Adaptive Robust Optimization Research Articles

Adaptive robust economic dispatch and real-time control of distribution system considering controllable inverter air-conditioner clusters

With a tremendous number of renewable energy sources (RES) integrated into the distribution system, the inherent uncertainty of RES power generation brings about significant challenges in distribution and power balance within the distribution system. This article proposes an adaptive robust economic dispatch (ARED) model and a real-time control strategy for distribution systems as countermeasures, which make full use of the adjustable capabilities of controllable inverter air-conditioner (IAC) clusters. Firstly, the concept of the adjustable capacity curve (ACC) is developed to accurately quantify the adjustable capacity of an IAC cluster. Afterward, a two-stage adaptive robust optimization is formulated for ARED, which comprehensively takes the adjustable capacity of the IAC cluster and the uncertainty of RES into consideration. Meanwhile, the solution methodology of ARED is also designed based on the column and constraint generation (C&CG) algorithm, where the master problem is quadratic programming with quadratic constraints (QCQP), and the max-min sub-problem is reformulated to a mixed integer linear programming (MILP) form by taking advantage of linear duality theory and big-M method. Finally, a novel real-time decentralized control strategy for IAC clusters is also proposed for purpose of hedging against stochastic RES power fluctuation after every round of ARED decisions. The results of the case study validate the effectiveness of ARED model and real-time control strategy under different uncertainty scenarios of RES power generation.

A robust flexibility evaluation method for distributed multi-energy microgrid in supporting power distribution system

Today’s power system is facing various challenges brought by large-scale renewable energy (RE) integration, which brings higher demand for flexibility. With the energy network gradually showing its distributed structural characteristics, multi-energy microgrids (MEMG) become an important component to effectively utilize distributed energy sources and supplement the flexibility of power distribution system (PDS). To effectively harness the operational flexibility of distributed MEMGs, we propose in this paper an evaluation method to quantify the flexibility capability of MEMG. A virtually established MG flexibility bus (MG-FB) is endowed with MG flexibility parameters (MG-FPs), which can reflect the flexibility characteristics of MEMG. To consider the impact of operational uncertainty on MG-FPs, a two-stage adaptive robust optimization (ARO) model is proposed, which can be solved by the C&CG algorithm. The results of a typical test system show the influence of system configuration, operator’s risk preference, and other factors on the values of MG-FPs. Besides, we illustrate the effectiveness and applicability of the proposed framework in modeling and quantifying the operational flexibility of MEMG to support the operation of the upstream network.

Robust Scheduling with Temporal Decomposition of Integrated Electrical-Heating System Based on Dynamic Programming Formulation

To realize the potential of the integrated electrical-heating systems (IEHS) for coping with the uncertain renewable energy, a reserve scheduling model based on the dynamic programming transformed multi-stage adaptive robust optimization (DMRO) is proposed in this paper. The DMRO model enforces the non-anticipativity of scheduling process via temporal decomposed framework, and enhances the formulation flexibility for time-coupled equipment, such as energy storage systems (ESS). The whole scheduling model is split into two problems under the decomposition constructure, which consists of a reserve interval generation (RG) problem for making the unit commitment and systems' reserve plan, and a multi-stage economic operation (EO) problem aiming with simulating the worst-case scheduling process. To obtain the global optimal, a nested Benders decomposition algorithm is employed to solve the DMRO model. Case studies on the 6-Bus-5-node IEHS and the 118-Bus-24-node IEHS demonstrate the effectiveness of the proposed model and the solution methodology.

A Unified MILP Solution Framework for Adaptive Robust Scheduling Problems With Mixed-Integer Recourse Objective

To mitigate the high computational cost of adaptive robust optimization with a mixed-integer recourse objective in real-world power scheduling, a unified mixed-integer linear programming (MILP) solution framework is proposed in this paper. Numerical tests verify the superiority and practicability of the proposed solution framework over the existing decomposition-based solution algorithms.

Microgrid Operation Optimization Considering Transient Stability Constraints: A New Bidirectional Stochastic Adaptive Robust Approach

This article proposes a new bidirectional stochastic adaptive robust framework with transient stability constraints to optimally and securely operate microgrids (MGs). Within the proposed framework, the uncertainties of photovoltaic/wind turbine generation, load, and electricity price are modeled using adaptive robust optimization (ARO), while the uncertainty of unscheduled islanding occurrence is modeled by stochastic programming. Besides covering the uncertainties related to MG operation, the proposed approach considers transient stability uncertainties leading to a more secure MG operation scheduling framework. To the best of the authors’ knowledge, this is the first time that transient stability analysis along with stochastic adaptive robust optimization (SARO) is proposed for MG operational scheduling. In addition, a new bidirectional scenario generation method is proposed to model MG unscheduled-islanding occurrence and MG restoration to the grid-connected mode. Modeling both the grid-connected and unscheduled-islanded MG operation modes as well as their transitions further distinguishes the proposed approach from the previous ones. Furthermore, a new solution methodology is presented to solve the proposed MG scheduling problem. This methodology is capable of solving both SARO and transient stability constrained-optimal power flow problems. The proposed MG operation optimization framework and solution method are tested on the modified IEEE 33-bus network.

