Upper-level Problem Research Articles

RETRACTED ARTICLE: Enhanced multi-objective crisscross optimization for dynamic economic emission dispatch considering demand response and wind power uncertainty

In this paper, the dynamic economic emission dispatch problem in electric power system is formulated as a multi-objective optimization problem in a smart grid perspective. Accordingly, two additional subproblems are included in the dynamic economic emission dispatch formulation. Firstly, the wind power generation is penetrated into the system such that the uncertain power varies between a predicted upper and lower bounds. Secondly, a demand response program is implemented at the customer end, to modify the consumption pattern of electricity according to different electricity prices at valley, peak and off-peak periods. A two-level optimization is proposed to determine the optimal schedule of generating units such that the upper level solves for the minimization of cost and emission, whereas the lower level minimizes the wind power output interval reduction. An enhanced multi-objective crisscross optimization using non-dominated sorting approach is proposed as main optimizer to solve upper-level problem, and a linear programming is adopted to solve the lower-level problem. A 10-unit system is taken as a case study for demonstration, and the result shows the effectiveness of proposed formulation in terms of minimizing cost and emission.

Two‐level area‐load modelling for OPF of power system using reinforcement learning

Load modelling is essential to the planning and operation of a power system. This study proposes a two-level hierarchical framework of real-time area-load modelling for optimal power flow (OPF). The upper-level problem is a parameter identification for an area-load model improved via using a weighting strategy decayed with time, whereas the lower-level optimisation is a dynamic OPF considering N − 1 static security constraints. In the framework, the error of the lower-level optimisation is added into the upper-level model, which guides a search direction for the load modelling toward minimising the error between the online measurement and equivalent model output as much as possible. An improved method is proposed based on function optimisation by reinforcement learning to identify the parameters of the area-load model online in real time. Simulation studies verify the effectiveness of the proposed framework, algorithm and improved strategies.

A Cluster-Based Operation Model of Aggregated Battery Swapping Stations

Battery swapping stations (BSSs) have a potential of becoming a part of the electric vehicle charging infrastructure mix. A number of aggregated BSSs have a significant capacity and can act as strategic players in electricity market. This paper formulates a discrete cluster model for optimal operation of aggregated BSSs that take part in the day-ahead energy and reserve capacity markets and have sufficient combined capacity to affect market prices. In the presented bilevel structure the upper-level problem optimally defines charging and discharging schedules, as well as reserve capacity, of the aggregated BSSs, while the lower-level problem clears the market and provides locational marginal prices for each BSS, as well as the reserve capacity prices. This problem is converted into a single-level equivalent using the Karush-Kuhn-Tucker optimality conditions. The presented case study reveals that BSSs can collect significant revenues when providing reserve capacity. The main benefit of the proposed cluster model is computational tractability and solution accuracy, which depends on the resolution of the clusters.

A population‐based approach to the bi‐level multifollower problem: an application to the electricity retail market

AbstractDynamic tariffs are expected to be implemented as commercial offers of electricity retailers in smart grids, conveying price signals aimed at shaping usage patterns, with potential benefits to enhance grid efficiency and reduce end‐users’ costs. Retailers and consumers have divergent goals. While the objective of the retailer is to maximize profits, the purpose of consumers is to minimize their electricity bill. The interaction between the retailer and consumers can be modeled by means of bi‐level (BL) programming: the retailer sets the prices to be charged to consumers and these react by scheduling flexible appliances according to those prices and their comfort requirements. In this work, two hybrid BL optimization approaches are proposed to solve this problem, considering one leader (retailer) and multiple followers (consumers). Two population‐based approaches were developed, a genetic algorithm (GA) and a particle swarm optimization (PSO) algorithm, to deal with the upper level problem, both encompassing an exact mixed‐integer linear programming solver to address the lower level optimization problem. Different scenarios were generated, comprising one leader (retailer imposing different price schemes) and three followers (with different consumer profiles). Typical residential appliances were considered, with different operation cycles. Also, diverse tariff structures set by the retailer were analyzed. The performance of the two algorithms was compared. Results revealed a consistent superiority of PSO over GA.

Optimal wind energy bidding strategies in real‐time electricity market with multi‐energy sources

This study investigates optimal wind power generator bidding strategies in the real-time electricity market. The goal is to maximise its operating profit by determining the optimal amount of wind power to bid in the real-time market. A bi-level stochastic optimisation model is proposed in which the upper-level problem is minimising the negative profit of wind power producers, while the low-level problem clears the real-time market. The uncertainties in the wind power production, thermal power, hydro power, demand, and energy storage are considered in the stochastic model. This study utilises a mathematical programming problem with equilibrium constraints (MPEC) and Karush-Kuhn-Tucker (KKT) conditions to transform the bi-level problem into an equivalent single-level mixed-integer linear problem (MILP). Case studies obtained with the IEEE RTS-24 Bus system demonstrate the effectiveness of the proposed model and the effect of scenarios on the wind power producer's profit.

