Adaptive Robust Optimization Problem Research Articles

A comprehensive stochastic-based adaptive robust model for transmission expansion planning

To address various uncertainties appearing in the transmission expansion planning (TEP) problem, appropriate tools such as stochastic programming (SP) or robust optimization (RO) are required. On the other hand, both long-term (LT) and short-term (ST) uncertainties must be well captured to acquire the most reliable and robust solution. The current works model the LT uncertainty by RO and the ST uncertainty by SP, forming a two-stage adaptive robust optimization problem. In contrast, this paper presents a novel approach in which the SP and RO are simultaneously utilized to cope with each type of uncertainty, i.e., LT and ST uncertainty. Unlike the current methods with bilinear terms appearing in the subproblems, the trilinear terms appear in the objective function of the proposed approach, which are linearized accordingly. In addition, we propose a mixed-integer bilinear programming (MIBP) model to specify the rectangles used as uncertainty sets, which is further solved using Konno's algorithm. A standard column-and-constraint (CCG) generation solves the established tri-level structure. Two case studies are used to implement the formulation developed in this work. Results signify the effectiveness of the proposed method.

Acceleration techniques for adaptive robust optimization transmission network expansion planning problems

The computational burden of two-stage adaptive robust optimization transmission network expansion planning problems increases when accurately representing the operation of power systems, i.e., when operational variability, inter-temporal operational constraints, and non-convex operational constraints are considered in the decision-making model. This motivates the use of acceleration techniques in order to avoid computationally intractable problems. The adaptive robust optimization problem is specifically solved using the nested column-and-constraint generation algorithm. We propose the application of two new acceleration techniques to this algorithm. On the one hand, the variables that model the uncertainty realizations are initialized to the solution obtained by using the alternating direction algorithm. On the other, the master problems of the solution procedure are relaxed by considering only certain cutting planes and including more constraints if the evolution of the bounds of the algorithm is not appropriate. Numerical results show that the use of the proposed acceleration techniques leads to reductions in the computational time of over 93%.

Optimal sizing of a reliability-constrained, stand-alone hybrid renewable energy system using robust satisficing

This paper presents and compares three approaches to deal with the uncertain nature of energy supply and demand in the problem of optimal sizing of a stand-alone, reliability-constrained hybrid renewable energy system. The system is composed of wind turbines, photovoltaic solar panels, a battery bank, and a diesel generator. The proposed approaches are: (a) adaptive robust optimization with unmet demand as the (un)-reliability metric, (b) scenario-based robust satisficing, and (c) stochastic-free robust satisficing. A column-and-constraint generation algorithm is developed to solve the adaptive robust optimization problem. Furthermore, the theoretical connection between the proposed approaches is drawn, and their numerical performances are compared, in terms of optimal solution cost and reliability, through computational experimentation on a realistic case study. Numerical results show that both the computational times and the out-of-sample performances of the optimal solutions obtained from the two proposed robust satisficing approaches outperform that of the adaptive robust optimization approach, providing a strong justification for their application

Network-Cognizant Time-Coupled Aggregate Flexibility of Distribution Systems Under Uncertainties

Increasing integration of distributed energy resources (DERs) within distribution feeders provides unprecedented flexibility at the distribution-transmission interconnection. To exploit this flexibility and to use the capacity potential of aggregate DERs, feasible substation power injection trajectories need to be efficiently characterized. This letter provides an ellipsoidal inner approximation of the set of feasible power injection trajectories at the substation such that for any point in the set, there exists a feasible disaggregation strategy of DERs for any load uncertainty realization. The problem is formulated as one of finding the robust maximum volume ellipsoid inside the flexibility region under uncertainty. Though the problem is NP-hard even in the deterministic case, this letter derives novel approximations of the resulting adaptive robust optimization problem based on optimal second-stage policies. The proposed approach yields less conservative flexibility characterization than existing flexibility region approximation formulations. The efficacy of the proposed method is demonstrated on a realistic distribution feeder.

Impacts of Transmission Switching in Zonal Electricity Markets - Part II

In this paper, we present a two-stage model of zonal electricity markets with day-ahead market clearing, and real-time re-dispatch and balancing that accounts for transmission line switching at both stages. We show how the day-ahead problem with switching can be formulated as an adaptive robust optimization problem with mixed integer recourse, and present a new algorithm for solving the adversarial max-min problem that obeys the structure of an interdiction game. We apply the model on a realistic instance of the Central Western European system, and comment on the impacts of both proactive, and reactive transmission switching on the operating costs of the system. Part I presents day-ahead models of a short-term zonal electricity market with switching, and describes our algorithmic approach for solving these models efficiently. Part II describes variants of the real-time model, and presents the results of our case study on the Central Western European market.

