Multistage Robust Optimization Research Articles

Deterministic scenarios guided [formula omitted]-Adaptability in multistage robust optimization for energy management and cleaning scheduling of heat transfer process

The heat transfer process is of paramount importance for energy management and heat recovery, but the inevitable uncertain fouling poses significant challenges to sustainable energy-saving efforts. This study aims to solve the energy management and cleaning scheduling problems under fouling uncertainty. To achieve this, a novel multistage Robust Optimization (RO) method based on the Deterministic Scenarios Guided K-Adaptability (DSGKA) strategy is proposed. The process scheduling problem is initially tackled through long-period Mixed Integer Optimal Control Problems (MIOCPs) with deterministic scenarios. Subsequently, considering the slow transition characteristics of energy efficiency degradation caused by fouling, the candidate paths generation algorithm guided by deterministic MIOCPs is developed. Additionally, the K-Adaptability method is employed to segment decision-dependent uncertainty sets into finite partitions, facilitating the resolution of the multistage RO problem through worst-case analysis. In theory, the rationality of the proposed guidance algorithm is established. Simulation results on a heat exchanger network are also provided to demonstrate the effectiveness of the DSGKA scheme. By transforming the original multistage RO problem into a finite-dimensional optimization problem solvable with intelligent heuristic algorithms, the proposed method adeptly manages nonlinearity in constraint equations, thereby improving its applicability for equipment maintenance scheduling across various slow transition industrial processes.

Robust optimization for integrated energy systems based on multi-energy trading

Amid rising energy demands and environmental concerns, integrated energy systems (IESs) face conflicting interests. Conventional strategies of interest distribution face issues such as irrational resource allocation. Accordingly, establishing energy trading strategies with multiple stakeholders becomes essential. This paper proposes a robust optimization (RO) for IESs based on multi-energy trading to reduce energy trading cost. In this regard, a single-leader-multi-follower Stackelberg game is first modeled where IES acts as the leader, and the users and the electric vehicles (EVs) act as the followers. Secondly, this model is transformed into a single-layer linear model and integrated into the multi-stage RO. With this, the nonanticipativity in the two-stage RO can be effectively handled. Besides, by constructing multi-interval uncertainty sets for renewable energy, load, and electricity prices, the conservatism of the model is reduced, making the optimization results closer to actual condition. Noteworthy that the Nash bargaining method ensures a fair distribution of benefits among IESs and encourages them to participate in energy trading. Finally, the multi-energy trading model is solved using the prediction-correction-based alternating direction method with multipliers (PCB-ADMM) algorithm. The PCB-ADMM not only protects each IES’s privacy but also has less execution time than the ADMM algorithm. The effectiveness of the proposed strategy is validated through simulation using Matlab.

Distributed Cooperative Optimal Operation of Multiple Virtual Power Plants Based on Multi-Stage Robust Optimization

This paper develops a distributed cooperative optimization model for multiple virtual power plant (VPP) operations based on multi-stage robust optimization and proposes a distributed solution methodology based on the combination of the alternating direction method of multipliers (ADMMs) and column-and-constraint generation (CCG) algorithm to solve the corresponding optimization problem. Firstly, considering the peer-to-peer (P2P) electricity transactions among multiple VPPs, a deterministic cooperative optimal operation model of multiple VPPs based on Nash bargaining is constructed. Secondly, considering the uncertainties of photovoltaic generation and load demand, as well as the non-anticipativity of real-time scheduling of VPPs in engineering, a cooperative optimal operation model of multiple VPPs based on multi-stage robust optimization is then constructed. Thirdly, the constructed model is solved using a distributed solution methodology based on the combination of the ADMM and CCG algorithms. Finally, a case study is solved. The case study results show that the proposed method can realize the optimal scheduling of renewable energy in a more extensive range, which contributes to the promotion of the local consumption of renewable energy and the improvement of the renewable energy utilization efficiency of VPPs. Compared with the traditional deterministic cooperative optimal operation method of multiple VPPs, the proposed method is more resistant to the risk of the uncertainties of renewable energy and load demand and conforms to the non-anticipativity of real-time scheduling of VPPs in engineering. In summary, the presented works strike a balance between the operational robustness and operational economy of VPPs. In addition, under the presented works, there is no need for each VPP to divulge personal private data such as photovoltaic generation and load demand to other VPPs, so the security privacy protection of each VPP can be achieved.