Bi-Level Participatory Forest Management Planning Supported by Pareto Frontier Visualization

Abstract This research addresses the problem of forested landscape management planning in contexts characterized by multiple ecosystem services and multiple stakeholders. A new methodology for participatory landscape-level forest management is proposed. Specifically, a bilevel representation is used, whereas models of subsystems are used for constructing an integrated model of the master problem. Participatory workshops and interactive visualization of the Pareto frontier are used to support the solution of the multi-objective optimization upper- and lower-level problems. The visualization is implemented by a technique—Interactive Decision Maps—that displays interactively the Pareto frontier in the form of decision maps, that is, collections of the objectives’ tradeoff curves. Since the upper-level problem may be characterized by a large number of decision variables, we compare the Pareto frontier generated by the Interactive Decision Maps technique with the Pareto frontier generated by a decomposition approach that builds from the Pareto frontiers of the lower-level subproblems. The approach supports further the negotiation between upper- and lower-level goals. Results are discussed for a large-scale application in a forested landscape in northwest Portugal.

A Risk-Based Decision Framework for the Distribution Company in Mutual Interaction With the Wholesale Day-Ahead Market and Microgrids

One of the emergent prospects for active distribution networks ( DN ) is to establish new roles to the distribution company ( DISCO ). The DISCO can act as an aggregator of the resources existing in the DN , also when parts of the network are structured and managed as microgrids ( MG s). The new roles of the DISCO may open the participation of the DISCO as a player trading energy in the wholesale markets, as well as in local energy markets. In this paper, the decision making aspects involving the DISCO are addressed by proposing a bilevel optimization approach in which the DISCO problem is modeled as the upper-level problem and the MG s problems and day-ahead wholesale market clearing process are modeled as the lower-level problems. To include the uncertainty of renewable energy sources, a risk-based two-stage stochastic problem is formulated, in which the DISCO 's risk aversion is modeled by using the conditional value at risk. The resulting nonlinear bilevel model is transformed into a linear single-level one by applying the Karush–Kuhn–Tucker conditions and the duality theory. The effectiveness of the model is shown in the application to the IEEE 33-bus DN connected to the IEEE RTS 24-bus power system.

Joint design of multimodal transit networks and shared autonomous mobility fleets

Providing quality transit service to travelers in low-density areas, particularly travelers without personal vehicles, is a constant challenge for transit agencies. The advent of fully-autonomous vehicles (AVs) and their inclusion in mobility service fleets may allow transit agencies to offer better service and/or reduce their own capital and operational costs. This study focuses on the problem of allocating resources between transit patterns and operating (or subsidizing) shared-use AV mobility services (SAMSs) in a large metropolitan area. To address this question, a joint transit network redesign and SAMS fleet size determination problem (JTNR-SFSDP) is introduced, and a bi-level mathematical programming formulation and solution approach are presented. The upper-level problem modifies a transit network frequency setting problem (TNFSP) formulation via incorporating SAMS fleet size as a decision variable and allowing the removal of bus routes. The lower-level problem consists of a dynamic combined mode choice-traveler assignment problem (DCMC-TAP) formulation. The heuristic solution procedure involves solving the upper-level problem using a nonlinear programming solver and solving the lower-level problem using an iterative agent-based assignment-simulation approach. To illustrate the effectiveness of the modeling framework, this study uses traveler demand from Chicago along with the region’s existing multimodal transit network. The computational results indicate significant traveler benefits, in terms of improved average traveler wait times, associated with optimizing the joint design of multimodal transit networks and SAMS fleets compared with the initial transit network design.

Electricity wholesale market equilibrium analysis integrating individual risk-averse features of generation companies

With the deregulation of world power markets, the importance of power market equilibrium analysis tools becomes more prominent. Due to the increasing uncertainty in the power markets, the influences of risk attitudes on the participants’ strategic bidding behaviors and market equilibrium become more significant, which makes risk feature formulation more necessary in the market bidding decision-making process. This paper fully considers the risk-averse features of power generation companies based on the concept of the conditional value at risk and proposes a bilevel equilibrium model to practically simulate market behaviors. The mathematical model is essentially a modified equilibrium problem with equilibrium constraints. The upper-level problem of the model maximizes the profits minus the risks of each generation company, while the lower-level problem represents the market clearing processes of day-ahead markets by the independent system operator. A linearization method is proposed to transform the proposed model into a mixed-integer linear model. An illustrative example and a numerical example are performed to show the effectiveness, validity and adaptability of the proposed model. The comparisons with other equilibrium models demonstrate the better performances of the proposed method to simulate the market equilibrium with several oligopolies. The sensitivity analysis is proposed to analyze the phenomenon where different risk-averse levels and market ownership structures would result in different market equilibrium.

