Two-stage Adaptive Robust Optimization Research Articles

Expansion planning of the transmission network with high penetration of renewable generation: A multi-year two-stage adaptive robust optimization approach

This paper addresses the multi-year two-stage expansion planning of the transmission network of a power system with high penetration of renewable generation modeling long-term uncertainty by using adaptive robust optimization. The multi-year nature of the problem is modeled by considering a comprehensive view of the planning horizon. A cardinality-constrained uncertainty set is used to model the future worst-case uncertainty realization of the peak power consumption of loads, along with the capacity and marginal production cost of generating units. Unlike previous works, we model certain features of the operation that are typically ignored in multi-year robust transmission network expansion planning problems, namely, the operational variability of renewable generating units, the operational flexibility of conventional generating units, and the non-convex operational feasibility sets of storage facilities. The solution procedure employed for this multi-year two-stage robust problem, which is formulated as a three-level problem, is based on the combination of the nested column-and-constraint generation algorithm with two exact acceleration techniques. We analyze the performance of the proposed model through the use of the IEEE 24-bus Reliability Test System and the IEEE 118-bus Test System. Numerical results show that the use of the multi-year approach leads to reductions in the total worst-case cost of up to 7% in comparison with the static and sequential static procedures. Moreover, an underestimation of the total worst-case cost of more than 8% is attained when ignoring certain operational constraints of conventional generating units and storage facilities. Lastly, a sensitivity analysis is presented in order to illustrate the impact of the maximum deviations of the uncertain parameters on the total worst-case cost.

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%.

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.

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.

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.

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.

Two-Stage Robust Edge Service Placement and Sizing Under Demand Uncertainty

Edge computing has emerged as a key technology to reduce network traffic, improve user experience, and enable numerous Internet of Things applications. In this article, we study an optimal resource procurement problem for a service provider (SP), who can purchase resources from various edge nodes in the edge computing market to serve its users’ requests. How to jointly optimize the service placement, resource sizing, and workload allocation decisions is a challenging problem, which becomes even more complicated when considering demand uncertainty. To this end, we propose a novel two-stage adaptive robust optimization framework to help the SP optimally determine the locations for installing its service (i.e., placement) and the amount of computing resource to purchase from each location (i.e., sizing). The proposed placement and sizing solution can hedge against any possible realization within a predefined demand uncertainty set. Given the first-stage robust solution, the optimal resource and workload allocation decisions are computed in the second stage after the uncertainty is revealed. To solve the two-stage model, this article presents an iterative solution approach by employing the column-and-constraint generation method that decomposes the underlying problem into a master problem and a max–min subproblem associated with the second stage. Extensive numerical results are shown to illustrate the efficacy of the proposed model.

Pre-Perturbation Operational Strategy Scheduling in Microgrids by Two-Stage Adjustable Robust Optimization

A two-stage adaptive robust optimization is developed for pre-disturbance scheduling in microgrids (MGs) for handling uncertainties associated with electricity market prices, renewable generation, demand forecasts, and islanding events. The objective is to produce a reliable and optimal solution for MG operation that minimizes operational costs and the risk/failure in islanding events. In the literature, the uncertainty sets associated with islanding events cover a full scheduling period which results in a sub-optimal solution. In this paper, uncertainty sets corresponding to islanding events are modeled based on reliability/resiliency-oriented indexes of the MG/grid to achieve a more accurate/reliable solution. Besides, the Benders decomposition algorithm which is used to handle uncertainties in solving the optimization problem is time-consuming. Therefore, the column-and-constraint generation (C&CG) decomposition strategy is adopted to make the problem computationally tractable. Further, the uncertainty budget parameters are clarified to balance the conservatism and optimality (cost minimization) of the robust solution in uncertainty sets. The effectiveness of the proposed framework is evaluated and discussed by using a set of numerical studies with different scenarios in an MG. The simulations show that the proposed framework reduces operational costs by using the precise analysis of uncertainty budgets and a change in scheduling periods.

Robust Microgrid Scheduling Considering Unintentional Islanding Conditions

This work presents a novel microgrid scheduling model considering the stochastic unintentional islanding conditions as well as forecast errors of both renewable generation and loads. By optimizing the dispatch of distributed energy resources (DERs), utility grid, and demand, the proposed model is targeted to minimize total operating cost of the microgrid, including start-up and shut-down cost of distributed generators (DGs), operation and maintenance (O&M) cost of DGs, cost of buying/selling power from/to utility grid, degradation cost of energy storage systems (ESSs) and cost associated with load shedding. To capture the stochastic unintentional islanding conditions and conventional forecast errors of renewable generation and loads, a two-stage adaptive robust optimization is proposed to optimize the objective function in the worst case scenario of the modeled uncertainties. The proposed optimization is solved with the column and constraint generation (C&CG) algorithm. The result obtained ensures robust microgrid operation in consideration of all possible realization of renewable generation, demand and unintentional islanding condition. The proposed model is validated with results of case studies on a microgrid consisting of various DGs and ESSs.

