Tri-level Optimization Research Articles

A tri-level optimization model for interdependent infrastructure network resilience against compound hazard events

Resilient operation of interdependent infrastructures against compound hazard events is essential for maintaining societal well-being. To address consequence assessment challenges in this problem space, we propose a novel tri-level optimization model applied to a proof-of-concept case study with fuel distribution and transportation networks – encompassing one realistic network; one fictitious, yet realistic network; as well as networks drawn from three synthetic distributions. Mathematically, our approach takes the form of a defender-attacker-defender (DAD) model—a multi-agent tri-level optimization, comprised of a defender, attacker, and an operator acting in sequence. Here, our notional operator may choose proxy actions to operate an interdependent system comprised of fuel terminals and gas stations (functioning as supplies) and a transportation network with traffic flow (functioning as demand) to minimize unmet demand at gas stations. A notional attacker aims to hypothetically disrupt normal operations by reducing supply at the supply terminals, and the notional defender aims to identify best proxy defense policy options which include hardening supply terminals or allowing alternative distribution methods such as trucking reserve supplies. We solve our DAD formulation at a metropolitan scale and present practical defense policy insights against hypothetical compound hazards. We demonstrate the generalizability of our framework by presenting results for a realistic network; a fictitious, yet realistic network; as well as for three networks drawn from synthetic distributions. We also analyze the sensitivity of outputs on budget constraints through a detailed case study. Additionally, we demonstrate the scalability of the framework by investigating runtime performance as a function of the network size. Steps for future research are also discussed.

Traffic Toll Design for Dynamic Wireless Charging in Coupled Power-Transportation Networks: A Tri-Level Optimization Approach

Traffic Toll Design for Dynamic Wireless Charging in Coupled Power-Transportation Networks: A Tri-Level Optimization Approach

Resilience enhancement by line hardening for communication routing considering renewable energy sources in cyber‐physical power systems

AbstractCommunication routing of cyber‐physical power systems (CPPS) with the high penetration of renewable energy sources (RES) plays an important role in the resilience enhancement against disasters, natural and man‐made alike. Therefore, a trilevel optimization model is proposed for the CPPS resilience enhancement with the RES against extreme events. The upper‐level model identifies optimal hardening lines of both the transmission and communication networks with consideration of the communication routing constraints. The middle‐level model identifies the attacked lines to maximize load shedding of the power system. The effects of RES uncertainty and communication routings on resilience enhancement are analysed. The lower‐level model attains an optimal allocation strategy of power generation to minimize load shedding. The model is solved by the column‐and‐constraint generation algorithm. Case studies are conducted on the IEEE 14‐bus, RTS 24‐bus, and 118‐bus system. The results show that the proposed hardening strategy can effectively ensure the adaptiveness of the transmission network and communication routing to improve the resilience of CPPS to extreme events. Moreover, it is observed that the load loss and total investment cost are heavily affected without and with RES.

An exact method for trilevel hub location problem with interdiction

In this paper, we study the problem of designing a hub network that is robust against deliberate attacks (interdictions). The problem is modeled as a three-level, two-player Stackelberg game, in which the network designer (defender) acts first to locate hubs to route a set of flows through the network. The attacker (interdictor) acts next to interdict a subset of the located hubs in the designer’s network, followed again by the defender who routes the flows through the remaining hubs in the network. We model the defender’s problem as a trilevel optimization problem, wherein the attacker’s response is modeled as a bilevel hub interdiction problem. We study such a trilevel problem on three variants of hub location problems studied in the literature namely: p-hub median problem, p-hub center, and p-hub maximal covering problems. We present a cutting plane based exact method to solve the problem. The cutting plane method uses supervalid inequalities, which is obtained from the solution of the lower level interdiction problem. To solve the lower level hub interdiction problem efficiently, we propose a penalty-based reformulation of the problem. Using the reformulation, we present a branch-and-cut based exact approach to solve the problem efficiently. We conduct experiments to show the computational advantages of the above algorithm. To the best of our knowledge, the cutting plane approach proposed in this paper is among the first exact method to solve trilevel location–interdiction problems. Our computational results show interesting implications of incorporating interdiction risks in the hub location problem.