Bi‐level operational planning of microgrids with considering demand response technology and contingency analysis

Considering financial, ecological and also reliability challenges in planning and operation of microgrids is an indispensable issue. On the other hand, an appropriate design of microgrids in planning stages has a major impact on the daily operation of the system. According to these matters, in this study, a hierarchical decision-making framework has been presented for both planning and operation of microgrids with wide ranges of means and concepts. The proposed model has been formulated as a bi-level optimisation problem in which every level optimises its own objectives independently, however, in interaction with each other. The upper level (leader) problem, which is related to the planning of microgrids, minimises the utility's demand, investment, and emission costs, while the lower level (follower) problem minimises the operation and maintenance costs through the implementation of an energy management system. The mentioned model is a non-linear bi-level problem, which is transformed into a linear single-level problem through Karush–Kuhn–Tucker conditions. Moreover, the contingency based energy management, demand response programme and uncertain nature of renewable resources have been taken into account. Finally, the proposed method has been applied to a typical microgrid and its results are compared with the weighted-sum multi-objective approach to depict its efficiency.

Implications of bid structures on the offering strategies of merchant energy storage systems

Energy storage Systems (ESS) may play a pivotal role in the cost-efficient integration of renewable energy sources. Integrating large volumes of grid-scale energy storage into electricity markets, however, raises questions related to their profitability and impact on electricity prices. This paper explores the implications of different bid structures on the strategic behavior of a storage owner that participates in the wholesale market and is able to influence prices to maximize its profits. We conduct a comparative analysis on four bid structures present in existing markets and the scientific literature: (i) simple quantity-based bids, (ii) simple price-based bids, (iii) price-quantity pairs bids and (iv) complex bidding in which the ESS owner provides full information of its technical characteristics through the bids. The decision problem of the ESS owner is formulated as a bilevel programming model, where the upper-level problem represents the profit maximization of the ESS and the lower-level problem simulates possible market clearing outcomes. These bilevel models are reformulated as equivalent mixed-integer linear programming problems by means of the Karush-Kuhn-Tucker optimality conditions, the strong duality theorem and the Big-M method. In a case study, we show that when the ESS owner is uncertain about the market clearing outcome, the bid structure affects the ESS’s profitability and, to a lesser extent, the market’s efficiency and the system’s generation costs.

A novel evolutionary algorithmic solution approach for bilevel reliability-redundancy allocation problem

Reliability-redundancy allocation problem (RRAP) is an optimization problem with objectives such as maximization of the system reliability and minimizations of cost, weight etc. It considers component reliability and the number of redundant components as decision variables. RRAP was mostly solved as a single level optimization problem. However, this problem has numerous variables and constraints. In this paper, we report a novel mixed-integer non-linear bilevel formulation for the RRAP of the series-parallel systems considering reliability, cost and weight as the main objectives. Then we show a novel way of solving the newly formulated bilevel RRAP using a latest bilevel optimization algorithm called BLEAQ (bilevel evolutionary algorithm based on quadratic approximations). Here, optimization is needed at two separate levels, where one problem is encircled within another problem. The inner problem is known as lower-level problem and the external problem is called upper-level problem. We demonstrate the working of the model considering a real-life example. For a thorough comparison, we also solve the problem using two existing bilevel evolutionary approaches to show the efficacy of our proposed solution approach.

