers towards disruption exposures. First, we propose a two-stage distributionally robust model with adjustable uncertainty sets, which solves a supply system solution with the least possible disruption exposures. We show that this two-stage robust disruption exposure model can be reduced to a computationally attractive single-stage mixed-integer linear program. We then propose an extended almost-robust disruption guarantee model to account for the ambiguity in the risk preference of decision-makers. We demonstrate that this almost-robust guarantee model can reveal clear preferences of most decision-makers under limited distribution information, which however does not resort to any particular disutility function specification and can be solved efficiently using a binary search algorithm. A decision support framework is also developed to guide users on how to apply the proposed disruption exposure models. Finally, we apply the proposed models to a distributed energy supply system design problem. Numerical results show that our models significantly outperform a risk-neutral model in hedging against a broad set of supply distributions. Moreover, the almost-robust guarantee model exhibits its advantages in hedging against high disruption levels, and performs the best under the vast majority of distributions regarding all tested statistical criteria.

Adjustable Uncertainty Set Constrained Unit Commitment With Operation Risk Reduced Through Demand Response

In this article, the approach of an adjustable uncertainty set is proposed to deal with the uncertainty of renewable energy (RE) in unit commitment (UC). Demand response (DR) is co-optimized to reduce the operation risk of load shedding and RE curtailment when the RE falls out of the adjustable uncertainty set. In comparison with existing approaches with an adjustable uncertainty set, the proposed approach further incorporates DR requires no predefined parameters to constrain the deviation from the forecast RE. It divides the maximum RE set into subintervals, and bounds of the adjustable uncertainty set are determined among these subintervals with the consideration of DR in reducing the operation risk. The original mixed-integer nonlinear problem of UC scheduling is transformed to be a mixed-integer linear problem to be effectively solved. The performance of the proposed approach is verified on the IEEE 6-bus, 30-bus, and 300-bus systems. Through the comparison with existing methods, the effectiveness of the proposed approach in reducing the conservativeness is verified. The effectiveness of the proposed approach in the reduction of the operation risk of load shedding and RE curtailment is verified through the comparison between situations with and without DR.

Uncertainty Cost of Stochastic Producers: Metrics and Impacts on Power Grid Flexibility

The widespread presence of contingent generation, when coupled with the resulting volatility of the chronological net-load (i.e., the difference between stochastic generation and uncertain load) in today's modern electricity markets, engender the significant operational risks of an uncertain sufficiency of flexible energy capacity. In this article, we address several operational flexibility concerns that originate from the increase in generation variability captured within a security-constrained unit commitment (SCUC) formulation in smart grids. To quantitatively assess the power grid operational flexibility capacity, we first introduce two reference operation strategies based on a two-stage robust SCUC, one through a fixed and the other via an adjustable uncertainty set, for which the state-of-the-art techniques may not be always feasible, efficient, and practical. To address these concerns and to account for the effects of the uncertainty cost resulting from dispatch limitations of flexible resources, a new framework centered on the adjustable penetration of stochastic generation is proposed. Our hypothesis is that if the SCUC is scheduled with an appropriate dispatch level of stochastic generation, the system uncertainty cost will decrease, and subsequently, the system's ability to accommodate additional uncertainty will improve. Numerical simulations on a modified IEEE 73-bus test system verify the efficiency of the suggested assessment techniques.

MILP Based Robust Short-Term Scheduling for Wind–Thermal–Hydro Power System With Pumped Hydro Energy Storage

The uncertainty of the wind power generation and complex constraints of the hydropower pose challenges for the short-term scheduling of coordinated wind power, thermal power, and cascaded hydroelectric system (WTHS). In this paper, a robust security-constrained unit commitment model is established for a WTHS. The proposed model ensures the utilization of wind power and economic return from the scheduling. Conservative adjustable uncertainty sets are used to characterize the uncertainty of wind power over temporal and spatial dimensions. In this model, pumped hydro energy storage (PHES) is incorporated to cope with the wind power fluctuations. A simplified affine policy is developed for the decision making of the adjustable variables. Based on a series of linearization techniques, the proposed model is formulated as a single-level mixed-integer linear programming (MILP) problem, where the numerical tests performed on the modified IEEE 30-bus, IEEE 118-bus, and Polish 2736-bus systems verify the effectiveness of the model. The comparative analyses quantitatively evaluate the contributions of the PHES in terms of economic performance and wind power accommodation. The test results reveal that the robustness of scheduling plans is enhanced by the use of the PHES, and the proposed approach is applicable to the large-scale power systems.