Low-carbon optimal scheduling of park-integrated energy system based on bidirectional Stackelberg-Nash game theory

Multi-stakeholder participation is crucial in facilitating the development of park-integrated energy systems (PIES). Balancing the diverse interests of various stakeholders, each with its distinct requirements presents a notable challenge. Concurrently, the model's complexity increases due to the engagement of various stakeholders, posing challenges to its resolution through traditional methods. In this context, this paper aims to investigate an optimal scheduling model that incorporates shared energy storage (SES) system, microgrids operator (MGO), electric vehicles station (EVS), and user aggregator (UA) with multiple prosumers. To comprehensively address the interests of all stakeholders, this study introduces a tri-level optimization model. The proposed model integrates a bidirectional Stackelberg-Nash game framework, in which the SES acts as the leader, the MGO acts as the secondary leader, and the UA-EVS acts as the followers while allocating benefits based on the asymmetric Nash bargaining theory. The Stackelberg game model between MGO and UA-EVS is analyzed using the Karush-Kuhn-Tucker (KKT) condition, while the Stackelberg game model between SES and MGO is resolved using the bisection method. Meanwhile, the Nash bargaining method among users is solved using the alternating direction method of multipliers (ADMM) technique. The analysis indicates that the proposed strategy can reduce PIES's costs and carbon emissions, yielding a win-win situation for all stakeholders.

Optimal scheduling of a community Multi-Energy system in energy and flexible ramp markets considering Vector-Coupling storage Devices: A hybrid Fuzzy-IGDT/Stochastic/Robust optimization framework

Community multi-energy systems (CMESs) are emerging as a pivotal solution to address the pressing issues of the current energy crisis and environmental pollution. This paper introduces a comprehensive multi-horizon risk-averse scheduling strategy for CMESs, targeting participation in energy and flexible ramp markets while meeting the demands of electric, natural gas, and hydrogen vehicle charging stations, data centers, and residential and commercial buildings. The proposed CMES incorporates power-to-x vector coupling storage (PtX-VCS) systems, enabling the storage of electricity in various forms to cater to multi-energy demands. Additionally, an integrated demand response (IDR) program, incorporating incentive and load-shifting strategies, is applied to enhance operational efficiency and system flexibility. To manage uncertainties effectively, a hybrid stochastic/robust/fuzzy information gap decision theory (IGDT) method is employed. This approach, formulated as a tri-level optimization model, is transformed into a single-level model using strong duality theory. The results demonstrate a notable reduction in daily operational costs by approximately 5% and 4% through the implementation of the IDR program and PtX-VCS technologies, respectively. Moreover, the CMES’s potential to participate in the flexible ramping market leads to an approximately 7% reduction in total operational costs.

Preventing tacit collusion in deregulated electricity markets through trilevel optimization

Preventing tacit collusion in deregulated electricity markets through trilevel optimization

Unlocking the potential of rooftop solar panels: An incentive rate structure design

With the continuous maturity of distributed generation technology, rooftop solar panels have come to play an increasingly critical role in the construction of future energy systems. However, there remains an open question as to how to design adequate rate structure and provide proper incentives for distributed solar production. In this paper, we propose an incentive rate structure for rooftop solar panels to fairly unlock the potential of such distributed energy resources. The interaction of residential consumers and utilities is formulated as a tri-level optimization model to evaluate the impact of the rate structure design on the welfare transfer and investment behavior of residential consumers. An incentive rate structure is designed in which the supply-side effects of rooftop solar panel investment are considered by explicitly modeling wholesale market dynamics. Contrary to conventional belief, we indicate that the integration of rooftop solar panels can increase the expected profits of the utility company in some usual cases. Case studies demonstrate that through adequate rate structure design, the market operator can achieve the established social goals while balancing the welfare transfer effect among different market entities. Hopefully, our work can provide novel insights and a theoretical reference for the relevant policy design.

Provably Convergent Federated Trilevel Learning

Trilevel learning, also called trilevel optimization (TLO), has been recognized as a powerful modelling tool for hierarchical decision process and widely applied in many machine learning applications, such as robust neural architecture search, hyperparameter optimization, and domain adaptation. Tackling TLO problems has presented a great challenge due to their nested decision-making structure. In addition, existing works on TLO face the following key challenges: 1) they all focus on the non-distributed setting, which may lead to privacy breach; 2) they do not offer any non-asymptotic convergence analysis which characterizes how fast an algorithm converges. To address the aforementioned challenges, this paper proposes an asynchronous federated trilevel optimization method to solve TLO problems. The proposed method utilizes u-cuts to construct a hyper-polyhedral approximation for the TLO problem and solve it in an asynchronous manner. We demonstrate that the proposed u-cuts are applicable to not only convex functions but also a wide range of non-convex functions that meet the u-weakly convex assumption. Furthermore, we theoretically analyze the non-asymptotic convergence rate for the proposed method by showing its iteration complexity to obtain ϵ-stationary point is upper bounded by O(1/ϵ²). Extensive experiments on real-world datasets have been conducted to elucidate the superiority of the proposed method, e.g., it has a faster convergence rate with a maximum acceleration of approximately 80%.

