Bi-level Stochastic Programming Research Articles

A Novel Approach to Solve Multi-objective Fuzzy Stochastic Bilevel Programming Using Genetic Algorithm

A Novel Approach to Solve Multi-objective Fuzzy Stochastic Bilevel Programming Using Genetic Algorithm

Optimal defense strategies against intelligent cyber attacks

We propose a comprehensive game-theoretic model pertaining to the security of computer networks, specifically addressing the interaction between defenders and attackers. The model incorporates attack graphs to outline potential attacker strategies and defender responses. To account for the attacker's capacity to execute multiple attempts, we introduce a probabilistic element, wherein the success or failure at any arc of the attack graph is treated as stochastic. This characterization gives rise to a multi-stage stochastic network-interdiction problem. In this problem formulation, the defender strategically interdicts a set of arcs in anticipation of the likely actions of the attacker, who, in turn, can make multiple attempts to traverse the network. We mathematically articulate this scenario as a stochastic bilevel mixed-integer program with a "min-max" objective. The defender's aim is to minimize the probability of the attacker's success, while the attacker seeks to maximize the probability of successfully traversing the network across multiple attempts. The defender's stochastic bilevel optimization model is solved using the integer L-shaped method. Upon analyzing the defender's perspective, we observe the anticipated trend that the overall success probability of the attacker diminishes with an increasing level of defense. Notably, in the sensitivity analysis involving relatively small attack graphs, we discover that the optimal defense strategy against a myopic attacker often aligns with that against a non-myopic attacker. Furthermore, in instances where deviations exist, the disparity in performance is generally marginal. However, our findings demonstrate a potential divergence in optimal defense strategies when the available attack paths share numerous common arcs.

A Cooperative Agent-Based Traffic Signal Control for Vehicular Networks Under Stochastic Flow

A two-tier Cooperative Agent-based Traffic Signal control (CATS) is proposed to minimize total delay for independent-learning signal-controlled junctions. For vehicular networks with traffic congestion, a link traffic model is presented to estimate time-varying signal delay under stochastic travel demand. To capture essential features of signal-controlled junctions, an agent-based value function approximator is proposed. For the 1 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">st</sup> tier, common cycle time and offsets are explored to achieve collaboration among control agents. For the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> tier, green splits are exploited to ensure scalability over entire vehicular networks. A stochastic bi-level program is presented to minimize total delays. Numerical experiments are performed at a real-data vehicular network under stochastic flow. Comparisons are made with state-of-the-art traffic signal controls. As reported, the proposed CATS outperforms other alternatives in all cases.

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications

We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds with a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. We assume that only the information regarding the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and random constraints under proper conditions of ambiguity sets. Under continuous distributions, using linear decision rule approaches, we construct upper bounds on the pessimistic DR bilevel program based on (1) a 0-1 semidefinite programming (SDP) approximation and (2) an exact 0-1 copositive programming reformulation. When the ambiguity set is restricted to discrete distributions, an exact 0-1 SDP reformulation is developed, and explicit construction of the worst-case distribution is derived. To further improve the computation of the proposed 0-1 SDPs, a cutting-plane framework is developed. Moreover, based on a mixed-integer linear programming approximation, another cutting-plane algorithm is proposed. Extensive numerical studies are conducted to demonstrate the effectiveness of the proposed approaches on a facility location problem. History: Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods &amp; Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0168 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0168 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Two-timescale coordinated operation of wind-advanced adiabatic compressed air energy storage system: A bilevel stochastic dynamic programming method