Nested evolutionary algorithms for computationally expensive bilevel optimization problems: Variants and their systematic analysis

Abstract Bilevel optimization problems involve a hierarchical model where an upper level optimization problem is solved with a constraint on the optimality of a nested lower level problem. The use of evolutionary algorithms (EAs) and other metaheuristics has been gaining attention to solve bilevel problems, especially when they contain non-linear/black-box objective(s) and/or constraint(s). However, EAs typically operate in a nested mode wherein a lower level optimization is executed for each upper level solution. Evidently, this process requires excessive number of function evaluations, which might become untenable if the underlying functions are computationally expensive. In order to reduce this expense, the use of approximations (also referred to as surrogates or meta-models) has been suggested previously. However, the previous works have focused only on the use of surrogates for the lower level problem, whereas the computational expense of the upper level problem has not been considered. In this paper, we aim to make two contributions to address this research gap. The first is to introduce an improved nested EA which uses surrogate-assisted search at both levels in order to solve bilevel problems using limited number of function evaluations. The second is the revelation and a systematic investigation of a previously overlooked aspect of bilevel search – that the objective/constraints at the upper and lower levels may involve different computational expense. Consideration of this aspect can help in deciding a suitable strategy, i.e., in which level is the use of surrogates most appropriate for the given problem. Towards this end, four different nested strategies – with surrogate at either level, none or at both levels, are compared under various experimental settings. Numerical experiments are presented on a wide range of problems to demonstrate the efficacy and utility of the proposed contributions.

Optimal operation of a virtual power plant in frequency constrained electricity market

In this study, optimal offering strategy problem of a virtual power plant (VPP) as a price-maker player in day-ahead frequency constrained electricity market is presented. The optimal offering strategy problem is modelled as a bi-level optimisation problem. In the upper-level problem, the total profit of VPP is maximised. In the lower-level problem, the clearing conditions of frequency constrained electricity market are modelled. The proposed bi-level optimisation problem is reformulated as a mathematical programming with equilibrium constraints (MPEC) problem by using Karush–Kuhn–Tucker conditions. Then, the proposed MPEC problem, which is non-linear and hard to solve by commercial solvers, is transformed into a mixed-integer linear programming problem by using strong duality theorem and big-number mathematical technique. Here, stochastic optimisation is included in the modelling to enable the VPP for optimisation in the presence of uncertainties, e.g. renewable energy source generations, demands, and offering strategies of rivals. Finally, the effectiveness of proposed model is investigated by implementing it on various case studies.

An Improved Defender–Attacker–Defender Model for Transmission Line Defense Considering Offensive Resource Uncertainties

Developing efficient strategies for defending electric power systems against attacks is a major concern for contemporary power grids, especially when uncertainties are involved. This paper addresses the allocation of the defensive resource to minimize the damage when there are uncertainties regarding the resource that the attacker has. A multiple-attack-scenario (MAS) defender–attacker–defender (DAD) model is proposed by extending the conventional trilevel DAD model. The proposed model considers the uncertainties related to the offensive resource and the interactions involving the security personnel at the top-level, the attacker at the middle-level, and the power system operator at the bottom-level. The column-and-constraint generation algorithm is implemented by decomposing the MAS DAD model into an upper-level problem for the security personnel, and a lower-level problem for the attacker involving the optimal power flow analysis-based corrective power redispatch implemented by the power system operator. Case studies are performed based on the IEEE RTS79 and 57-bus systems, and the results validate that the proposed method is able to minimize the damage when uncertainties are involved in the offensive resource. This paper can offer meaningful insights into power system protection involving uncertainties.

Network Reconfiguration for Increasing Transportation System Resilience Under Extreme Events.

Evacuating residents out of affected areas is an important strategy for mitigating the impact of natural disasters. However, the resulting abrupt increase in the travel demand during evacuation causes severe congestions across the transportation system, which thereby interrupts other commuters' regular activities. In this article, a bilevel mathematical optimization model is formulated to address this issue, and our research objective is to maximize the transportation system resilience and restore its performance through two network reconfiguration schemes: contraflow (also referred to as lane reversal) and crossing elimination at intersections. Mathematical models are developed to represent the two reconfiguration schemes and characterize the interactions between traffic operators and passengers. Specifically, traffic operators act as leaders to determine the optimal system reconfiguration to minimize the total travel time for all the users (both evacuees and regular commuters), while passengers act as followers by freely choosing the path with the minimum travel time, which eventually converges to a user equilibrium state. For each given network reconfiguration, the lower-level problem is formulated as a traffic assignment problem (TAP) where each user tries to minimize his/her own travel time. To tackle the lower-level optimization problem, a gradient projection method is leveraged to shift the flow from other nonshortest paths to the shortest path between each origin-destination pair, eventually converging to the user equilibrium traffic assignment. The upper-level problem is formulated as a constrained discrete optimization problem, and a probabilistic solution discovery algorithm is used to obtain the near-optimal solution. Two numerical examples are used to demonstrate the effectiveness of the proposed method in restoring the traffic system performance.