Stacking Revenues From Flexible DERs in Multi-Scale Markets Using Tri-Level Optimization

Rapid proliferation of flexible Distributed Energy Resources (DERs) as a result of Net Zero Emissions objectives entails a profound shift in the paradigm of local and national energy systems. Currently, DERs' simultaneous participation in multiple markets is generally restricted, which undermines their profitability. With the aim of increasing the number of business cases for them, a tri-level optimization problem that seeks the maximisation of revenues from DERs is proposed. The optimization problem considers simultaneous participation of different flexible DERs, such as, Electric Vehicles (EVs), Battery Energy Storage Systems (BESSs) and Heating, Ventilation and Air Conditioning (HVACs), in national and local markets. Markets are cleared sequentially, and the model is recast into a tractable single-level problem using its dual formulation and strong duality condition. Results from a case study based on the IEEE 14 bus transmission network, a realistic distribution network and SimBench dataset demonstrate the effectiveness of the proposed approach in increasing profits compared with a baseline scenario.

Trilevel and multilevel optimization using monotone operator theory

AbstractWe consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term. Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters.

Power generation allocation of cyber–physical power systems from a defense–attack–defense perspective

Cyber–physical power systems have emerged as intelligent devices integrate into traditional power systems. These systems, featuring an integration of the physical and cyber components, are exposed to compounded risks from both realms. This paper explores the optimization of generation node allocation with a defense–attack–defense perspective to bolster the system’s resistance to external deliberate attacks. A trilevel optimization model is introduced, addressing the complications arising from communication line failures. This model encompasses upper-level defenders orchestrating generation allocation, middle-level attackers executing deliberate attacks, and lower-level operators optimizing system performance. The model leverages the Column-and-Constraint Generation (C&CG) approach to derive the optimal generation allocation strategy. Empirical validation using two IEEE test cases demonstrates the model’s effectiveness. Results highlight a significant escalation in total load loss upon communication line failures, and underscore the critical role of decentralized generation allocation in enhancing system resilience. These findings hold the potential to inform strategies for safeguarding cyber–physical power systems and reinforcing the architecture of resilient smart grids.

Look-ahead scheduling of energy-water nexus integrated with Power2X conversion technologies under multiple uncertainties

Co-optimizing energy and water resources in a microgrid can increase efficiency and improve economic performance. Energy-water storage (EWS) devices are crucial components of a high-efficient energy-water microgrid (EWMG). The state of charge (SoC) at the end of the first day of operation is one of the most significant variables in EWS devices since it is used as a parameter to indicate the starting SoC for the second day, which influences the operating cost for the second day. Hence, this paper examines the benefits and applicability of a look-ahead optimization strategy for an EWMG integrated with multi-type energy conversion technologies and multi-energy demand response to supply various energy-water demands related to electric/hydrogen vehicles and commercial/residential buildings with the lowest cost for two consecutive days. In addition, a hybrid info-gap/robust optimization technique is applied to cover uncertainties in photovoltaic power and electricity prices as a tri-level optimization framework without generating scenarios and using the probability distribution functions. Duality theory is also used to convert the problem into a single-level MILP so that it can be solved by CPLEX. According to the findings, the implemented energy-water storage systems and look-ahead strategy accounted for, respectively, 4.03% and 0.43% reduction in the total cost.

A Multi-Objective Tri-Level Algorithm for Hub-and-Spoke Network in Short Sea Shipping Transportation