Renewable energy is a promising solution to address the energy crisis and environmental issues, but it comes with challenges due to its inherent volatility and limited dispatchability. Advanced adiabatic compressed air energy storage (AA-CAES) is a favorable partner for centralized renewable integration, due to its numerous benefits, such as large capacity, long lifetime, fast response capability, and zero carbon emissions. For the economic management of a wind-AACAES system, a battery-like AA-CAES dispatch model is proposed, where the charging/discharging efficiencies and capacities are dependent on the operating power and state-of-charge (SoC) to capture the part-load characteristics. A hybrid control strategy is proposed to enhance the flexibility and efficiency of the compression side under off-design conditions caused by simultaneous changes in back pressure and load. To provide dispatch and control strategies of AA-CAES, a multi-timescale optimization problem is established. The slow timescale determines the baseline output scheduling on an hourly basis to maximize total revenue, while the fast timescale updates the real-time adjustment every five minutes to minimize the penalty of tracking error. A bi-level stochastic dynamic programming (SDP) framework is formulated to incorporate the wind uncertainties and multi timescale coordination, where terminal SoCs and value functions are essential to integrate decisions across slow and fast timescales. A non-iterative parametric programming-based method is proposed to solve the slow-timescale SDP, and a simulation-based rollout algorithm is applied to extract the fast-timescale actions, which yields evident improvement over any heuristic base policy that is sequentially consistent. The numerical results demonstrate that the proposed method has a performance gap of approximately 7.1%, which outperforms the rule-based method by 32.6%. The installation of AA-CAES improves the total revenue by 11.9%. Additionally, the hybrid control strategy enhances the charging flexibility and efficiency by 30.7% and 4.3%, respectively, resulting in a 3.9% reduction in tracking deviation penalty.

Learning-Based Coordinated Operation of Multiple Microgrids With Hydrogen Systems: A Novel Bilevel Framework

This article provides a framework for coordinating the operation of multiple microgrids with hydrogen systems in a distribution network considering the uncertainties of wind and solar power generation as well as load demands. The model is based upon a bilevel stochastic programming problem. On the upper level, the distribution system is the leader with a profit-maximization goal, and the microgrids are followers with cost-minimization goals on the lower level. The problem is solved by transforming the model to a single-level model using Karush–Kuhn–Tucker (KKT) conditions and linearized using McCormick’s relaxation and Fortuny–Amat techniques. Unlike previous studies, both levels are modeled as scenario-based stochastic problems. Moreover, the scenarios associated with uncertain variables are obtained from a real data set. After preparing the data set, scenarios are reduced using a machine learning-based clustering approach. An application of the coordinated operation model is developed for a distribution network containing several microgrids. By solving the problem, the optimal amount of power exchange and the clearing price between microgrids and distribution systems are determined. Moreover, the proposed bilevel model made 13% more profit for the distribution system than the centralized model. Also, the effects of integrating hydrogen systems with microgrids on increasing the flexibility of operators are investigated.

Profit-Oriented BESS Siting and Sizing in Deregulated Distribution Systems

Within the deregulation process of distribution systems, the distribution locational marginal price (DLMP) provides effective market signals for future unit investment. In that context, this paper proposes a two-stage stochastic bilevel programming (TS-SBP) model for investors to best allocate battery energy storage systems (BESSs). The first stage obtains the optimal siting and sizing of BESSs on a limited budget. The second stage, a bilevel BESS arbitrage model, maximizes the arbitrage revenue in the upper level and clears the distribution market in the lower level. Karush-Kuhn-Tucker (KKT) optimality conditions, strong duality theory, and the big-M method are utilized to transform the TS-SBP model into a tractable two-stage stochastic mixed-integer linear programming (TS-SMILP) model. A novel statistics-based scenario extraction algorithm is proposed to generate a series of typical operating scenarios. Then, scale reduction strategies for BESS candidate buses and inactive voltage constraints are proposed to reduce the scale of the TS-SMILP model. Finally, case studies on the IEEE 33-bus and 123-bus systems validate the effectiveness of the DLMP in incentivizing BESS planning and the efficiency of the two proposed scale reduction strategies.

Optimal microgrid programming based on an energy storage system, price-based demand response, and distributed renewable energy resources

Microgrids (MGs) in distribution systems can be operated in far regions at lower investment costs using renewable distributed energy resources (DERs). The present paper introduces a stochastic model for optimal energy-heat programming and the daily storage of an MG. Bi-level stochastic programming is presented for integrated energy-heat scheduling and storage in the presence of an energy storage system (ESS) and demand response (DR) based on social welfare maximization. Out of the incentive-based DR programs, the tender and redemption and the ancillary services market programs were selected and applied to the given model. Besides, the time of use (TOU) -based DR and real-time pricing (RTP) were considered as the price-based demand response (PBDR) programs in optimal programming. The PBDR programs have been included in the objective function using a linear function based on consumer benefits. The proposed bi-level stochastic model was solved using a developed metaheuristic optimization algorithm called the lightning search algorithm (LSA) in the present work. The Latin hypercube sampling (LHS) and KMEANS methods were used to produce and reduce the scenario. The proposed framework was investigated in a 33-bus test model. The obtained simulation results were evaluated from different aspects. The TOU and RTP effects and ESS are shown in obtained numerical analysis by considering the operating cost, total social welfare, and the client's utility function.