From Viewpoint of Reserve Provider: A Day-ahead Multi-stage Robust Optimization Reserve Provision Method for Microgrid with Energy Storage

From Viewpoint of Reserve Provider: A Day-ahead Multi-stage Robust Optimization Reserve Provision Method for Microgrid with Energy Storage

Multi-Stage Robust Optimization of Real-Time Dynamic Dispatch with Fast-Acting Units in Resilient Power Systems

Multi-Stage Robust Optimization of Real-Time Dynamic Dispatch with Fast-Acting Units in Resilient Power Systems

On the complexity of robust multi-stage problems with discrete recourse

We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known.In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove Σ3p-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant difference in computational complexity between column-wise and row-wise uncertainty.

Improved Decision Rule Approximations for Multistage Robust Optimization via Copositive Programming

Improved decision rule approximations for multistage robust optimization via copositive programming Previous research in the field has proposed several approaches to tackle multistage robust optimization problems, but they are often limited in their applicability. These existing methods either fail to handle cases where recourse matrices are uncertain or struggle to handle large-scale problems effectively. In their paper titled “Improved decision rule approximations for multistage robust optimization via copositive programming,” Guanglin Xu and Grani A. Hanasusanto contribute to the robust optimization literature by presenting a novel solution method. Their approach utilizes convex conic techniques and aims to address the general case of multistage robust optimization, where uncertainty exists in the recourse matrices. One significant advantage of their proposed method is its ability to scale well with large-sized instances, overcoming a common limitation faced by previous methods. Through numerical experiments on various simulated applications, Xu and Hanasusanto demonstrate the superiority of their algorithm over existing state-of-the-art methods.

Multistage robust optimization for the day-ahead scheduling of hybrid thermal-hydro-wind-solar systems

Multistage robust optimization for the day-ahead scheduling of hybrid thermal-hydro-wind-solar systems

Multi-stage risk-based assessment for wind energy accommodation capability: A robust and non-anticipative method

The volatility and uncertainty of wind energy has brought great challenges to its accommodation in electric power systems. To determine a wind power trading strategy in the real-time power market, the accurate assessment of wind energy accommodation capability (WAC) is of great significance. However, most current studies investigate the wind energy accommodation potential from the perspective of dispatch, but rarely provide quantitative assessment results. Additionally, the non-anticipativity of decisions has not been considered in the assessment yet. This paper proposes a novel quantitative assessment method for WAC based on the multi-stage robust optimization (RO), which addresses the anticipativity issues in the traditional two-stage method while maintaining the decision-making process' robustness. In the assessment method, the operational risk is integrated via a tractable scheme to obtain dynamic admissible WAC boundaries. To obtain the optimal solution of the proposed multi-stage RO problem, a fast robust dual dynamic programming (FRDDP) algorithm is employed, where the adaptive techniques are developed to accelerate the computation. Numerical studies on the modified IEEE 14-Bus system and a real-world system in Zhejiang Province of China validate the effectiveness of the proposed assessment model and adaptive acceleration techniques. The simulation results demonstrate the presented method brings a 18.87% reduction in operational cost, and reduces 29.34% curtailment of wind energy. Compared with the original FRDDP, the adaptive technique significantly reduces the computational consumption by 73.34% on the real-world test system.