Hub-and-Spoke (H&S) network modeling is a form of transport topology optimization in which network joins are connected through intermediate hub nodes. The Short Sea Shipping (SSS) problem aims to efficiently disperse passenger flows involving multiple vessel routes and intermediary hubs through which passengers are transferred to their final destination. The problem contains elements of the Hub-and-Spoke and Travelling Salesman, with different levels of passenger flows among islands, making it more demanding than the typical H&S one, as the hub selection within nodes and the shortest routes among islands are internal optimization goals. This work introduces a multi-objective tri-level optimization algorithm for the General Network of Short Sea Shipping (GNSSS) problem to reduce travel distances and transportation costs while improving travel quality and user satisfaction, mainly by minimizing passenger hours spent on board. The analysis is performed at three levels of decisions: (a) the hub node assignment, (b) the island-to-line assignment, and (c) the island service sequence within each line. Due to the magnitude and complexity of the problem, a genetic algorithm is employed for the implementation. The algorithm performance has been tested and evaluated through several real and simulated case studies of different sizes and operational scenarios. The results indicate that the algorithm provides rational solutions in accordance with the desired sub-objectives. The multi-objective consideration leads to solutions that are quite scattered in the solution space, indicating the necessity of employing formal optimization methods. Typical Pareto diagrams present non-dominated solutions varying at a range of 30 percent in terms of the total distance traveled and more than 50 percent in relation to the cumulative passenger hours. Evaluation results further indicate satisfactory algorithm performance in terms of result stability (repeatability) and computational time requirements. In conclusion, the work provides a tool for assisting network operation and transport planning decisions by shipping companies in the directions of cost reduction and traveler service upgrade. In addition, the model can be adapted to other applications in transportation and in the supply chain.

A tri-level optimization model for facility location-protection problem considering design and redesign decisions under disruption

PurposeFortification-interdiction models provide system designers with a broader perspective to identify and protect vital components. Based on this concept, the authors examine how disruptions impact critical supply systems and propose the most effective protection strategies based on three levels of decision-makers. This paper aims to investigate location and fortification decisions at the first level. Moreover, a redesign problem is presented in the third level to locate backup facilities and reallocate undisrupted facilities following the realization of the disruptive agent decisions at the second level.Design/methodology/approachTo address this problem, the authors develop a tri-level planner-attacker-defender optimization model. The model minimizes investment and demand satisfaction costs and alleviates maximal post-disruption costs. While decisions are decentralized at different levels, the authors develop an integrated solution algorithm to solve the model using the column-and-constraint generation (CCG) method.FindingsThe model and the solution approach are tested on a real supply system consisting of several hospitals and demand areas in a region in Iran. Results indicate that incorporating redesign decisions at the third level reduces maximum disruption costs.Originality/valueThe paper makes the following contributions: presenting a novel tri-level optimization model to formulate facility location and interdiction problems simultaneously, considering corrective measures at the third level to reconfigure the system after interdiction, creating a resilient supply system that can fulfill all demands after disruptions, employing a nested CCG method to solve the model.

Optimal Defensive Strategy for Power Distribution Systems Against Relay Setting Attacks

Improving the cyber security of critical protection devices is of paramount importance for reducing the impact of cyber attacks on power distribution system. This paper proposes a game-theoretic approach to identify the optimal defensive strategy and scheme to minimize the impact of relay setting attack on power distribution systems. To this end, two types of attacks, namely active and passive relay setting attacks (RSA), are introduced and modeled as the potential threats to the protection relays. The active RSA impacts the distribution system immediately, while passive RSA disrupts the relay coordination and escalates the outages resulted from future system faults. The optimal budget allocation problem for strengthening the security of relays is modeled by formulating the interaction between the defender, the attacker and the distribution system operation as a tri-level optimization problem, which is then cast into a single-level operation and a bi-level defender-attacker problem. The bi-level problem is further reduced to an expected energy loss minimization problem using an auxiliary constrained parameter. The proposed approach is tested on the IEEE 123-node test feeder, the results of which exhibit the effectiveness of the proposed model on determining the most vulnerable relays and reducing the expected energy not served.

Siting and sizing of energy storage for renewable generation utilization with multi-stage dispatch under uncertainty: A tri-level model and decomposition approach

For grids suffering from large-scale renewable generation curtailment, the reasonable allocation of energy storage can smooth renewable generation fluctuation for better utilization. This paper analyzes the optimal non-profit planning of energy storage in a grid rich in renewable generation from the perspective of third-party investors. First of all, to model the prediction errors of wind and solar output, a multi-stage scenario tree of intraday uncertainty is established to describe the stochastic output of wind and photovoltaics. In addition, scenario reduction techniques and nonanticipativity constraints are adopted to realize multi-stage optimization. Second, aimed to improve social welfare with an acceptable cost, a tri-level optimization model is established to describe storage siting and sizing, storage dispatch, and market clearing respectively to reach a high level of renewable generation utilization. Furthermore, Karush–Kuhn–Tucker conditions and linearization techniques are used to reform the tri-level model into a linear bi-level one. Third, a decomposition algorithm and a differential cut are proposed to solve the bi-level problem. The optimum is gradually approached via iterating by adding differential cuts and integer cuts. Finally, the model and method are verified based on a region of the HRP-38 system. The results show that storage investment aimed at social welfare can effectively achieve a much higher goal of renewable generation utilization than individual profit-seeking storage investment.