Incentive-Based Integrated Demand Response Considering S&amp;C Effect in Demand Side With Incomplete Information

Incentive-based integrated demand response (IDR) has been recognized as a powerful tool to mitigate supply-demand imbalance in integrated energy systems with high penetration of distributed renewable energy sources (RESs). However, incomplete information as well as demand-side substitute and complementary effect (S&C effect)—the change in demand for a type of energy carrier as a result of a change in the relative incentive rates of other kind of energy carriers—have been central challenges to estimate accurately the response behavior of consumers. This paper proposed an improved incentive-based IDR model to effectively cope with the S&C effect and incomplete information. Besides, both output uncertainty of RESs and responsiveness uncertainty of consumers are taken into account, with an improved model of energy storage unit to deal with balancing power deviation and measure corresponding risk costs caused by the uncertainties. The proposed IDR model is formulated as a bi-level stochastic programming problem, which is converted equivalently into a nonlinear convex optimization problem to solve it efficiently. Simulation results validate the effectiveness of the proposed method to cope with the S&C effect and incomplete information, verifying merits of the model in reducing both comfort loss of consumers and the total cost/risk cost of multi-energy aggregator (MEA), decreasing balancing power deviation, enhancing MEA’s ability of risk management, and promoting calculation efficiency.

Bayesian Nash Equilibrium in Electricity Spot Markets: An Affine-Plane Approximation Approach

This article proposes a system of stochastic mixed complementarity problems (MCP) for calculating the Bayesian Nash equilibrium (BNE) in an electricity spot market. The generators submit strategic multilevel price-offer functions under incomplete information about their rivals’ marginal cost. This strategic multilevel price offering of each generator is modeled as a standard stochastic bilevel program which is then reformulated as its equivalent MCP. The merger concept is employed and formulated to model multilevel price-offer functions. We then propose the BNE-MCP model which solves the stochastic MCP models of all generators together. A scenario reduction technique is also developed to accurately model the wind and demand uncertainty in the proposed BNE-MCP model. The proposed BNE-MCP model is carefully studied on two illustrative examples. The modified IEEE 14-, 30-, and 57-node systems are also examined in our article.

Coordinating harvest planning and scheduling in an agricultural supply chain through a stochastic bilevel programming

AbstractThis paper addresses a selective harvest planning problem in the context of a hierarchical agricultural supply chain, which integrates the definition of management zones for harvest planning, and the coordination between the producer and the wholesaler. The problem is represented through a stochastic bilevel program that allows the representation of the hierarchy between the producer (leader) and a wholesaler (follower). The producer decides planning and scheduling of the selective harvest for each homogeneous management zone into the resulting partition, and the wholesaler decides the amount to be acquired to satisfy demand requirements. At each decision level, a stochastic optimization model is proposed for representing the uncertainty in future crop yields, prices, and demands, using a finite set of scenarios. A reformulation of the bilevel model into a mixed‐integer linear program is provided using Karush–Kuhn–Tucker conditions and replacing the nonlinear complementary constraints allowing for the introduction of auxiliary binary variables and a big‐M term. This model was applied in a case study for selective harvesting of grapes with data collected from a farm located in the south central zone of Chile. Our research shows valuable results of the proposed methodology from a set of instances representing the behavior of both decision makers under uncertainty.