Data-driven adaptive robust optimization for energy systems in ethylene plant under demand uncertainty

The operational optimization of energy systems is of great significance for improving the overall efficiency of industrial processes. Facing new challenges brought by widespread uncertainties, a data-driven adaptive robust industrial multi-type energy systems optimization framework was proposed by bridging robust optimization and machine learning methods in this paper. The industrial data were used to capture the demand uncertainty of the actual process. Hybrid models of units were first developed considering the operational characteristics, and the energy system optimization model was then formed as a mixed-integer nonlinear programming problem. The uncertain parameter set of process power demands was formed by the process models using historical data of a whole operating period. Afterward, the uncertainty set was constructed by applying the robust kernel density estimation method, which can reduce conservatism by considering the distributional information. By integrating the derived data-driven uncertainty set, a two-stage adaptive robust optimization model aiming at minimizing the weighted total energy consumption was developed. The multi-level robust optimization model was reformulated as a tractable single-level model by employing the affine decision rule. A case study on a plant-wide industrial energy system in the ethylene plant was performed, and the minimum optimal energy consumption was 25,350 kg/h, whose price of robustness was only 2.18%. The robust optimization results can guide the operational optimization of energy systems under uncertainty for the operators of the ethylene plant.

Leveraging Two-Stage Adaptive Robust Optimization for Power Flexibility Aggregation

Adaptive robust optimization (ARO) is a well-known technique to deal with the parameter uncertainty in optimization problems. While the ARO framework can actually be borrowed to solve some special problems without uncertain parameters, such as the power flexibility aggregation problem studied in this paper. To effectively harness the significant flexibility from massive distributed energy resources (DERs), power flexibility aggregation is performed for a distribution system to compute the feasible region of the exchanged power at the substation over time. Based on two-stage ARO, this paper proposes a novel method to aggregate system-level multi-period power flexibility, considering heterogeneous DER facilities, network operational constraints, and an unbalanced power flow model. This method is applicable to aggregate only the active (or reactive) power, and the joint active-reactive power domain. Accordingly, two power aggregation models with two-stage optimization are developed: one focuses on aggregating active power and computes its optimal feasible intervals over multiple periods, and the other solves the optimal elliptical feasible regions for the aggregate active-reactive power. By leveraging the ARO technique, the disaggregation feasibility of the obtained feasible regions is guaranteed with optimality. Numerical simulations on a real-world distribution feeder with 126 multi-phase nodes demonstrate the effectiveness of the proposed method.

Adaptive robust vulnerability analysis of power systems under uncertainty: A multilevel OPF-based optimization approach

With the growing level of uncertainties in today’s power systems, the vulnerability analysis of a power system with uncertain parameters becomes a must. This paper proposes a two-stage adaptive robust optimization (ARO) model for the vulnerability analysis of power systems. The main goal is to immunize the solutions against all possible realizations of the modeled uncertainty. In doing so, the uncertainties are defined by some pre-determined intervals defined around the expected values of uncertain parameters. In our model, there are a set of first-stage decisions made before the uncertainty is revealed (attacker decision) and a set of second-stage decisions made after the realization of uncertainties (defender decision). This setup is formulated as a mixed-integer trilevel nonlinear program (MITNLP). Then, we recast the proposed trilevel program to a single-level mixed-integer linear program (MILP), applying the strong duality theorem (SDT) and appropriate linearization approaches. The efficient off-the-shelf solvers can guarantee the global optimum of our final MILP model. We also prove a lemma which makes our model much easier to solve. The results carried out on the IEEE RTS and modified Iran’s power system show the performance of our model to assess the power system vulnerability under uncertainty.

A two-stage model for asynchronously scheduling offshore wind farm maintenance tasks and power productions

This paper studies the model for scheduling both offshore wind farm maintenance tasks and power productions in consideration of their distinct scheduling timescales and the wind power uncertainty. A two-stage adaptive robust optimization model is formulated to derive the optimal schedule, which well handles wind turbine maintenance tasks and maximizes wind farm power productions by anticipating the worst wind power output scenario based on a wind power uncertainty set and a set of metocean conditions. In two decision stages, we schedule maintenance tasks daily and power productions hourly respectively. The proposed model is equivalently transformed to a form which can be efficiently solved by the column-and-constraint generation algorithm. Numerical experiments demonstrate the effectiveness and applicability of the proposed model considering wind farms of different large sizes. Results show that the developed model can maximize the total power production of the wind farm and enhance the robustness of obtained maintenance schedules against the wind power uncertainty.