Rolling Horizon Robust Real-Time Economic Dispatch with Multi-Stage Dynamic Modeling

A multi-stage robust real-time economic dispatch model (MRRTD) for power systems is proposed in this paper. The MRRTD takes the dynamic form of multi-stage robust optimization as the framework to naturally simulate the operation of equipment that is temporally coupled, e.g., utility-level energy storage systems. For normal systems, the MRRTD can work directly in short time slots with a rolling horizon. For large-scale systems, the MRRTD expands the time-slot scale and generates optimal dispatch policies. With this guidance, the real-time dispatch decision can be swiftly made thereafter. In addition, a dynamic uncertainty set based on deep learning is proposed, which can dynamically refine the covering ability for probable occurred wind power scenarios. To efficiently solve the MRRTD, a novel fast robust dual dynamic programming method is employed. The effectiveness of the proposed model and solution algorithm, especially the improved scalability compared to several other dynamic economic dispatch methods, are demonstrated by simulation results from six benchmark test cases ranging from a modified IEEE 6-bus system to a 6495-bus system.

Robust multi-stage economic dispatch with renewable generation and storage

In this paper, an economic dispatch problem for power grids involving renewable generation and storage units is studied. A multi-stage robust optimization algorithm for computing base dispatches and re-dispatches is proposed. Unlike the other approaches in the literature, the proposed method finds an exact solution to the multi-stage robust optimization problem considered when there are no storage devices or storage devices are ideal (i.e. charging/discharging inefficiencies are neglectable). In the presence of non-ideal storage devices, the algorithm can produce upper-bounding approximations. The devised approximation scheme has two distinguishing features. Firstly, the base dispatches and, especially, multi-stage re-dispatch policies computed are guaranteed to be free of simultaneous charges/discharges. Secondly, it produces lower bounds allowing one to assess the quality of solutions. Optimality and convergence properties of the proposed method are proven and it is compared with other alternatives in the literature through numerical experiments. Results show that significant economic benefits can be achieved compared to the other approaches and solutions could be obtained within acceptable time limits for large-scale problems.

Multistage Mixed-Integer Robust Optimization for Power Grid Scheduling: An Efficient Reformulation Algorithm

The penetration of renewables into power systems is gradually increasing, but its inherent uncertainty poses tremendous challenges to the coordinated operation of power grids. To overcome the demerits of canonical two-stage mixed-integer robust optimization (RO) method that neglects the nonanticipativity requirement and the computational burden in engineering applications, this paper offers a reformulated multistage mixed-integer RO method for regional power grid scheduling. Firstly, a multistage mixed-integer RO scheduling model is established considering the scheduling requests in real-world projects. The scheduling plans under the nominal scenario are determined ahead of uncertainty in the first-stage optimization, and the multistage max-min optimization with binary recourse variables is then executed for feasibility-checking with respect to the nonanticipative realization of uncertainty. Secondly, a dedicated reformulation algorithm is proposed for this intractable multistage mixed-integer RO model. Based on the implicit affine strategy, the multistage max-min optimization is equivalently encapsulated to a single-stage max-min problem, and the dual Fourier-Motzkin elimination is put forward to eliminate both continuous and binary recourse variables in the resulting feasibility-checking optimization. Therefore, the original multistage mixed-integer RO is finally recast as a mixed-integer linear programming that can be solved directly. Numerical tests on an actual 16-bus grid, the IEEE 118-bus system and the 319-bus provincial grid verify the superiority and applicability of the proposed reformulated mixed-integer RO scheduling method, which is of great significance in guiding system operations.

Decentralized Optimization of Multiarea Interconnected Traffic-Power Systems With Wind Power Uncertainty

The high proliferation of electric vehicles has intensified the interdependency between traffic networks (TNs) and power distribution networks (PDNs). Accordingly, this article proposes a decentralized optimization framework for the multiarea optimal traffic-power flow (OTPF) problem, in which the models of PDNs and TNs are established separately in a distributed manner. A multistage distributionally robust optimization (MDRO) model is formulated for PDNs to address wind power uncertainty, with the introduction of a traffic assignment problem to describe the distribution of traffic flows in TNs. Furthermore, an improved alternative direction method of multipliers (I-ADMM) algorithm is developed to solve the multiarea OTPF problem. Numerical results from a three-area traffic-power coupled system demonstrate that the proposed MDRO model bears a cost $89.50 lower than that of the multistage robust optimization model, while the I-ADMM algorithm yields a solving time only one-third that of the traditional ADMM.