Reliability-Based Expansion Planning Studies of Active Distribution Networks With Multiagents

In this paper, a multi-agent framework is proposed to address the expansion planning problem in a restructured active distribution network. In this framework, the objective and techno-economic constraints of participating agents are addressed in the expansion planning of power network and DER assets as well as the network and DERs optimal operation management. The agents include distributed generator owners and load aggregators which participate along with the distribution network operator (DNO) in the active distribution network planning. The proposed framework is formulated as a bi-level optimization problem with multi-lower levels in which the DNO optimizes the network expansion planning and operation in the upper level where the network reliability is also modeled. In the lower level problems, other participating agents optimize the expansion planning and operation of their assets. The model addresses the optimal operation of resources by clearing the local energy market. The uncertainties in the upper and the lower level problems are effectively addressed using a stochastic adaptive robust optimization approach. The proposed model is implemented on the 54-bus and 138-bus distribution networks and the results demonstrate the effectiveness and scalability of the proposed framework.

Robust Allocation of FACTS Devices in Coordinated Transmission and Generation Expansion Planning considering Renewable Resources and Demand Response Programs

Nowadays, the use of demand response programs (DRPs) in a variety of long-term and short-term planning problems has been explored. In this paper, a generation and transmission expansion planning (GTEP) model along with FACTS device allocation is presented. Furthermore, demand response programs are taken into account for more load flexibility. The proposed model is presented as a multi-objective minimizing problem considering emission, cost, and voltage security index. Furthermore, the conventional Pareto optimization is adopted using fuzzy weighted sum method (FWSM) to achieve a single-objective model. The final problem is constrained by equations of alternative current power flow, operation and voltage security limits, planning model of shunt FACTS devices, and operation of the DRP. Adaptive robust optimization (ARO) is used to reach suitable models for the active power of renewable resources and power consumption. As a main search algorithm, a hybrid combination of water cycle algorithm (WCA) and ant lion optimization (ALO) is proposed to find the optimum solution with a small standard deviation. The problem is tested on different standard IEEE systems, and the results validate the operation and network security improvement due to optimal location of FACTS devices. According to the results, the economic and environmental status of the network has also improved.

Multistage adaptive robust optimization for the hydrothermal scheduling problem

The current water scarcity faced by many countries increases the need to consider an appropriate representation of future hydro inflows in power system operation and planning models. Hydrothermal scheduling is the problem that seeks to use the water stored in reservoirs throughout time in order to find an optimal dispatch policy between hydro and thermal power plants. Due to both the inherent randomness of water inflows and the intertemporal decision process, this problem has been typically approached through multistage stochastic optimization, minimizing the total expected operational cost over the entire planning horizon. However, this approach has some practical disadvantages. Among the main ones we highlight (i) the complexity of balancing the statistical representativeness of the stochastic processes and the computational efficiency of the optimization model; (ii) the need to employ computationally intensive decomposition methods for its solvability; and (iii) the need to carry out network simplifications to tackle tractability issues arising in large networks. As an alternative, we propose a multistage adaptive robust optimization model for the hydrothermal scheduling problem. Robust optimization is useful to prevent the previous disadvantages because it does not make any distributional assumption and it works with the so-called uncertainty sets instead of carrying out sampling processes. In particular, we propose an efficient formulation based on linear decision rules and vector autoregressive models to represent the uncertainty in hydro inflows. Our experiments, based on the Chilean electric power system with hundreds of hydro nodes and connections, show the proposed model’s efficiency for large-scale systems and provide insights into the adequate balance between cost-effectiveness and reliability that robust optimization models guarantee.