Robust Day-Ahead Scheduling of Electricity and Natural Gas Systems via a Risk-Averse Adjustable Uncertainty Set Approach

The requirement for energy sustainability drives the development of renewable energy technologies and gas-fired power generation. The increasing installation of gas-fired units significantly intensifies the interdependency between the electricity system and natural gas system. The joint scheduling of electricity and natural gas systems has become an attractive option for improving energy efficiency. This paper proposes a robust day-ahead scheduling model for electricity and natural gas system, which minimizes the total cost including fuel cost, spinning reserve cost and cost of operational risk while ensuring the feasibility for all scenarios within the uncertainty set. Different from the conventional robust optimization with predefined uncertainty set, a new approach with risk-averse adjustable uncertainty set is proposed in this paper to mitigate the conservatism. Furthermore, the Wasserstein–Moment metric is applied to construct ambiguity sets for computing operational risk. The proposed scheduling model is solved by the column-and-constraint generation method. The effectiveness of the proposed approach is tested on a 6-bus test system and a 118-bus system.

Data‐driven distributionally robust reserve and energy scheduling over Wasserstein balls

In order to address the wind power uncertainty in the security constrained economic dispatch (SCED), this study proposes a two-stage data-driven distributionally robust reserve and energy scheduling (DDRRES) model that considers the loss of wind spillage and load shedding. This model aims to minimise the total cost while ensuring that the operating constraints are satisfied on the adjustable uncertainty set, of which the boundaries are decision variables. Unlike the previous approaches, which assumed that the underlying true probability distribution (PD) of uncertainty is known, the proposed model does not rely on the specified distribution but extracts the information from historical data directly. The ambiguity sets, i.e., Wasserstein balls are constructed to contain the possible PDs. Fixing the boundaries of adjustable uncertainty set, the operational risk is the expected loss under the worst-case PDs over Wasserstein balls. Thus the operational cost and operational risk can be balanced by adjusting the adjustable uncertainty set. After the tractable formulation of DDRRES is obtained, such a model is solved with the column-and-constraint generation method. The performance of the proposed approach is verified on a 6-bus test system and a 118-bus system.

Distribution Network Electric Vehicle Hosting Capacity Maximization: A Chargeable Region Optimization Model

To coordinate electric vehicle (EV) charging, the EV aggregator (EVA) is usually assumed to obtain the privilege from EV owners (EVOs) to determine the EV charging profile, and complex communication between EVA and EVOs is demanded, which poses difficulties for practical applications. In contrast, this paper proposes the concept of an EV chargeable region to evaluate the distribution network (DN) EV hosting capacity, i.e., how much EV charging demand can be accommodated in a DN, within which the technical constraints of DN (e.g., voltage deviation) are guaranteed and EVOs’ charging requests are maximally ensured. The optimization of the EV chargeable region is formulated as a two-stage robust optimization model with adjustable uncertainty set. The EV chargeable region and DN decision variables are optimized in the first stage and the feasibility in the real-time worst-case scenario is checked in the second stage, considering the uncertainty of EV charging demand and DN active and reactive power. A modified column and constraint generation and outer approximation method is adopted to address the proposed problem. Simulations on an IEEE 123-node DN demonstrate the effectiveness of the proposed model.

Robust optimal control with adjustable uncertainty sets

In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets. In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. Nevertheless, we demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system.

Robust Look-Ahead Power Dispatch With Adjustable Conservativeness Accommodating Significant Wind Power Integration

Robust look-ahead power dispatch is an effective and secure approach to account for uncertainties in wind power. Based on the robust look-ahead power dispatch model proposed in our previous work, in this paper, we focus on coping with adjustable uncertainty sets to reduce the conservativeness of robust dispatch (RD). Robust look-ahead power dispatch with adjustable uncertainty sets is formulated using robust optimization (RO) and is transformed into biconvex programming. Several strategies exploiting different levels of uncertainty information are developed to adjust the conservativeness by varying the desired confidence level. Monte Carlo simulations are carried out to compare the performance of the proposed approaches with another popular strategy. Test results show that the proposed method is effective in reducing the conservativeness of RD and ensuring system security with controllable risk. Experiments on large-scale benchmark systems show the scalability of the proposed method.