Multi-objective two-stage adaptive robust planning method for an integrated energy system considering load uncertainty

An integrated energy system (IES) is considered as an effective approach to reduce carbon emissions, lower energy supply costs and increase system flexibility. As numerous energy conversion and storage devices with various features have been developed, determining their types and capacity size and optimizing the synergic action of all selected energy devices under load uncertainty have become challenging issues during the planning and designing of a new IES. To address these issues, a load uncertainty model is constructed by integrating the regional equivalent standard building-integrated load prediction method, the robust uncertainty set method and the fuzzy C-means clustering algorithm. Based on this uncertainty model, this paper proposes a planning method that incorporates the fuzzy multi-objective decision and two-stage adaptive robust optimization methods for handling these challenging issues. The multi-objective decision method enables comprehensive optimization based on the system's economic performance, carbon emissions and grid integration level. The two-stage adaptive robust optimization method aims to optimize the single-objective problem transformed by the multi-objective decision method. The planning method is implemented in an industrial park in the Baoding City of China. The influence of load budget uncertainty, annual load growth rate and energy purchase price is then investigated. The results show that the load budget uncertainty and annual load growth rate only affect the optimal capacity size of the selected devices, whereas the energy purchase price influences both the optimal device types and their capacity size. The optimal capacity size remains constant when the load budget uncertainty exceeds 3. This study provides a methodology to select the suitable device types and determine their capacity size for IES under load uncertainty, which will benefit the design of a new IES.

A Two-stage Adaptive Robust Model for Residential Micro-CHP Expansion Planning

This paper addresses the planning problem of residential micro combined heat and power (micro-CHP) systems (including micro-generation units, auxiliary boilers, and thermal storage tanks) considering the associated technical and economic factors. Since the accurate values of the thermal and electrical loads of this system cannot be exactly predicted for the planning horizon, the thermal and electrical load uncertainties are modeled using a two-stage adaptive robust optimization method based on a polyhedral uncertainty set. A solution method, which is composed of column-and-constraint generation (C&CG) algorithm and block coordinate descent (BCD) method, is proposed to efficiently solve this adaptive robust optimization model. Numerical results from a practical case study show the effective performance of the proposed adaptive robust model for residential micro-CHP planning and its solution method.

Robust transmission expansion planning with uncertain generations and loads using full probabilistic information

Recent advances in Transmission Network Expansion Planning (TNEP) have demonstrated that two-stage adaptive robust optimization (ARO) renders the expansion planning problem tractable for real systems and, at the same time, constitutes a relevant approach to deal with uncertain demand and generation capacity in the TNEP problem. However, the use of cardinality and polyhedral uncertainty sets does not consider the correlation of uncertain parameters and makes the interpretation of final results difficult from a probabilistic view-point. The recent use of level II structural-reliability and decomposition techniques allows the incorporation of ellipsoidal uncertainty sets, which permits the use of expected values and the variance-covariance matrix of uncertain parameters (First-Order Second-Moment, FOSM). However, it also constitutes an approximation. This paper presents an iterative method which, using FOSM and appropriately modifying the reliability index associated with the desired objective function quantile, transforms the model into a level III structural-reliability method that allows the consideration of the complete probabilistic structure of the uncertain parameters. Numerical results from a realistic case study are presented and discussed.

Decentralized robust energy and reserve Co-optimization for multiple integrated electricity and heating systems

The growing uncertainty caused by the increasing wind power penetration brings an urgent need of increased reserve capacity to ensure the secure operation of power systems. The interconnection of multiple integrated electricity and heating systems (IEHSs) allows resource sharing among them and the district heating system can provide additional reserves to the electric power system. In this paper, a decentralized robust energy and reserve co-optimization for multiple IEHSs is proposed to handle wind power uncertainty in a more economic and efficient way. In each IEHS, the day-ahead energy and reserve scheduling is formulated as a two-stage adaptive robust optimization, where the feasibility of day-ahead reserve deployment in the real-time operation is guaranteed. Then, a decentralized methodology is developed based on the alternating direction method of multipliers (ADMM) to achieve the synergistic yet independent operation by exchanging tie-line power flows with adjacent IEHSs. In order to accelerate the convergence of the decentralized operation and reduce the communication demand, the improved ADMM with locally adaptive penalty parameters is adopted. Simulation results demonstrate the effectiveness of the proposed decentralized robust energy and reserve optimization scheme in terms of improving the economic efficiency and ensuring the secure operation for IEHSs, as well as preserving information privacy.

Two-Stage Robust Sizing and Operation Co-Optimization for Residential PV–Battery Systems Considering the Uncertainty of PV Generation and Load

This article presents a two-stage adaptive robust optimization (ARO) for optimal sizing and operation of residential solar photovoltaic (PV) systems coupled with battery units. Uncertainties of PV generation and load are modeled by user-defined bounded intervals through polyhedral uncertainty sets. The proposed model determines the optimal size of PV–battery system while minimizing operating costs under the worst-case realization of uncertainties. The ARO model is proposed as a trilevel min–max–min optimization problem. The outer min problem characterizes sizing variables as “here-and-now” decisions to be obtained prior to uncertainty realization. The inner max–min problem, however, determines the operation variables in place of “wait-and-see” decisions to be obtained after uncertainty realization. An iterative decomposition methodology is developed by means of the column-and-constraint technique to recast the trilevel problem into a single-level master problem (the outer min problem) and a bilevel subproblem (the inner max–min problem). The duality theory and the Big-M linearization technique are used to transform the bilevel subproblem into a solvable single-level max problem. The immunization of the model against uncertainties is justified by testing the obtained solutions against 36 500 trial uncertainty scenarios in a postevent analysis. The proposed postevent analysis also determines the optimum robustness level of the ARO model to avoid over/under conservative solutions.

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%.