A transformation-proximal bundle algorithm for multistage adaptive robust optimization and application to constrained robust optimal control

This paper presents a novel transformation-proximal bundle algorithm for multistage adaptive robust optimization problems. By partitioning recourse decisions into state and control decisions, the proposed algorithm applies affine control policy only to state decisions and allows control decisions to be fully adaptive, thus transforming the original problem into an equivalent two-stage Adaptive Robust Optimization (ARO) problem. Importantly, this multi-to-two transformation is general enough to be employed with any two-stage ARO solution algorithms, thus opening a new avenue for a variety of multistage ARO algorithms. The proximal bundle method is developed for the resulting two-stage problem along with convergence analysis. In an inventory control application, the affine disturbance-feedback control policy suffers from a severe suboptimality with an average gap of 34.88%, while the proposed algorithm generates an average gap of merely 1.68%.

Day-Ahead Self-Scheduling of a Virtual Power Plant in Energy and Reserve Electricity Markets Under Uncertainty

This paper proposes a novel model for the day-ahead self-scheduling problem of a virtual power plant trading in both energy and reserve electricity markets. The virtual power plant comprises a conventional power plant, an energy storage facility, a wind power unit, and a flexible demand. This multi-component system participates in energy and reserve electricity markets as a single entity in order to optimize the use of energy resources. As a salient feature, the proposed model considers the uncertainty associated with the virtual power plant being called upon by the system operator to deploy reserves. In addition, uncertainty in available wind power generation and requests for reserve deployment is modeled using confidence bounds and intervals, respectively, while uncertainty in market prices is modeled using scenarios. The resulting model is thus cast as a stochastic adaptive robust optimization problem, which is solved using a column-and-constraint generation algorithm. Results from a case study illustrate the effectiveness of the proposed approach.

Adaptive Distributionally Robust Optimization

We develop a modular and tractable framework for solving an adaptive distributionally robust linear optimization problem, where we minimize the worst-case expected cost over an ambiguity set of probability distributions. The adaptive distributionally robust optimization framework caters for dynamic decision making, where decisions adapt to the uncertain outcomes as they unfold in stages. For tractability considerations, we focus on a class of second-order conic (SOC) representable ambiguity set, though our results can easily be extended to more general conic representations. We show that the adaptive distributionally robust linear optimization problem can be formulated as a classical robust optimization problem. To obtain a tractable formulation, we approximate the adaptive distributionally robust optimization problem using linear decision rule (LDR) techniques. More interestingly, by incorporating the primary and auxiliary random variables of the lifted ambiguity set in the LDR approximation, we can significantly improve the solutions, and for a class of adaptive distributionally robust optimization problems, exact solutions can also be obtained. Using the new LDR approximation, we can transform the distributionally adaptive robust optimization problem to a classical robust optimization problem with an SOC representable uncertainty set. Finally, to demonstrate the potential for solving management decision problems, we develop an algebraic modeling package and illustrate how it can be used to facilitate modeling and obtain high-quality solutions for medical appointment scheduling and inventory management problems. The electronic companion is available at https://doi.org/10.1287/mnsc.2017.2952 . This paper was accepted by Noah Gans, optimization.

Coordination of Wind Farm and Pumped-Storage Hydro for a Self-Healing Power Grid

The increasing penetration of wind energy poses great challenges to the operation of power systems in normal and emergency states. However, energy storage technologies can help accommodate wind power uncertainty and variability due to their flexible characteristics. This paper focuses on the restoration phase, and provides a novel coordination strategy of wind and pumped-storage hydro (PSH) units for a faster and reliable self-healing process. The wind-PSH assisted power system restoration is formulated as a two-stage adaptive robust optimization problem. The first-stage problem determines the start-up sequence of generators and the energization times of transmission paths; and the second-stage problem decides load pickup sequences, wind power dispatch levels, and PSH units’ operating modes. The column-and-constraint generation decomposition algorithm is applied to solve the two-stage adaptive robust optimization problem, which has a mixed-integer optimization in the inner-level problem. The developed coordination strategy is tested on the modified IEEE 39-bus system. Numerical results demonstrate that the coordinated wind and PSH units can increase the total energy served, enhance wind power dispatchability, and reduce wind power curtailment.

Robust Transmission Expansion Planning Representing Long- and Short-Term Uncertainty

This paper addresses the transmission expansion planning problem under long- and short-term uncertainty. Long-term uncertainty pertains to changes across years (demand growth and future increase in production capacity), whereas short-term uncertainty pertains to changes within a year (demand and wind/solar power production variability, and equipment availability). This expansion problem is formulated as an adaptive robust optimization problem that provides protection against long-term uncertainty while carefully representing short-term uncertainty via scenarios. The problem is solved via a tailored implementation of the primal Benders’ decomposition algorithm that focuses on an efficient solution of the subproblem. The effectiveness of the proposed algorithm to identify robust expansion plans and its computational efficiency are illustrated through a realistic case study.