Data-Driven Optimization with Distributionally Robust Second Order Stochastic Dominance Constraints

This paper presents the first comprehensive study of a data-driven formulation of the distributionally robust second order stochastic dominance constrained problem (DRSSDCP) that hinges on using a type-1 Wasserstein ambiguity set. It is, furthermore, for the first time shown to be axiomatically motivated in an environment with distribution ambiguity. We formulate the DRSSDCP as a multistage robust optimization problem and further propose a tractable conservative approximation that exploits finite adaptability and a scenario-based lower bounding problem. We then propose the first exact optimization algorithm for this DRSSDCP. We illustrate how the data-driven DRSSDCP can be applied in practice on resource-allocation problems with both synthetic and real data. Our empirical results show that, with a proper adjustment of the size of the Wasserstein ball, DRSSDCP can reach acceptable out-of-sample feasibility yet still generating strictly better performance than what is achieved by the reference strategy.

Multistage Scheduling of Regional Power Grids Against Sequential Outage and Power Uncertainties

Aiming at the scheduling problem for regional power grids under both outage and power uncertainties, this paper puts forward a multistage robust optimization (RO) method that ensures the reliable power supply in system operations. Firstly, a multistage RO model is proposed according to the practical scheduling request of the regional power grid. The startup and shutdown of slow-acting generators are optimized in the first stage. The optimal topology is then determined in the second stage in preparation to the outage uncertainty of grid-connected lines. In the third to 2+ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${T}$ </tex-math></inline-formula> stages, the scheduling plans of generation units in each time period are further formulated regarding the sequential realizations of renewable-load power uncertainties, which ensures the nonanticipativity of the obtained robust schedules. Secondly, a tailored and efficient transformation-decomposition algorithm is exploited for the solution of multistage RO problem. Based on the affine control function, the nested max-min optimization of the third to 2+ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${T}$ </tex-math></inline-formula> stages is explicitly transformed to a single-stage max-min model via the multi-to-one strategy. A column and constraint generation (C&CG) algorithm with looped alternative optimization procedure (AOP) handles the equivalent three-stage RO problem with binary recourses, thus avoiding numerous forward and backward iterations as well as speeding up the modeling solution. Numerical experiments of real-world regional power grids verify the effectiveness, superiorities and scalability of the proposed multistage RO scheduling method, which indicates great guidance for engineering applications.

Optimality-feasibility-aware multistage unit commitment considering nonanticipative realization of uncertainty

The intrinsic uncertainty in source-load power and line outage poses huge obstacles to unit commitment (UC) optimization in power grids. To satisfy the practical requests in power grids, this paper exploits a multistage robust UC method to promise both optimality and feasibility in nonanticipative scheduling. Firstly, a multistage robust UC model is established accounting for the sequential realization of uncertainty. This model makes decisions with the goals of economy in the normal condition and power balance in the emergent condition. Secondly, a customized multi-cut decomposition algorithm is offered to address the intractable multistage robust optimization problem. The original multistage problem is decoupled into the form of one master and several slave problems (MP-SPs), where the robust dual dynamic programming (RDDP) solves the resulting multistage max–min SPs, and the column-and-constraint generation (C&CG) algorithm successively constructs multiple cutting planes to ensure the optimality and convergence of the alternative optimizations between the MP and SPs. Computational tests on real-size power grids validate the practicability and superiority of the proposed multistage robust UC method, which is of great significance in guiding system operations.