Adaptive robust optimization for European strategic gas infrastructure planning

The European natural gas market is undergoing fundamental changes that foster uncertainty on both supply and demand sides. This uncertainty has elicited questions about the value of the Projects of Common Interest—strategic infrastructure investments supported by public funding from the European Union. This paper addresses this matter by deploying the adaptive robust optimization framework to consider long-term uncertainties in the gas infrastructure expansion planning problem. This framework confronts the drawbacks of mainstream methods of incorporating uncertainty in gas market models, such as stochastic scenario trees, in which the modeler predefines the probabilities and realization paths of unknown parameters. Our mathematical model endogenously identifies the unfortunate realizations of unknown parameters and suggests the optimal investment strategies to address them. We use this feature to assess which Projects of Common Interest are valuable in maintaining system resilience amid cold-winter demand spikes and supply shortages, while also considering budget constraints. The robust solutions indicate consistent preferences for specific projects. We find that real-world construction efforts have been focused on the most promising projects from a business perspective. However, we also find that the vast majority of projects are unlikely to be realized without financial support, even if they would serve as a hedge against stresses in the European gas system.

An adaptive robust optimization model for parallel machine scheduling

Real-life parallel machine scheduling problems can be characterized by: (i) limited information about the exact task duration at the scheduling time, and (ii) an opportunity to reschedule the remaining tasks each time a task processing is completed and a machine becomes idle. Robust optimization is the natural methodology to cope with the first characteristic of duration uncertainty, yet the existing literature on robust scheduling does not explicitly consider the second characteristic the possibility to adjust decisions as more information about the tasks duration becomes available, despite that re-optimizing the schedule every time new information emerges is standard practice. In this paper, we develop an adaptive robust optimization scheduling approach that takes into account, at the beginning of the planning horizon, the possibility that scheduling decisions can be adjusted. We demonstrate that the suggested approach can lead to better here-and-now decisions and better makespan guarantees. To that end, we develop the first mixed integer linear programming model for adaptive robust scheduling, and a two-stage approximation heuristic, where we minimize the worst-case makespan. Using this model, we show via a numerical study that adaptive scheduling leads to solutions with better and more stable makespan realizations compared to static approaches.

Block-Coordinate-Descent Adaptive Robust Operation of Industrial Multi-layout Energy hubs under Uncertainty

This paper presents an adaptive robust optimization approach to optimal operation of multi-layout energy hubs under uncertainty. In the first step, the multi-layout energy hub concept is presented and discussed comprehensively followed by its required energy management model, but in the deterministic form. In the next step, an adaptive robust optimization approach is developed for the energy management model of multi-layout energy hubs. The uncertainties of energy hub load as well as upstream energy market prices are considered through bounded intervals using polyhedral uncertainty sets. The proposed adaptive-robust multi-layout EHS optimizer (ARMEO) is developed as a tri-level min-max-min optimization problem which cannot be solved directly. To do so, column-and-constraint (C&C) technique is used to recast the tri-level model into a "min" master problem and a "max-min" sub-problem. However, the "max-min" sub-problem is still a bi-level model and cannot be solved directly. To cope, block coordinate descent (BCD) methodology is applied to the sub-problem to iteratively solve the "max-min" sub-problem. An industrial-based case study is conducted to show the effectiveness of the proposed model in 1) managing multi-layout energy hubs, and 2) provide immunized operational solutions against uncertainties. Based on the results, it is observed that the ARMEO model is subject to a higher operation cost (compared to deterministic model), however, the obtained operating solutions are immunized against the uncertainties. Moreover, it has been shown that the proposed multi-layout EHS model can provide reasonable operating solutions for all layouts of the system as a whole.

Stochastic adaptive robust approach for day-ahead energy market bidding strategies in hydro dominated sequential electricity markets

This paper presents a novel methodological approach for the optimal day-ahead energy market bidding behavior of a cascaded hydropower plants (HPPs) portfolio in the sequential electricity markets. The understudy markets are day-ahead energy market and manual frequency restoration reserve (mFRR) markets in both capacity and energy setups. The introduction of the mFRR capacity market ensures transmission system operators (TSOs) about the availability of energy bids in the real-time market, which acts as binding constraints in the mFRR energy markets. As a determining factor, the active-time duration of mFRR energy bids is uncertain at the time of day-ahead bidding, which is modeled as the intervals in our robust optimization, while the electricity prices are considered as the scenarios in the stochastic optimization. Hence, we have proposed a novel stochastic adaptive robust optimization to address the bidding problem in the face of uncertainties accurately. The results show a considerable improvement compared to the conventional fully-stochastic approach in the case study of Swedish cascaded hydropower plants.