The r‐interdiction selective multi‐depot vehicle routing problem

AbstractThe protection of critical facilities has been attracting increasing attention in the past two decades. Critical facilities involve physical assets such as bridges, railways, power plants, hospitals, and transportation hubs among others. In this study we introduce a bilevel optimization problem for the determination of the most critical depots in a vehicle routing context. The problem is modeled as an attacker–defender game (Stackelberg game) from the perspective of an adversary agent (the attacker) who aims to inflict maximum disruption on a routing network. We refer to this problem as the r‐interdiction selective multi‐depot vehicle routing problem (RI‐SMDVRP). The attacker is the decision maker in the upper level problem (ULP) who chooses r depots to interdict with certainty. The defender is the decision maker in the lower level problem (LLP) who optimizes the vehicle routes in the wake of the attack. The defender has to satisfy all customer demand either using the remaining depots or through outsourcing to a third party logistics service provider. The ULP is solved through exhaustive enumeration, which is viable when the cardinality of interdictions does not exceed five among nine depots. For the LLP we implement a tabu search heuristic adapted to the selective multi‐depot VRP. Our results are obtained on a set of RI‐SMDVRP instances synthetically constructed from standard MDVRP test instances.

A Computationally Efficient Optimization Method for Battery Storage in Grid-connected Microgrids Based on a Power Exchanging Process

Battery storage (BS) sizing problems for grid-connected microgrids (GCμGs) commonly use stochastic scenarios to represent uncertain natures of renewable energy and load demand in the GCμG. Though taking a large number of stochastic scenarios into consideration can deliver a relatively accurate optimal result, it can also highly deteriorate the computational efficiency of the sizing problem. To make an accuracy-efficiency trade-off, a computationally efficient optimization method to optimize the BS capacities based on the power exchanging process of the GCμG is proposed in this paper. According to the imbalanced power of the GCμG, this paper investigates the power exchanging process between the GCμG, BS and external grid. Motivated by the BS dynamics, a forward/backward sweep-based energy management scheme is proposed based on the power exchanging process. A heuristic two-level optimization model is developed with sizing BS as the upper-level problem and optimizing the operational cost of the GCμG as the lower-level problem. The lower-level problem is solved by the proposed energy management scheme and the objective function of the upper-level is minimized by the pattern search (PS) algorithm. To validate the accuracy and computational efficiency of the proposed method, the numerical results are compared with the mixed integer linear programming (MILP) method. The comparison shows that the proposed method shares similar accuracy but is much more time-efficient than the MILP method.

A community microgrid architecture with an internal local market

This work fits in the context of community microgrids, where members of a community can exchange energy and services among themselves, without going through the usual channels of the public electricity grid. We introduce and analyze a framework to operate a community microgrid, and to share the resulting revenues and costs among its members. A market-oriented pricing of energy exchanges within the community is obtained by implementing an internal local market based on the marginal pricing scheme. The market aims at maximizing the social welfare of the community, thanks to the more efficient allocation of resources, the reduction of the peak power to be paid, and the increased amount of reserve, achieved at an aggregate level. A community microgrid operator, acting as a benevolent planner, redistributes revenues and costs among the members, in such a way that the solution achieved by each member within the community is not worse than the solution it would achieve by acting individually. In this way, each member is incentivized to participate in the community on a voluntary basis. The overall framework is formulated in the form of a bilevel model, where the lower level problem clears the market, while the upper level problem plays the role of the community microgrid operator. Numerical results obtained on a real test case implemented in Belgium show around 54% cost savings on a yearly scale for the community, as compared to the case when its members act individually.

Wind Versus Storage Allocation for Price Management in Wholesale Electricity Markets

This paper investigates the impacts of installing regulated wind and electricity storage on average price and price volatility in electricity markets. A stochastic bi-level optimization model is developed, which computes the optimal allocation of new wind and battery capacities, by minimizing a weighted sum of the average market price and price volatility. A fixed budget is allocated on wind and battery capacities in the upper-level problem. The operation of strategic/regulated generation, storage, and transmission players is simulated in the lower-level problem using a stochastic (Bayesian) Cournot-based game model. Australia's national electricity market, which is experiencing occasional price peaks, is considered as the case study. Our simulation results quantitatively illustrate that the regulated wind is more efficient than storage in reducing the average price, while the regulated storage more effectively reduces the price volatility. According to our numerical results, the storage-only solution reduces the average price at most by 9.4%, and the wind-only solution reduces the square root of price volatility at most by 39.3%. However, an optimal mixture of wind and storage can reduce the mean price by 17.6% and the square root of price volatility by 48.1%. It also increases the consumer surplus by 1.52%. Moreover, the optimal mixture of wind and storage is a profitable solution unlike the storage-only solution.