Multi-stage robust clearing model considering renewable energy output uncertainty and unit effective reserve calculation in electricity market

With the advancement of China’s carbon policy, the proportion of renewable energy, mainly wind and solar, has increased, which brings greater challenges to the reservation of power systems. Due to the complexity of China’s power systems, adopting the method of the zonal reserve to ensure stable system operation is very difficult. Among the existing clearing results for calculating reservation, the unit reservation will be restricted by the network security constraints, leading to system operation risks. To obtain the unit reservations efficiently without breaking the security constraints, this paper proposes an effective reserve calculation method for engineering implementation. A box robust optimization is further utilized for Security Constrained Unit Commitment in the method to ensure the consumption of renewable energy, and the stability and clearing efficiency of the power system. Furthermore, a data-driven robust random optimization is employed for Security Constrained Economic Dispatch optimizing the economy of power systems. This multi-stage robust optimization model, which has good extensibility, is in line with China’s clearing process. The simulation analysis based on the actual operation data of a province in China and the IEEE 300-bus system verifies the correctness and feasibility of the proposed model and theory.

Multi-stage online robust parameter design based on Bayesian GP model

Online robust parameter design (RPD) for the complex production process has recently attracted increasing attention among researchers and practitioners. However, the existing online RPD methods usually ignore the model uncertainty of initial steps, which may lead to the overestimated optimal solutions in the early stage of online RPD. This paper proposes a multi-stage robust optimization approach based on the Bayesian Gaussian process (BGP) model to improve the robustness of the optimal solutions of the online RPD process. First, the Gibbs sampling method is used to estimate the hyperparameters of the BGP model. Second, the global optimization and clustering analysis techniques are combined to determine the optimal design region of input variables. Consequently, the Bayesian posterior probability analysis technique is used to obtain the optimal robust design region for performing the online parameter optimization. Finally, an online RPD model is constructed by integrating the global optimization algorithm, parameter update strategy, and quality loss function. The proposed approach is validated through a simulation example and a laser drilling case study. The comparison results show that the proposed approach obtains more robust optimal solutions than the existing ones.

Integrated Planning of Cyber-Physical Active Distribution System Considering Multidimensional Uncertainties

The integration of cyber technologies is transforming the power distribution network into a cyber-physical active distribution system (CPADS), and brings more flexibility to its operation. However, various uncertainties from the cyber space and physical space are emerging and coupled with each other, which results in significant challenges for the distribution network planning and operation. Thus, a bilevel integrated planning model accounting for multidimensional uncertainties from both the cyber space and physical space is proposed for the CPADS in this paper. With the goal of minimizing the annual investment and operation costs, the model comprehensively considers the location and selection of active-management elements (e.g., electrical storage systems, capacitor banks, and switches), investment constraints, power flow constraints, and operation constraints for all the active-management elements, etc. Moreover, based on the typical cyber system failure scenario represented by information link failure (ILF), a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N-k</i> uncertainty set considering the failure probability distribution is proposed, while the uncertainties of renewable energy and load demand are characterized by box-like uncertainty sets. Finally, the proposed planning model is recast into a multi-stage robust optimization model which can be divided into a master problem and two-step subproblems (SP-1 and SP-2) to be solved by the column and constraint generation (CCG) algorithm. Case studies and comparative analysis on the PG 69 distribution system are performed to verify the effectiveness of the proposed method.

Multistage Robust Optimization of Routing and Scheduling of Mobile Energy Storage in Coupled Transportation and Power Distribution Networks

Mobile energy storage systems (MSSs) manifest a significant potential for enhancing the reliable and economic operations of distribution systems with high photovoltaic (PV) penetrations. This article proposes a robust and dynamic MSS scheduling method, which includes MSS mobility and its power management, in a coupled transportation and power distribution network. A rolling-horizon multistage robust optimization model with integer and linear decisions is proposed to schedule the MSS mobility and its charging and discharging strategies, considering uncertain traffic conditions, PV power output, and load demands, as well as nonanticipativity MSS operation constraints. The proposed model is transformed into a mixed-integer linear programming (MILP) problem to achieve computational tractability based on mixed-linear and binary decision rules, and the duality theory. The proposed dynamic MSS scheduling method is tested on the augmented 33- and 123-bus distribution systems with actual solar irradiance, load, and traffic data; the simulation results are compared with those considering day-ahead MSS scheduling and stationary energy storage systems to verify the effectiveness of the proposed dynamic MSS operation.