Resilience Constrained Scheduling of Mobile Emergency Resources in Electricity-Hydrogen Distribution Network

This paper proposes a trilevel mixed-integer formulation to coordinately schedule the electricity-hydrogen distribution network (EHDN) with resilience consideration. The rescue hydrogen distribution system is first studied and modeled based on the roadway transportation and vehicle-to-grid power supply of mobile hydrogen energy resources (MHERs). Various practical constraints on traffic capacity, travel distances and time delaying of hydrogen transportation are incorporated. Then, to improve the economic viability of EHDN to fulfill the load survivability requirement in disastrous situations, we propose a risk-constrained trilevel optimization formulation. In particular, the resilience constraint with a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\min -\max$</tex-math></inline-formula> mixed-integer program structure is introduced, which ensures the post-event service preservation by co-optimizing the re-routing of MHERs and network reconfiguration under worst-case damage scenario. Then, by making use of its structural properties, this complex formulation is solved by a customized nested column-and-constraint generation (NC&CG) algorithm with effective enhancements. Numerical results on an exemplary EHDN indicate that the proactive deployment and adaptive relocation of MHERs provide an economically feasible solution to achieve grid resiliency target. Also, comparing to the standard NC&CG, our improved customization demonstrates a superior performance that drastically reduces the computation time, and thus enables efficient emergency response and disaster relief with synergistic network operations.

A tri-level optimal defense method against coordinated cyber-physical attacks considering full substation topology

In recent years, power systems have been facing an increasing risk of malicious attacks. As modern power systems have a greater reliance on communication and automatic control systems, coordinated cyber-physical attacks will cause more serious consequences for the reliable power supply than traditional malicious attacks. To enhance the resilience of power systems against coordinated attacks, a defense method to allocating hardening resources is proposed in this paper considering the full substation topology and the multi-stage response process of power systems. The problem is formulated as a tri-level optimization model involving interactions among the defender, the attacker, and the operator, in which the synergy impact of three attack forms is considered, i.e., physical attacks causing short-circuiting, cyber attacks causing protective relay misoperation, and cyber attacks causing protective relay maloperation. A graph-based analytical model is established to characterize the impact process of such coordinated attacks on power systems. Case studies are conducted on the IEEE RTS 24-bus system. The influence mechanism of coordinated cyber-physical attacks involving protective relays on power systems is systematically revealed. Results show that the proposed method can effectively defend against the above-mentioned coordinated attacks, and enhance the resilience level of power systems. In addition, a sensitivity analysis is implemented on the influence of attack resources and defense resources on the unserved energy at each stage of the system.

Impact of demand side management approaches for the enhancement of voltage stability loadability and customer satisfaction index

This research work presents the tri-level optimization framework for the optimal scheduling of grid-connected and autonomous microgrids to diminish power losses and maximize loadability. Since the network's voltage profile depends on the loading level, the flexible load shaping-based demand-side management strategy is incorporated to investigate its impact on microgrid loadability. With the consideration of uncertain parameters related to renewable power generation, load demand, and power loss, voltage limit constraints, the resultant problem is formulated as a stochastic mixed-integer non-linear problem to enhance microgrid loadability and optimize daily operating costs. The interdependency of demand side management program and microgrid loadability is investigated. The seasonal load profiles covering the weekend and weekday loads in winter, summer, and spring/fall seasons are examined in this research work. The enhanced versions of the distribution networks IEEE-33 and IEEE-69 based microgrid test systems are chosen to evaluate the proposed framework in both off-grid and autonomous modes of operation. Simultaneously, the overall customer satisfaction index is evaluated and improved according to the seasonal load profiles winter weekday, winter-weekend, summer-weekday, summer-weekend, spring-weekday, and spring-weekend by 8.68%, 7.97%, 16.7%, 19.62%, 17.14%, 20.50% respectively. The recently reported Whale Optimization Algorithm is adopted to solve the proposed optimization problem, and the obtained simulation results are validated by comparing them with popular metaheuristic algorithms. The computational burden on the utility is reduced for optimal scheduling of grid-integrated microgrid to extract maximum power by maintaining network voltage profile.