Enabling inter-area reserve exchange through stable benefit allocation mechanisms

The establishment of a single European day-ahead market has accomplished the integration of the regional day-ahead markets. However, reserve provision and activation remain an exclusive responsibility of regional operators. This limited spatial coordination and the separated structure hinder the efficient utilization of flexible generation and transmission, since their capacities have to be ex-ante allocated between energy and reserves. To promote reserve exchange, recent work proposed a preemptive model that withdraws a portion of the inter-area transmission capacity available from day-ahead energy for reserves by minimizing the expected system cost. This decision-support tool, formulated as a stochastic bilevel program, respects the current architecture but does not suggest area-specific costs that guarantee sufficient benefits for areas to accept the solution. To this end, we formulate a preemptive model in a framework that allows application of game theory methods to obtain a stable benefit allocation, i.e., an outcome immune to coalitional deviations ensuring willingness of areas to coordinate. We show that benefit allocation mechanisms can be formulated either at the day-ahead or the real-time stages, in order to distribute the expected or the scenario-specific benefits, respectively. For both games, the proposed benefits achieve minimal stability violation, while allowing for a tractable computation with limited queries to the bilevel program. Our case studies, based on an illustrative and a more realistic test case, compare our method with well-studied benefit allocations, namely, the Shapley value and nucleolus, and analyze the factors that drive these allocations (e.g., flexibility, network structure, wind correlations). We show that our method performs better in stability and tractability.

Risk-averse stochastic bilevel programming: An application to natural gas markets

This paper presents a risk-based integer bilevel programming model that represents two players (leader and follower) who interact strategically over an infrastructure network. Both players seek to maximize their individual profits in the face of demand uncertainty. Variations of the basic model are proposed that allow us to identify optimal strategies under risk-neutral and risk-averse decision-making. To solve the resulting stochastic bilevel problem, we develop an algorithm that relies on iteratively solving a restricted version of the problem to obtain feasible solutions to the original model. Extensive computational experiments are performed to evaluate algorithmic tractability and solution quality. The methodology is demonstrated using an example that addresses emerging strategic issues in natural gas markets. The model considers two players, a liquefied natural gas (LNG) operator as the leader and a natural gas (NG) producer as the follower. The LNG operator makes decisions on the locations of LNG facilities and the amount of gas to buy from the producer. The NG producer decides how much gas to produce and how to ship it through the existing pipeline network. Both players face demand uncertainty in their respective markets and have to make investment, production, and sales decisions. Our case study focuses on the Gulf-Southwest region of the United States and demonstrates the impact of risk-averse decision-making on investment and operational decisions.

Locating and protecting interdependent facilities to hedge against multiple non-cooperative limited choice attackers

This paper studies an interdependent facility location and protection problem against multiple non-cooperative limited choice attackers. In a supply network, the defender and attackers have completely opposite objectives by optimizing their respective patterns. This interaction is represented with a game-theoretic bilevel framework. The defender as a leader in the upper-level makes location and protection decisions while the attackers as followers in the lower-level carry out attack strategies following a limited choice rule. A real supposition is that the effects of protection and attack are imperfect, making the decision-dependent uncertainty in the post-disruption states of the facilities. We incorporate the uncertainty into a two-stage model and formulate the resulting model as a two-stage stochastic bilevel programming problem. The first stage problem involves a bilevel defender–attacker model that locates and protects facilities against attacks, while the second stage problem involves allocation decisions around customers that optimizes a scenario-based minimization problem of allocating demand capacity. Due to the NP-hardness of the considered problem, we develop a hybrid solution method based on genetic algorithm and column generation to solve the model. Computational experiments show the efficiency of the proposed algorithm and demonstrate the effects of limited choice and risk propagation on the optimal solutions.

A two‐stage stochastic bilevel programming approach for offering strategy of DER aggregators in local and wholesale electricity markets

A two-stage stochastic programming scheme is proposed in order to evaluate the offering strategy of a distributed energy resource aggregator in both wholesale and local electricity markets and appropriately cope with uncertainties associated with its decision-making problem. In this regard, the aggregator combines a broad range of virtual and real distributed energy resources to simultaneously participate in the local electricity market as a price-maker or strategic player and the wholesale electricity market as a price-taker or non-strategic player. To model the studied aggregator as a strategic entity in the local market, a bilevel programming approach is exploited in this work. Accordingly, at the upper level of the raised problem, the aggregator tends to promote its expected profit through taking part in the wholesale and local electricity markets, while at the lower level, the considered local market is cleared in a way to maximise the social welfare. In the end, the effectiveness of the proposed framework for the simultaneous participation of the distributed energy resource aggregator in these two markets has been explored utilising a case study.