Adaptive robust optimization with minimax regret criterion: Multiobjective optimization framework and computational algorithm for planning and scheduling under uncertainty

Regret is defined as the deviation of objective value from the perfect information solution, and serves as an important evaluation metric for decision-making under uncertainty. This paper proposes a novel framework that effectively incorporates the minimax regret criterion into two-stage adaptive robust optimization (ARO). In addition to the conventional robustness criterion, this ARO framework also simultaneously optimizes the worst-case regret to push the performance of the resulting solution towards the utopia one under perfect information. By using a data-driven uncertainty set, we formulate a multiobjective ARO problem that generates a set of Pareto-optimal solutions to reveal the systematic trade-offs between the conventional robustness and minimax regret criteria. The resulting multi-level mixed-integer programming problem cannot be solved directly by any off-the-shelf optimization solvers, so we further propose tailored column-and-constraint generation algorithms to address the computational challenge. Two applications on process network planning and batch process scheduling are presented to demonstrate the applicability of the proposed framework and the efficiency of the proposed solution algorithms.

Robust dynamic transmission and renewable generation expansion planning: Walking towards sustainable systems

Recent breakthroughs in Dynamic Transmission Network Expansion Planning (DTNEP) have demonstrated that the use of robust optimization, while maintaining the full temporal dynamic complexity of the problem, renders the capacity expansion planning problem considering uncertainties computationally tractable for real systems. In this paper an adaptive robust formulation is proposed that considers, simultaneously: (i) a year-by-year integrated representation of uncertainties and investment decisions, (ii) the capacity expansion lines have and (iii) the construction and/or dismantling of renewable and conventional generation facilities as well. The Dynamic Transmission Network and Renewable Generation Expansion Planning (DTNRGEP) problem is formulated as a three-level adaptive robust optimization problem. The first level minimizes the investment costs for the transmission network and generation expansion planning, the second level maximizes the costs of operating the system with respect to uncertain parameters, while the third level minimizes those operational costs with respect to operational decisions. The method is tested on two cases: (i) an illustrative example based on the Garver IEEE system and (ii) a case study using the IEEE 118-bus system. Numerical results from these examples demonstrate that the proposed model enables optimal decisions to be made in order to reach a sustainable power system, while overcoming problem size limitations and computational intractability for realistic cases.

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

A computational framework and solution algorithms for two‐stage adaptive robust scheduling of batch manufacturing processes under uncertainty

A novel two‐stage adaptive robust optimization (ARO) approach to production scheduling of batch processes under uncertainty is proposed. We first reformulate the deterministic mixed‐integer linear programming model of batch scheduling into a two‐stage optimization problem. Symmetric uncertainty sets are then introduced to confine the uncertain parameters, and budgets of uncertainty are used to adjust the degree of conservatism. We then apply both the Benders decomposition algorithm and the column‐and‐constraint generation (C&CG) algorithm to efficiently solve the resulting two‐stage ARO problem, which cannot be tackled directly by any existing optimization solvers. Two case studies are considered to demonstrate the applicability of the proposed modeling framework and solution algorithms. The results show that the C&CG algorithm is more computationally efficient than the Benders decomposition algorithm, and the proposed two‐stage ARO approach returns 9% higher profits than the conventional robust optimization approach for batch scheduling. © 2015 American Institute of Chemical Engineers AIChE J, 62: 687–703, 2016

A robust optimization approach to energy and reserve dispatch in electricity markets

To a large extent, electricity markets worldwide still rely on deterministic procedures for clearing energy and reserve auctions. However, increasing shares of the production mix consist of renewable sources whose nature is stochastic and non-dispatchable, as their output is uncertain and cannot be controlled by the operators of the production units. Stochastic programming models allow the joint determination of the day-ahead energy and reserve dispatch accounting for the uncertainty in the output from these sources. However, the size of these models gets quickly out of hand as a large number of scenarios are needed to properly represent the uncertainty. In this work, we take an alternative approach and cast the problem as an adaptive robust optimization problem. The resulting day-ahead energy and reserve schedules yield the minimum system cost, accounting for the cost of the redispatch decisions at the balancing (real-time) stage, in the worst-case realization of the stochastic production within a specified uncertainty set. We propose a novel reformulation of the problem that allows considering general polyhedral uncertainty sets. In a case-study, we show that, in comparison to a risk-averse stochastic programming model, the robust optimization approach progressively trades off optimality in expectation with improved performance in terms of risk. These differences, however, gradually taper off as the level of risk-aversion increases for the stochastic programming approach. Computational studies show that the robust optimization model scales well with the size of the power system, which is promising in view of real-world applications of this approach.