Robust Stochastic Facility Location: Sensitivity Analysis and Exact Solution

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

Optimal scheduling of a hybrid AC/DC multi-energy microgrid considering uncertainties and Stackelberg game-based integrated demand response

The emergence of solid-state transformers (SSTs) enriches the system structure and operation mode of the multi-energy microgrid (MEMG). This paper depicts a novel hybrid AC/DC MEMG structure based on a three-stage SST, and proposes a two-level optimal scheduling model considering generation and load uncertainties, as well as Stackelberg game-based integrated demand response. In the upper level model for the energy hub operator (EHO), the pricing strategy and equipment dispatch are optimized to maximize profit, and the two-stage adaptive robust optimization and reserve allocation are used to address the uncertainties of renewable generation and loads, respectively. In the lower level model for the demand response aggregator (DRA), the response strategy of different loads, including shiftable, adjustable and replaceable loads, is optimized for surplus maximization. To solve the proposed scheduling model, the two-level optimization problem is transformed to a single-level min–max-min problem according to Karush-Kuhn-Tucker conditions, then it is divided into two subproblems and solved by the column-and-constraint generation algorithm and CPLEX solver. Furthermore, a case study is carried out to validate the effectiveness of the proposed model, and it shows that the total benefit of EHO and DRA can be improved by 5.75% due to the demand response strategy.

Transactive Energy Sharing in a Microgrid via an Enhanced Distributed Adaptive Robust Optimization Approach

The rapid growth of microgrids with various distributed energy resources (DERs) brings new opportunities for local energy sharing in the microgrids. However, the uncertainties of renewable distributed generation and loads pose a great technical challenge for a microgrid operator (MGO). Thus, this paper proposes a transactive energy sharing (TES) approach for the MGO and DER aggregators to minimize the total social cost, considering network operating constraints. Accordingly, a two-settlement transactive energy (TE) market with an incentive energy pricing scheme is developed to encourage the participation in the energy sharing. Besides, the real-time energy transactions of the aggregators are considered in a day-ahead optimization stage to address their negative impacts on microgrid operation. To solve the proposed TES problem, an alternating direction method of multipliers (ADMM) is applied. The uncertainties in each ADMM local problem are further addressed by an adaptive robust optimization (ARO) method solved by a column-and-constraint generation (C&CG) algorithm. The updated dual and coupling variables at each ADMM iteration could interact with the C&CG algorithm, impairing ADMM convergence. To solve this issue, an alternating uncertainty-update procedure is developed. The simulation results verify the high efficiency and solution robustness of the proposed TES method.

Transmission Expansion Planning Considering Resistance Variations of Overhead Lines Using Minimum-Volume Covering Ellipsoid

Reliable planning of electricity networks is a crucial challenge to maintain the continuous power supply. This can be ensured through careful planning which requires a number of parameters as inputs such as electricity demand, generation capacity, fuel price, renewable intermittent generation, among others. The uncertain behaviors of such parameters have been well studied in literature. Another important parameter which is a vital requirement for reliable planning is the resistance of overhead transmission lines. Traditionally, the current models ignore to consider the resistance variations due to Joule heating and ambient temperature changes in transmission expansion planning (TEP) problem. In this sense, this paper presents an adaptive robust optimization (ARO) framework for TEP to model the uncertain resistance through ellipsoidal uncertainty set. Moreover, a data-driven selection of the ellipsoidal uncertainty set is proposed. In this regard, Khachiyans algorithm (KA) is used to identify the minimum-volume covering ellipsoid (MVCE) that contains all uncertain parameters. A case study based on IEEE 118-bus power system is presented to demonstrate the effectiveness of the proposed method. Simulation results show that resistance uncertainty is of serious concern in the TEP problem since the solutions are highly sensitive to fluctuations in the line resistances.

Robust Scheduling of Networked Microgrids for Economics and Resilience Improvement

The benefits of networked microgrids in terms of economics and resilience are investigated and validated in this work. Considering the stochastic unintentional islanding conditions and conventional forecast errors of both renewable generation and loads, a two-stage adaptive robust optimization is proposed to minimize the total operating cost of networked microgrids in the worst scenario of the modeled uncertainties. By coordinating the dispatch of distributed energy resources (DERs) and responsive demand among networked microgrids, the total operating cost is minimized, which includes the start-up and shut-down cost of distributed generators (DGs), the operation and maintenance (O&M) cost of DGs, the cost of buying/selling power from/to the utility grid, the degradation cost of energy storage systems (ESSs), and the cost associated with load shedding. The proposed optimization is solved with the column and constraint generation (C&CG) algorithm. The results of case studies demonstrate the advantages of networked microgrids over independent microgrids in terms of reducing total operating cost and improving the resilience of power supply.