Planning a Water–Food–Energy–Ecology Nexus System toward Sustainability: A Copula Bi-level Fractional Programming Method

A copula bi-level fractional programming (CBFP) method is developed for planning the water–food–energy–ecology (WFEE) nexus system. CBFP has advantages in (i) dealing with ratio-objective problems, (ii) balancing the conflicts between hierarchical decision levels, and (iii) reflecting joint risks of correlated uncertain variables. Then, a CBFP–WFEE model is formulated to the Ili-Balkhash basin in Central Asia, in which 108 scenarios associated with different irrigation efficiencies, ecological-flow demands (EBW), and water–land resources have been examined. Solutions of water allocation, hydropower generation, and land-use pattern are obtained. Our major findings are as follows: (i) from 2021 to 2050, water allocation to livestock in East Kazakhstan would remarkably increase (by 40.9%) when water allocation to food is satisfied; (ii) hydropower generation would increase with the rising ecological flows and arable land resources; (iii) EBW is the key factor influencing the water inflow to the Balkhash Lake; (iv) the share of ecosystem water allocation would exceed 17.5% by 2050, and grassland area would rank in the first place of the ecosystem. Compared with conventional bi-level stochastic programming and single-level fractional programming models, the CBFP–WFEE model can achieve a higher ecosystem service value and higher efficiency of water–land resources, which can provide more feasible and sustainable alternatives for the WFEE nexus system. The obtained results can help balance the contradictions among water shortage, economic development, and ecology protection, as well as provide synergic management strategies for regional sustainability.

Stochastic Bi-level Programming Model for Home Healthcare Scheduling Problems Considering the Degree of Satisfaction with Visit Time

Stochastic Bi-level Programming Model for Home Healthcare Scheduling Problems Considering the Degree of Satisfaction with Visit Time

Stochastic bilevel programming with multiple followers: a solution approach using the systematic sampling evolutionary method

A stochastic bilevel programming problem with multiple followers is presented in this article. Such kinds of problem are computationally difficult and efficient algorithms are lacking thanks to the randomness properties in the problem setting, its hierarchical structure and the expected simultaneous decision at the followers' level for each strategy of the leader. This article proposes a systematic sampling evolutionary algorithm that is established on a sample average approximation, a systematic sampling technique and particle swarm optimization integrated with an iterated method. The solution procedure is implemented and its effectiveness is tested on a variety of illustrative examples from the literature and on carefully constructed problems. The simulation results show that the proposed method is promising and can be used to solve a variety of complex stochastic bilevel programming problems with multiple followers.

On continuity in risk-averse bilevel stochastic linear programming with random lower level objective function

We study bilevel stochastic linear programs with fixed lower level feasible set and random follower's goal function and show that the parametrized random variable arising as the upper level outcome depends continuously on the leader's decision w.r.t. any Lp-norm with p∈[1,∞). This entails continuity of the objective function for a class of models involving convex risk measures defined on appropriate Lp-spaces and allows to formulate verifiable sufficient conditions for the existence of optimal solutions.

About a regional development model that takes into account environmental problems with budgeting uncertainty

Raw-materials base (hereinafter RMB) is one of the largest industries for financial investments in Russia. There are various mathematical descriptions for the development of regions with resource-based economy. Earlier in [1] the researchers considered the model based on bilevel integer stochastic programming problems with Boolean variables. This paper proposes a new approach to public-private partnership modelling, including a bilevel linear stochastic programming problem. This model assumes that budget constraints of the state and the investor can vary in a random manner with a specific probability distribution. We put forward two methods to solve this problem: problem’s reduction to the deterministic bilevel one and formulation of deterministic problems sequence with help of Monte Carlo methods. In order to solve deterministic problems of integer programming, we suggest two approaches: direct enumeration and heuristic “Game” approach. The numerical experiments for proposed algorithms validation are conducted on the basis of actual data of Zabaykalsky Krai development. Multiple input parameters of the model vary in these experiments. Finally, we present a brief analysis of the obtained solutions to the stochastic linear programming problems with Boolean variables.