Min-Sup-Min Robust Combinatorial Optimization with Few Recourse Solutions

In this paper, we consider a variant of adaptive robust combinatorial optimization problems where the decision maker can prepare K solutions and choose the best among them upon knowledge of the true data realizations. We suppose that the uncertainty may affect the objective and the constraints through functions that are not necessarily linear. We propose a new exact algorithm for solving these problems when the feasible set of the nominal optimization problem does not contain too many good solutions. Our algorithm enumerates these good solutions, generates dynamically a set of scenarios from the uncertainty set, and assigns the solutions to the generated scenarios using a vertex p-center formulation, solved by a binary search algorithm. Our numerical results on adaptive shortest path and knapsack with conflicts problems show that our algorithm compares favorably with the methods proposed in the literature. We additionally propose a heuristic extension of our method to handle problems where it is prohibitive to enumerate all good solutions. This heuristic is shown to provide good solutions within a reasonable solution time limit on the adaptive knapsack with conflicts problem. Finally, we illustrate how our approach handles nonlinear functions on an all-or-nothing subset problem taken from the literature. Summary of Contribution: Our paper describes a new exact algorithm for solving adaptive robust combinatorial optimization problems when the feasible set of the nominal optimization problems does not contain too many good solutions. Its development relies on a progressive relaxation of the problem augmented with a row-and-column generation technique. Its efficient execution requires a reformulation of this progressive relaxation, coupled with dominance rules and a binary search algorithm. The proposed algorithm is amenable to exploiting the special structures of the problems considered as illustrated with various applications throughout the paper. A practical view is provided by the proposition of a heuristic variant. Our computational experiments show that our proposed exact solution method outperforms the existing methodologies and therefore pushes the computational envelope for the class of problems considered.

Robust Scheduling of Virtual Power Plant Under Exogenous and Endogenous Uncertainties

Virtual power plant (VPP) provides a flexible solution to distributed energy resources integration by aggregating renewable generation units, conventional power plants, energy storages, and flexible demands. This paper proposes a novel stochastic adaptive robust optimization (SARO) model for determining the optimal self-scheduling plan for VPP’s participation in the day-ahead energy-reserve market. We consider exogenous uncertainties (or called decision-independent uncertainties, DIUs) associated with market clearing prices and available wind generation, as well as endogenous uncertainties (or called decision-dependent uncertainties, DDUs) pertaining to real-time reserve deployment requests. A tractable solution methodology based on modified Benders dual decomposition is developed to effectively solve the proposed SARO model with both DIUs and DDUs. Case studies are conducted to verify the efficiency and applicability of the proposed approach. Comparative results show that the proposed method can mitigate the conservatism of robust strategy by capturing a satisfactory trade-off between profitability and real-time operation feasibility.

Risk-Averse Stochastic Programming vs. Adaptive Robust Optimization: A Virtual Power Plant Application

This paper compares risk-averse optimization methods to address the self-scheduling and market involvement of a virtual power plant (VPP). The decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast. We focus on two methods: risk-averse two-stage stochastic programming (SP) and two-stage adaptive robust optimization (ARO). We investigate both methods concerning formulations, uncertainty and risk, decomposition algorithms, and their computational performance. To quantify the risk in SP, we use the conditional value at risk (CVaR) because it can resemble a worst-case measure, which naturally links to ARO. We use two efficient implementations of the decomposition algorithms for SP and ARO; we assess (1) the operational results regarding first-stage decision variables, estimate of expected profit, and estimate of the CVaR of the profit and (2) their performance taking into consideration different sample sizes and risk management parameters. The results show that similar first-stage solutions are obtained depending on the risk parameterizations used in each formulation. Computationally, we identified three cases: (1) SP with a sample of 500 elements is competitive with ARO; (2) SP performance degrades comparing to the first case and ARO fails to converge in four out of five risk parameters; (3) SP fails to converge, whereas ARO converges in three out of five risk parameters. Overall, these performance cases depend on the combined effect of deterministic and uncertain data and risk parameters. Summary of Contribution: The work presented in this manuscript is at the intersection of operations research and computer science, which are intrinsically related with the scope and mission of IJOC. From the operations research perspective, two methodologies for optimization under uncertainty are studied: risk-averse stochastic programming and adaptive robust optimization. These methodologies are illustrated using an energy scheduling problem. The study includes a comparison from the point of view of uncertainty modeling, formulations, decomposition methods, and analysis of solutions. From the computer science perspective, a careful implementation of decomposition methods using parallelization techniques and a sample average approximation methodology was done . A detailed comparison of the computational performance of both methods is performed. Finally, the conclusions allow establishing links between two alternative methodologies in operations research: stochastic programming and robust optimization.